Statistical Techniques for Sampling and Monitoring Natural Resources

Transcription

1 Uted States Departmet of Agrculture Forest Servce Statstcal Techques for Samplg ad Motorg Natural Resources Rocky Mouta Research Stato Geeral Techcal Report RMRS-GTR-6 Has T. Schreuder, Rchard Erst, ad Hugo Ramrez-Maldoado Aprl 004

2 CORRECTIONS Page 6: I Equato (3), a should be added to the deomator,.e., Page 7: The fourth equato from the top ad the le followg t should be The wth varace estmate

3 Schreuder, Has T.; Erst, Rchard; Ramrez-Maldoado, Hugo Statstcal techques for samplg ad motorg atural resources. Ge. Tech. Rep. RMRS-GTR-6. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Research Stato. p. Abstract We preset the statstcal theory of vetory ad motorg from a probablstc pot of vew. We start wth the bascs ad show the terrelatoshps betwee desgs ad estmators llustratg the methods wth a small artfcal populato as well as wth a mapped realstc populato. For such applcatos, useful ope source software s gve Appedx 4. Varous sources of acllary formato are descrbed ad applcatos of the samplg strateges are dscussed. Classcal ad bootstrap varace estmators are dscussed also. Numerous problems wth solutos are gve, ofte based o the expereces of the authors. Key addtoal refereces are cted as eeded or desred. Ackowledgmets We owe a cosderable measure of grattude to revewers for valuable commets. Chstopher Kle, Stee Magusso, Ray Czaplewsk, Rudy Kg, ad Keth Reolls revewed the etre book, Geoff Wood revsed a earler draft, ad Jule Caylor took the tme to check the remote sesg secto ad added some updates to t. Mke Wllams revewed the coarse woody debrs secto, Jm Baldw ad Paul Gessler the wldlfe secto, Ro McRoberts ad Nck Crooksto the modelg secto, Frak Roesch ad Paul Patterso the secto o dscrete varable samplg, Charles Scott the secto o multlevel samplg, ad Gretche Mose the secto o small area estmato. Gary Boyak revewed ad partally rewrote the sectos o GIS ad GPS, Hery Lachowsk revewed the secto o GIS, ad Jeff Goebel revewed the secto o samplg for rare evets. Tm Gregore ad Km Iles made useful suggestos o how to hadle boudary trees. Lae Eskew dd a excellet job of revewg the mauscrpt from a edtoral pot of vew ad makg sure that t coformed wth acceptable stadards of publcato. We owe a cosderable debt of grattude to Ray Czaplewsk for suggestg updatg Frak Freese s book ad facltatg the subsequet wrtg of ths book. The Authors Has T. Schreuder s a Mathematcal Statstca (retred) wth the USDA Forest Servce s Rocky Mouta Research Stato Fort Colls, CO. Rchard Erst s a Mesuratost wth the USDA Forest Servce s Forest Maagemet Servce Ceter (Washgto Offce) Fort Colls, CO. Hugo Ramrez-Maldoado s a Drector Geeral wth the Natoal Isttute o Forestry, Agrculture ad Amal Husbadry Research Mexco Cty, Mexco. You may order addtoal copes of ths publcato by sedg your malg formato label form through oe of the followg meda. Please specfy the publcato ttle ad seres umber. Fort Colls Servce Ceter Telephoe (970) FAX (970) E-mal [email protected] Web ste Malg address Publcatos Dstrbuto Rocky Mouta Research Stato 40 West Prospect Road Fort Colls, CO 8056 Rocky Mouta Research Stato Natural Resources Research Ceter 50 Cetre Aveue, Buldg A Fort Colls, CO 8056

4 Cotets I. Itroducto... II. Objectves of Samplg ad Motorg... Why Sample?... Plag Your Survey... 3 Objectves... 3 Iformato to be collected... 4 Developg the samplg approach... 5 III. Samplg Cocepts ad Methodologes... 6 Samplg Frame... 6 Purposve ad Represetatve Samplg... 6 Populatos, Parameters, Estmators, ad Estmates... 7 Bas, Accuracy, ad Precso... 8 Varables: Cotuous ad Dscrete... 9 Dstrbuto Fuctos... 0 Tools of the Trade... 0 Notato... 0 Characterzg a dstrbuto usg measures of cetral tedecy ad dsperso... Stadard errors ad cofdece lmts... 4 Expaded varaces ad stadard errors... 5 Coeffcet of varato... 6 Covarace, correlato, ad regresso... 6 Idepedece... 8 Varaces of products, ratos, ad sums... 9 Trasformato of varables... IV. Samplg Strateges... Desgs Wth the Horvtz-Thompso Estmator... Varace Estmato Geeral Regresso ad Rato Estmators Some Specfc Forestry Samplg Methods Sample Sze Determato... 4 Groud Samplg Edge Effects Whe Samplg at Stad Boudares Desg Issues Istrumetato Samplg for Coarse Woody Debrs (CWD) Wldlfe Samplg V. Samplg Methods for Dscrete Varables... 5 Smple Radom Samplg (SRS) for Classfcato Data... 5 Cluster Samplg for Attrbutes Cluster Samplg for Attrbutes Wth Uequal-Szed Clusters Samplg of Cout Varables VI. Remote Sesg ad Other Acllary Iformato Remote Sesg ad Photography Accuracy of Remotely Sesed Iformato... 6 Global Postog System for Spatal Locato Needs Geographc Iformato System (GIS) Small Area Estmato... 65

5 VII. Samplg for Rare Evets VIII. Multple Level Samplg Multstage Samplg Multphase Samplg IX. Motorg Over Tme X. Buldg Models ad Cause-Effect XI. Forest Samplg Stuatos Ptfalls...79 Suggestos XII. Refereces... 8 XIII. Glossary Appedx. Iferece Appedx. Dstrbutos Cotuous Dstrbutos Normal dstrbuto Gamma dstrbuto... 9 Multvarate dstrbutos... 9 Dscrete Dstrbutos... 9 Bomal dstrbuto... 9 Hypergeometrc dstrbuto Posso dstrbuto... 9 Multomal dstrbuto Multvarate hypergeometrc dstrbuto Laws of large umbers Appedx 3. Tables Appedx 4. Statstcal Aalyss Worked Examples... 0 Aalyss Software... 0 Data Sets... 0 Results Idex... 08

6 I. Itroducto The purpose of ths book s to serve as a complete troducto to the statstcal techques of samplg atural resources startg at a very basc level ad progressg to more advaced methods. We descrbe supplemetary tools ad materals ad detfy key refereces for readers wshg to pursue the subject further. Cosderable materal s based o drect expereces of the authors. We clude troductory materal, much of whch s take from the excellet troductory book of Freese (96). These sectos Freese s book are expressed a compellg ad stll relevat maer. A good example s Chapter V: Samplg Methods for Dscrete Varables. More advaced readers ca skp these sectos. To facltate readg geeral, we dspese wth the proof of propertes of estmators, such as ther ubasedess ad how the varaces of estmators are derved. Schreuder spet most of hs career wth the USDA Forest Servce workg for Forest Ivetory ad Aalyss Program (FIA). Erst teaches umerous short courses forest vetory ad samplg ad provdes cosultato o such methods to vetory ad measuremet specalsts the Natoal Forest System (NFS). Ramrez-Maldoado has cosderable experece teachg courses forest samplg, vetory, ad modelg as well as cosultg for Mexca ageces forest vetory ad motorg. There are several good troductory books avalable o samplg. The book by Johso (000) s very basc ad gves extesve formato. It s dated, however, that t does ot cover more recet advaces the feld. The books by Freese (96) ad devres (986) are stll useful provdg several of the basc cocepts, the latter gog further afeld what s avalable. Freese s book has the addtoal advatage that t s avalable Spash. Shver ad Borders (996) provde a moderzed verso of Freese (96) materal wth some emphass o typcal forestry methods. More advaced books are avalable, too, forestry. To a large degree ths book represets a smplfcato of the book by Schreuder ad others (993). The book by Iles (003) reveals why he s such a good speaker ad wrter; t s a delght to read ad s worth examg for practcal suggestos. Gregore ad Valete (004), as judged from the outle of ther book, appear to have cosderable overlap wth ours, but t s more lkely to be tghtly wrtte ad amed at a more sophstcated audece. It s also more lmted ts objectves but helpfully cotas umerous proofs showg the propertes of varous estmators. Arvats ad Rech (004) provde the most complete descrpto of geostatstcal methods forestry, methods whch rely heavly o models at ths tme. For readers terested obtag a full uderstadg of how ad why probablstc samplg methods work, the classcal books Sardal ad others (99) ad Cassel ad others (977) are madatory readgs ad surprsgly easy to read gve ther strog theoretcal oretato. The book by Cochra (977) s stll qute popular wth may practtoers ad presets the basc samplg theory well (wth a few exceptos). It too s avalable Spash. USDA Forest Servce RMRS-GTR

7 II. Objectves of Samplg ad Motorg Before dscussg the methodology of survey samplg, some bref commets about statstcs are desrable. What s t? Geerally, statstcs should be thought of as systematzed commo sese. It protects us from jumpg to coclusos. A good example s the classcal expermet o the effect of aspr o headaches. Ital tests showed that t helped 80 percet of the people who tred t, certaly a pheomeal success rate. The someoe had the dea of tryg a placebo. It showed a 60 percet success rate, dcatg that although aspr was useful, may people apparetly dd ot eed t to releve ther headache. Because so may thgs are ope to dfferet terpretatos ad because the USA s such a ltgous socety, statstcs have become crtcal may felds of edeavor cludg atural resources. Hece statstcally vald samplg methods ad surveys have become mportat geeratg relable ad legally defesble estmates. Surveys, also called vetores, are the bass for strategc, maagemet, ad project plag by geeratg a relable data base. Sce a cesus or complete cout of resources would be prohbtvely costly ad tme cosumg, samplg of forest resources bega aroud the begg of the 0 th cetury (Schreuder ad others 993). Before desgg a sample survey, the objectves must be clearly defed. May surveys are started wth a sgle lmted objectve, e.g., we just wat to kow how much tmber s avalable for harvestg from a certa area or what areas may support umerous (ofte uspecfed) rare plat speces. May of these surveys are subsequetly used for other purposes. Ofte the ovce sampler speds much moey collectg data o a large umber of tems ad the caot aswer specfc questos. If a survey s plaed, partcularly a large-scale oe, t s hghly desrable to crtcally exame the data to be collected to be sure that the survey truly addresses the requremets of the proposed users. Questos to be asked may be: Are objectves well defed ad attaable? Are measuremets o weeds really eeded? If tree qualty s cosdered a mportat varable but caot be measured relably, s t stll worth measurg? Remember: you may be blamed for falure to pla ahead eve though your users may have assured you that they had oly lmted objectves or that tmber really was ot more mportat tha other formato or that they dd ot have eough moey to face the survey properly. Why Sample? The purpose of samplg s to draw fereces about a populato of terest such as what s the average heght of trees a forest. The overall feld of ferece s very broad ad techcal ad s dscussed more detal the secto o ferece Appedx. There are may ways of makg fereces ad people ca dffer vehemetly o how to get the ecessary formato ad how to draw fereces/coclusos o the bass of that formato. We focus o a lmted part of the feld of ferece, the drawg of probablstc samples from fte populatos, ad the fereces typcally made wth such data. Most decsos lfe are made wth complete kowledge. Your physca may dagose dsease from a few drops of blood or mcroscopc sectos of tssue; a cosumer judges watermelos by the soud they emt whe thumped; ad we select toothpaste, surace, vacato spots, mates, ad careers wth but a fragmet of the total formato ecessary or desrable for complete uderstadg. Our hope s that the drops of blood or the tssue samples represet the o-sampled portos of the body, the souds of the watermelos dcate the maturty of the melo, ad that the advertsg clams preset a hoest represetato of the truth. Partal kowledge s ormal. The complete cesus s rare; the sample s more usual. A rager advertses tmber sales wth estmated volume, estmated grade yeld ad value, estmated cost, ad estmated rsk. Bdders take the accuracy ad relablty of ths formato at ther ow rsk ad judgmet. The ursery specalst sows seed whose germato s estmated from a ty fracto of the seed lot, ad at harvest estmates the seedlg crop wth sample couts the ursery beds. Maagers pla the mateace of recreato areas based o past use ad experece. USDA Forest Servce RMRS-GTR

8 Typcally we collect formato from a populato. We call ths a sample. We the summarze ths formato some maer. Probably the most wdely used sample summarzato s the sample mea. Assume that we ca take samples of 3 uts from some populato, the our judgmet ofte s based o the mea of the three,.e., y+ y + y3 y = wth y the value for the varable o sample ut, =,,3. 3 However desrable a complete cesus may seem, there are good reasos why samplg s ofte preferred. I the frst place, complete measuremet or eumerato may be mpossble, e.g., determg the exact amout of wood a forest would cost may tmes ts value; the ursery specalst would be better formed f the germato capacty of all the seed to be sow was kow, but the destructve ature of the germato test precludes testg every seed. Clearly where testg s destructve, some sort of samplg s escapable. Use of a recreato area s ot kow utl the seaso s over; judgg what resources are eeded to maage a area has to be based o prevous formato. Samplg frequetly provdes the essetal formato at a far lower cost tha complete eumerato. For large populatos especally, the data collected s ofte more relable. There are several reasos why ths mght be true. Wth fewer observatos to be made ad more tme avalable, crews wll become less tred ad rema more commtted to careful measuremet. I addto, a porto of the savgs cost could be used to buy better strumets ad to employ or tra hghly qualfed persoel. Certaly careful measuremets o 5 percet of the uts a populato could provde more relable formato tha careless measuremets o 00 percet of the uts. Fally, sample data ca usually be collected ad processed a fracto of the tme requred for a complete vetory, so that the formato obtaed s tmely. Objectves Plag Your Survey The frst step s to defe the objectves, also cosderg the possblty that they may be amplfed, modfed, or exteded over tme. Successful surveys ofte are cotued over tme wth addtoal objectves added o. For large-scale forest surveys such as the oe the USA, where a atoal survey s coducted by the Forest Ivetory ad Aalyss (FIA) staff of the FS, the objectves have chaged over tme. Ths s what oe should expect wth successful surveys. The objectves of most surveys are covered by the followg set of objectves for large-scale surveys such as FIA:. Geerate curret status estmates such as acreage forest area, amout of wood volume by speces groups, mortalty, tmber volume avalable for harvest, etc.. Motor chage the above ad other parameters over tme. 3. Establsh procedures requred for detfyg possble cause/effect relatoshp hypotheses. 4. Establsh procedures requred to prove or documet cause-effect. Sce cause-effect ca rarely be establshed wth survey data ad usually requres followup expermetato, t s mportat to dcate what ca ad caot be doe ths regard. 5. Provde -place formato for maagers by proper developmet of such techques as usg maps cojucto wth small area estmato. 6. Provde tmely formato for decso makers. 7. Mata a relable database wth comprehesve documetato ad relable archvg, ad ecourage better ad more aalyses. Orgally FIA was establshed for objectve #. Over tme as cocer for tmber supples became more crtcal, # became as mportat. #3 became mportat the 980s wth the cotroversy of apparet declg forest growth the state of Georga. #4 almost always requres both survey samplg ad expermetato. Much research s beg doe o #5 rght ow. #6 wll be a crtcal oe for maagemet areas, ad #7 has always bee mportat but wll become eve more so wth a aualzed vetory where dustry ad the states ca ad wat to aalyze the data much USDA Forest Servce RMRS-GTR

9 more frequetly ad depedetly. FIA has chaged from a perodc to a aual approach, so stead of collectg data every 0 years, betwee 0 ad 0 percet of the atoal plots wll be measured every year wth reports o a state bass every 5 years. Ths was doe to meet the creasg eed for more curret formato. A classc o plag surveys s Hush (97). Ths provdes useful supplemetary readg to help such plag. Partcularly, hs appedx Sample outle for preparg vetory plas would be a very useful startg pot for people cotemplatg a brad ew survey. Iformato to be collected For most objectves, exstg probablstc survey desgs ca be used or modfed readly to satsfy oe s objectves. Cause-effect s a dfferet ssue, dealt wth more detal later. Ofte the credblty of the results from a vetory ad motorg system s of paramout mportace. Ths requres strget crtera the survey. Some or all of the followg crtera ad thoughts should geerally be cosdered for ay reasoable survey (Schreuder ad Czaplewsk 99):. Oly varables wth eglgble measuremet errors or oes that ca be effcetly calbrated wth varables wth such eglgble measuremet errors should be used. Subjectve observatos have hgh rates of measuremet error ad upredctable bases that ca compromse ther utlty; objectve measuremets ca readly be justfed usually eve f more expesve to collect.. Destructve samplg caot be allowed o permaet sample plots. Off-plot destructve samplg mght be acceptable the mmedate vcty of the plot. 3. Exact locatos of permaet sample plots eed to be kept secret to avod based treatmet by ladowers or vstors or excess vstatos that damage vegetato or sol ad make t urepresetatve of the populato. 4. Defe all varables so o cofuso s possble. 5. Defe some varables as truth beg measured from remote sesg sources rather tha by groud samplg. Remote sesg terpretato ca be more effcet ad accurate tha feld measuremet for some varables, prevetg the advertet dsturbace of plots by feld crews ad deyg access to plots by ladowers. I some cases there s some flexblty the defto of a varable of terest; for example, caopy cover measured from low-alttude photos as opposed to estmated from groud samples. 6. Do t protect sample plots dfferetly from the remader of the populato as s ofte doe for growth ad yeld plots. Closely related to ths s the mportace of defg varables wth the followg characterstcs:. Those that ca be accurately measured o aeral photos so that feld samplg s ot ecessary. For example, some cases ths may be possble wth percet caopy cover or chage area of mature forest but ot chage commercally sutable forest.. Varables that ca easly be measured the feld such as tree mortalty ad umber of trees. Such varables may ofte be correlated wth varables measured o aeral photos. 3. Varables dffcult or expesve to measure the feld. Examples are tree volume, tree crow codto, ad horzotal ad vertcal vegetato profles. Surrogates should be sought such as basal area for volume. 4. Varables for whch a wth-growg seaso dstrbuto may be desred such as rafall amouts ad dstrbuto, ozoe cocetratos, chemcal composto of tree compoets, ad symptoms of arthropod ad mcrobal effects o trees. Ths requres more tha oe vst a seaso, somethg we ofte caot afford surveys. 5. Varables for whch destructve samplg s requred such as sol ad eedle samples for chemcal composto ad tree cores for tree growth ad dedrochroologcal studes. How ths may affect remeasuremet over tme eeds to be cosdered carefully. 4 USDA Forest Servce RMRS-GTR

10 The followg desg objectves are crtcal:. Collect data o explaatory/stress varables such as rafall defcecy, low sol mosture, exposure to polluto, etc. Ths type of data usually caot be collected o plots yet but are essetal buldg relable models.. Smplcty desg. Ths provdes flexblty over tme ad ease aalyss. 3. Cosstecy of desg over tme. Ths smplfes chage estmato ad detfyg possble cause-effect hypotheses. 4. Flexblty to address ew evrometal or maagemet ssues whle matag desg cosstecy. 5. Flexblty to corporate ew measuremet techologes whle matag desg cosstecy. 6. Ablty to treat each sample ut as a populato. Ths s mportat, for example, classfyg each sample to estmate acreage forest types. Ths meas, for example, o mssg data for a sample ut because of the desg used. 7. Use terpeetratg samplg or smlar methods so samplg testy ca be readly creased tme ad space f eeded. Ths s a ce feature of aualzed vetores f hadled properly. 8. Provde flexblty to accommodate replacemet of plots to deal wth damage caused by the measuremet process (for example, tramplg or destructve samplg) or deal of access to plots by prvate ladowers for example samplg wth partal replacemet. 9. Ablty to hadle mssg data such as plots beg accessble or ladowers deyg access (as oted by C. Kle, accessblty may also be caused by lad mes or wldlfe such as elephats ad los). Iaccessblty s best hadled by settg asde a separate stratum for such plots ad clearly statg the estmated sze of that stratum ad how estmates f ay are geerated for t. Implemet a strog qualty assurace program so that true chages sample plots over tme wll ot be cofouded wth chages measuremet error or subtle detals measuremet protocol. 0. Cosder use of several plot desgs at the same sample locatos. Although ths complcates data collecto, t may well be requred whe a large sute of parameters s of terest. For example, for umber of trees ad total basal area of trees, very dfferet plot desgs are effcet (fxed area ad varable radus plots respectvely). Developg the samplg approach Gve the objectves of oe s survey, the dea s to develop the most cost effcet approach to reach those objectves. Most of the remader of the book s devoted to desgg such effcet samplg approaches by gvg the reader sght to methods avalable ad how ad whe to use them. Bascally what we are developg are samplg strateges, whch cosst of how to collect the data what we call the desg ad how to use the data to estmate the quatty of terest,.e., the estmato process. USDA Forest Servce RMRS-GTR

11 III. Samplg Cocepts ad Methodologes Samplg Frame We all make fereces about populatos based o what s typcally a based sample. Kowg ths ofte drves us crazy talkg wth people. For example: Perso A: teeagers are terrble drvers! Perso B: Oh, o what do you base that statemet? Perso A: Well, whe I was drvg last week I got cut off twce by teeagers. A samplg frame s a complete lst of the sample uts that ca potetally be selected the populato. To avod based fereces such as the oe above about the teeagers, make sure that the populato defed for samplg s the oe of terest as well as the sample uts t cossts of. For example, suppose we are terested the followg two parameters about the cty of Colma the state of Colma, Mexco, ad Fort Colls, Colorado, USA:. The average come of each household.. The average area of lad owed by ladowers. I these examples, a household would tally be the possble sample ut the frst ad a ladower the secod. How do we proceed to lst the two populatos of terests? Is ths mportat? It s crtcal that each sample ut the populato has a postve probablty of beg selected for the sample ad that we kow what that probablty s. Usg a lst of people wth phoes s clearly ot a complete lst of all people ether cty, but t s certaly less complete Colma. Sce a lst of households may ot be avalable for ether cty, dfferet sample uts may be cosdered such as cty blocks for whch there geerally would be a lst (how to make fereces about households whe the sample ut s a cty block wll be explaed later uder cluster samplg). Obtag a lst of all ladowers would probably be farly easy for both ctes sce all ladowers presumably pay taxes ad hece ca be foud o a tax lstg for the ctes. Selectg a represetatve sample from a populato s easest whe we have a complete lst of all uts (sample uts) from whch to draw a represetatve sample. For example, assume N = umber of ha ad N = umber of trees the same forest. Clearly we ofte kow N but rarely N. If all N ha are lsted, we ca take a smple radom sample (SRS) of ha plots. The we have a radom sample of plots but wth a dfferet umber of trees per plot usually. It s geerally ot easy (ad effcet) to draw a radom sample of trees from a populato of trees. It s possble to draw a radom sample wthout havg a samplg frame. The a lst of uts s avalable oly after samplg (see Sect 3.4 p.7-73 Schreuder ad others 993, where a procedure, descrbed by Pkham 987 ad Chao 98, s dscussed). However, the procedure s awkward to mplemet. Purposve ad Represetatve Samplg I purpose samplg, also called o-probablstc or model-based samplg, samples are selected more or less delberately. Ths ca be doe o the bass of the judgmet of the sampler of what s a desrable sample or whatever sample happes to be coveet to collect. Ths s geerally ot cosdered a represetatve sample of the populato of terest. The dea of selectg a represetatve sample from a populato was extesvely dscussed the lterature datg from early the 0 th cetury (Johso ad Kotz 988, Vol. 8, p. 77-8). Eght methods of selecto have bee descrbed, but the method of radom or probablty samplg dscussed below s geerally favored. The basc dea s to select a sample completely by chace selecto to esure that there s o persoal bas volved selectg t. To do ths, we use radomzato selectg the sample,.e., select a sample from a delberately haphazard arragemet of observatos. To mplemet ths, we use probablstc samplg, whch volves samplg such a way that:. Each ut the populato s potetally selected wth a kow postve probablty of selecto.. Ay two uts the populato have a jot postve probablty of selecto. 6 USDA Forest Servce RMRS-GTR

12 Problem: A property comprses 00,000 ha of forest, rage, ad water ad the ower wshes to fd out what s there. Develop a samplg method that satsfes the two codtos above. Aswer: There are umerous ways of dog so. Oe approach: dvde a map of the property 00,000 -ha plots ad radomly select 0 of these -ha plots for classfcato to the categores forest, rage, ad water. Ths satsfes the two codtos. Estmato may be dffcult because some of the plots may cota more tha oe of the classes; but how to deal wth that wll be covered the estmato theory later o. A samplg strategy s comprsed of a samplg desg ad assocated estmato theory where a samplg desg states formally how samples are selected. Potetal sample uts ca have equal or uequal probabltes ad jot probabltes of selecto meetg the above two crtera. Ths flexblty leads us to the desgs dscussed later,.e., SRS, stratfed samplg, cluster samplg, ad varable probablty samplg. To complete the pcture of our samplg strategy, we eed estmators accordat wth the desg selected. Samplg desgs wth the smplest estmator, addtoal estmators, ad some geeral samplg procedures are dscussed below. Populatos, Parameters, Estmators, ad Estmates The cetral oto ay samplg problem s the exstece of a populato. It s helpful to thk of a populato as a collecto of uts wth values of terest attached. The uts are selected some way ad the values of terest are obtaed from the selected oes some maer, ether by measuremet or observato. For example, we may mage a 40-ha tract of tmber whch the ut beg observed s the dvdual tree ad the value beg observed s tree heght. The populato s the collecto of trees wth heghts o the tract. The aggregate umber of braches o these same trees would be aother populato as would the umber of trees wth hollows sutable for amal estg. To characterze the populato as a whole, we ofte use certa costats of terest that are called parameters, usually symbolzed wth Greek letters. Some examples are the mea umber of trees per plot a populato of plots; the proporto of lvg seedlgs a pe platato; the total umber of shrub speces a populato; ad the varablty amog the ut values. The objectve of sample surveys s usually to estmate such parameters. I the past, we teded to estmate the populato mea or total of oe or more varables. Nowadays, we are ofte also terested possble explaatos of why a parameter s a certa value. The value of the parameter as estmated from a sample wll hereafter be referred to as the sample estmate or smply the estmate, symbolzed by Roma letters. The mathematcal formula geerally used to geerate a estmate s called a estmator. Wheever possble, matters wll be smplfed f the uts whch the populato s defed are the same as those to be selected the sample. If we wsh to estmate the total weght of earthworms the top 5 cm of sol for some area, t would be best to thk of a populato made up of blocks of sol of some specfed dmeso wth the weght of earthworms the block beg the ut value. Such uts are easly selected for cluso the sample, ad projecto of sample data to the etre populato s relatvely smple. If we thk of dvdual earthworms as the uts, selecto of the sample ad expaso from the sample to the populato may be very dffcult f ot mpossble. Problem: How would you go about samplg a at est to estmate the umber of ats t? Aswer: If the est ca be destroyed, oe ca scoop t up, take samples of a certa volume from t at radom, ad cout the umber of ats each of these samples. If t caot be destroyed there s really o obvous way to take a represetatve sample from the est to cout the ats. USDA Forest Servce RMRS-GTR

13 Bas, Accuracy, ad Precso Whe estmatg populato parameters, oe wshes to obta good estmates close to the true values at a reasoable cost. Whe oly a part of a populato s measured, some estmates may be hgh, some low, some close, ad some far from the true value. A estmate that s close to the true value s cosdered accurate. If the perso selectg or measurg a sample s prejudced some way, the the estmate may be based. For example, f oe was terested the recreatoal prefereces of vstors to a park ad tervewed a sample of 99 wome ad ma, oe mght feel ueasy about bas the results, just as oe mght f the results for a survey of 50 me ad 50 wome showed a heavy preferece for fshg a park ot oted for fshg ad also kowg that the tervewer was a avd fsherma. Though most people have a geeral oto as to the meag of bas, accuracy, ad precso, statstcas have well-defed expressos for them because they are crucal ther area of expertse, as follows: Bas Bas s a systematc dstorto that ca arse whe selectg a sample, durg ts measuremet, or whe estmatg the populato parameters. Bas due to samplg selecto arses whe certa uts are gve a greater or lesser represetato the sample tha the populato. Ths s ot compesated for estmato. Assume for example that we are estmatg the recreatoal prefereces of vstors to a park ad we oly tervew people o weekeds. The results wll be based because weekday users had o opportuty to appear the sample. Measuremet bas ca result. For example, f seedlg heghts are measured wth a ruler from whch the frst half-cm s mssg, all measuremets wll be oe-half cm too large ad the estmate of mea seedlg heght wll be based. I studes volvg tree couts, some observers may always clude a tree that s o the plot boudary whle others may cosstetly exclude t. Both routes are sources of measuremet bas. I tmber crusg, the volume table selected or the maer whch t s used may result bas. For example, a volume table costructed from data of tall trees wll gve based results whe used wthout adjustmet o short trees. Smlarly, f a cruser cosstetly overestmates tree heghts, volume tables usg heghts as put wll be based. The oly practcal way to mmze measuremet bas s by metculously trag the crews measuremet procedures ad the use, care, ad calbrato of strumets. The techque used to estmate the populato parameters from the sample data s also a possble source of bas. For example, f the recreato preferece o two atoal forests s estmated by takg a smple arthmetc average of the prefereces recorded o each forest, the result may be serously based f the area of oe forest s 500,000 ha ad t has a mllo vstors aually ad the other s 00,000 ha sze wth oly 0,000 vstors aually. A better overall estmate would be obtaed by weghg the estmates for the two forests proporto to ther szes ad/or ther umber of vstors. Aother example of ths type of bas occurs the commo forestry practce of estmatg the average dameter of trees a forest from the dameter of the tree of mea basal area. The latter procedure actually gves the square root of the mea squared dameter, whch s ot the same as the arthmetc mea dameter uless all trees are exactly the same sze. Selecto ad measuremet bases are rarely acceptable, partcularly f the data are of terest to several users. However, estmato bas ca ofte be acceptable sce some based estmators are much better tha ubased oes, the bas beg ofte trval ad the results more precse tha those acheved usg the ubased procedures. Acceptable based estmators are usually asymptotcally ubased estmators, defed as follows: Asymptotcally ubased If the bas of a estmator approaches 0 as the sample sze approaches the populato sze, the estmator s cosdered to be asymptotcally ubased. Such a estmator used to be called cosstet, for example Cochra (977). Precso ad accuracy A based estmate may be precse but t ca ever be accurate. Amog statstcas accuracy refers to the success of estmatg the true value of a quatty; precso refers to the extet of clusterg of sample values about ther ow average, whch, f based, caot be the true value. 8 USDA Forest Servce RMRS-GTR

14 A target shooter who puts all of hs shots the er crcle of a target mght be cosdered both accurate ad precse; hs fred who puts all of her shots the outer crcle ad ear the o clock posto would be cosdered equally precse but owhere ear as accurate; aother fred who always hts the target at some radom locato would be ubased but ether accurate or precse. Ths s llustrated Fgure below. A forester makg a seres of careful dameter measuremets at a fxed posto o the bole of a tree wth a calper, oe arm of whch s ot at rght agle to the graduated beam, would acheve precse but ot accurate results. Sce the calper s ot properly adjusted, the measured values wll be off the true value (bas) ad the dameter estmate wll be accurate. If the calper was properly adjusted but used carelessly, the measuremets would be ubased but ether accurate or precse. Geerally we strve to use estmators that predct a parameter more relably tha competg estmators where relablty s usually measured by the rato of the mea square errors of the estmators. Such estmators are called effcet estmators. Varables: Cotuous ad Dscrete Varato s a fact of lfe. Copg wth some of the samplg problems created by varato s a mportat part of makg vald fereces. All objects have characterstcs such as sze, shape, ad color. A characterstc that vares from ut to ut s called a varable. I a populato of trees, tree heght s a varable, as are tree dameter, umber of coes, volume, form class, ad speces. The umber of people each recreato group s a varable, as are ther sex, ther age, etc. A varable that s expressed a umercal scale of measuremet, ay terval of whch may, f desred, be subdvded to a fte umber of values, s sad to be cotuous, e.g., tme recreatg, heght, weght, precptato, ad volume. Qualtatve varables ad those that are represeted by tegral values or ratos of tegral values are sad to be dscrete. Two forms of dscrete data may be recogzed: attrbutes ad couts. A attrbute refers to uts classfed as havg or ot havg some specfc qualty; examples of attrbutes mght be speces or whether trees are alve or dead. Results are ofte expressed as a proporto or percet, e.g., cdece of rust slash pe seedlgs, survval of plated seedlgs, ad the percetage of users from a partcular coutry of a recreato area. A cout refers to uts descrbed by a umber, such as umber of people a recreato group, umber of weevls a coe, ad umber of sprouts o a tree stump. Imprecse Ubased (Iaccurate) Based (Iaccurate) Ubased (accurate) Based (accurate) Precse Fgure. A example of bas, precso, ad accuracy f average dstace to plot ceter s used estmatg dstace to ceter of target for fve shots. USDA Forest Servce RMRS-GTR

15 A dstcto s made betwee cotuous ad dscrete varables because the two types of data may requre dfferet statstcal procedures. Most of the samplg methods ad computatoal procedures dscussed ths book are for use wth cotuous varables. The procedures for dscrete varables are geerally more complex. Ofte dscrete varables ca be treated as cotuous, especally for larger sample szes ad a large umber of classes. Dstrbuto Fuctos Dstrbuto fuctos for populatos show the relatve frequecy wth whch dfferet values of a varable occur. Gve such a fucto, the proporto of uts wth certa sze lmts ca be estmated. Each populato has ts ow dstct dstrbuto fucto but these ca ofte be approxmated by certa geeral types of fucto such as the ormal, bomal, ad Posso. The bell-shaped ormal dstrbuto, famlar to most foresters, s ofte used whe dealg wth cotuous varables. The bomal dstrbuto s used wth attrbutes. The Posso dstrbuto s used wth couts havg o fxed upper lmt, partcularly f zero or very low couts ted to predomate. Several of the more mportat dstrbutos are descrbed Appedx. The form of the dstrbuto fucto dctates the approprate statstcal treatmet of a set of data whle ts exact form wll seldom be kow. Some dcatos may be obtaed from the sample data or from a geeral famlarty wth the populato. The methods of dealg wth ormally dstrbuted data are smpler tha most of the methods that have bee developed for other dstrbutos. Fortuately, t has bee show that, regardless of the dstrbuto of a varable, the meas of large samples ted to follow a dstrbuto that approaches the ormal. Ths approach to ormalty s ofte used for assessg the relablty of sample-based estmates. Notato Tools of the Trade I descrbg varous samplg methods, frequet use wll be made of subscrpts, brackets, ad summato symbols. These devces are, lke the more famlar otatos of +, -, ad =, a cocse way of expressg deas that would be cumbersome f put to covetoal laguage. Usg ad uderstadg them s just a matter of practce. Subscrpts The appearace of x, z jk, or y lm s aoyg to dvduals ot accustomed to them. Yet terpretg ths otato s smple. I x, the subscrpt meas that x ca take o dfferet forms or values. Isertg a partcular value for tells whch form or value of x we are cocered wth. mght mply a partcular characterstc of a dvdual ad x mght be ts heght, x ts weght, x 3 ts age, ad so forth. Or the subscrpt mght mply a partcular dvdual. I ths case, x could be the heght of the frst dvdual, x that of the secod, x 3 that of the thrd, ad so forth. Whch meag s teded wll usually be clear from the cotext. A varable (say x) wll ofte be detfed more tha oe way. Thus, we mght wat to refer to the age of the secod dvdual or the heght of the frst dvdual. Ths dual classfcato s accomplshed usg two subscrpts. I x k, mght detfy the characterstc (for heght, = ; for weght, = ; ad for age, = 3) ad k could be used to desgate the dvdual we are dealg wth. The, x, 7 would tell us that we are dealg wth the weght ( = ) of the seveth (k = 7) dvdual. Ths procedure ca be carred to ay legth eeded. If the dvduals the above example were from dfferet groups we could use aother subscrpt (say j) to detfy the group. The symbol xjk would dcate the th characterstc of the th th k dvdual of the j group. Summatos To dcate that several (say 6) values of a varable ( x ) are to be added together we wrte ( x+ x + x3 + x4 + x5 + x6) or, somewhat shorter, ( x+ x x6). The three dots ( ) dcate that we cotue to do the same thg for all the values from x 3 through x 5. The same operato ca be expressed more compactly by 6 x. = 0 USDA Forest Servce RMRS-GTR

16 I words ths tells us to sum all values of x, lettg go from up to 6. The symbol s the Greek letter sgma, dcatg that a summato should be performed. The x tells what s to be summed ad the umbers above ad below dcates the lmts over whch the subscrpt wll be allowed to vary. If all of the values a seres are to be summed, the rage of summato s frequetly omtted from the summato sg gvg x, x, or sometmes x. All of these mply that we would sum all values of x. The same prcple exteds to varables that are detfed by two or more subscrpts. A separate summato sg may be used for each subscrpt. Thus, we mght have 3 4 = j= Ths tells us to add up all the values of x j, j havg values from to 4 ad from to 3. Wrtte the log way, ths meas ( x + x + x3 + x4 + x + x + x3 + x4 + x3 + x3 + x33 + x34 ). As for a sgle subscrpt, whe all values a seres are to be summed, the rage of summato may be omtted, ad sometmes a sgle summato symbol suffces. The above summato mght be symbolzed by xj, j, x j, or maybe eve x. If a umercal value s substtuted for oe of the letters the subscrpt, the summato s performed by lettg the letter subscrpt vary but holdg the other subscrpt at the specfed value. As a example, Aalogously, 4 x j. x3j = ( x3+ x3 + x33 + x34), ad x = ( x + x + x3 + x4 + x5) j= 3 j 5 = y y dcates that we wat to sum both ad j from to 3 but omt the values whe = j. The sum log had the s yy + yy 3+ yy + yy 3 + yy 3 + yy 3. Brackets Whe other operatos are to be performed alog wth the addto, bracketg may be used to dcate the order of operatos. For example, x tells us to square each value of x ad the add up these squared values. But x dcates to add all the x values ad the square the sum. The expresso j x tells us to square each x j value ad the add the squares. But xj j j j USDA Forest Servce RMRS-GTR

17 dcates that for each value of we should frst add up the x j over all values of j. Next, ths xj s squared ad these squared sums are added up over all values of. If the rage of j s from j to 4 ad the rage of s from to 3, the ths meas 3 4 = j= xj = x + x + x + x + x + x + x + x + x + x + x + x ( ) ( ) ( ) The expresso xj j tells us to add up the x j values over all combatos of ad j ad the square the total. Thus, 3 4 xj = ( x + x + x3 + x4 + x + x + x3 + x4 + x3 + x3 + x33 + x34 ) = j= Where operatos volvg two or more dfferet varables are to be performed, the same prcples apply. 3 x y = x y + x y + x y but, = = = = x y ( x x x3)( y y y3). Note that x s ot usually equal to x. Smlarly, x y s ot usually equal to x y. Factorals For coveece we use the followg mathematcal otato for factorals,.e.,! = ( )( )... where s a teger ad where 0!=. Characterzg a dstrbuto usg measures of cetral tedecy ad dsperso The dstrbuto of values for a populato varable s characterzed by costats or parameters such as the mea ad the varace. The measure of cetral tedecy gves some dea of the typcal or mddle value of the dstrbuto of a varable. The prcpal measures used are the mea, meda, ad mode. Measures of dsperso dcate how much heterogeety there s the dstrbuto of the varable. They summarze the degree to whch values of the varable dffer from oe aother. The most commo oes used are the varace or ts square root, the stadard devato, ad the rage. Measures of cetral tedecy Probably the most wdely kow ad used populato parameter s the mea. Gve a sample where all uts have the same probablty of selecto, the populato mea s estmated by y = y = ().. USDA Forest Servce RMRS-GTR

18 wth sample sze ad y the value for the varable o sample ut. For example, f we have tree heghts for 5 out of 0 trees wth heghts 0, 0, 5, 30 ad 35 m, the our estmated mea heght for the 0 trees s y = = 6 m. 5 Other estmators of cetral tedecy ca be useful occasoally too. For example, the meda s the value so that half are larger ad half are smaller tha ths value. I ths example the meda would be 5. The mode s the most commo value occurrg the data set, whch would be 0 ths case. The meda fds some utlty a estmate of cetral tedecy for hghly skewed populatos, the classcal oe beg come of people. Clearly the fact that there s a umber of people the USA, for example, that have comes of several mllo dollars a year ad others wth less tha $0,000 makes the sample mea rather a poor dcator of cetral tedecy; the meda would be more approprate. Smlarly a stad geerated by seed trees, the presece of some huge dameter seed trees may make the meda a more meagful estmate as a measure of the cetral tedecy for such a stad. If terest s detfyg beetle-fested stads where oly recetly attacked trees may be saved, t may be desrable to detfy stads where such trees are the most commo ad the mode would be the best dcator of that. Johso (000) gves a detaled descrpto of the above three measures of cetral tedecy plus several others. We focus o the mea ad the correspodg total Y = Ny ths book, where N s the total umber of sample uts the populato. A measure of dsperso Although there are several measures, we wll oly dscuss the varace or ts square root, the stadard devato, because t s used by far the most statstcs. I ay populato, such as a stad of trees, the characterstc of terest wll usually show varato. For example, there wll be varato tree heght. Older trees wll be cosderably taller tha youger oes ad both wll vary from a overall mea stad heght. More observatos would be eeded to get a good estmate of the mea heght of a stad where heghts vary from to 80 m tha where the rage s from 0-5 m. The measure of varato most commoly used by statstcas s the varace. The varace of a populato characterstc such as tree heght s a measure of the dsperso of dvdual values about ther mea. A large varace dcates wde dsperso ad a small varace lttle dsperso. Ths varace s a populato characterstc (a parameter) ad s usually deoted by σ. Most of the tme we do ot kow the populato varace so t has to be estmated from sample data. For most types of measuremets, the estmate of the varace from a smple radom sample s gve by ( y ) y = s = () where s s the sample estmate of the populato varace, ad y s the arthmetc mea of the sample, as defed () above. Sometmes, computato of the sample varace s smplfed by rewrtg the above equato as y y = y y = = s. (3) = = Suppose we have observatos o three uts wth the values 7, 8, ad. For ths sample our estmate of the varace s 7 ( ) s = = = 7. USDA Forest Servce RMRS-GTR

19 Note that the uts o varace are the square of the uts of the observatos. If terest s 3 heght meters (m) the the varace wll be m. If terest s tree volume m the the 3 varace would be m squared. To avod puzzlemet we wll ot show the uts of the varaces. Also, we do ot dstgush betwee populato values Y ad sample values y. It has bee our experece that ths dstcto s uecessary ad cofusg for the objectves of ths book. The stadard devato s the square root of the varace ad s expressed the same uts as the mea ad the varable. It s symbolzed by s, ad the above example would be estmated as s = 7 = Both the terms varace ad stadard devato are used extesvely statstcs. Stadard errors ad cofdece lmts Sample estmates are subject to varato just lke the dvdual uts a populato. The mea dameter of a stad as estmated from a sample of three trees wll frequetly dffer from that estmated from other samples of three trees the same stad. Oe estmate mght be close to the mea but a lttle hgh, aother mght be much hgher, ad the ext mght be below the mea. The estmates vary because dfferet uts are observed the dfferet samples. Ad oe would expect that geerally, a sample of sze sx would geerate better estmates tha a sample of sze three. It s desrable to have some dcato of how much varato mght be expected amog sample estmates. A estmate of mea tree dameter that would ordarly vary betwee ad cm would spre more cofdece tha oe that mght rage from 7 to 6 cm, eve though the average s the same. As dscussed above, the varace ad the stadard devato ( σ = stadard devato = varace ) are measures of the varato amog dvduals a populato. Measures of the same form are used to dcate how a seres of estmates mght vary. They are called the varace ad the stadard error of the estmate σ y = stadard error of the estmate of y = varace of the estmate of y ). The term stadard error of estmate s usually shorteed to stadard error whe the estmate referred to s obvous. The stadard error s merely a stadard devato but amog estmates rather tha dvdual uts. I fact, f several estmates were obtaed by repeated samplg of a populato, ther varace ad stadard error could be computed from equato (3) above. However, repeated samplg s uecessary; the varace ad the stadard error ca be obtaed from a sgle set of sample uts. Varablty of a estmate depeds o the samplg method, sample sze, ad varablty amog the dvdual uts the populato, ad these are the peces of formato eeded to compute the varace ad stadard error. For each of the samplg methods descrbed later o, a procedure for computg the stadard error of estmate wll frequetly be gve. Computato of a stadard error s ecessary because a sample estmate may be meagless wthout some dcato of ts relablty. If t takes 00 brds of a rare speces to mata or grow ts populato a forest, we may feel good whe the maager tells us that he estmates there to be 50. But how useful s that formato? If we subsequetly fd out that the actual estmate s betwee 0 ad 300, we have a much more realstc pcture of the real stuato ad realze that we stll do ot kow whether the populato wll survve or ot ad that we have to obta better formato. Fgure from Czaplewsk (003) llustrates the mportace of a good sample sze costructg cofdece tervals. Gve the stadard error of estmate, t s possble to estmate lmts that suggest how close we mght be to the parameter beg estmated. These are called cofdece lmts. For large samples we ca take as a rough gude that, uless a --3 chace has occurred samplg, the parameter wll be wth oe stadard error of the estmated value. Thus, for a sample mea tree dameter of 6 cm wth a stadard error of.5 cm, we ca say that the true mea s somewhere wth the lmts 4.5 to 7.5 cm. I makg such statemets we wll, over the log ru, be rght o average two tmes out of three. Oe tme out of three we wll be wrog because of atural samplg varato. The values gve by the sample estmate plus or mus oe stadard error are called the 67-percet cofdece lmts. By spreadg the lmts we ca be more cofdet that they wll clude the parameter. Thus, the estmate plus or mus two stadard errors wll gve lmts that wll clude the parameter uless a --0 chace has occurred. These are called the 95-percet cofdece 4 USDA Forest Servce RMRS-GTR

20 Frequecy True value N = 4 = 4 N = 4 = N = 403 = 40 N = 40 = 4 Frequecy 0 % 4% 6% 8% 0% % 4% Natoal Scale (bass for Tucker ad Towshed's cocluso) 0 % 4% 6% 4% Sub-cotetal Scale 0 % 4% 4% Cotetal Scale, Ecoflorstc Zoes 0 % 4% 4% Global Scale Fgure. Estmated extet of tropcal deforestato wth a 0 percet sample of Ladsat satellte scees (Czaplewsk 003). lmts. The 99-percet cofdece lmts are defed by the mea plus or mus.6 stadard errors. The 99-percet cofdece lmts wll clude the parameter uless a --00 chace has occurred. It must be emphaszed that ths method of computg cofdece lmts wll gve vald approxmatos oly for large samples. The defto of a large sample depeds o the populato tself but, geeral, ay sample of less tha 30 observatos would ot qualfy. Some techques of computg cofdece lmts for small samples wll be dscussed later for a few of the samplg methods. Expaded varaces ad stadard errors Very ofte a estmate s multpled by a costat to geerate estmates of other parameters, for example gog from a estmate of the mea to a estmate of the total for a populato. If a survey has bee made usg oe-ffth ha plots ad the mea volume per plot computed, ths estmate would be multpled by 5 order to express t o a per-ha bass, or, for a tract of 800 ha, multpled by 4,000 to estmate the total volume. Sce expadg a estmate ths way must also expad ts varablty, t wll be ecessary to compute a varace ad stadard error for these expaded values. Ths s easly doe. If the varable y has varace s ad ths varable s multpled by a costat (say k), the product (ky) wll have a varace of k s. Suppose the estmated mea volume per oe-ffth ha plot s 4 m 3 wth a varace of the mea of 5 gvg a stadard error of 3 5 = 5m. The mea volume per ha s: 5(4) = 70 m 3 ad the varace of ths estmate s 5 5 = 65 wth stadard error 3 65 = 5m. Note that f the stadard devato of y s s or the stadard error of y s s y, the the stadard devato of ky s ks ad the stadard error of ky s ks y. Ths makes sese sce costats have o varablty. So, the above case, sce the stadard error of the estmated mea per oe-ffth ha s 5, the stadard error of the estmated mea volume per ha s 5 5 = 5. A costat may also be added to a varable. Such addtos do ot affect varablty ad requre o adjustmet of the varace or stadard errors. Thus f z = y + k wth y a varable ad k a costat, the sz = sy. Ths stuato arses where for computatoal purposes the data have bee coded by the subtracto of a costat. The varace ad stadard error of the coded ad ucoded values are the same. Suppose we have the three observatos 7, 04, ad 4. For ease of computato, these could be coded by subtractg 00 from each, to make 7, 4, ad 4. (Ths was a mportat advatage the past whe computers had lmted capabltes ad had trouble dealg wth very large values especally whe used computg varaces.) The varace of the coded values s: 45 ( ) s = 3 = 33, the same as the varace of the orgal values USDA Forest Servce RMRS-GTR

21 s 345 ( ) = 3 = 33. Coeffcet of varato The coeffcet of varato, C, s the rato of the stadard devato to the mea. s C = y, (4) Thus, for a sample wth a mea of y =0 ad a stadard devato of s = 4, 4 C = = 0.4 or 40 percet. 0 Varace, our measure of varablty amog uts, s ofte related to the mea sze of the uts; large tems ted to have a larger varace tha small tems. For example, the varace a populato of tree heghts would be larger tha the varace of the heghts of a populato of shrubs. The coeffcet of varato expresses varablty o a relatve bass. The populato of tree heghts mght have a stadard devato of 4.4 m whle the populato of shrubs mght have a stadard devato of m. I absolute uts, the trees are more varable tha the shrubs. But, f the mea tree heght s 40 m ad the mea heght of the shrubs s 5.9 m, the two populatos may have the same relatve varablty,.e., a coeffcet of varato of C = 0.. Varace also depeds o the measuremet uts used. I our example above, the stadard devato of shrub heghts was m. Had the heghts bee measured dm, the stadard devato would have bee 0 tmes as large (f z = 0y, sz = 0sy ) or 6.49 dm. But the coeffcet of varato would be the same regardless of the ut of measure. I ether case, we would have C s 0.649m 6.49dm = = = = 0. or percet y 5.9 m 59 dm. I addto to puttg varablty o a comparable bass, the coeffcet of varato smplfes the job of estmatg ad rememberg the degree of varablty of dfferet populatos. I may of the populatos wth whch foresters deal, the coeffcet of varato could be 00 percet or more. Because t s ofte possble to guess at the sze of the populato mea, we ca roughly estmate the stadard devato. Such formato s useful plag a sample survey. Covarace, correlato, ad regresso Covarace ad correlato are measures of how two varables vary relatoshp to each other (covarablty). I some samplg applcatos two or more varables are measured o each sample ut. I measurg forage producto, for example, we mght record the gree weght of the grass clpped to a heght of cm from a crcular plot m dameter. Later we mght record the ovedry weght of the same sample. We would expect that there would be a postve relatoshp betwee these two varables. Suppose the two varables are labeled y ad x. If the larger values of y ted to be assocated wth the larger values of x, the covarace wll be postve. If the larger values of y are assocated wth the smaller values of x, the covarace wll be egatve. Whe there s o partcular assocato of y ad x values, the covarace approaches zero. Lke the varace, the covarace s a populato characterstc, a parameter. For smple radom samples, the formula for the estmated covarace of x ad y ( s xy ) s ( x x)( y y) = sxy = Computato of the sample covarace s smplfed by rewrtg the formula (5) 6 USDA Forest Servce RMRS-GTR

22 s xy x y = = xy xy xy = = = =. (6) Suppose a sample of = 6 uts produced the followg x ad y values, say gree weght ad ovedry weght of the forage example above: Totals y x The, 54 4 ( ) + ( 4) (8 7) s xy = = = The egatve value dcates that the larger values of y ted to be assocated wth the smaller values of x. Clearly we should be dubous about ths result ad exame more carefully what happeed sce oe would expect larger values of gree weght wth larger values of ovedry weght. The magtude of the covarace, lke that of the varace, s ofte related to the sze of the ut values. Uts wth large values of x ad y ted to have larger covarace values tha uts wth smaller x ad y values. A measure of the degree of lear assocato betwee two varables that s uaffected by the sze of the ut values s the smple correlato coeffcet. A sample-based estmate of the correlato coeffcet, R, s covarace of x ad y sxy sxy r = xy varace(x) varace(y) = ss = ss x y. (7) The correlato coeffcet ca vary betwee ad +. As covarace, a postve value dcates that the larger values of y ted to be assocated wth the larger values of x. A egatve value dcates a assocato of the larger values of y wth the smaller values of x. A value close to + or dcates a strog lear assocato betwee the two varables. Correlatos close to zero suggest that there s lttle or o lear assocato. For the data gve the dscusso of covarace we foud s xy = 4.4. For the same data, the sample varace of x s s x =.0, ad the sample varace of y s s y = 8.4. The the estmate of the correlato betwee y ad x s r xy x y = = = The egatve value dcates that as x creases y decreases, whle the earess of r to dcates that the lear assocato s very close. I ths example we would become eve more suspcous of the results ad hypothesze for example that sample labels were swtched somehow, sce we would expect a strog postve relatoshp betwee gree ad ovedry weght. A mportat thg to remember about the correlato coeffcet s that t s a measure of the lear assocato betwee two varables. A value of r close to zero does ot ecessarly mea that there s o relatoshp betwee the two varables. It merely meas that there s ot a good lear (straght-le) relatoshp. There mght actually be a strog olear relatoshp. Remember that the correlato coeffcet computed from a set of sample data s a estmate, just as the sample mea s a estmate. Lke the sample mea, the relablty of a correlato coeffcet creases wth the sample sze (see Appedx 3, Table 5). Regresso aalyss deals prmarly wth the relatoshp betwee varables of terest ad other varables cosdered to be covarates. The dea s to use formato o the covarates to mprove estmato for the varables of terest ether because formato o the covarates s avalable or USDA Forest Servce RMRS-GTR

23 ca be collected more readly/cheaply tha o the varables of terest. For ths reaso we establsh a lear relatoshp betwee the varable of terest y ad the covarate x such that y = α + βx + e, =,...,N where e, =,..., N are the resduals for the populato wth the average resdual over the populato deoted by Ee ( ), where Ee ( ) = 0 ad the covarate of resduals ad j s deoted by Eee ( j) where Eee ( j) = σ v f = j or Eee ( j) = 0 otherwse; α ad β are regresso coeffcets that are estmated from the data so that we ca predct the y for the x that were ot sampled as well as estmate the mea or total for the varable y; σ v deotes the varace of y at x ( v s ofte represeted as a fucto of x such k as v = x ). The value k s usually assumed kow where k = 0 deotes a costat varace ad k = or are ofte used whe the varace of y s expected to crease learly wth some fucto of x ). σ s usually estmated from the data. Idepedece Whe o relatoshp exsts betwee two varables they are sad to be depedet; the value of oe varable tells us othg about the value of the other. The commo measures of depedece (or lack of t) are the covarace ad the correlato coeffcet. As prevously oted, whe there s lttle or o assocato betwee the values of two varables, ther covarace ad correlato approach zero (but keep md that the coverse s ot ecessarly true; a zero correlato does ot prove that there s o assocato but oly dcates that there s o strog lear relatoshp). Completely depedet varables are rare bologcal populatos, but may varables are weakly related ad may be treated as f they were depedet for practcal purposes. As a example, the aual heght growth of pole-szed loblolly pe domats s relatvely depedet of (8) m /ha). There s also cosderable the stad basal area wth farly broad lmts (say to 30 evdece that perodc cubc volume growth of loblolly pe s poorly assocated wth (.e., almost depedet of) stad basal area over a farly wde rage. The cocept of depedece s also appled to sample estmates. I ths case, however, the depedece (or lack of t) may be due to the samplg method as well as to the relatoshp betwee the basc varables. Two stuatos ca be recogzed: two estmates have bee made of the same parameter or estmates have bee made of two dfferet parameters. I the frst stuato, the degree of depedece depeds etrely o the method of samplg. Suppose that two completely separate surveys have bee made to estmate the mea volume per ha of a forest. Because dfferet sample plots are volved, the estmates of mea volume obtaed from these surveys would be regarded as statstcally depedet. But suppose a estmate has bee made from oe survey ad the addtoal sample plots are selected ad a secod estmate s made usg plot data from both the frst ad secod surveys. Sce some of the same observatos eter both estmates, the estmates would be depedet. I geeral, two estmates of a sgle parameter are ot depedet f some of the same observatos are used both. The degree of assocato wll deped o the proporto of observatos commo to the two estmates. Problem: Two radom samples of sze are take wthout replacemet from a populato. By pure chace, the two samples are detcal. Are the estmates depedet? Aswer: Yes, they are. Problem. I the above example, how would you go about combg the two samples? Aswer. The most sesble soluto would probably be to treat t as a sample of sze wth replacemet eve though each sample tself was take wthout replacemet. The advatage of ths s that the varace estmate would typcally be a overestmate of the actual varace. 8 USDA Forest Servce RMRS-GTR

24 I the secod stuato (estmates of two dfferet parameters) the degree of depedece may deped o both the samplg method ad the degree of assocato betwee the basc varables. If mea heght ad mea dameter of a populato of trees were estmated by radomly selectg a umber of dvdual trees ad measurg both the heght ad dameter of each tree, the two estmates would ot be depedet. The relatoshp betwee the two estmates (usually measured by ther covarace or correlato) would, ths case, deped o the degree of assocato betwee the heght ad dameter of dvdual trees. O the other had, f oe set of trees were used to estmate mea heght ad aother for estmatg mea dameter, the two estmates would be statstcally depedet eve though heght ad dameter are ot depedet whe measured o the same tree. A measure of the degree of assocato betwee two sample estmates s essetal evaluatg the samplg error for estmates from several types of surveys. Procedures for computg the covarace of estmates wll be gve whe eeded. Varaces of products, ratos, ad sums Earler, we leared that f a quatty s estmated as the product of a costat ad a varable (say Q = kz, where k s a costat ad z s a varable) the varace of Q wll be sq = k sz. Thus, to estmate the total volume of a stad, we multply the estmated mea per ut ( y, a varable) by the total umber of uts (N, a costat) the populato. The varace of the estmated total wll be N s y. Its stadard devato (or stadard error) would be the square root of ts varace or Ns y. The varace of a product I some stuatos the quatty whch we are terested wll be estmated as the product of two varables ad a costat. Thus Q = kyx (9) where: k = a costat ad y ad x = varables havg varaces s y ad s x ad covarace s xy. For large samples, the varace of Q s estmated by s y s s x xy sq = Q + + = k x s y + y sx + xys xy y x xy. (0) As a example of such estmates, cosder a forest survey project whch uses a dot cout o aeral photographs to estmate the proporto of a area that s forest ( p ), ad a groud cruse to estmate the mea volume per ha ( v ) of forested lad. To estmate the forested area, the total lad area (N) s multpled by the estmated proporto forested. Ths tur s multpled by the mea volume per forested ha to gve the total volume. I formula form Total volume = Npv where: N = the total lad area ha (a kow costat), p = the estmated proporto of the area that s forested, ad v = the estmated mea volume per forested ha. The varace of the estmated total volume would be s p s s v pv s = ( Npv) + + p v pv. If the two estmates are made from separate surveys, they are assumed to be depedet ad the covarace set equal to zero. Ths would be the stuato where p s estmated from a photo dot USDA Forest Servce RMRS-GTR

25 cout ad v from a depedetly selected set of groud locatos. Wth the covarace set equal to zero, the varace formula would be s p s v s = ( Npv) + p v. Varace of a rato I other stuatos, the quatty we are terested s estmated as the rato of two estmates multpled by a costat. Thus, we have y Q = k. () x For large samples, the varace of Q ca be approxmated by s y s s x xy sq = Q + y x xy () as s stll ofte used, for example, by Freese (96) or Cochra (977). A more robust estmator s ( ) ( ) ( j) j= N f X D v ( ˆ J Yrm) =, (3) where f = /N, X s the populato mea for varable x, ad for every j the sample D( j) s the dfferece betwee the rato ( y y ) j ad the average of these ratos (Schreuder ad others ( x x j ) 993). Varace of a sum Sometmes we mght wsh to use the sum of two or more varables as a estmate of some quatty. Wth two varables we mght have wth k ad k costats ad x ad x varables havg varace The varace of ths estmate s Q3 = kx + kx (4) Q3 s ad s ad covarace s. s = ks + ks + kks. (5) If we measure the volume of sawtmber ( x ) ad the volume of poletmber ( x ) o the same plots (ad the same uts of measure) ad fd the mea volumes to be x ad x, wth varaces s ad s ad covarace s, the the mea total volume pole-sze ad larger trees would be x x. The varace of ths estmate s + s = s + s + s. (6) The same result would, of course, be obtaed by totalg the x ad y values for each plot ad the computg the varace of the totals. Ths formula s also of use where a weghted mea s to be computed. For example, we mght have made sample surveys of two tracts of tmber. Example: Tract Sze = 3,00 ha Estmated mea volume per ha = 48 3 m Varace of the mea =.5 Tract Sze =,00 ha Estmated mea volume per ha s 74 3 m Varace of the mea =.4 0 USDA Forest Servce RMRS-GTR

26 I combg these two meas, to estmate the overall mea volume per ha, we mght wat to weght each mea by the tract sze before addg ad the dvde the sum of the weghted meas by the sum of the weghts. Ths s the same as estmatg the total volume o both tracts ad dvdg by the total acreage to get the mea volume per ha. Thus 300(48) + 00(74) x = = Because the two tract meas were obtaed from depedet samples, the covarace betwee the two estmates s zero, ad the varace of the combed estmate would be s x (300) (.5) + (00) (.40) = (.5) + (.40) = = (4400) The geeral rule for the varace sq of a sum = = (7) = Q kx kx kx kx s sq = ks + ks ks + kks + kks k ks ( ) = ks + kks j j = j where: (8) k, =,..., are costats, x are varables wth varaces s ad covarace s j, for =,, ad j( ) =,...,. Trasformato of varables Some of the statstcal estmato procedures descrbed already ad later sectos mply certa assumptos about the ature of the varable beg studed. Whe a varable does ot ft the assumptos for a partcular procedure, some other method must be used or the varable must be chaged to ft the assumptos or, as we say statstcs, trasformed. Oe of the commo assumptos s that varablty s depedet of the mea. Some varables (e.g., those that follow a bomal such as proporto of trees that are of a partcular speces or Posso dstrbuto such as a cout of umber of trees) geerally have a varace that s some way related to the mea,.e., populatos wth large meas ofte havg large varace. I order to use procedures that assume that there s o relatoshp betwee the varace ad the mea, such varables are frequetly trasformed. The trasformato, f properly selected, puts the orgal data o a scale whch ts varablty s depedet of the mea. Some commo trasformatos are the square root, arcs, ad logarthm. If a method assumes that there s a lear relatoshp betwee two varables, t s ofte ecessary to trasform oe or both of the varables so that they satsfy ths assumpto. For example the relatoshp betwee total tree volume ad dbh s usually curvlear whereas the relatoshp betwee volume ad dbh squared s usually lear. A varable may also be trasformed to covert ts dstrbuto to the ormal o whch may of the smpler statstcal methods are based. Good dscussos o trasformatos are gve Kuter ad others (003) ad Carroll ad Rupert (988). Fally, ote that trasformato s ot syoymous wth codg (say dvdg all umbers by,000), whch s doe to smplfy computato. Nor s t a form of mathematcal magc amed at obtag aswers that are agreemet wth precoceved otos. But terpretato of results becomes more complcated wth trasformatos. We mght uderstad a relatoshp betwee umber of brds per ha ad the desty of a desrable plat speces, but explag a lear relatoshp betwee log (umber of brds) ad log (plat desty) s hard to grasp eve f the latter s requred for vald statstcal estmato. Whe possble, estmates should be trasferred back to the orgal scale of terest. Ths s ot always straghtforward as ca be see the refereces cted above. USDA Forest Servce RMRS-GTR

27 IV. Samplg Strateges Desgs Wth the Horvtz-Thompso Estmator We dscuss oly sgle-phase probablty samplg desgs ths chapter,.e., we assume that there s a samplg frame avalable from whch we ca select a sample drectly. Ths could be a samplg frame of trees, feld plots, recreato users, campgrouds, or samplg days for recreato use. Recall that a samplg strategy comprses both the samplg desg ad the estmator(s) used. For clarty of uderstadg, we start wth the smplest probablty desg: smple radom samplg (SRS) to llustrate the cocepts uderlyg probablstc samplg. Ths s combed wth the smple estmator of the total or the mea of the varable of terest to gve us a smple samplg strategy. Ths allows us to pot out that the smple mea ad total estmators are specal cases of the geeral ubased uequal probablty samplg estmator, called the Horvtz-Thompso estmator. We the go o to look at the geeral case of uequal probablty samplg ad ote how SRS, stratfed samplg, cluster samplg, samplg wth probablty proportoal to sze (pps), ad to some degree systematc samplg wth a radom start are specal cases ad why they are good samplg desgs to use uder specfc crcumstaces. I the text a small populato of sze 0 s used wth the data show Table. For computer oreted readers, Appedx 4 uses a more realstc large mapped populato called Suram wth some worked examples. Readers ca use the examples the text ad others drectly wth program R as show that Appedx. Ths data set cossts of a 60 ha stem-mapped populato of trees from a tropcal forest Suram used ad descrbed by Schreuder ad others (997). Ths populato of 6,806 trees has the relatve spatal locato of the trees ad ca be used to llustrate the effcecy of several samplg strateges. Smple radom samplg (SRS) Ths s the smplest probablstc approach. All samples of sze (samples cludg sample uts) have the same probablty of selecto. All sample uts have ( ) probablty of selecto /N ad each set of two uts have jot probablty of selecto N( N ) the most usual stuato of samplg wthout replacemet. Ths may appear to be dffcult to mplemet because there are possble samples f samplg s wthout replacemet (so that all N!!( N )! uts are dstct). For example, for a small populato of sze 0 as Table wth two uts selected, there are 0!/(! 8!) possble dstct samples. Selectg dstct uts s more effcet tha selectg wth replacemet, where a ut ca be selected ad measured more tha oce. Ths should be tutvely reasoable, sce remeasurg a ut already the sample does ot provde us ay ew formato as would the measuremet of a ew ut for the sample. Ted to ths s the N cocept of the fte populato correcto (fpc) = =, based o the samplg fracto N N (/N) = f. The fpc s usually part of the varace estmate ad dcates that as sample sze goes to Table. A small populato used for llustrato of some of the deas dscussed, where y = varable of terest ad x ad x are covarates. Ut Age y = tree volume x = basal area (ba) x = remotely measured ba Y = 75 X = 85 X = 30 USDA Forest Servce RMRS-GTR

28 populato sze N, the varace estmate becomes zero. Ths s true because bascally we are measurg the etre populato as the samplg fracto goes to oe, or stated aother way, as the fpc goes to zero. We ofte gore the fpc because may populatos are qute large ad sample szes are small so that the fpc s essetally. SRS s ot hard to mplemet coceptually f there s a lst of the populato uts avalable. Oe oly has to make sure that the selecto of ay oe ut s ot flueced by the others selected or to be selected. For example, oe ca gve each of the uts a dstct umber from to N ad the select dstct radom umbers betwee ad N. Tradtoally oe could use a radom umber table (Appedx 3, Table ) but t s ofte more coveet ow to use a radom umber geerator, also gve the Appedx. SRS samplg also has the advatage that sce all uts have the same probabltes of selecto, applcable aalyss techques are easy to mplemet ad estmato s straghtforward ad uderstadable, e.g., whe estmatg the mea µ or total Y of a populato. The ubased estmator of the total s: N y = Y = (9) wth sample sze ad y the value for varable of terest o sample ut. The varace of the sample mea s: N ( y Y) N ( N ) = N ( N ) S S V ( y) = = = N ( f). (0) N ( N ) N A ubased estmator of the varace of the estmated total s: where: ( y y) ( ) = ( ) ( ˆ N N N N s v Y ) = = N ( ) N N = umber of sample uts the populato ad s s the sample varace. A estmator of the mea µ, y would be obtaed by dvdg Y ˆ by N, so y = Y ˆ / N ad ts varace would be v( y) = v( Yˆ )/ N. Example: Assume we have the small populato show Table ad are terested estmatg ether the average tree volume, µ = Y, or total volume Y, for ths m-forest. A possble sample of sze = 4 s: Sample Ut 3 4 Value 3 4 The the estmated average tree volume for the populato of trees s: ( ) y = =.5 ad the varace s: 4 (.5) + (.5) + (3.5) + (4.5) (.5) + (.5) s = = =.67 (4 ) 3 ad 0 4 v( y) =.67 = If terest s the total Y, our estmate would be Y ˆ = 0.5 = 5 wth estmated varace v( Y ˆ) = = () USDA Forest Servce RMRS-GTR

29 Note that ths s ot a good sample sce the actual Y = 75. But for all SRS samples, the average value of Y ˆ would be 75. To llustrate how samplg estmates ca vary dramatcally wth SRS, take aother radom sample of sze = 4 from ths populato, say (,, 9, 0). Sample Ut 9 0 Value 0 0 The the estmated average tmber volume for the populato of trees s: y = = = ( 8.5) + ( 8.5) + (0 8.5) + (0 8.5) s = = (4 ) ad 0-4 v( y) = 77.58= Y ˆ= 0 8.5= 8.5 ad v( Y ˆ) = = 64. Therefore, the frst, accurate estmate shows a small estmated varace whereas the secod estmate s much more accurate but shows a large estmated varace. Ths s somethg that ca happe wth probablty samplg, especally wth SRS, whch s why we have other desgs that typcally perform much better o average. Problem: What are the advatages of SRS? Idetfy at least oe key drawback. Aswer: The overrdg advatage of SRS s the smplcty aalyss. The equally serous dsadvatage s that t ofte s qute effcet estmato sce more relable ad formatve probablstc samples ca usually be collected. Note that for the smple populato of sze 0 above, there are 0! 0 4!6! = wthout-replacemet samples of sze 4, but 75 wth-replacemet samples (gorg the order of uts selected). Clearly t would be advatageous f we ca mprove the chaces of favorg the selecto of some of those samples over others the probablstc samplg cotext f more s kow about the populato. For example, t makes sese to have the uts selected be dfferet to ga maxmum formato about the populato. Hece selectg a wthout-replacemet sample s clearly better tha a wthreplacemet sample f we ote that for samples of sze four there are oly 0 completely dstct samples out of 75 wth-replacemet samples, 360 wth three dstct uts, 35 wth two dstct uts, ad 0 wth oly oe dstct ut. Hece oly 0/75 = 0.34 of the wth-replacemet samples cota the maxmum of formato for 4 uts them. Problem: Show that for large populatos wth small sample sze, t does ot make a dfferece whether or ot wth or wthout replacemet samplg s used. Aswer: Especally for small sample szes, the maxmum formato s desred for the sample take. So a sample cosstg of all dfferet uts s better tha oe cotag duplcates. The the probablty of dstct uts a sample of uts out of a populato of N uts s P( out of dstct) = N(N - )(N - )...(N - + )/N. For example, for a populato of 0 uts wth a sample sze of 4 ths s: 5,040/0,000 = For a populato of 0 uts wth = 4, ths becomes: For a populato of 00 wth a sample of = 4, ths becomes: Clearly ths probablty s essetally for large N holdg = 4. 4 USDA Forest Servce RMRS-GTR

30 Problem: A fuels researcher comes to you rather upset. He measured the depth of the orgac matter layer o both a bured ad a ubured stad. For radom samples of sze 5 o each stad, he obtaed the followg results: for the bured plot, average depth was 0 cm; for the ubured plot t was 8 cm. Could you expla what mght have happeed? Aswer: Ecourage hm to compute the stadard errors of estmates for both estmates! It s lkely that the varablty was so hgh that oe or both of the samples were ulucky samples the sese that they ether yelded estmates too hgh or too low for the sampled plots. The cofdece lmts for the two estmates are lkely overlappg cosderably dcatg that the sample estmates are ot statstcally dfferet whch would make the researcher feel somewhat better, at least, sce he would certaly expect the bured area to have smaller average depth tha the ubured area. We ca ofte do eve better tha smple radom samplg wthout replacemet. Sometmes, we may have complete kowledge o a covarate assocated wth the varable of terest for whch we kow all the values the populato; or we ca ofte get these wth relatve ease. Ths formato whe combed wth the formato o the varable of terest measured o a sub-sample of the uts ca be used varous ways sample selecto ad estmato. Deotg by y = varable of terest ad x = covarate, umerous sample selecto schemes ad estmators are possble. Uequal probablty samplg Oe bg advatage of uequal probablty samplg s that, for a fxed sample sze, t s a geeralzato of the other sgle-phase probablstc procedures. Uderstadg the cocept of uequal probablty samplg greatly facltates uderstadg of the other procedures ad why t s advatageous to use them certa crcumstaces. Let π be the probablty of selectg ut ad π j the jot probablty of selectg uts ad j. The the Horvtz- Thompso estmator of the populato parameter Y s: y Y = HT π. () = Y s a ubased estmator of Y wth varace: ˆHT N y y j V ( YHT ) = ( ππ j πj ) j π π (3) j or N y y j V ( YHT ) = wj j π π (4) j wth π the probablty of selectg ut, π j the probablty of selectg uts ad j, ad w = ππ π. j j j Note that (9), (0), ad () are specal cases of (), (3), ad (4), respectvely. I the followg we wll ot gve the actual varace for the dfferet samplg strateges sce they are all specal cases of (4). Ubased varace estmators based o a sample are: ππ j π j y y j v ( YHT ) = (5) j π j π π j ad ( π ) π j ππ j yy j v ( YHT ) = y + π π ππ. (6) = j j j If samplg s wth replacemet (wr) ad the probablty of sample ut at a sgle draw s p, the the estmated total Y s: wr y Y = wr p (7) = USDA Forest Servce RMRS-GTR

31 wth varace ad ubased varace estmator ( ) N y V Ywr = p Y = p v Y ( wr ) y Ywr = p = ( ) (8). (9) ( ) Note that f all the π = ad all π j = the equato () reduces to the smple mea N N( N ) (9) for SRS ad, smlarly, ubased varace estmators (3) ad (4) reduce to the ubased varace estmator for SRS (). Let us exame equatos () ad (4) more detal. If π = ky wth k a costat, the Y s costat, actually Y, ad clearly V ( Y ˆ ) would be ˆHT HT 0, the deal stuato. Ths s oly a dealzed codto that wo t happe practce but we ca approxmate t. For example, the small populato show Table, we are terested total volume. If we ca select trees proportoal to ther basal area the the ratos ( y = volume for tree )/( x = basal area for tree ) are essetally costat over the 0 trees so that y y j s close to 0. Sce we are approxmatg volume rather well wth basal area, such a π π j procedure should be effcet, ad ths s cofrmed the varace computato sce the varace estmates wll be close to 0. Smlarly, f our terest s umber of trees, gvg each tree a equal weght selecto s effcet ad the procedure of selectg proportoal to basal area would ot be. Selectg as closely as possble wth probablty proportoal to the varable of terest s the dea behd uequal probablty samplg. Stratfed samplg I ths method, the populato of terest s dvded to subpopulatos or strata of terest. I ths case, the covarable x represets strata, e.g., say x = represets a oldgrowth stratum, x = pole tree stratum, x = 3 clearcut areas, ad x = 4 agrcultural/wooded lads. Ths s a smple but powerful exteso of SRS. We smply mplemet SRS wth each stratum. The dea behd stratfcato s fourfold: To provde formato o subpopulatos as well as the total populato. To dvde the populato to more homogeeous subpopulatos or strata ad mprove the effcecy estmato by a more effectve dstrbuto of the sample. To eable oe to apply dfferet samplg procedures dfferet strata; e.g., samplg the Amazo jugle s lkely to be very dfferet from that doe the pampas or other less forested areas. For coveece; e.g., samplg may be doe from dfferet feld statos. I stuatos where a populato s relatvely homogeeous, SRS may be more ecoomcal tha stratfed samplg. A ubased estmator of the populato mea s wth estmated varace of the mea where: y v( y ) h y = sample mea for stratum h, k = umber of strata, st k st Nh yh N h = = (30) N ( N ) k h h h = s h (3) h= N Nh 6 USDA Forest Servce RMRS-GTR

32 ad Nh ad h are umber of sample uts the populato ad sample respectvely stratum h. I Table, f we stratfed o the bass of the remote sesg varable, x, we mght put the frst 5 uts stratum ad the last 5 stratum. It s clear that the wth-strata varablty s much less tha that betwee strata. The suppose we decded to select a sample of sze 4, samples each stratum, such as uts, 3, 8, 0. Thus we have: Stratum : Stratum : ad The ( + 3) ( ) + (3 ) =, y = =, s = =, N = 5 ( ) (0+ 0) (0 5) + (0 5) =, y = = 5, s = = 50, N = 5 ( ) y st ( 5 ) + ( 5 5) = = = = = ( ) v y st The Y ˆ st = = 85 wth varace estmate v( Y ˆ st ) = = 780. Problem: Where would you ad would you ot use stratfed samplg? Aswer: Use f terest s dfferet subpopulatos (strata) or f strata are more homogeeous tha the overall populato; also, use f dfferet samplg schemes are desrable for dfferet parts of the populato. Do ot use f smplcty s desred, for example whe dffereces probabltes of selecto are ot desred. Geerally stratfcato s desrable. I stratfed samplg, dfferet samplg testes ca be used each stratum. I proportoal samplg the samplg testy s proportoal to the umber of uts each stratum. I optmal allocato, samplg the overall varace estmated by (3) or the overall samplg cost, C, s mmzed. Clearly ths requres kowledge or a estmate of the wth-strata varaces ad a cost fucto, so optmal allocato s usually at best a approxmato (see Schreuder ad others 993 for detals o proportoal ad optmal allocato). Cluster samplg Ths s aother exteso of SRS that ow clusters of (say) trees are sampled by smple radom samplg. Cluster samplg s useful whe o lst of sample uts s avalable or a lst s costly to obta, whch s ofte the case wth trees; ad t s usually cheaper to vst ad measure clusters of trees rather tha dvdual trees as SRS. I cluster samplg there are actually two covarates, for example, the area of each cluster whch s kept equal (say -ha plots) ad the umber of trees each cluster whch s rarely kow ad becomes kow oly for the clusters sampled. For maxmum effcecy, clusters should be heterogeeous rather tha homogeeous as wth strata. Cluster samplg s most useful whe o lst of sample uts s avalable or s very costly to obta ad the cost of obtag observatos creases as the dstace betwee them creases. A mechasm for radomly selectg ad locatg the clusters must be avalable. Suppose we select out of N clusters at radom for samplg ad each cluster s measured completely for the varable of terest. The for clusters of dfferet szes a based estmator, y cl, of the mea per ut s: My. = ycl = (3) M = USDA Forest Servce RMRS-GTR

33 where M s the umber of uts cluster, wth a estmator of the varace: M ( y. y cl) ( N ) = M v( ycl ) = N ( ) M wth N = umber of clusters the populato, = umber of clusters selected by SRS, = M =, the average umber of uts per cluster the sample, ad y. = the total for all observatos cluster. Ths estmator s asymptotcally ubased, whch meas that as N, the bas goes to 0. Usg the data Table let us take a cluster sample. Ths s ot somethg that ca usually be doe practce but we assume t ca be doe here to llustrate a pot. Let us frst do t a udesrable way,.e., have mmal varablty the clusters. If we put uts - cluster, 3-4 cluster,..., ad 9-0 cluster 5, we would have lttle wth-cluster varablty ad cosderable varablty betwee clusters. To mplemet cluster samplg wth = 4, we set up 5 clusters of uts each as dcated above, ad select of those clusters at radom. If the followg clusters were selected Cluster Sample uts Volume y.,.5 5 9, 0 5 M (33) the y cl ( 5.5) + ( 5 5) = = ad v( y cl ) = + = (.5 8.5) (5 8.5) 8.5. The Y ˆ cl = = 8.5ad v( Y ˆ ) = = 8. cl Recall that for cluster samplg we would lke cosderable varablty wth clusters. If we put uts ad 0 cluster, ad 9 cluster,..., ad 5 ad 6 cluster 5, we would have maxmum varablty wth the clusters. Suppose the followg two clusters are ow selected: Cluster sample uts volume y., , 9 6 The y cl ( 5 0.5) + ( 5 6) = = v y cl = (6 8.5) = 5 5. ad ( ) ( ) The Y ˆ cl = = 8.5ad v( Y ˆ cl ) = = USDA Forest Servce RMRS-GTR

34 Clearly, based o the results of the two samples, the secod set of clusters was much more effectve tha the frst oe effcet estmato of the mea or total volume. Problem: Assume you wsh to estmate the average age of the 0 trees Table. You are allowed to core oe tree each of three clusters to determe age ad you ca set up the clusters as you lke. How would you go about assgg trees to the three clusters? How would you assg the trees to three strata selectg oe tree from each stratum radomly? Aswer: To maxmze the formato collected t would be best to group the 0 trees to maxmze the varablty wth clusters for cluster samplg ad to mmze the varablty wth groups for stratfed samplg. Although o formato s gve o the age of the trees, t s most reasoable to assume that age s postvely correlated wth ether volume or basal area. Ths meas that for cluster samplg cluster mght be (,,9,0), cluster : (3,4,8) ad cluster 3: (5,6,7). For stratfed samplg: stratum : (,,3), stratum : (4,5,6) ad stratum 3: (7,8,9,0). pps samplg I samplg wth probablty proportoal to sze (pps samplg), we sample proportoal to the covarate (or depedet varable). Ths s effcet whe y ad x are hghly postvely ad learly correlated. For example, basal area, x, s a excellet covarate whe samplg for total tree volume, y. I Table, tree 0 would have 0 tmes the probablty of selecto of tree f trees were selected proportoal to basal area. The formato collected o the covarate ad o the varable of terest s the combed the ubased Horvtz-Thompso estmator to geerate a estmate of, say, total volume. It s usually best to sample wthout replacemet rather tha wth replacemet. The problem wth pps samplg wthout replacemet s that whe the sample sze s larger tha, the jot probabltes of selecto eeded for varace estmato are usually ot computable. There are also questos of ease of mplemetato, fxed sample sze, ad selecto probabltes exactly proportoal to sze. May procedures have bee developed to avod such problems pps samplg, e.g., Brewer ad Haff (983) dscuss 50, ad more have bee developed sce. Some of the dffcultes ad some of the key methods are also dscussed by Schreuder ad others (993, p. 57-6). Oe advatage of pps samplg s that the other procedures dscussed (SRS, stratfed samplg, cluster samplg) are specal cases of t. A ubased estmator of the populato mea s: y y = HT N = π (34) wth a ubased varace estmator: ππ j π j y y j v ( yht ) = = N π j π π (35) j where: = umber of uts sample ad N = umber of uts populato. To llustrate pps samplg, assume usg Table, that the sample cossts of uts 3, 6, 9,0 selected proportoal to basal area, x. The: y HT = = 3.5 = ad Y ˆ HT = We have ot computed the varace estmate (35) because t requres the jot probabltes of selecto for the four uts selected. We ca compute that ths case but t s ot easy. We dd compute the bootstrap varace estmate as show Table. USDA Forest Servce RMRS-GTR

35 Table. Comparso of results from samplg the small populato Table usg fve samplg methods wth a sample sze of 4 uts. Method Estmated total Estmated varace of the total SRS case 5 5 case Stratfed samplg Cluster samplg case case pps samplg Systematc samplg Problem: Thk of a stuato atural resources samplg where pps samplg would really be effcet! Aswer: The classcal aswer s the selecto of trees proportoal to basal area f terest s volume. Ths s doe usg a prsm curretly. Geerally, we would ot recommed pps samplg actual practce. I multvarate vetores, t s ulkely that there s a covarate that s postvely correlated wth all or several varables. Eve whe terest s oly oe varable, ofte tmes stratfed samplg ca guaratee us a effcet allocato of sample uts to dfferet szes of uts. O the other had, wth pps samplg eve the less desrable samples cosstg of the smallest or largest uts are probablstcally possble. Coectvty of the above desgs To get a feelg for whe to use the above samplg strateges from a theoretcal pot of vew, cosder the varace (4) called V here for coveece. If all uts have the same probablty of samplg ad all sets of sample uts have equal ( ) probablty of selecto, the wth all jot probabltes of selecto beg equal,.e., π j =, N( N ) N ( ) the weghts wj are wj = ππ j πj = for all uts ad j so that all ½ N(N-) terms the N ( N ) summato cotrbute to the varace (4). As oted earler, ths s smple radom samplg (SRS). For SRS usg the data Table wth = 4, all π j = / 90 = /5 = 0.33 ad all w j = 4 / 5 /5 = / 75 = I geeral, assumg all the π are equal to /N ad makg some of the π j equal to so that N the correspodg w j = 0, mples that those ad j have to be selected depedetly. For such ( ) uts ad j, the π j creased from NN ( ), so some of the other π j have to be reduced correspodgly because the sum of all jot probabltes s π N j = ( ). To reduce the varace V, t would be advatageous f w j = 0 for large values of j y y j π π or equvaletly y yj j for equal probabltes of selecto eve f ths creases w j for small values. Ths s the dea behd stratfed samplg, where we try to put uts that are dssmlar to separate strata to maxmze y y j ad smlar uts to the same oes. For example, Table f terest s volume, we ca use remotely sesed basal area x as a covarate. It the makes sese f wth two strata we put uts - 5 to stratum ad 6-0 stratum because (say) for = 4, wth uts per stratum, the N j π j = 4 3= wth the jot probabltes of selecto of uts wth each stratum beg 30 USDA Forest Servce RMRS-GTR

36 π =(/5)*/4=/0=0.0 j ad the jot probablty of uts dfferet strata beg π j = (/5)(/5) = 4/5= 0.6. The w 0 j = for uts dfferet strata ad w j = 3/ 50 = 0.06 for uts the same stratum. Clearly ths s a effectve strategy relatve to SRS because we have attached the hgher jot probabltes of selecto to uts the same stratum (whch are qute homogeeous) ad the lower probabltes to uts the two separate strata. The deal cluster samplg s that egatve weghts wj should be attached to larger values of y y j for reductos V. No π j ca exceed π π π orπ j, so that for example f all π = / N j the all π / N. π = / N mples that f s selected the so s j. Thus all sample uts for j j whch π j = /N are all selected together. Ths s the dea of a cluster. To make some of the w j < 0, y y we wat the π j j that are equal to /N to be attached to the largest dffereces, whch π π j mples that the members wth each cluster deally should vary as much as possble. For example, Table for estmatg volume assume there are 5 clusters of sze each ad take a sample of = 4. The, as oe good opto, put uts ad 0 cluster ; 3 ad 9 cluster ; 3 ad 8 cluster 3; 4 ad 7 cluster 4; ad 5 ad 6 cluster 5. The the probabltes of selecto for each ut s /5 = 0.40 but ow the jot probablty of uts the same cluster s π j = /5= 0.40 ad separate clusters s π j = (/5)*/4=/0 = 0.0 so that w j = / 5 = 0.04 for uts the same clusters ad w j = 3/ 50 = 0.06 for uts dfferet clusters. For the example of = 4 above usg the data Table, we have for SRS that all π j = 0.33 wth w j = For stratfed samplg we have π j = 0.40 wth w j = 0 for uts dfferet strata ad π =0.0 j wth w j = 0.06 for uts wth the same stratum. For cluster samplg, π j = 0.40 ad w j = 0.04 for uts the same clusters ad π = 0.0 ad w 0.06 j j = for uts dfferet clusters. As the results Table show, stratfcato ad cluster samplg ca reduce the varace of the estmates dramatcally relatve to SRS. The dea behd cluster samplg s atthetcal to the dea behd stratfcato. Cluster samplg s more rsky tha stratfed samplg. There wll be sharp gas f the clusters are chose well y y but sharp losses f the egatve w j j are assocated wth small values of. I stratfed π π j samplg the w j are chaged much less typcally tha cluster samplg because few sample uts wll be selected wth jot probablty as the latter. Ths s all elegatly explaed Stuart (964). I probablty proportoal to sze (pps) samplg, a verso of uequal probablty samplg, t s assumed that there s a covarate that s postvely correlated wth the varable of terest, the ultmate dream beg that y ad x are essetally the same so that V s essetally 0. We do reasoably well that respect wth x regards to estmatg volume y Table as oted before. Pps samplg s eve more rsky tha cluster samplg. For example, f the w j are held costat, t s y y j clear that π π ca be very large f the probabltes π are egatvely correlated wth the y. j Systematc samplg wth a radom start I ths type of samplg, a radom sample ut s frst selected as a startg ut ad the every k th ut thereafter s selected. Systematc samplg assumes that the populato ca be arrayed some order, whch may be atural such as days of the week recreato samplg or artfcal, such as umbered plot locatos o a map. The orderg must be carefully cosdered the frst case but may be haphazard the latter. For example, whe samplg the use of a recreato area, we probably would ot wat to sample every seveth day, say USDA Forest Servce RMRS-GTR

37 every Suday. I the past systematc samplg has ot bee geerally edorsed by theoretcal statstcas but practtoers ad appled statstcas have prevaled because t s a practcal way of collectg formato the feld ad avods the problem of poorly dstrbuted samples as ca happe radom samplg. I geeral, SRS estmato procedures are used systematc samplg (wth a radom start), the assumpto beg that the varace estmate for SRS should geerally gve a overestmate of the varace actually acheved wth systematc samplg. Systematc samplg wth a radom start should ot be used whe the populato s dstrbuted a grd patter ad the sample patter may match t! For example, samplg recreato use of a area t may ot be desrable to select every seveth day sce clearly a sample cosstg of every Moday could yeld qute dfferet results from a sample of every Suday. Problem: What s a practcal stuato forestry where systematc samplg would really be effcet? Aswer: I most feld stuatos, t s usually practcally more effcet to put a grd of plots or select a systematc sample of trees the forest. A ubased estmator of the populato mea s: y syst = = y (36) wth based varace estmator: ( syst ) v y N s =. (37) N Note that these formulas are the same as for SRS. From Table, assume we decde to select our startg pot at radom from uts,, ad 3 ad ut s selected. The, f = 4, we would take uts, 5, 8, ad. We do ths by usg mode 0 umberg,.e., we select uts, 5, 8, ad so that becomes. The our estmate would be: y syst = =4.5 4 wth ( 4.5) + ( 4.5) + (5 4.5) + (0 4.5) 49 s = = = so v ( y syst ) The = 0 = Y = 45 ad v( Y ˆ ) = 45. ˆsyst syst Problem: Assumg the populato s vsted the above order wth a systematc sample of sze three, startg wth ut, what samples of sze three caot occur? Aswer: Oe example: uts, 3, ad 4 caot occur together. I Table we show the results from the examples above for the varous samplg methods. It s clear that both estmated totals ad ther varace estmates vary cosderably from sample to 3 USDA Forest Servce RMRS-GTR

38 sample. Beg qute effcet ths case, oe would expect SRS to vary much more tha the others, ad the table certaly dcates tremedous dffereces the results for the two SRS samples. Oe would expect the other methods to vary much less. The cluster samplg results show dramatcally the dffereces betwee effectve clusterg, as case, vs. poor clusterg, as case table. Pps samplg should be especally effcet here sce we are samplg proportoal to basal area, whch s qute closely learly related to volume ths small populato. Problem: Show how stratfed samplg wth optmal allocato s a uequal probablty samplg procedure. Show that eve wth proportoal allocato t should be cosdered such. Aswer: I optmal samplg the uts dfferet strata would have dfferet probabltes of selecto. I proportoal allocato two uts the same stratum would typcally have dfferet jot probabltes of selecto tha two uts dfferet strata. Problem: Assume a populato of 5 grzzly bears that the meat cosumpto for bear 3 s typcal. Bear eats oly ½ the average of the 5 bears, ad bear 4 eats as much as the other 4 combed, accordg to a local wldlfe specalst. She s wllg ad able to gve you good estmates of the amout eate by each bear. If due to a lmted budget, we ca sample the actual cosumpto of oly bear ad we eed to make sure that eough meat s provded to mmze maulg of customers, how would you pck the sample bear? Aswer: If you decded to sample proportoal to estmated cosumpto as gve by the wldlfe specalst, you faled! It s clearly best to select bear 3. Ths s a example of usg commo sese rather tha applyg theory. Oe has to make a mmedate decso ad selectg ether bear 4 or bear for example would yeld useless results for makg such a decso. Ths example s a modfcato of a crcus elephat example gve by Basu (97) to llustrate the bld use of probablstc samplg. I hs example, the statstca recommeded usg pps samplg ad was promptly fred by the crcus drector for gvg such bad advce. Problem: Show how systematc samplg wth a radom start ca be cosdered a specal case of: stratfed samplg cluster samplg Aswer: It s stratfed samplg where oe ut s selected per stratum or t ca be cosdered cluster samplg where all selected uts form a cluster. Varace Estmato Geeral Classcal varace estmato was dscussed earler. The varaces are typcally dervable ad usually gve ubased or at least cosstet estmates of the actual varace. I may cases, however, the actual samplg strategy used s qute complex ad such classcal varace estmators caot be derved ad, hece, varace estmates caot be computed. For such stuatos ad eve cases where the actual varaces ca be derved ad computed, other methods are avalable, the two best kow beg jackkfg ad bootstrappg. We oly dscuss bootstrappg sce t s the easest to mplemet most stuatos. Bootstrappg s a clever techque takg full advatage of the computg power that we ow have worldwde. Ths computer-based method allows oe to calculate measures of precso to statstcal estmates. Cofdece tervals ca be costructed wthout havg to make ormal theory assumptos. The basc cocept s most easly uderstood for smple radom samplg. Suppose we have a sample of uts of y, wth sample mea y ad varace v( y ). Bootstrappg s accomplshed by takg a sample of uts wth replacemet from the sample uts. Ths s doe B USDA Forest Servce RMRS-GTR

39 tmes. The for each of the B samples we compute yb, b=,..., B wth average y B. The varace betwee these bootstrap estmates s: B ( yb y B) b= v( yb ) =. (38) B Ths varace estmator ca also be used for y. I addto the B sample estmates geerate a dstrbuto of estmates for easy cofdece terval costructo. The selecto of the bootstrap samples should mmc the actual sample selecto method used. Usg smple radom wth replacemet bootstrap samplg from a sample selected by uequal probablty samplg s uacceptable. So s the applcato of bootstrappg to a purposve sample. There are varous ways of bootstrappg descrbed, for example, Schreuder ad Wllams (000). Whe both the bootstrap ad classcal varace estmates ca be computed t s ot yet clear whch s best to use. The bootstrap method yelds mmedate, o-symmetrc cofdece tervals whereas the classcal varace s easer to compute. Regresso ad Rato Estmators Although the Horvtz-Thompso estmator s effcet may stuatos, t ca be qute urelable some. For ease of uderstadg we lmt ourselves to oe covarate; those terested several covarates should cosult Sardal ad others (99). Cosder a populato where some of the covarate values, x, are qute small relatve to the values of the varable of terest, y. It s clear that f some of the sample uts cota y ad x values where x s qute small, these ratos the estmator, y/x could be qute large. For example f x = 0 for oe or more uts, ts rato would be udefed. Uts wth x = 0 would ot be selected by pps samplg (causg bas the estmato) but would be wth SRS. Havg extreme ratos ca cause serous problems wth the mea-of-rato estmators (oly the Horvtz-Thompso oe was dscussed here) ad ther use s geerally ot recommeded at all wth SRS. Regresso ad rato-of-meas estmators, lke stratfcato, were developed to crease the precso or effcecy of a sample by makg use of supplemetary formato about the populato beg studed. The crtcal dfferece of whe to use the regresso or the rato-of-meas estmator s llustrated Fgure 3. Cosder the lear relatoshps betwee two varables x ad y show wth the le marked A passg through the org ad the oe marked B tersectg the ordate y. B A Y B A X Fgure 3. Postulated relatoshps betwee varables y ad x. 34 USDA Forest Servce RMRS-GTR

40 If le B s the relatoshp expected betwee the varables, so that clearly the relatoshp does ot go through the org, oe should use regresso. Wth relatoshp A through the org, rato estmato s dcated. Mathematcally both regresso ad mea-of-rato estmators are based o the followg model beg reasoable for the data y = α + βx + e, =,... N where Ee ( ) = 0 ad Eee ( j) = σ v f =j ad Eee ( ) = 0 otherwse. (39) j Here Ee ( ) dcates the average error for the regresso model over the populato of y ad x values, Eee ( j) deotes the covarace of the errors gve that the average error s zero ad k σ v deotes the varace of y at x ( v s ofte represeted as a fucto of x such as v = x where k = 0 deotes a costat varace ad k = or are ofte used whe varace of y s expected to crease wth x ). The f α 0 use a regresso estmator ad f α = 0 approxmately use a rato estmator. Whe doubt, t s geerally better to use the regresso estmator. Ordarly the questo s aswered based o our kowledge of the populato ad by specal studes of the varablty of y at varous values of x. If we kow the way whch the varace chages wth chages the level of x, a weghted regresso procedure may be used by settg k to kow values such as k = or. Regresso estmato Assumg a straght le relatoshp betwee y ad x wth costat varace (.e., ν =, =,..., N) s stll the most geerally accepted approach at ths tme. The equato for the le ca be estmated from where: R ( ) y = y+ b X x = a+ bx (40) y R = the mea value of y as estmated at a specfed value of the varable x, x. y = the sample mea of y, x = the sample mea of x, b = = ( y y)( x x) = ( x x) a = y bx = the tercept of y o x., the lear regresso coeffcet of y o x, ad As oted Sardal ad others (99), the regresso estmator s equal to the Horvtz-Thompso estmator plus a adjustmet term. The regresso estmator works well whe the adjustmet term s egatvely correlated wth the error of the Horvtz-Thompso estmator. For large errors the Horvtz-Thompso estmator, the adjustmet terms wll be about equal to the errors but of the opposte sg for large samples wth a strog lear relatoshp betwee the varables y ad x. Stadard error for regresso I computg stadard errors for smple radom samplg ad stratfed radom samplg, t was frst ecessary to obta a estmate ( sy ) of the varablty of dvdual values of y about ther mea. To obta the stadard error for a regresso estmator, we eed a estmate of the varablty of the dvdual y-values about the regresso of y o x. A measure of ths varablty s the stadard devato from regresso ( syx, ) computed by s yx. = SS y ( SPxy ) SSx (4) USDA Forest Servce RMRS-GTR

41 where y ( ), = SS = y y SSx = ( x x), ad = SP = ( y y)( x x). xy = The the stadard error of y R s ( X x) N sy = s R y, x +. (4) SSx N So for y = volume ad x = basal area Table for a sample of = 4 wth observatos (,,9,0) we have: Y = x so our estmated mea volume s y R = = 8.5 ad the estmated total volume s Y ˆ R = 0(8.5) = 8.5 wth stadard devato from regresso s = ad stadard error from regresso: s =.5. yx. 5.0 It s terestg to compare s y wth the stadard error that would have bee obtaed by estmatg the mea volume by SRS from the y-values oly. A estmated mea volume per tree s R y = 8.5 wth stadard error of s = 8.8, ad stadard error of the estmate of s y = 4.4. The famly of regresso estmators The regresso procedure the above example s vald oly f certa codtos are met. Oe of these s, of course, that we kow the populato mea for the supplemetary varable (x). As wll be show a later secto (double samplg for regresso), a estmate of the populato mea ca ofte be substtuted. Ofte the x varable ca be measured o a much larger sample tha the y-varable so that our estmate for the x-varable s much better ad ca be used to mprove the estmate for the y-varable. The lear regresso estmator that has bee descrbed s just oe of a large umber of related procedures that eable us to crease our samplg effcecy by makg use of supplemetary formato about the populato. Two other members of ths famly are the rato-of-meas estmator ad the mea-of-ratos estmator. The Horvtz-Thompso estmator ca be cosdered a example of the mea-of-ratos estmator. It s very dagerous to use wth equal probablty samplg such as SRS, ad we wll oly dscuss rato-of-meas estmato here. The rato-of-meas estmator s approprate whe the relatoshp of y to x s the form of a straght le passg through the org ad whe the stadard devato of y at ay gve level of x s proportoal to the square root of x. Ths meas that equato (39) we assume that α 0 ad that v = x approxmately for all =,,N uts the populato. The rato estmate ( y rm ) of mea y s where ˆR = the rato of meas obtaed from the sample X = the kow populato mea of x. y R y = Rˆ X (43) rm y = = x y ad x The stadard error of ths estmate ca be reasoably approxmated for large samples by the jackkfe varace estmator: D( j) ( ˆ = vj Yrm) = N ( f) X ( ), (44) y y j where for every j the sample, D( j) s the dfferece betwee the rato x x ad the average of j these ratos. Ths robust estmator ofte provdes a overestmate of the actual varace (Schreuder ad others 993). 36 USDA Forest Servce RMRS-GTR

42 It s dffcult to say whe a sample s large eough for the stadard error formula to be relable, but Cochra (977) has suggested that must be greater tha 30 ad also large eough so that the s y ratos y ad sx are both less tha 0.. x From ths sample the rato-of-meas usg the same sample of four trees as for the regresso estmator s: The rato-of-meas estmator s the _ rm y = R X R ˆ = 33/ 43 = v Y ˆ =.5. = 0.77*8.5 = 6.5 ad the stadard error of the estmated total s J ( rm) Ths computato s, of course, for llustratve purposes oly. For both the regresso ad the rato-of-meas estmators, a stadard error based o less tha 30 observatos s usually of questoable value. Warg! The reader who s ot sure of hs kowledge of rato ad regresso estmato techques would do well to seek advce before adaptg regresso estmators hs samplg. Determato of the most approprate form of estmator ca be very challegg. The rato estmators are partcularly troublesome. They have a smple, fredly appearace that begules samplers to msapplcatos. The most commo mstake s to use them whe the relatoshp of y to x s ot actually the form of a straght le through the org (.e., the rato of y to x vares stead of beg the same at all levels of x or α 0 ). To llustrate, suppose that we wsh to estmate the total acreage of farm woodlots a couty. As the total area farms ca probably be obtaed from couty records, t mght seem logcal to take a sample of farms, obta the sample rato of mea forested acreage per farm to mea total acreage per farm, ad multply ths rato by the total farm acreage to get the total area farm woodlots. Ths s, of course, the rato-of-meas estmator, ad ts use assumes that the rato of y to x s a costat (.e., ca be graphcally represeted by a straght le passg through the org). It wll ofte be foud, however, that the proporto of a farm that s forested vares wth the sze of the farm. Farms o poor lad ted to be smaller tha farms o fertle lad, ad, because the poor lad s less sutable for row crops or pasture, a hgher proporto of the small-farm acreage may be left forest. The rato estmator may be serously based. The total umber of dseased seedlgs a ursery mght be estmated by gettg the mea proporto of fected seedlgs from a umber of sample plots ad multplyg ths proporto by the kow total umber of seedlgs the ursery. Here aga we would be assumg that the proporto of fected seedlgs s the same regardless of the umber of seedlgs per plot. For may dseases ths assumpto would ot be vald, for the rate of fecto may vary wth the seedlg desty. I geeral, more complex but also more robust estmators should be used. The followg geeralzed regresso ad rato-of-meas estmators are geeralzatos of the above smple lear ad rato-of-meas estmators. There are of course other estmators possble, for example regresso estmators based o olear relatoshps betwee y ad x, but those are oly applcable very specfc stuatos especally sce trasformatos may ofte make the relatoshp betwee varables lear so that lear regresso estmato ca be used o the trasformed scale. A very geeral effcet estmator, the geeralzed regresso estmator (Sardal 980), s: where: Yˆ y a N b X x y e N ˆ gr = + gr + gr = + = π = π = π = = π (45) yˆ = a + b x, e = y yˆ, gr gr USDA Forest Servce RMRS-GTR

43 a gr = y bgr = πv = πv π v = x b gr = xy y x π v π v vπ vπ = = = = x x π v vπ vπ = = = wth varace: N ( ˆ e e j V Ygr ) = ( ππ j πj ) j π π (46) j ad two possble varace estmators ad ( ˆ gr ) v Y ( ππ π ) j j e e j = j π j π π (47) j where: v ( Yˆ gr ) ' ' ( ππ j πj) e e j = j π j π π (48) j e = y y b ( x x ), s gr s x x ' e = e e + N N + X X ˆ l ˆ l ( N N) ( X X) l= vlπl l= πlv l ˆ xl ˆ x { ( ) ( ) } v l= vlπl l= πlvl v x x = = = π v π v π v, ˆ ˆ x = π v N =, N s =, X =, x s = = π = πv = π N s x y, ad = π v y s = N s. Schreuder ad others (993) gve some alteratve varace estmators. 38 USDA Forest Servce RMRS-GTR

44 Problem: Show how the wdely used smple lear regresso estmator (40): ˆ ( ) ˆ ( ˆ b = Y = Na+ bx = Y+ bx X) wth lr Aswer: Set all = ( y y)( x x) = ( x x) v = ad select uts by SRS,.e., all π = / N. ad a = y bx s a specal case of Y ˆgr. The geeralzed regresso estmator (45) takes to accout both the probabltes of selecto ad the varace structure the relatoshp betwee y ad x. The latter s usually ot kow, but ca ofte be approxmated based o exstg kowledge. A geeralzato of the rato-of-meas estmator s: wth approxmate varace ( ) ˆ y x Y ˆ ˆ grm = X = YHT XHT X = π = π (49) ˆ ˆ ˆ ˆ ˆ grm HT HT HT HT V ( Y ) = V( Y ) RCov( Y, X ) + R V( X ). (50) There s a good dscusso o varace estmators for ths rato-of-meas estmator Schreuder ad others (993). We recommed the use of the bootstrap varace estmator for both the geeralzed regresso estmator (45) ad the geeralzed rato estmator (49). Both the geeralzed regresso ad the rato-of-meas estmators are based but asymptotcally ubased the sese that whe N, the bas goes to 0. Problem: Show that both the geeralzed regresso ad rato-of-meas estmators are based but asymptotcally ubased. Aswer: Provg that the estmators are based s ot easy. It requres dervg approxmate formulas for the bas, somethg beyod the capabltes of most readers. The easest way s to look at the formulas for the bas books lke Schreuder ad others (993). Provg that the estmators are asymptotcally ubased ca be show by lettg N (45) ad (49). The the sample estmators become the populato parameter. Problem. I the state of Jalsco, Mexco, all farmers of agave have to regster wth a dustry cooperatve terms of acreage grow, whe agave s plated ad at what desty. The cooperatve wats to fd out how much des each year for each age ad how much s stole each year from the felds (agave s a very lucratve crop ad each head o a harvestable plat s worth qute a bt of moey). Preset two alteratves to the cooperatve. Aswer: We actually have a complete samplg frame of the populato of terest ad the soluto s straghtforward. We offer two possbltes: We ca stratfy the populato to age classes of agave ad select a radom sample from each stratum. Sce theft should oly be a problem harvestable agave, we should take a larger sample from the harvestable age classes. I addto to the stated objectves, we mght ask the cooperatve f they may wat the formato by sze of owershp too. If yes, we mght mpose addtoal stratfcato based o owershp ad take a radom sample from all such strata. Note that the dsadvatage s that umber of strata could easly get out of had. If we have 9 age classes ad 5 owershp sze classes, we already have 45 strata. So we have a tradeoff betwee formato by strata, each of whch s presumably of terest, ad possble lmtatos o sample sze. Note that both cases we could also use pps samplg, such as pps samplg proportoal to age of the felds or sze of owershp. We prefer the stratfed samplg geerally because the pps samplg ca gve udesrable sample sze allocato due to radom chace. We may also be able to use regresso estmato rather tha the Horvtz-Thompso estmator f we thk some regstered varable such as sze of owershp mght be learly related to ether mortalty or cdece of theft. USDA Forest Servce RMRS-GTR

45 Some Specfc Forestry Samplg Methods Almost all samplg methods that have proved useful other dscples have bee used forestry. However, oly three methods uque to or of cosderable terest to atural resources vetores are dscussed here. For other methods see Schreuder ad others (993, 990), ad Hajek (957). The three methods are varable radus plot samplg (VRP), fxed area plot, ad Posso samplg: VRP samplg Ths method was troduced forestry by Btterlch (947) to estmate total basal area, G, of a forest by a smple coutg techque varously kow as agle cout samplg, pot samplg, plotless crusg, ad Btterlch samplg. The method works as follows: A assessor vsts a umber of locatos, m, the forest ad couts the umber of trees at each whch, whe vewed at a gve heght o a tree, usually breast heght, subted a agle greater tha some fxed crtcal agle α geerated by a agle gauge. Ths gauge could be oe s thumb held at a gve dstace from oe s eye, a smple rod wth a cross pece, or for precse work, a Spegel Relaskop or a prsm. Trees are selected proportoal to ther cross-sectoal area at the sghted pot. If terest s basal area, the trees are vewed at breast heght. Sce the trees are selected proportoal to the varable of terest, a smple cout of those selected multpled by a kow costat gves a estmate of the total basal area the forest. I geeral, aalogy wth equato (34), the estmator s ˆ N Y m k k yk k= YHT = = (5) m k= = π k m where π k = g k / (FA) wth g k the basal area of tree at pot k, F the basal area factor that determes the sze of the agle α, A the area of the populato of terest, ad Y k the estmated total basal area from pot k. The varace s: wth a ubased varace estmator: m N y N yy jπ = j V ( YHT ) = + Y m m j ππ j v Y ( HT ) = m k = ( Yˆ ˆ ) k YHT mm ( ) (5). (53) For volume estmato the geeral recommedato s to select a prsm (or basal area factor) resultg a cout o average of 6-0 trees at each sample pot. VRP samplg has the bg advatage especally to tmber-oreted people that trees are selected proportoal to ther sze ad so mmzes the selecto of umerous small trees. Problem: If VRP samplg terest s basal area, why s the varace, V, ot zero? Aswer: Because the sample sze s radom so that the varace sample sze s ot zero. The varace of the basal area estmate s a combato of the varablty basal area estmates ad varablty sample sze. The frst part s zero but the secod oe s ot. Problem: Several people had the dea of takg prsms of dfferet szes to the feld ad the selectg the oe that gave them the desred umber of trees at each pot. What s wrog wth ths procedure? (See Schreuder ad others 98.) Aswer: It ca be serously based. I fact, that s how t came to the authors atteto. Estmates based o the approach were so much larger tha prevous estmates that estmates of growth were clearly urealstc ad forest maagers suspected somethg had goe wrog. 40 USDA Forest Servce RMRS-GTR

46 The basc prcple used VRP samplg s applcable other forestry dscples, e.g., samplg a area for amout of recreatoal use. A stat cout of the umber of users at radom tmes durg the day gves a estmate of the amout of use for that day sce users are selected proportoal to ther use. For example, a fsherma who s there the whole day would be couted every tme a sample s take whereas a famly who speds oly a few mutes would most lkely be mssed. Clearly, f we are terested umber of users, we eed to adjust the estmated cout of people by ther use (.e., ther probablty of selecto). Fxed area plot samplg Ths procedure s usually appled usg crcular plots ad subplots. Wth the geeral terest ow ecologcal as opposed to tmber formato, t s dffcult to optmze for ay specfc varable sample selecto as oe does wth VRP samplg for volume. Because of ts smplcty, fxed area plot samplg s easy to uderstad ad mplemet relatve to VRP samplg. I tropcal areas, log rectagular plots are stll ofte used because of ease of establshmet dese forest ad rough terra (Wood 990). Posso samplg Ths form of samplg, developed by Hajek (957), was troduced to the forestry lterature as 3-P samplg by Grosebaugh (964). Grosebaugh proposed the method for tmber sales where trees to be cut must be selected ad marked ad some of them ca be sampled for volume at that tme too. I the orgal applcato, samplg was doe proportoal to a covarate, whch could be the ocularly estmated basal area or volume of a tree. To be effcet, the cruser eeded to be sklled. Oe way to mplemet Posso samplg s to vst every ut the populato ad whle dog that obta the covarate value x for each tree (say ocular estmate of volume). Each estmate x s the compared to a radom umber geerated betwee 0 ad X / t where X s the populato total for the populato ad t s the target sample sze. If the radom umber for ut s less tha or equal to x, the ut s part of the sample to be measured; otherwse t s ot. Clearly f x > X / t, the ut s selected wth certaty. I actual mplemetato, X s ot kow * ad has to be estmated beforehad by X so radom umbers have to be used betwee 0 ad * * L= X / t. Here L s set by estmatg X by X ad the determg the desred sample sze t. Wood (988) clarfes procedures for how to select Posso samples. Note the that the acheved sample sze a s a radom varable wth varace: N N a = = =. V ( ) π π Hajek (957) troduced the ubased Horvtz-Thompso type estmator: a y Yu = y / π = X = = x t *. (54) The varace of Y s: ˆu N ˆ y ( π ) V ( Yu ) = (55) π = ad a ubased varace estmator s: a ˆ y ( π ) v( Yu ) = (56) π = where a s the acheved sample sze. Y s ubased but ca be a spectacularly effcet estmator. ˆu USDA Forest Servce RMRS-GTR

47 Grosebaugh (967) suggested a slghtly based but geerally more effcet estmator for Posso samplg called the adjusted estmator, Y, where: ˆa ˆ ˆ Y u e Ya = (57) a wth e = X / L, the expected sample sze. A approxmate varace of Y s: ˆa N ( ˆ y V( a) V Ya) p Y / e + = p (58) e where p = x / X. A relable varace estmator s: a a y y j x j axj ( ˆ X = = v Ya ) = (59) e a (Schreuder ad others 993). A specal case of ths where every ut has a equal probablty of selecto s called bomal samplg. Problem: Show how the ubased Posso estmator ca be very effcet ad urelable. Aswer: Substtutg y Y L X t * a a a a y y * y y = toyu = = X = L t = L shows that whe π x x x = = t = t = = x for all =,,N uts, our estmate ca stll be far from Y sce substtuto yelds y X a * a u = L = = x e pretty rough.. Clearly eve f o average e a =, our tal guess X * of Y ca ofte be Problem: A lad maagemet agecy sampled a large forest area for volume usg several strata based o expected tmber volume the strata. Te years later they wated to resample the forest for volume ad chage volume but had lost track of the probabltes of selecto used earler. They would lke to treat ther orgal sample as a smple radom sample from the forest ad remeasure those same plots for both volume ad chage volume. Is ths advsable? (See Schreuder ad Alegra 995.) Aswer: No! The refereced paper derves a formula for the bas of ths procedure. It ca be qute severe. A mportat lesso s to save the probabltes of selecto of uts for future use case a radom sample of these plots are to be revsted for remeasuremet. Sample Sze Determato The most frequetly asked statstcal questo by users of sample surveys s, what sample sze do I use? A frst step s to specfy well-defed objectves for the samplg. More moey has bee wasted because a perso has poorly defed objectves. Ths ofte leads to umet objectves wth the sample collected. Oce clear objectves are specfed, the decso about sample sze s much easer to make. I geeral, the recommedato wll be to take the largest sample possble cosstet wth the moey avalable. If ths s ot a satsfactory aswer, a systematc statstcal approach s called for. Typcally oe wats cofdece tervals of a certa acceptable wdth to estmate a parameter Y,.e., we would lke a cofdece terval: ˆ zs y ˆ zs y P Y Y Y + = α 4 USDA Forest Servce RMRS-GTR

48 where z s the stadard ormal percetle, to esure a hgh probablty ( α) ad s the stadard error of estmate of the estmate Y ˆ we would lke to geerate. Ths equato mples that the parameter of terest Y s lkely to be wth the terval o average ( α) 00% of the tme. The problem s that usually we do ot kow what S y s ad sce we also do ot kow the sample sze, the t dstrbuto rather tha the z dstrbuto should be used. To estmate sample sze, do as follows for SRS: Develop a equato that expresses terms of the desred precso of estmate. For SRS, tsy α λ where s the desred sample sze, t s the α / quatle of the cetral t dstrbuto wth - degrees of freedom that ca readly be foud t-tables (Appedx 3, Table α t ), s y s the estmated varace for varable of terest y, usually based o a prelmary sample ts α y of some sort, ad λ t = s the desred wdth of the cofdece terval specfed. Estmate the ukow populato parameters the equatos used to estmate the desred sample sze. If ths s ot possble, a rule of thumb s to take a sample of sze 50. Set prortes o the objectves of samplg. For example, f you have more tha oe characterstc of terest the populato, compromse s probably requred to determe the optmal sample sze desred to satsfy the dfferet requremets. Is tree mortalty as mportat as volume, etc.? Esure that the value of chose s cosstet wth the resources avalable to take the sample. Ofte s determed solely o ths bass ad t may well be that f oe goes through the above exercse, oe may recommed ot samplg at all because the feasble sample sze s too small. Usually ths recommedato s gored. S y Problem: A research group wats to sample pollutats the ar above a fre usg a arplae. The group has a budget of $,000. You estmate that to make a relable estmate, t takes a sample of sze 50 to sample carbo doxde ad 60 to sample troge. The group ca oly afford a sample of to sample both carbo doxde ad troge ad s also terested aother 5 chemcals. What would you recommed? Aswer: The sesble aswer s to recommed ot samplg at ths tme utl more moey s avalable. The more lkely outcome s that the group wll actually do the samplg. A stuato very smlar to ths was actually ecoutered by the seor author. Oe could argue that wth the tremedous varablty oe ca expect ths stuato that a sample of sze oe could be worse tha ot samplg at all sce the sample of sze oe could ofte geerate a very msleadg estmate of the actual parameters to be estmated. Example: We are terested estmatg eedle legth o a tree wth a cofdece terval of o more tha 0 mm at the 95 percet cofdece level. Based o a small sample from aother tree earby we estmated mea leaf legth to be y = 9.8 ad s = 4. mm. To acheve our objectve the t.05s we eed = =.69. Hece we would probably take a sample of 3 eedles from the tree to (0 / ) esure that the sample obtaed s suffcet ad hope that the prelmary sample o whch we based our sample sze determato was vald for our tree of terest. Problem. A orgazato tells you that for a populato of 00,000 ha t foud that a sample of sze 40 ha was eough to gve a relable estmate for a gve varable. It wats you to sample 0,000 ha for ths varable ad wats you to take a sample of sze 4 sce t s /0 of the orgal populato ad hece ts opo should gve a equally precse estmate for the smaller populato. Do you agree? Aswer: No, you should ot! The result s lable to be much less relable for the smaller populato. See for example Czaplewsk (003) for a actual example of a smlar stuato. See also Table. USDA Forest Servce RMRS-GTR

49 Groud Samplg What approach of locatg plots ad what types of plots should be used? The am samplg s to obta a represetatve sample of the populato of terest. Frequetly large-scale surveys, samplg s based o a grd sample wth a radom start. Strctly speakg, ths s ot a radom sample sce some locatos wll have jot probabltes of selecto of 0. But t s justfed as beg SRS sce the estmator s ubased ad the varace estmator for systematc samplg wth a radom start wll usually be a overestmate of the varace assumg SRS. It s lkely that at some pot the future, mult-resource vetores wll requre dfferet plot szes ad shapes for dfferet varables of terest but sharg the same plot ceters. But ths s ot true curretly where ofte samplg for resources other tha tmber s supermposed o tradtoal tmber surveys. For example, the USA, FIA uses four crcular 0.07 ha (/4 acre) subplots samplg a -ha (.5-acre) plot for most ecologcal tree varables ad use trasects for dow woody materals ad uderstory varables. Plot ad trasect samplg techques Ubased estmates of forest populato parameters ca be obtaed from ay combato of shape ad sze of sample uts f doe properly but the optmum combato vares wth forest codto. The shapes of fxed area plots forestry are commoly rectagular, square, crcular, ad arrow-wdth rectagular wth the crcular plot beg by far the most commo. Clusters of plots are ofte more effcet tha sgle plots ad are used commoly forestry. If there s a clear gradet, rectagular plots lad out across t are effcet (remember that cluster samplg s more effcet f clusters are heterogeeous) but oretato should be decded the offce pror to samplg. Rectagular ad square plots are usually lad out by startg wth a corer pot located by survey (compass ad tape) usg a aeral photo or map. The secod corer s the located ad at both corers, rght agles are establshed to locate corers three ad four. Crcular plots are defed by the plot ceter ad radus. Establshg a crcular plot s usually smpler tha other plot types because dstaces from the plot ceter have to be checked oly for those trees wth a perpheral strp of wdth approxmately.5 to 3.0 m. The legth of the strp ad hece the umber of boudary trees creases wth crease the radus of the plot. Sometmes exact measuremets are eeded to determe whether a tree s or out of the plot. Narrow rectagular plots are most coveet f formato o topography ad forest composto s also requred as part of the survey ad f dese udergrowth or dffcult terra ecesstates spedg a large amout of tme o plot establshmet. The wdth of the strps, determed beforehad the offce, usually rages from 5 to 40 m depedg o samplg testy, topography, forest composto, desty of udergrowth, ad varablty ad value of the forest. For a gve sample testy, a strp survey may be faster tha a survey based o plots because the rato of workg tme o the uts to travelg tme betwee them s greater for strps. Strps ad plots may be combed what are called le plots. Wth these, topographcal ad forest-type data are gathered o the strps ad quattatve formato o the forest s obtaed from plots located at tervals alog ther legth. I forestry three procedures have bee popular for samplg tmber attrbutes such as volume, growth, mortalty, etc.: Varable radus plot (VRP) samplg usually cosstg of a cluster of four or fve VRP subplots samplg a certa area such as a acre or ha. Ths s a verso of uequal probablty samplg where trees are selected proportoal to basal area. It s effcet for samplg for volume ad basal area, sce tree basal area s of course hghly correlated wth volume. VRP samplg was veted by W. Btterlch, a Austra forester, the 930s although he dd ot publsh hs work utl the 940s presumably because of the terveg war. Ths method s stll used qute a few Europea coutres. I the USA, the Chef of the FS madated that the procedure ot be used aymore by FIA. But ths s clearly stll a hghly desrable procedure for a tmber sale ad for some other uses. 44 USDA Forest Servce RMRS-GTR

50 Fxed area plot samplg. Geerally a large plot s subsampled by a cluster of small crcular plots. Trees are selected wth equal probabltes. Ths s ow used by FIA ad NFS of the USFS ad by several Europea coutres. Rectagular plots could also be used but are ot popular at ths tme although they mght be hghly desrable tropcal regos or cojucto wth remote sesg. Le tercept or le tersect samplg. Ths s used ofte for dow woody materal o the groud ad uderstory vegetato. For the former, the cluso probablty s ls w / L where l s the legth of the log, w the acute agle betwee the log ad the survey trasect, ad L the spacg betwee the les. FIA ad the curret vegetato system (CVS) plots used by Rego 6 (Orego ad Washgto) of the USFS (Max ad others 996) are compact, samplg a crcular -ha plot. Although they ca be establshed the feld faster tha log rectagular plots, they are less effcet for estmato because of spatal correlatos ad the smlarty of adjacet compact subplots. Measurg them duplcates much of the work already doe ad yelds relatvely lttle ew formato. Log subplots spread out over the observato area reduce the effect of spatal correlato relatve to crcular or square subplots samplg the same sze area. To crease the precso of the estmates for large areas, oe seeks to make the plot estmates as smlar as possble. To do ths, oe cludes as much of the varablty as possble wth the plot, thus creasg effcecy. However, log rectagular or large square plots have a large permeter that creases the umber of decsos requred o whether trees o the boudary are or out. Log plots are advatageous for remote sesg, especally low-level aeral photography ad vdeography. Numerous tree ad stad varables, e.g., stockg (trees/ha) ad mortalty ca be measured wth a hgh degree of relablty usg remotely sesed magery. However, samplg subplots o the groud s desrable at ths tme to verfy the remote sesed measuremets ad adjust them by regresso estmato f ecessary. Characterstcs of plot types used the USA are summarzed Table 3. The followg s a overvew of the advatages of dfferet subplot szes ad shapes (Schreuder ad Gessler 999): Log rectagular plots are advatageous for low alttude photography measuremets ad plat bodversty estmates. Table 3. Characterstcs of plot types. Plot/subplot FIA CVS 40 x 50 m 5 x 400 m 0 x 500 m Plot Area(ha) Radus/dmesos (m) m 40 x 50 m 5 x 400 m 0 x 500 m Permeter (m) Large subplot Area (ha) Radus/dmesos (m) x 40 5 x 40 0 x 50 Permeter (m) Medum subplot Area (ha) Radus/dmesos (m) x 0 0 x 0 0 x 0 Permeter (m) Small subplot Area Radus/dmesos (m) x 0 0 x 0 x 5 Permeter (m) USDA Forest Servce RMRS-GTR

51 Rectagular plots are easer to fly ad terpret, ad a -ha plot s a coveet sze to fly ad phototerpret. Log arrow plot or trasects are desrable to assess plat bodversty (speces rchess ad detfcato of speces) because oe wshes to cover as may habtat codtos ad as large a area as possble to fd rarer speces. Boudary ssues are relatvely less mportat because oe oly has to check to see f the occasoal speces ot foud the subplots s or out of the plot. Crcular subplots are advatageous for VRP samplg ad for measurg other varables where boudary ssues are mportat, as regeerato subplots. Trasects are advatageous for traversg a large area to measure scattered or rare objects such as woody debrs o the groud. A seres of small area samples such as sol cores are best for certa destructve ad expesve measuremets such as cores for assessg sol qualty ad sol seres measuremets. Plot desgs for amals are more geeral tha for plats. The seres of artcles edg wth Schwarz ad Seber (999) ht at the possblty that wth creasg techologcal mprovemet, amal populatos may be sampled some day wth the same ease as plat populatos. Rado taggg, recordg devces, ad traps ca smplfy amal samplg ad are ofte eeded. Brds ad large mammals cover large areas because of ther moblty so samplg for them requres large plots. Slow movg amals such as worms, sals, ats, ad may sects ca be sampled o mcroplots smlar to those used for plats descrbed above but are ofte hard to observe ad traps may be requred to fd them. Brds are partcularly dffcult to sample because they mgrate so ther populatos are also reflected by codtos elsewhere. Couts of brds are also flueced by seaso, the tme of the day, ad weather. Problem: You are charged wth developg a samplg strategy for the states of Chapas Mexco ad Colorado the USA to estmate tmber volumes those states. What kd of groud plot(s) would you recommed the feld? Aswer: Chapas has cosderable tropcal forest wth dffcult travel codtos. It s lkely that log arrow plots, say 5 m x 00 m, mght be best there. I Colorado travel the forests would be easer ad VRP plots may be the best way so that trees are selected proportoal to basal area. Edge Effects Whe Samplg at Stad Boudares Radomly selected plots may fall close to a stad boudary, ad part of such plots may cross over to a dfferet stad. These boudary plots have bee dealt wth may ways, eve to the pot of movg the plots away from the boudary or etrely elmatg them. Some practces ca serously bas stad estmates, partcularly log sky stads or fragmeted ladscapes where there s a lot of edge. Trees alog the edge may grow very dfferetly dameter ad form, for example where the borderg area s ope, so gorg boudary codtos s clearly wrog. Irregular shaped boudares ca troduce addtoal problems. For a complete techcal treatmet of the ssues, see Schreuder ad others (993), sec 7..3, ad Iles (003), chapter 4. I a practcal applcato, probably the most commoly used ad accepted method to deal wth boudary plots s the mrage plot (Avery ad Burkhart 983, p. ). To use the mrage techque, place the plot wthout bas where t would fall, ad f part of the plot falls outsde the stad boudary, stall a mrage plot. From the orgal plot ceter, tally all of the trees o the plot that are also wth the stad boudary. Measure the dstace from the plot ceter to the boudary ad stall the mrage plot the same dstace o the other sde of the boudary. Tally all of the trees o the mrage plot that are wth the stad boudary. I effect, the area of the plot that exsts outsde the stad boudary s mrrored back sde the stad boudary, resultg coutg some trees twce from pots that are orthogoal projectos of ( ls, l s) across the stad boudares that trucate the area of cluso a. Formally the mrage method works as follows: 46 USDA Forest Servce RMRS-GTR

52 A samplg locato ( ls, ls) s radomly located wth a area A. If r s the dstace betwee ths locato ad tree ad R s the lmtg dstace for beg cluded the sample, dbh the R = or R = R. Depedg o whether VRP or fxed area crcular plots are used, ut u s F cluded the sample f r R. The cluso area a s a crcular area cocetrc wth u but trucated by the area boudary f t s wth R of the tree. The weght attached to y s a teger multple of A/ a (0) where the multpler depeds o whether u ca also be talled. The mrage method has problems wth rregular boudares ad wth accessblty, for example clffs, swamps, water, or trespass. For such areas, a method called walkthrough (Ducey ad others 004) has bee troduced to address these shortcomgs. For trees betwee the plot ceter ad the boudary, measure the dstace from the plot ceter to the tree ceter. Followg alog the same le, measure that same dstace from the tree ceter to the boudary. If you are outsde the boudary, the tree s couted twce; otherwse, oly oce. The advatage s that you ever eed to cross the boudary or worry about rregular shaped boudares. A dsadvatage may be that defg the boudary for each tree ca be eve more subjectve ofte tha for plots. The followg desg ssues are crtcal: Desg Issues Collect data o explaatory/stress varables such as rafall defcecy, low sol mosture, exposure to polluto, etc. Ths type of data caot usually be collected o plots but are essetal buldg relable models. Smplcty desg. Ths provdes flexblty over tme ad ease aalyss. Cosstecy of desg over tme. Ths smplfes chage estmato ad detfyg possble cause-effect hypotheses. Flexblty to address ew evrometal or maagemet ssues whle matag desg cosstecy. Flexblty to corporate ew measuremet techologes whle matag desg cosstecy. Ablty to treat each sample ut as a populato. Ths s mportat for example classfyg each sample to estmate acreage forest types. Ths meas, for example, o mssg data for a sample ut because of the desg used. Of course ths s frequetly ot feasble. Use terpeetratg samplg or smlar methods so samplg testy ca be readly creased tme ad space f eeded. Ths s a ce feature of aualzed vetores f hadled properly. Provde flexblty to accommodate replacemet of plots to deal wth damage caused by the measuremet process (for example, tramplg or destructve samplg) or deal of access to plots by prvate ladowers for example, samplg wth partal replacemet. Ablty to hadle mssg data such as plots beg accessble or ladowers deyg access (as oted by C. Kle, accessblty may also be caused by lad mes or wldlfe such as elephats ad los). Iaccessblty s best hadled by settg asde a separate stratum for such plots ad clearly statg the estmated sze of that stratum ad how estmates f ay are geerated for t. Implemet a strog qualty assurace program so that true chages sample plots over tme wll ot be cofouded wth chages measuremet error or subtle detals measuremet protocol. Cosder use of several plot desgs at the same sample locatos. Although ths complcates data collecto, t may well be requred whe a large sute of parameters s of terest. For example, for umber of trees ad total basal area of trees, very dfferet plot desgs are effcet: fxed area ad VRP plots, respectvely. USDA Forest Servce RMRS-GTR

53 Istrumetato Measuremet techques are covered great detal Schreuder ad others (993), Chapter 7. Ths secto wll serve as a supplemetal update to that chapter. Although the strumets used today by the forest practtoer are very dfferet tha the past, the uderlyg prcples rema the same. I geeral, measuremets are take for the easly measured legths ad agles, ad basc trgoometrc relatoshps are used to calculate the harder to measure elemets. Techologcal advaces electrocs allow these measuremets to be made easly, quckly, ad accurately. I addto, rugged hadheld computers allow ot oly capturg these measuremets, but also audtg ad processg them. New dameter measuremet tools The tool of choce for most remas the d-tape or calper. Two ew tools however provde for coveece: the electroc calper from Haglof ad a ew electroc dameter measuremet devce, fuctoally equvalet to the Relaskop, from Laser Techology. The calper looks lke the tradtoal beam calper, but t also has a dgtal readout of the dameter as well as a data recorder; after a day s feld work, the data s dowloaded to a computer for processg. A promsg ew strumet, although ot yet avalable commercally, s the electroc dameter measuremet devce. A lghted bar s supermposed o the tree, ad the wdth of the bar s mapulated wth the cotrols to cocde wth the dameter of the tree. A dstace s etered ether maually or captured from a coected laser dstace devce. The dstace to the tree, together wth the wdth of the bar, allows the dameter to be calculated terally. Wth ths strumet s X magfcato ad vertcal agle ecoder, t ca also be used for upper stem dameters. New heght measuremet tools The key to determg heght s a accurate measuremet of horzotal dstace to the tree. Laser dstace measuremet devces have prove themselves to be very effectve over the past few years. Laser Techology, Newco Optk, LaserAce, Hadlaser, Opt-Logc, ad others offer laser dstace measuremet. As wth ay ew techology that s cotually chagg, search the World Wde Web for the latest formato. Some have bult vertcal agle ecoders, ad alog wth the teral logc they ca dsplay the heght. A optoal, addo flux-gate compass s avalable for some models. Aother recet addto to the practtoer s toolbox s the Haglof Vertex Hypsometer, a ultrasoc dstace measurg devce. Ths system has two parts, a traspoder ad the hypsometer; the traspoder ca be placed at the plot ceter or hug o a tree, ad the the hypsometer s used to determe the dstace to the traspoder, ad optoally a vertcal agle. Dstace ad heght are dsplayed o the scree. The problem of boudary trees, that s, those that occur at or ear the boudary of a plot, always arses whe establshg sample uts the feld. Measuremet error assocated wth such trees ca be a source of cosderable error dervg plot estmates forest vetory. Ultrasoc dstace measurg devces should make t easer to mplemet the mragg or walkthrough methods descrbed earler for samplg boudary areas. New data recordg Source pot data collecto o a hadheld portable data recorder (PDR) has may advatages over hadwrtte forms, partcularly lght of the ease of data commucato betwee the hadheld ad other electroc measuremet devces. Drect capture of strumet output by the PDR avods the commo put errors ofte ecoutered. Eve wth mechacal measuremet processes, keyg the data to the PDR avods the possblty of trascrpto errors. I addto, the PDR ca be programmed to look for mssg or llogcal data etry values. As the Mcrosoft Wdows CE platform matures, may hardware ad software solutos for forestry are avalable as replacemets for the DOS ad other propretary operatg systems. There are may choces of software for crusg, scalg, ad samplg. Commercal software s avalable through most hardware vedors, ad s also avalable through publc ettes. The ready avalablty of expesve persoal data assstats (PDA) has made automated feld data collecto much more affordable. Wth the addto of a hardshell case, the PDA has become a very servceable feld ut. For producto feld use, however, the truly rugged uts wth tegrated keypads are preferable. 48 USDA Forest Servce RMRS-GTR

54 Samplg for Coarse Woody Debrs (CWD) I CWD vetores, oe may be terested both stadg ad falle woody materal. Sce assessmet of stadg lve ad dead trees s usually doe as part of a tradtoal tmber vetory, oly the samplg of falle woody debrs s dscussed here. The dscusso draws heavly o the revew of Stahl ad others (00). We assume terest s total volume ad umber of peces. As oted by Stahl ad others, there s o obvous best way of samplg CWD. But le wth the emphass o smplcty ths book, strp or le samplg are favored. Strp samplg s the same as the other fxed area samplg techques dscussed elsewhere ad hece does ot eed further elaborato here except that oe eeds to clearly decde whe a log s or out of the sample. Usually t s best to call the log f the ceter of the butt ed s the strp for both volume ad umber of logs estmato. Oe could cout a log for volume f part of the log s but the butt ceter s ot, but ths ca lead to such complcatos as possbly havg volume wth a zero estmate of umber of logs. The advatage of ths techque s that t s smple to mplemet sce such plots are easly lad out geerally ad materal o the groud s readly accessble for measuremets. There are also o problems wth logs ot lyg horzotally or how crooked the stems ad braches are (the latter have to be cosdered for estmatg umber of CWD uts). I le tercept, also called le tersect samplg, all uts tersected by a vetory le are sampled. Usually the les are lad out segmets wth a specfc spacg ad oretato. Assumg the les are lad out a fxed drecto, the cluso probablty of selecto for a ut requres measuremet of the projecto of the legth of the ut perpedcular to the oretato of the survey le. The the estmators for varable y, ether total volume or umber of uts, s: m ˆ y Y = L = l s w (6) where L s the spacg betwee survey les lad out systematcally across the etre populato, m s the umber of les, l s the legth of the ut, ad w the acute agle betwee the ut ad the survey le. If m les of szes s are used, the the followg rato estmator should geerally be more effcet: Yˆ = A m y l s w = m wth A the area beg sampled. A complcato of ths samplg desg ca be sample logs parallel to the drecto of samplg. Such logs have a probablty of selecto of close to zero ad as dcated earler wth the Horvtz-Thompso estmator ths ca create serously flated estmates f such logs are couted eve f they are a vald part of the sample. If they are couted out whe they should have bee couted, ths clearly causes a bas estmato. See Wllams ad Gove (003) for more detals about the potetal bas. Ths method has the cosderable advatage that establshg ad walkg a le the feld s easy but suffers from the problems of havg to measure agles, havg to compesate for logs ot lyg horzotally or for crooked stems ad braches, ad decdg whether logs parallel to the le of samplg are or out. A comprehesve dscusso of the theory ad hstory behd le tersect samplg s gve Chapter 3 of DeVres (986). = s (6) Problem: Cosder strp samplg where a log s couted for volume but ot for umber of logs. If part of the log s the strp but the butt ceter s ot, s t possble to a. Have volume estmates but a zero cout of umber of logs? b. Have a postve estmate of umber of logs but wth zero volume? Aswers: a. Yes b. No USDA Forest Servce RMRS-GTR

55 Wldlfe Samplg Much of the theory of samplg fte plat populatos s ot applcable to samplg may wldlfe populatos (Schreuder ad others 993, p. 36). May amal speces are moble ad hde, makg detecto or measuremet dffcult ad so samplg may affect ther locato. There s usually o samplg frame ad probabltes of selecto have to be estmated usually after the sample s draw. The exstece of a specfc selecto probablty for a dvdual the populato s ofte maly coceptual. As a result, samplg amal populatos s usually more expesve tha samplg plat populatos ad more statstcal assumptos have to be made to make estmato possble, so errors are more lkely (Burham 980). The prmary parameters of terest wldlfe samplg are usually populato sze ad rates of brth, mmgrato, emgrato, ad mortalty. Populatos are classfed ofte as ether closed or ope. A closed populato s assumed to have a costat sze wth the same members except for kow removals durg a study. I a ope populato, brths, mmgratos, emgratos, ad deaths ca occur. Tradtoally, oly a sgle vst s made to a prmary sample ut (psu). However, t s dffcult to obta repeatable amal observatos wth oe vst, because couts are flueced by weather, tme of day, ad other factors. Leavg recordg equpmet the feld for a few weeks would eable samples to be take at all tmes, day ad ght, ad uder varyg weather codtos, makg the observatos much more repeatable. A mportat advatage of automatc recorders s that octural ad shy amals ca be observed. As oted, samplg strateges for amals are cosderably more complex tha for vegetato. Such devces as rado tags, classfcato of DNA samples from hars ad pellets ecoutered o sample locatos, ad hgh-detal remote sesg should make amal samplg easer the future. Detaled procedures for samplg amal populatos are gve Schwartz ad Seber (999) ad Thompso ad others (998). 50 USDA Forest Servce RMRS-GTR

56 V. Samplg Methods for Dscrete Varables Smple Radom Samplg (SRS) for Classfcato Data Assume that for a populato of a gve rare tree speces t s mportat to determe the proporto of male ad female trees, ad the sex of a tree ca oly be obtaed easly the fall. From a radom sample of 50 trees, the umber of females s 39. The the estmate, p, of the proporto that s female s: p = Number havg the specfed attrbute/number observed (63) 39 = = Stadard error of estmate The estmated stadard error of p s s p p p = N (64) where: = umber of uts observed. I ths example N s extremely large relatve to, ad the fte-populato correcto (-/N) ca be gored, so that s _ p = ( )( ) ( 50 ) = Cofdece lmts For certa sample szes, cofdece lmts ca be obtaed from Appedx 3, Table 3. I ths example we foud that a sample of = 50 trees, 39 were female. The estmated proporto of females was 0.78 ad, as show Table 3, the 95-percet cofdece lmts would be 0.64 ad For samples of 00 ad larger the table does ot show the cofdece lmts for proportos hgher tha These ca easly be obtaed, however, by workg wth the proporto of uts ot havg the specfed attrbute. Thus suppose that, a sample of =,000, the 95- percet cofdece terval for a observed fracto of 0. s 0.9 to 0.5. If the true populato proporto of males s wth the lmts of 0.9 ad 0.5, the populato proporto of females must be wth the lmts of 0.75 ad 0.8. Cofdece tervals for large sample For large samples, the 95-percet cofdece terval ca be computed as p± s p +. (65) Assume that a sample of = 50 uts has bee selected ad that 70 of these uts are foud to have some specfed attrbute. The, 70 p = = Ad, gorg the fte-populato correcto, ( 0.8)( 0.7) s _ = = p 49 USDA Forest Servce RMRS-GTR

57 The, the 95-percet cofdece terval s: 0.80 ( ) ( ) ± = ± = 0. to Thus, uless a --0 chace has occurred, the true proporto s betwee the lmts 0. ad For a 99-percet cofdece terval we multply s p by.6 stead of. (For samples of = 50 or,000, the cofdece terval could be obtaed from Appedx 3, Table 3. For ths example the table gves 0. to 0.34 as the lmts.) The above equato gves the ormal approxmato to the cofdece lmts. Ths approxmato ca be used for large samples. What qualfes as a large sample depeds o the proporto of tems havg the specfed attrbute. As a rough gude, the ormal approxmato wll be good f the commo (base 0) logarthm of the sample sze () s equal to or greater tha ( P 0.5 ) where: P = our best estmate of the true proporto of the populato havg the specfed attrbute ad P 0.5 = the absolute value (.e., algebrac sg gored) of the departure of P from 0.5. Thus, f our estmate of P s 0. the P 0.5 s equal to 0.3. To use the ormal approxmato, the log of our sample sze should be greater tha.5 + 3(0.3) =.4 so that must be 5 (.4 = log 5). Sample sze Appedx 3, Table 3 may also be used as a gude to the umber of uts that should be observed a SRS to estmate a proporto wth a specfed precso. Suppose that we are samplg a populato whch about 40 percet of the uts have a certa characterstc ad we wsh to estmate ths proporto to wth ± 0.5 (at the 95-percet level). The table shows that for a sample of sze 30 wth p = 0.40, the cofdece lmts would be 0.3 ad Sce the upper lmt s ot wth 0.5 of p = 0.40, a sample of sze 30 would ot gve the ecessary precso. A sample of = 50 gves lmts of 0.7 ad As each of these s wth 0.5 of p = 0.40, we coclude that a sample of sze 50 would be adequate. If the table suggests that a sample of over 00 wll be eeded, the sze ca be estmated by = E + for 95-percet cofdece, ad 4 P P N ( )( )( ) = E + P P N ( 6.76)( )( ) for 99-percet cofdece where: E = the precso wth whch P s to be estmated ad N = total umber of uts the populato. The table dcates that to estmate a P of about 0.4 to wth E = ± 0.05 (at the 95-percet cofdece level) would requre somewhere betwee 50 ad,000 observatos. Usg the frst of the above formulae (ad assumg N = 5,000) we fd, = = 357 ( 0.05) , 000 ( )( )( ) If we have o dea of the value of P, we wll have to make a guess at t order to estmate the sample sze. The safest course s to guess a P as close to 0.5 as t mght reasoably occur. 5 USDA Forest Servce RMRS-GTR

58 The followg problem shows how dagerous t ca be to sample for attrbutes wthout realzg the mplcatos exactly. Problem. Idustry ad a evrometal group are argug about how much old growth there s a certa large forest. They agree upo the followg defto of old growth: A hectare of forest s cosdered old growth f t cotas at least oe tree wth a dameter breast heght of 00 cm. A cosultacy group s selected to make a vetory of the forest ad decdes to select 00 -ha plots radomly from the forest. Because t s expesve to measure all trees o the sample plots they propose radomly selectg 4 subplots of 0. ha each ad the classfy each hectare as to whether t s old growth or ot. Both dustry ad the evrometal group wat a ubased estmate of old growth for the forest. Would they get t wth ths approach? Aswer: No. Wth ths approach oe ca oly err oe way. A hectare ca be classfed as ot beg old growth whe fact t s but t ca ever be classfed as beg old growth whe t s ot. Serous bas ca result such a estmate of old growth. See Wllams ad others (00) for a extesve treatmet of the ssue volved. To obta a ubased estmate, all 00 -ha plots would have to be cesused. How to select a tree or a seed at radom If we try to estmate the proporto of trees a stad havg a certa dsease, we could do t by bomal samplg but ths requres vstg every tree the populato ad at that tme determg whether t s a sample tree or ot. Ths s SRS but s tme cosumg ad results a radom sample sze. Selectg trees completely at radom the s dffcult to do a practcal maer, whch explas why systematc samplg wth a radom start s popular such stuatos as a practcal alteratve. I some populatos, the dvduals themselves are radomly located or ca easly be made so. A batch of seed s such a populato. By thoroughly mxg the seed pror to samplg, t s possble to select a umber of dvduals from oe posto the batch ad assume that ths s equvalet to a completely radom sample. Those who have sampled seed war agast mxg such a maer that the lght empty seeds ted to work towards the top of the ple. As a precauto, most samplers select samples from several places the ple wth a scoop, combe them, ad treat that sample as a SRS. Cluster Samplg for Attrbutes I attrbute samplg the cost of selectg ad locatg a ut s ofte very hgh relatve to the cost of determg whether or ot the ut has a certa attrbute. I such stuatos, cluster samplg s usually preferred over SRS. I cluster samplg, a group becomes the ut of observato, ad the ut value s the proporto the group havg the specfed attrbute. I estmatg the survval percetage of trees a platato, t s possble to choose dvdual trees for observato by radomly selectg pars of umbers ad lettg the frst umber stad for a row ad the secod umber desgate the tree wth that row. But t s effcet to gore all of the trees that oe walks by to get to the oe selected. Istead, survval couts are made a umber of radomly selected rows ad averaged to estmate the survval percet f the same umber of trees occur each row. Ths s a form of cluster samplg, the clusters beg rows of plated trees. The germato percet of a batch of seed ca also be estmated by cluster samplg. Here the advatage of clusters comes ot the selecto of uts for observato but from avodg some hazards of germato tests. Such tests are commoly made small covered dshes. If all the seeds are a sgle dsh, ay mshaps (e.g., excess waterg or fugus attack) may affect the etre test. To avod ths hazard, t s commo to place a fxed umber of seeds (oe or two hudred) each of several dshes. The dvdual dsh the becomes the ut of observato ad the ut value s the germato percet for the dsh. Whe clusters are farly large ad all of the same sze, the procedures for computg estmates of meas ad stadard errors are much the same as those descrbed for measuremet data. To llustrate, assume that 8 samples of 00 seeds each have bee selected from a thoroughly mxed batch. USDA Forest Servce RMRS-GTR

59 The 00-seed samples are placed eght separate germato dshes. After 30 days, the followg germato percetages are recorded: Dsh umber Sum of percetages Germato (percet) If p s the germato percet the th dsh, the mea germato percet would be estmated by p 664 p = = = = The varace of p would be computed by s = p p p = ( ) = = 7 ad the stadard error of p ca be obtaed as ( 664) 8 = s p s p sp = N sp = = =.35, f the fte-populato correcto s gored. 8 Here stads for the umber of clusters sampled ad N s the umber of possble clusters the populato. As smple radom samplg of measuremet data, a cofdece terval for the estmated percetage ca be computed by Studets t 95-percet cofdece terval: p _ ± tsp wth t = the value of Studet s t at the 0.05 level wth degrees of freedom. Thus, ths example, t has 7 degrees of freedom ad t. 05 s.365. The 95-percet cofdece terval s: 83.0 ± (.365) (.35) = 83.0 ± 3.9 = 79.8 to 86.. Trasformato of percetages If clusters are small (less tha 00 uts per cluster) or f some of the observed percetages are greater tha 80 or less tha 0, t may be desrable to trasform the percetages before computg meas ad cofdece tervals. Ths s doe to approxmate the ormal dstrbuto better so that the cofdece tervals should be more relable. The commo trasformato s arcs percet. Appedx 3, Table 4 gves the trasformed values for the observed percetages. For the data the prevous example, the trasformed values are Dsh No. Percet Arcs percet Total USDA Forest Servce RMRS-GTR

60 56.0 The mea of the trasformed values s = The estmated varace of these values s: s = ( ) ( 56) = ad the stadard error of the mea trasformed value s gorg the fte populato correcto s y = =.086 = So the 95-percet cofdece lmts would be (usg t. 05 for 7 df s =.365) (.365)(.009) = CI = ± ± = to Referrg to the table aga, we see that the mea of from the acrs trasformato correspods to a percetage of 83.. The cofdece lmts correspod to percetages of 79.9 ad 86.. I ths case the trasformato made lttle dfferece the mea or the cofdece lmts, but geeral t s safer to use the trasformed values eve though some extra work s volved. Other cluster-samplg desgs If we regard the observed or trasformed percetages as equvalet to measuremets, t s easy to see that ay of the desgs descrbed for cotuous varables ca also be used for cluster samplg of attrbutes. I place of dvduals, the clusters become the uts of whch the populato s composed. Stratfed radom samplg mght be appled whe we wsh to estmate the mea germato percet of a seed lot made up of seed from several sources. The sources become the strata, each of whch s sampled by two or more radomly selected clusters of 00 or 00 seeds. Smlarly we mght stratfy a platato to sectos (strata), oes wth hgh expected mortalty ad oes wth lower expected mortalty order to assess survval percetage of trees by secto. Two or more rows would be radomly selected each secto. I both cases ot oly mght ths be more effcet estmatg overall germato or survval percetages but we also ca geerate estmates for the strata, whch mght be of terest ther ow rght. Wth seed stored a umber of casters of 00 kg each, we mght use two-stage samplg, the casters beg prmary sample uts ad clusters of 00 seeds beg the secodary sample uts. If the casters dffered volume (or the dfferet sectos the platato were of dfferet mportace), they (or the sectos) could be sampled at dfferet testes, a form of uequal probablty samplg. Cluster Samplg for Attrbutes Wth Uequal-Szed Clusters Frequetly whe samplg for attrbutes, t s coveet to let a plot be the sample ut. O each plot we cout the total umber of dvduals ad the umber havg the specfed attrbutes. Eve though the plots are of equal area, the total umber of dvduals may vary from plot to plot; thus, the clusters wll be of uequal sze. I estmatg the proporto of dvduals havg the attrbute, we deftely do ot wat to average the proportos for all plots because that would gve the same weght to plots wth few dvduals as those wth may. I such stuatos, we mght use the rato-of-meas estmator. Suppose that a pestcde has bee sprayed o a area of small scrub oaks ad we wsh to determe the percetage of trees klled. To make ths estmate, the total umber of trees ( ) o 0 plots, each 0.04-ha sze. x ad the umber of dead trees ( ) y s determed USDA Forest Servce RMRS-GTR

61 Plot No. trees (x ) No. dead trees (y ) Total Mea The rato-of-meas estmate of the proporto of trees klled s y 48.0 p = = = x The estmated stadard error of p s s p sy + p sx ps yx = x N where: y x s = varace of dvdual y values, s = varace of dvdual x values, s yx = covarace of y ad x, ad = umber of plots observed. I ths example s yx s y s x = ( ) = ( ) = 0 =, ( )( 5) ( 3)( 4 )... ( 59)( 84) ( 960)( 35) = 0 =, USDA Forest Servce RMRS-GTR

62 Wth these values (but gorg the fpc), s p ( ) ( ) ( )( ) = = 0.06 ( 67.55) 0. As ay use of the rato-of-meas estmator, the results may be based f the proporto of uts a cluster havg a specfed attrbute s related to the sze of the cluster. For large samples, the bas wll ofte be trval. Samplg of Cout Varables Statstcal complcatos ofte arse hadlg data such as umber of weevls a coe, umber of seedlgs o a ha plot, ad smlar cout varables havg o fxed upper lmt. Small couts ad those wth umerous zeroes are especally troublesome. They ted to follow dstrbutos (Posso, Negatve Bomal, etc.) wth whch t s dffcult to work. If cout varables caot be avoded, the sampler s best course may be to defe the sample uts so that most of the couts are large ad to take samples of 30 uts or more. It may the be possble to apply the procedures gve for cotuous varables. I order to estmate the umber of larvae of a certa sect the ltter of a forest tract, 30 cm samples were take at 600 radomly selected pots (Freese 96). The ltter was carefully examed ad the umber of larvae recorded for each sample. The couts vared from 0 to 6 larvae per plot. The umber of plots o whch the varous couts were observed gave the followg results: Cout Total Number of plots The couts are close to followg a Posso dstrbuto (see Appedx ). To permt the applcato of ormal dstrbuto methods, the uts were redefed. The ew uts cosst of 5 of the orgal uts selected at radom from the 600. There are a total of 40 of the ew uts, ad ut values are the total larvae cout for the 5 selected observatos. The values for the 40 redefed uts are Total = 504 By the procedures for smple radom samplg of a cotuous varable, the estmated mea y per ut s _ 504 y = = The varace s y s s y = 504 ( ) ( ! + 3 ) = Igorg the fte populato correcto, the stadard error of the mea s s _ = = y 40 USDA Forest Servce RMRS-GTR

63 The ew uts have a total area of.35 m ; hece to estmate the mea umber of larvae per ha the mea per ut must be multpled by 0000/.35 = Thus, the mea per ha s (666.67) (.6) = ad the stadard error of the mea per ha s (666.67) (0.47) = As a approxmato we ca say that uless a --0 chace has occurred samplg, the mea cout per ha s wth the lmts ± (33.33) or to USDA Forest Servce RMRS-GTR

64 VI. Remote Sesg ad Other Acllary Iformato Remote Sesg ad Photography Remote sesg ca be defed as the scece ad art of obtag formato about objects, areas, ad pheomea uder vestgato through aalyss of data acqured by some devce ot cotact wth these objects, areas, or pheomea (Lllesad ad Kefer 987). Remote sesg has a umber of sgfcat advatages ot attaable by groud samplg from a vetory ad motorg pot of vew. It provdes a syoptc vew of the study area, ca be collected quckly over a large area, provdes formato about lad cover vsble ad ovsble portos of the electromagetc spectrum, s creasgly acqured ad processed dgtally, ad provdes a permaet record of the stuato at the tme. Remote sesg sesors are ether passve or actve. Passve oes receve sgals from the target tself, ad actve oes trasmt a kow sgal. Passve remote sesg techologes useful atural resource applcatos today clude photographc ad electro-optcal magg systems such as satellte bore sesors ad arbore scaers. Ther sesg capabltes exted from ultravolet to well to the mcrowave. Actve sesors cludg RADAR ad LIDAR are just begg to prove useful for selected applcatos. Bref descrptos of the three types of sesors follow:. Photographc systems clude camera, flm, ad a platform (usually a arcraft) to carry them. These systems ow are ofte tegrated wth geographc postog systems (GPS) ad other electrocs to help detfy ad record the locato ad posto of the camera over the target to be photographed. Image resoluto s prmarly a fucto of camera les resoluto, flm resoluto, degradato due to mage moto (forward moto, ptch ad roll, ad vbrato), ad flm processg. If all goes well, a egatve or reversal flm postve wll resolve 50 le pars per mllmeter or more. Sxty or eve 70 le pars per mllmeter resolved o the flm are ot ucommo. Paper prts stll are lmted to about 5-30 le pars per mllmeter. The sze of groud features resolved s a fucto of the above, plus the camera les focal legth ad the flyg heght above the terra. Ths results photographs of a gve scale, whch, together wth the mage resoluto resdet the system, determes what wll be possble to see ad terpret from the fal photographs. Improvemets cameras ad camera mouts, ad tegrato of GPS wth the arcraft ad camera system, make ppot plot photography at very large scale (:3000 to :000) operatoally feasble. Use of computer equpmet ad geographc formato systems (GIS) make flght plag relatvely easy ad relable. Recorded formato o the photo ceter fles makes t possble to plot out a map of a photo flght shortly after the msso s completed. Because of these acqusto ad dsplay techology mprovemets, very low alttude photography may have real possbltes as a prcpal source of formato for hard-to-sample areas such as wlderess or t ca at least decrease the amout of groud samplg eeded. A old but stll mportat referece o the use of photography for vetory s Aldrch (979).. Electro-optcal magg sesors collect data as arrays of pxels. A pxel s defed as the smallest ut or cell of a raster mage. It s usually assumed to be square shape ad cossts of a dgtal umber that represets the brghtess value recorded for that pxel wth a sgle spectral bad. The groud resoluto of the pxel s usually uderstood to be the dstace that oe sde of the pxel represets o the groud. The key ssue s to extract useful formato from the spectral bad data usg mage aalyss (Holmgre ad Thuresso 998). Such sesors clude multspectral scaers, the ma oe of whch curretly s the thematc mapper (TM). The spatal resoluto for TM s 30 m for sx of the seve bads carred by Ladsats 4 ad 5. Ladsat 7 has eght bads, oe of whch s a black ad whte bad wth 8 m resoluto. Bad 6, the thermal IR bad, has a spatal resoluto of 0 m. Advaced Very Hgh Resoluto Radometer (AVHRR) s used regularly by the US Natoal Oceac ad Atmospherc Admstrato. Ths was desged for daly hgh spatal-resoluto mages of regoal cloud patters for USDA Forest Servce RMRS-GTR

65 weather forecastg. The bads were desged to dscrmate betwee clouds, water, ce, sow, ad lad. Oe bad was subsequetly modfed o the operatoal sesor pror to NOAA-6 to allow also for observatos sutable for vegetato studes. AVHRR s very coarse ts coverage. Ths s advatageous for gettg a smaller umber of observatos for a very large area but s qute lmted ts spatal resoluto (from. km to a maxmum of 3.5 km at adr) for the same reaso. The Frech system SPOT (System probatore d observato de la terre) s a commercal alteratve to TM. It has hgher resoluto (0 m) but s much more expesve to acqure. Newer satellte systems wth much hgher resoluto are ow avalable, geerally from commercal sources. Czaplewsk (999) revews remote sesg sources avalable ad soo to be avalable for vetory purposes. He dstgushes the followg categores: Low-resoluto satellte data clude AVHRR, MODIS, Orb Vew-, ERS-, ad SPOT 4. Such data are expesve ad have a,000-,900 km (600-,800 mle) swath wdth. Because of ths wde swath, spatal resoluto s poor wth a pxel represetg 64-8 ha (58-36 acres) sze. Such data have bee useful for very large-scale maps of forested ladscapes for global chage models, ad to detect hot spots of serous deforestato heavly forested ladscapes. But they are too coarse to relably measure ad motor most forest codtos. Medum-resoluto satellte data clude Ladsat 5 ad 7, Radarsat, SPOT ad 4, IRS C ad D, P ad 5, Sp, EOS AM-m, ad CBERS ad wth pxel szes of 0-30 m (33 to 98 feet) wde. They are more expesve wth a km (30 to 00 mle) swath wdth. Such systems ca separate forest from o-forest, ad ca detfy some forest types ad desty classes. Ladsat ca geerally detfy clearcuts but ot most partal cuts. Advaced regeerato after lad clearg, urba ceters, ad sze, shape, ad coectvty of forest patches ca also be measured. Hgh qualty data wthout clouds are geerally avalable every - years except humd tropcal areas ad may boreal forests. Hgh-resoluto satelltes clude Ikoos-, OrbVew 3 ad 4, EROS B ad B, SPOT 5, ad Quckbrd ad wth km (-6 mle) swath wdth ad pxel sze of from -3 m (3-0 feet) wde. These sesors have capabltes, lmtatos, ad costs smlar to hgh alttude 56.5 cm square (9-ch square) :40000 small scale aeral photography as avalable from the USA Geologcal Servce (USGS) atoal aeral photography program (NAPP), whch covers a area about 8 km (5 mles) wde. Such satellte ad photo data ca be used to relably dstgush some forest types, several stages of stad developmet, clearcuts ad may partal cut areas, regeerato after lad clearg, ad cocetrated tree mortalty. Forest stads, lad use, dstace to adjacet roads, water bodes, forest fragmetato, ad varous types of urbazato ca be photo terpreted. Large scale aeral photography wth scales from :500 to :000 s routely acqured by aeral survey compaes for small stes. Each photo covers a area km (0. to mles) wde. Iterpreters ca relably detfy may forest cover codtos such as 0 broad forest types, 5 stages of stad developmet, 3 stad desty classes, clearcut ad partal cut areas, regeerato success rates, atural or artfcal stad org, 3-5 severty levels of tree mortalty, most dcators of urbazato ad fe-scale forest fragmetato, ad stad sze, shape, ad edge measuremets. Aldrch (979) otes that forest dseases are less easly detected ad evaluated tha sect damage wth aeral photography because t takes a log tme for vsble symptoms of dsease to show up. The symptoms are ofte ot uform over the forest ad are more subtle tha sect damage. Dwarf mstletoe, Dutch elm dsease, oak wlt, basal caker of whte pe, ash deback, Fomes aosus, sulfur doxde damage, ad ozoe damage are detectable wth some degree of success. Large-scale color ad color frared (CIR) flm (:584) are eeded to ascerta the degree of damage, whle :8000-:6000 scales of CIR photography ca be used to defe ad deleate the boudares of the dsease. 70 mm color ad CIR photography ca be used as part of a samplg strategy wth susceptble forest types for damage assessmet. Of course dsturbaces to the 60 USDA Forest Servce RMRS-GTR

66 vegetato caused by wdstorm, flood, fre, or huma actvtes are relatvely easy to detect o aeral photography as s chage these characterstcs f the photography s repeated. Aeral color vdeo system magg may have real utlty especally for aualzed vetores to fd out more about chages observed certa areas of specal terest. Vdeocameras ca be easly mouted o a varety of arcraft for ether vertcal or oblque sesg. Images ca be dgtzed easly for computer-aded terpretato ad ca be used mmedately sce developmet s ot eeded. Vdeo equpmet s portable, versatle, easy to use, ca tolerate dfferet lght codtos, ad s cheaper to operate tha photographc systems. Also, the operator ca vew the magery o a motor the plae at the tme of acqusto, ca adjust the exposure settgs teractvely, ad ca record commets flght. Also, the hgh rate of pcture acqusto (30 frames/sec) provdes extra data. Its dsadvatages are ts low spatal resoluto relatve to flm, the dffculty of obtag hard copy mages from the data, practcal lmts o feld-of-vew because of the small tape format, dffculty calbratg the cameras because of the automatc exposure cotrol, ad vgettg problems wth the may ear-ir vdeo sesors sce the camera optcs are ot desged for ths wave legth bad. The value of commo color vdeo systems for atural resources ad agrcultural applcatos s lmted because of the dffculty of extractg dscrete spectral data from a composte vdeo sgal ad the lack of spectral bads outsde the vsble resoluto. 3. Mcrowave sesors are geeratg cosderable terest at ths tme but the applcatos forestry are stll lmted (see for example Lefsky ad others 00). The ma data sets of ophotographc mages curretly comprse those collected by Ladsat -5 ad 7, SPOT, ad AVHRR. These are avalable o computer-compatble meda ad electrocally recosttuted photographs. Both computer-aded meda ad covetoal phototerpretato methods are used to terpret such data. Key refereces regardg vetory ad motorg usg remote sesg are: Holmgre ad Thuresso (998), USDA Forest Servce (998), ad Lefsky ad others (00). Accuracy of Remotely Sesed Iformato Maagemet of lads by ageces, such as those of two federal ageces the USA, the Natoal Forest System of the USDA Forest Servce, ad the Bureau of Lad Maagemet of the U.S. Departmet of the Iteror (USDI), requres relable maps of varables such as percet forest cover, stad structure, ad vegetato types. Such maps also requre frequet updatg ad geeratg them s expesve. It s atural that remote sesg sources such as TM are used for ths purpose sce t s facltated by the frequet, large-scale, dgtal acqusto. Cosderable work has goe to makg such maps. However, although TM cotas useful formato, the amout s lmted. For example, t s ulkely to be useful for stad structure, a dffcult to measure varable eve o the groud. Smlarly, vegetato types are dffcult to defe ad terpretato may vary from oe user to aother. Ideally, TM formato should be combed wth geo-refereced feld vetory ad other mapped data to provde the ecessary formato for maagemet decsos. Remote sesg researchers desre a sgle coeffcet to represet the accuracy of a thematc map ad of each category dsplayed (Rosefeld ad Ftzpatrck-Ls 986). Usually the results of a accuracy assessmet of a map are dsplayed a matrx called a cotgecy table (called error matrx remote sesg) where the colums dcate the classes defed by the stadard of comparso ad rows dcate the mapped oes. The elemets the cotgecy table are the couts the row/colum classes wth the umber the last row the total cout that row class ad the umbers the last colum the total cout that colum class. A obvous frst estmator of overall accuracy s the rato of the sum of all correct couts over the total umber of couts the cotgecy table. But deally we also wat estmators of the errors of commsso (the proportos of dagoal values to colum sums = user s accuracy) ad of omsso (the proportos of dagoal values to row sums = producer s accuracy). A wdely accepted coeffcet of agreemet s the Kappa statstc (K) estmated by: ˆ p0 pc K = (66) p c USDA Forest Servce RMRS-GTR

67 where: p p k 0 = pjwj = weghted proporto of uts that agree,, j k = w p p = weghted proporto of uts wth expected chace agreemet, ad c j.. j, j k p = p, p = p. j. j j j= = k where wj s the assged weght of mportace of agreemet for (,j) wth w j = for all,j for the smple uweghted Kappa statstc ad 0 w j for the weghted Kappa. Uequal weghts ca be assged f the accuracy of some classes s more mportat tha for others, wth the dsadvatage heret ths that such weghts would be subjectve. Here K ˆ = 0 dcates that obtaed agreemet equals chace agreemet, K ˆ > 0 dcates greater tha chace agreemet, K ˆ < 0 less tha chace agreemet, ad K ˆ = s perfect agreemet. The user s accuracy = (umber correctly classfed to be correct the dagoal/umber that row) ad producer s accuracy = (umber classfed to be correct the dagoal/umber that colum). To assess accuracy, we eed a probablstcally selected sample of sze o whch both truth ad the map values to be assessed are avalable (Schreuder ad others 003). For specfcty we oly dscuss the stuatos dscussed that paper here. We assume plots are used. Truth should be defed exactly for each varable ad measured accordgly. It should ot be defed as the best readly avalable formato as s doe frequetly remote sesg. For percet tree cover ( y ), very low alttude photography may best be used; for speces composto ( y ) such photography should be combed wth groud samplg; for stad structure ( y 3) more emphass s lkely requred o groud samplg wth the photography provdg some utlty; for other varables such as uderstory vegetato especally dese stads, relace may have to be placed completely o groud samplg. Call the values of the varable of terest o plot, y,( =,,3) ad the correspodg value of the varable o the map to be assessed for accuracy x,( =,,3). For a smple radom sample of plots: We have plots wth y from truth coverage. For may varables, t s lkely that some plots wll cota more tha oe class. Truth may be obtaed from photography aloe or a combato of photo ad groud formato, depedg o the varable of terest. However, f oly photo formato s used for percet cover, the truth s obtaed error-free for the whole plot whereas f groud samplg s volved, the plot formato may have samplg error. Locato error for the mapped formato s assumed eglgble sce we do ot kow what t s. If preset t wll lkely lead to a uderestmate of the actual accuracy. Uless we have detaled formato about errors plot locatos, we caot correct for them. For a certa umber of the plots, all the formato falls wth oe or more categores for the varable of terest for both the truth ad the mapped formato. The followg treats the case of both x ad y labelg oly the same two truth classes occurrg o a truth plot. The exteso to more tha two classes s straghtforward. For a gve plot, assume that the part x labeled x j s part of or covers the part y j called that by the truth plot. If the truth plot ad mapped plot could be overlad completely, ths assumpto s ot eeded. But the truth plot may oly provde estmated areas of the plot area the classes of terest for y ad y 3 ; ths assumpto s requred sce we wo t kow what part of the plot belogs to the category estmated. That s the stuato we curretly have to lve wth. Geerally volato of ths assumpto wll result hgher estmates of accuracy tha actually obtaed. 6 USDA Forest Servce RMRS-GTR

68 Cotuous varables wll have to be put to classes order to determe whether mapped correctly or ot. Ths ca be doe objectvely; for example, for percet tree cover use the 0 classes 0 to 0%, 0+ to 0%,.90+ to 00%. We the have the followg determato for each of the plots k (k =,,): y y If truth calls t y j ad y j wth plot area weghts wj, w j such that w y y y j + wj + wother = ad x x x calls t the same wth plot area weghts wj, w j such that w x x x j + wj + wother =, the f x y x y x x wj wj, wj wj, correct classfcato for the plot gets a value of ( wj + wj )/ for p 0. y x wjother, w jother are the percet (weghts) of plot areas for whch y or x or both defe a codto o the plot ot recogzed by the other. The weght gve to all partally or totally correctly classfed plots s ( wj + wj )/ for pj where z ad z are the smaller of wj, wj ad wj, wj, respec- z z y x y x z z tvely. The respose varable for each plot k s the pj = ( wj + wj )/ where 0 pj /. Plots classfed correctly or 0.80 correct are couted as / ad 0.80/, respectvely. k Calculate p0 = pjwj ad the compute the Kappa statstc equato (66). By repeatedly, j takg plots wth replacemet from the sample plots B (say),000 tmes ad applyg the above computato of the p j to each sample, we geerate a seres of B estmates for each cell of our cotgecy table as well as for producer ad user accuracy ad a Kappa statstc for each. Wth ths bootstrappg we the ca costruct cofdece lmts aroud all the cells the table as well as for the Kappa statstc by treatg the B samples as depedet estmates of the same quattes. A example of a cotgecy table wth user, producer, overall accuraces, ad the Kappa statstc based o results Table 4 for a sample of = 00 plots are: User s accuracy for class s 60.3/0 = 0.60, for class s 44./53 = 0.83, ad for class 3 s 30.8/46 = Producer s accuracy for class s 60.3/70 = 0.86, for class s 44./73 = 0.6, ad for class 3 s 30.8/57 = 0.54, wth overall accuracy 35.3/00 = The p 0 = 0.68 ad p c = 0.33 ad ˆ K = = = The by repeatedly takg say,000 wth replacemet samples from the 00 sample plots, we compute for each sample the accuraces ad the Kappa statstcs aga ad costruct cofdece lmts aroud the above producer s, user s, ad overall accuracy as well as aroud the Kappa statstc. The cotgecy tables are the basc product from the accuracy assessmet. Users should study those tables order to attempt to expla the causes of msclassfcatos. Some are obvous whle others eed vestgatg. Msclassfcatos may result from problems wth the techology used, user errors, regstrato errors, errors the fal preparato of map products, or calculatos the accuracy assessmets. Studyg the results s essetal that t may expla or ucover errors that ca be corrected. Table 4. A umercal example of a cotgecy table for forest cover class. Cover class Cover class Cover class 3 Row totals Map class Map class Map class Colum totals USDA Forest Servce RMRS-GTR

69 It s also desrable for a maager to kow how serous a msapplcato of a treatmet to a area may be expected to be f the area s thought to belog oe category whe fact t belogs to aother oe. There would be dfferet cosequeces applyg a treatmet to a category close to the desred oe tha to a very dfferet oe. Summary of what s eeded for accuracy assessmets: Defe truth for the varables of terest ad where ad how to measure t. Mmze, or f possble, elmate measuremet errors truth by observers. Decde o usg ether pxel accuracy assessmets or polygo accuracy assessmets. Esure a adequate sample sze each of the categores of terest for the varables of terest. Defe dfferet types of accuracy or gve some of these a dfferet label tha accuracy. Determe the mplcatos of achevg a stated accuracy terms of makg correct or correct maagemet decsos. Combe/tegrate the accuracy assessmets for the varables of terest ad use the formato also to mprove the maps developed. Global Postog System for Spatal Locato Needs Spatal locato s crtcal for success forest vetory ad motorg because preset eeds requre mappg ths formato. Tradtoal methods of determg geographc locato are stll used, but more ofte tha ot, these former methods ow supplemet the Global Postog System (GPS) whe used atural resources aalyss. The GPS uses satelltes to locate groud postos, usually wth 50 meters, ad ofte to less tha 0 meters. Wth ths system the locato of arcraft ad plots ca be establshed rapdly wth reasoably accuracy. GPS does ot requre the use of kow geodetc markers for autoomous observato. Also, measuremets ca be made ay tme ay weather, wth the excepto of possbly large solar storms. However, because the measuremets requre a clear le of sght to the satelltes, establshg locato s dffcult a forest wth a very dese caopy, deep valleys or gorges, or smlar stuatos. Addtoally, a umber of errors do happe f the user does ot use the GPS recever correctly such as usg the correct datum. Moreover, atmospherc ad satellte-sgal-path-errors occur aturally ad eed to be recogzed ad adjusted or reported sde the accuracy statemets by the user. GPS has bee used successfully as the bass of a sophstcated avgato ad flght recordg system cotrollg the acqusto of large-scale aeral photographs West Australa (Bggs ad others 989, Bggs ad Specer 990). Remote sesg has bee oversold the USA but wll oe day fulfll ts promse partally because of relable GPS systems. Accurate GPS wll be eve more relable ad precse the future. For example, algorthms are beg developed that correct satellte sgals ad make postog more accurate. It behooves potetal users to keep abreast of the techology because t s creasgly crtcal for vetory ad motorg. Geographc Iformato System (GIS) A Geographc Iformato System provdes for eterg, storg, mapulatg, aalyzg, ad dsplayg spatal data. Data ca be represeted by pots, les, or polygos wth the assocated varables. Such data ca be represeted by raster or grd data o the oe had ad by vector data o the other. The raster system stores the data a grd or pxel format wth bouded geodetc values such as lattude ad logtude, whle the vector system uses a seres of x, y coordates to defe the lmts of the attrbute of terest. Grd data are computatoally easer to mapulate but usually requre large amouts of storage space. Vector data requre less storage ad usually represet dscrete data more accurately. Though the vector system may reta the shape of a dscrete feature more correctly (has better resoluto), t s computatoally more tme-cosumg ad dffcult to reder 64 USDA Forest Servce RMRS-GTR

70 ad aalyze. Satellte magery, dgtal pctures, ad dgtal elevato models are examples of grd data; property boudares, structure outles, utlty poles, ad utlty les are examples of vector data. A GIS s a computerzed system that ca play a crtcal role vetory, mapulatg ad processg data, ad assessg lad use ad lad cover. It has emerged as a effectve tool defg ad focusg dscusso relatve to the merts of alteratve lad use allocatos. For example, t gves the aalysts the ablty to smulate the effects of chages maagemet (Gree 993). The GIS should have the capabltes to: Iput may forms of data such as: aalog ad dgtal maps, textual or tabular formato, ad mages. Store ad mata formato wth the ecessary spatal relatoshps. Mapulate data, as search ad retreval, ad to do computatos effcetly o the data. Provde levels of modelg takg to accout data terrelatoshps ad possble cause-adeffect resposes of the relevat factors. Preset tabulatos, vdeo dsplays, ad computer geerated maps of exstg or derved formato. A good GIS depeds prmarly o good data. I addto hgh-speed computers, a varety of perpheral put-output devces, ad powerful software are requred. Artcles by Cogalto ad Gree (99), Gree (993), ad Bolstad ad Smth (99) gve a good overvew of GIS ad Lachowsk ad others (99) preset a useful example of tegratg remote sesg ad GIS. Small Area Estmato There s cosderable terest maagemet ageces to have relable spatal formato. I the past foresters ad other lad maagers crused or sketch mapped a area usually to decde what s where. Maagers avoded statstcal samplg because t mght gve relable data o how much was there but ot where. Frequet legal challeges chaged ths the USA. Now terest s obtag relable (defesble) mapped ad statstcal data together. Oe such area of curret research s referred to as small area estmato, bascally a model-buldg approach usg statstcal data combato wth acllary data such as TM, GIS, topographc maps, ad other related formato. Small area estmato techques represet a substatal mprovemet terms of qualty of data, especally defesblty of data-based maagemet decsos relatve to what used to be doe whe maagers reled o subjectve formato. Small area estmates have bee clamed to have stadard errors smlar to those for classcal samplg. The trouble s the comparso s made for the etre populato of terest whereas maagers are also terested predctos for much smaller areas such as polygos used as a bass for maagemet. Stadard errors for dvdual predctos ca be large, as oe would expect, gve the varablty ecoutered o the groud forests. For successful small area estmato, two codtos eed to be met. Frst of all there should deally be a good correlato betwee sampled ad o-sampled areas ether earby or from other acllary sources such as remote sesg. Ths usually requres a much more tesve grd tha the 5,000-m grd ow used by FIA. Secodly the spatal locatos for both the sampled areas ad the acllary data eed to be accurate. Gve these codtos t should be possble to develop good predcto models. Cosderable work small area estmato of forest resources s ow beg doe. For example, there s ovatve work beg doe Flad where the more homogeeous codtos relatve to other coutres may make small area methods more useful. Multple mputato methods (cludg regresso models) ad k-earest eghbor techques have bee proposed for cotuous varables. I these techques, feld sample formato s extrapolated to the etre populato where formato o sample locatos s put to o-sampled locatos by some crtera such as smlar USDA Forest Servce RMRS-GTR

71 TM readgs for the sampled ad o-sampled locatos. I multple mputatos for each ut wthout sample data, a seres of say 00 predctos are made usg radomly selected data ad a uderlyg model ad database. The the data sets are aalyzed separately ad pooled to a fal result, usually a average of the results. Fraco-Lopez (999) revews methods for projectg ad propagatg forest plot ad stad formato. As he otes, cosderable effort has bee expaded Nordc coutres combg forest motorg formato, remote sesg, ad geographc formato systems (GIS) to develop maps for forest varables such as cover type, stad desty, ad tmber volume wth emphass o the k-earest eghbor techque. He cofdes that whle hs results are poor for Mesota, they are represetatve of those obtaed by other methods ths rego. L (003) presets a sem-parametrc bootstrap method for estmatg the precso of estmates. I geeral:. Ft the best fttg model say y = α + βx + ε resultg the estmated model: y = αˆ + ˆ βx.. Compute the resduals ˆ ε ad calculate the scaled resduals = ε = ˆ ε ˆ ε /, =,...,. 3. Bootstrap the resduals ε,,..., =,.e. take a sample of resduals wth replacemet from the resduals. Do ths say,000 tmes, each sample costtutg a bootstrap sample. 4. For each bootstrap sample compute yˆ ˆ ˆ = α + βx + ε. 5. Reft the model to each of the bootstrap samples usg the sample pots yˆ, x ad predct for each of the samples at the desred locatos; the varablty betwee these estmates s the used for the bootstrap varace for that locato. L s results for predctg mortalty, total basal area, ad umber of lve trees o the Suslaw Natoal Forest Orego showed errors averagg early 00 percet for plots o a.36-km (0.85-mle) grd base usg data from sampled -ha plots o a.7-km (.7-mle) grd. To obta relable predctos today, addtoal formato s requred, such as that avalable from mproved remote sesors or large-scale photos ad varous maps combed wth expertse from local ecologsts. Also, at preset t s stll ecessary to correct for locato errors wth models. Makg such correctos requres cosderable formato o the extet ad locato of the errors. Hopefully, mprovemets GPS-type sesors wll reduce locato errors the future ad mprove results from small area estmato techques. 66 USDA Forest Servce RMRS-GTR

72 VII. Samplg for Rare Evets Samplg rare populatos s a order of magtude more dffcult tha samplg commo oes. Yet, assessg such populatos ca be crtcal. For example, the world s losg may plat ad amal speces ad people may wat to preserve at least some of these speces. Kowg how may of a speces exst ad where ad why specfc areas s crtcal for ther preservato. The costs of locatg rare populatos are cosderable ad ofte exceed the costs of measuremet. The fudametal problem s that several attempts may be eeded to detfy sample uts wth the rare trat the overall populato such as rare mushroom or tree speces. Also, detfyg the actual speces or other attrbutes may requre very specalzed kowledge that oly a few people have. Possble approaches are:. Screeg. A large sample from the total populato s examed to detfy members of the rare populato or at least areas where t s more lkely to occur. If the latter s possble the such areas are sampled wth much hgher testy tha other areas for frequecy of occurrece.. Multplcty samplg ad adaptve samplg. Bascally, these are techques that rely o locatg some of the uts wth the rare attrbute ad the obtag addtoal formato about them, whch s the used to locate others, thus reducg the cost of the survey. a. I multplcty samplg a selected sample ut yelds formato about tself as well as about other uts. Obvously ths s more applcable to huma surveys tha vegetato surveys. b. I adaptve samplg a sample of uts s selected probablstcally ad uts the eghborhood of a sampled ut are added f the attrbutes of terest for those uts satsfy a gve crtero. The cleveress of the approach s that all uts the populato are put o-overlappg clusters ad all uts the sample clusters are measured. Clusters ca vary greatly sze ad shape. Adaptve samplg s a probablstc procedure but s hard to mplemet, ad aalyss of the results s dffcult. 3. I multple frame samplg, a sample s take from a exstg partal lst ad a addtoal oe from the total populato to scree for uts wth the characterstcs of terest. The weakess of ths approach s the overlap of frames (for whch Kalto ad Aderso, 977 gve some solutos) ad the expese of screeg ad samplg the screeed part of the populato. 4. I sowball samplg, a ecessary codto s that uts cota formato about each other. The a frame of uts s created the rare populato by samplg a few uts ad through them detfyg others. Clearly aga, ths s more lkely to be frutful wth huma populatos tha wth vegetato. Oce a frame has bee developed, a probablstc sample s draw. The weakess of ths approach s the degree of completeess of the frame. A advatage s that rare uts are detfed more quckly tha wth other methods. 5. Sequetal samplg. Select a tal probablstc sample of suffcet sze to gve the desred sample sze () of members of the rare populato based o the rate of cdece observed. Ths wll yeld members of the rare populato ad a estmate of cdece. If <, a secod sample s selected to produce the remag members of the rare populato based o the cdece obtaed the frst sample. Ths procedure s geerally expesve ad hece s ot practcal most vegetato surveys. Several of these techques may become more useful vegetato ad amal populato surveys as rapd, smplfed DNA detfcato methods are developed. USDA Forest Servce RMRS-GTR

73 VIII. Multple Level Samplg Prevously, we dscussed sgle-level surveys where ether the varables of terest or these varables plus covarates o the same sample uts were measured. Whe covarates were measured, we assumed that the covarate values for all uts the populato or, at least the populato total, were kow. Ofte the covarates are useful for estmato ad are cheaper to collect but ukow. It ofte pays to collect covarate formato o a large sample ad the varable of terest o a subsample. Ths approach s referred to as multlevel samplg. For example, a tmber sale we mght obta ocular estmates of dameter at breast heght D (ad hece D ) for a large sample of trees ad measure actual volume of some of these trees. Or, estmatg recreatoal use of a area, we may use traffc couters recordg couts of vehcles at the etrace to the park o a large umber of days ad actually cout the umber of users o a subset of these days. Multlevel samplg ca be separated to multphase ad multstage samplg. Multstage Samplg Refers to samplg desgs where the ultmate sample uts, called elemets, are selected stages. Samples at each stage are take from the sample uts comprsg clusters of uts selected the prevous stage. Iterest s estmato of attrbute totals or meas per elemet, such as bomass per tree rather tha per ha. The populato s frst dvded to a umber of prmary sample uts (PSU), some of whch are selected as the frst stage sample. These selected PSUs are the subdvded to a seres of secodary sample uts (SSU), some of whch are radomly selected as the secod stage sample. Ths process ca be repeated of course wth addtoal stages f ecessary. The procedure has the advatage of cocetratg work o a relatvely small umber of PSUs after whch much less effort s usually eeded to obta the secod ad later samples. The ma reasos for selectg a multstage sample are:. Drawg a set of uts from a populato such as trees a large forest or recreato users of a park over a full seaso s expesve. It s dffcult to obta a lst of all the trees ad eve more dffcult to determe all users of a park.. Eve f a lst of populato uts was avalable, effcecy mght dctate that groups of uts (clusters) rather tha sgle uts be chose ad that oly some uts each cluster are measured. For example, t s usually cheaper to sample 0 radomly located clusters of 30 trees a forest tha 600 radomly located trees ad we may wat to oly sample 0 out of the 30 trees each cluster because of the homogeety the cluster or the expese of measurg all 30 trees. I samplg recreato users, t s clearly easer to select ad subsample radom days o whch to tervew all users rather tha attempt to radomly sample dvdual users or days, respectvely. Geerally, though as dcated earler, there s a defte tradeoff effcecy betwee cluster samplg ad radom samplg of uts because uts close together are ofte more smlar tha those further apart ad t ofte pays to measure oly some of them each selected cluster. Samplg ca be a large umber of stages. We llustrate how ths works wth the smple ad ofte practcally useful stuato of -stage samplg wth SRS at each stage. Assume N groups or clusters wth M uts ( =,, N) the th cluster. Our total of terest ca ow be wrtte as: N M N j. (67) = j= = Y = y = Y I -stage samplg a radom sample of s selected out of the N clusters but stead of measurg all uts the cluster, a radom sample of m uts s chose each. Thus, the cluster total Y s frst estmated by. 68 USDA Forest Servce RMRS-GTR

74 Yˆ. y m = j M j= m (68) for each of the clusters sampled. Our estmated total s: ˆ N y N Y = M = Y m m j ˆ. (69) = j= = wth varace N ˆ NMa( f) σb N M ( f) σw V ( Y ) = + (70) m = where: σ b = N = M Y Y M a ( N ), the betwee-cluster varace, ad σ w = M ( y ) j Y. j= M, the wth-cluster varace. m Y Y M Here f = f = Y = Y = M = N M M N N per PSU. Smlarly, a ubased varace estmator v( Yˆ ) s: N N..,,.,, ad a, = = the average umber of SSUs ˆ sb N M ( f) sw v( Y) = N Ma ( f) + (7) m = where: ( My y) m y ˆ = j Y sb =, y. =, y = m NM j= a ad s w = ( y y ) m j.. j= m There s a cosderable lterature o multstage samplg, but ths subject s stll best dscussed the book by Murthy (967). Multphase Samplg I multphase samplg the same sze of sample uts are retaed at each level (phase) but wth fewer sample uts selected at each cosecutve oe. I the last phase the varable of terest s measured ad s combed wth covarate formato from the early phases ether desg (stratfcato or pps samplg) or estmato (regresso or rato estmato). I multphase samplg a complete frame of uts s requred sce a sample of uts s selected at each phase. The ma reaso for usg multphase samplg s to reduce the cost of samplg by collectg a large amout USDA Forest Servce RMRS-GTR

75 of relatvely cheap formato o covarates that are correlated wth the varables of terest ad the measurg the varables of terest o a smaller sample. Stratfed double samplg ad double samplg for regresso or rato estmato are two examples. Specfcally:. For stratfed double samplg, the large (frst phase) sample formato s used to costruct strata from whch the secod phase samples are selected. Typcally ths s doe f terest s specfc subpopulatos (strata) or the strata are more homogeeous tha the overall populato so that effcecy s gaed by stratfcato. For example, tradtoal large-scale tmber surveys we mght have a large sample of say ' -ha plots from remote sesg or photos classfed to prmarly large tmber, pole tmber, ad regeerato. Clearly, f terest s volume, those three strata would be of terest ther ow rght ad are lkely to be much more homogeeous (f suffcetly well doe by remote samplg) tha the overall populato. A subsample of those -ha plots would the be sampled o the groud for volume by stratum. Smlarly samplg a large park for recreato use, we mght take a large sample of photos o sample days to cout users, use that formato to dvde the park to strata of heavy, moderate, ad low use days, ad the sample these three strata o a subset of those same sample days. The estmator of the total both cases s: h (7) y Yˆ N w N y Ny K h K = ˆ dst = h = h h = st h= h h= ' where K = umber of strata, ˆ Nh Nh = ' s the estmated umber of sample uts stratum h, ' h ' wh = s the estmated weght for stratum h for the frst phase sample wth h ad the frst ' phase sample szes for stratum h ad overall respectvely, h s the secod phase sample stratum h, ad yst s the estmated mea for stratum h for the sample of h uts that stratum. The varace of ths estmator s: ( dst ) ˆ WS V Y = N S + N (73) K h h N h= vh wth S the populato varace of y, S h the varace of y stratum h, vh = h / ad yh ad y st the sample mea for stratum h ad overall sample mea for stratfed samplg respectvely. A almost ubased sample estmator of V ( Y ˆ dst ), f both /N ad are eglgble, s: ( ) ˆ ws g v Y N N w s N w ( y y ) K K K h h dst = h h + h h st h= h h= h= N where g = N. (74) Strata may be of dfferet degrees of terest ad vary homogeety, so varyg samplg rates may be desrable. Ths requres kowledge of or a estmate of the varablty wth the strata order to allocate. If such kowledge s avalable, oe ca the optmally allocate the sample to the strata. Assume that there s formato avalable or easly collectable o a varable x correlated wth y. The applyg the smple cost fucto: C = C + Ch K h= h 70 USDA Forest Servce RMRS-GTR

76 where C s the cost of classfyg a ut for the frst phase ad Ch s the cost of measurg a ut stratum h, the expected cost E(C) s: K EC ( ) = C + CvW h h h. (75) h= The the optmal ' ca be computed from by substtutg C vˆ s s w s K h = yh y 0h yh C h h= for vh where sy ad syh are the estmated varace for varable y the populato ad stratum h respectvely ad w0h s the estmated stratum weght for stratum h based o the prelmary formato. More complex cost fuctos are dscussed the lterature, especally Hase ad others (953), but usually suffcet formato s avalable to assume a better cost fucto, so t makes sample sze computatos more dffcult ad sample sze determato seems farly sestve to mproved cost fuctos.. For double samplg wth rato or regresso estmators, a lear relatoshp s assumed betwee the covarates ad the varables of terest as show the geeral lear model (39). For stace the tmber example above, oe may have cofdece that the formato o the -ha remotely sesed or photo plots s learly related to the same formato as measured o the groud. Or, smlarly, the photo couts of recreatoal users mght be learly related to the actual couts o the groud. Clearly, whether such a lear relatoshp exsts as a useful approxmato or there s a useful but ukow relatoshp betwee the remote sesg ad the groud formato both cases determes whether stratfed double samplg or double samplg wth rato/ regresso estmato s more effcet ad relable. The regresso estmator of the overall total s: ˆ y ˆ ˆ x Ygr = + agr N + bgr X = π = π = π (76) where: a gr = y bgr = πv = πv π v = x b gr = xy y x π v π v vπ vπ = = = = x x π v vπ vπ = = = x wth ˆ ˆ ˆ x ˆ x = π v N =, N =, N s =, X =, X =, x s = π ja π π v π π N j= = = = ja = s USDA Forest Servce RMRS-GTR

77 where π ja s the probablty of selectg ut j the sample of ' uts ad π the probablty of y selectg ut the sample of uts ad = π v y s = N s. The π may ot always be computable. Dervg a classcal varace estmator for ths estmator s dffcult ad ths s a example where bootstrap varace estmato would be the method of choce. Illustrato: A large sample of plots s measured for plot basal areas o aeral photos. These could be stratfed to K strata, selectg ether a subsample of plots the K strata or a SRS of out of the plots whch are the measured o the groud. Usg BAT ad as measured o the photo plots ad volume as measured o the groud plots, we the have: VT to deote basal area o plot plots wth BAT, =,..., plots wth BATh, h =,..., K, =,..., h ad VTh, h=,..., K, =,..., h for stratfed double samplg or BAT, =,..., ad VT, =,..., for double samplg wth regresso. Whether stratfed double samplg or double samplg wth a regresso estmator would be used, depeds o the relatoshp expected betwee BAT ad VT. If there s expected to be a lear relatoshp, regresso estmato would be used, otherwse double samplg for stratfcato s dcated. For stratfed double samplg oe would use (7) ad (74) to estmate total volume ad ts varace. For double samplg wth regresso oe would use (76) wth a bootstrap varace estmator. If oe expects the relatoshps betwee the covarates ad the varables of terest to go through the org approxmately, a double samplg wth a rato of meas estmator ca be used: y ˆ ˆ π ˆ Y = HT ˆ drm = = x ˆ X HT = π Y X X. (77) I ths case too t s best to use bootstrappg to estmate the varace of Y. Here too the ˆdrm may ot always be computable. Multlevel samplg methods forestry are commo especally for large scale surveys. For example:. Double samplg for stratfcato s used large-scale surveys such as FIA. Areas are stratfed usually to forested vs. o-forested areas by ether photography or more commoly ow by data collected from remote sesg sources such as the Ladsat Thematc Mapper Satellte (TM) ad the groud plots are measured those strata. I the past, wth prmary terest tmber, prestratfcato was used. Now post-stratfcato s used because plots are grd-based. Newer remote sesg sources wll defe small features o the groud better ad locatos of both the groud ad remote sesg formato ca be ppoted more accurately wth mproved GPS uts. It s lkely that more detaled stratfcato ad regresso estmato wll mprove estmato the future.. VRP samplg wth subsequet selecto of trees by ether Posso samplg proportoal to estmated tree heghts or aother subsamplg scheme were frequetly used tmber sales. Clearly combatos of multphase ad multstage samplg ca be desrable too. For stace, example above we mght select a radom sample of trees o the selected groud plots. Ths desg the would be double samplg for stratfcato wth radom subsamplg. π 7 USDA Forest Servce RMRS-GTR

78 IX. Motorg Over Tme Maagers of bologcal resources are always terested chage over tme tmber volume, mortalty, wldlfe habtat, degree of urbazato, chage from forest lad to agrcultural lad, etc. There are three major samplg optos to cosder samplg over tme:. Complete replacemet samplg (CRP). Complete remeasuremet samplg (CR) 3. Samplg wth partal replacemet (SPR), a combato of a ad b. I CRP samplg, a completely ew set of sample plots s used each tme. Such a desg s smple ad cheap to mplemet sce plot locatos do ot have to be moumeted for future use ad oe does ot have to worry about the plots beg treated dfferet from other parts of the populato or chages the uderlyg populato. CRP samplg s effcet for estmatg curret attrbutes but ot effcet for estmatg chage relatve to CR ad SPR. I CR samplg all sample plots are remeasured perodcally. Ths requres that they rema represetatve of the populato over tme, so that the plots should ot be vsted excessvely, ad should be treated o dfferetly from other parts of the populato. CR samplg s the most effcet of the methods avalable for chage estmato. I SPR a radom subset of the permaet plots s remeasured as well as a ew set of plots,.e., t s a combato of CRP ad CR samplg. Regresso estmato betwee the remeasured ad ew plots s used to update the plots that were ot remeasured. SPR ca be effcet whe tryg to balace precso betwee curret ad chage estmato. Duca ad Kalto (987) summarzed the propertes of the three optos cely. They also lst aother method that s a combato of the other three (Table 5). Both CR ad CRP samplg are specal cases of SPR samplg from a estmato pot of vew so we oly preset SPR samplg for two occasos here: If sample uts are selected from N uts at both occasos wth m uts commo to both, the u = m uts are ot shared. Let Yˆ, ˆ m Y, ad u Y equal the estmates of Y ˆ, the populato total o the th occaso ( =,), based o the m, u, ad uts respectvely, ˆ β = the regresso coeffcet estmator based o the m commo uts, σ ad σ the varaces of y at tmes ad, σ the covarace of y betwee tmes ad ad ρ the correlato betwee measuremets at tmes ad. The a ubased estmator of Y based o the u ew uts at tme s: u y ˆ = Y u = N, (78) u wth varace: V ( Yˆ ) = N σ / u, (79) u ad varace estmator: v( Yˆ ) = s ()/ u (80) u wth s () the wth-sample varace of the u y measuremets. Equvaletly, a regresso-based estmator of Y usg the m commo uts at tmes ad to update the total from tme s: Yˆ = Yˆ + ˆ β ( Yˆ Yˆ ) (8) mr m m USDA Forest Servce RMRS-GTR

79 Table 5. Objectves ad propertes of four remeasuremet desgs (adapted from Duca ad Kalto 987). Complete replace- Complete remeasuremet samplg met samplg Samplg wth partal Combato of CR wth Samplg objectve (CRP) (CR) replacemet (SPR) CRP or SPR a) Estmate populato Automatcally takes Needs mechasm for Needs mechasm for Remeasuremet compoet parameters at dstct to accout takg populato takg to accout eeds mechasm for takg tmes. populato chages. chages to accout. populato chages populato chages to b) Estmate average durg lfe of replace- accout. values of populato met group. Composte parameters. estmates ca be used to produce effcet estmates. c) Estmate et chage. Estmates combed Needs mechasm for Needs mechasm for Remeasuremet compoet effect of chagg takg populato takg to accout eeds mechasm for takg values ad chagg chages to accout. populato chages populato chages to populato. Varace of chage durg lfe of accout. Varace of chage reduced by postve remeasuremet. remeasuremet compoet correlato of values Composte estmato reducedby postve correlato betwee surveys. ca be used to produce of values betwee surveys. effcet estmates. d) Estmate compoets Not possble. Well-suted for these Ca be used for chage Complete remeasuremet of chage over tme. populatos. estmato or aggregate compoet s well-suted for e) Aggregate data for formato over tme these purposes. Not possble dvduals over tme. perods shorter tha the for complete replacemet tme a sample s to be surveys compoet. replaced sample. Oly the sample to be replaced ca be used. f) Collect data o evets Not possble. Ca costruct log- Ca costruct log-term Ca costruct lmted logoccurrg specfc tme term hstory of evets hstory of evets but o term hstory of evets. perods. by combg data a more lmted from several surveys. bass tha complete remeasuremet surveys. g) Cumulate samples Excellet for statc Not useful for statc Of some use for statc Complete remeasuremet over tme. characterstcs ad characterstcs, but characterstc ad survey compoet s for ew evets. useful for ew useful for ew evets. excellet. Complete evets. remeasuremet survey compoet s useful for ew evets but ot for statc characterstcs. wth varace: ˆ σ uρ V ( Y mr ) = N. (8) m ˆ u ˆ mr Combg estmates Y ad Y to obta a mproved estmate of Y s usually doe by weghg them versely proportoal to ther sample varaces, so we obta: Yˆ wth approxmate varace V ( Y ˆ ) / w ( wy ˆ ˆ ˆ ˆ u + wy mr) = (83) wˆ 74 USDA Forest Servce RMRS-GTR

80 wth ad varace estmator: w = w + w = / σ + / σ ˆ 4 wˆ ( ˆ ˆ w w) v( Y ˆ ) = + w wˆ = d (84) wˆ =, wˆ =, wˆ = wˆ + wˆ, d = m, d = u, ad w s estmated by ŵ. wth vy ( ˆ ˆ u) vy ( mr) Two estmators of chage Y are possble, the most obvous oe beg: Yˆ = Yˆ Yˆ. (85) A desrable property of such a estmate s that t s cosstet wth the estmates at the two occasos. A more effcet estmator tha Y ˆ geeral takes advatage of the regresso based o the m commo uts as Y ˆ above. Ths estmator s: ˆ ˆ ˆ wˆ Y ˆ + w Y Y =. (86) wˆ Here Y ˆ ad Y ˆ are the estmators of chage from the m remeasured ad u umatched plots respectvely, where wˆ are ubased estmators respectvely of: w s s s s s m m m u u = +, wˆ = + = σ + σ σ = σ + σ, m m m u u, w wˆ wˆ wˆ wth approxmate varace estmator: = + estmates w w w = +, ad V ( Yˆ ) = / w (87) where: ( ˆ ) 4 wˆ ( ˆ ˆ w w) v Y ˆ = + w wˆ (88) = d d = m, d = u. Problem: Show how CR ad CPR samplg are specal cases of SPR samplg. Aswer: Set ρ = 0 (8) to get the varace for CR samplg ad ρ = to get the varace for CPR samplg. The matchg proporto SPR samplg depeds o the correlato ρ betwee the measuremets at the two tmes. It should ot exceed 0.5 for optmzg the estmator Y ˆ. SPR samplg quckly becomes much more complcated estmato whe more tha two occasos are measured (Schreuder ad others 993). But a serous dsadvatage of SPR samplg, that of varace USDA Forest Servce RMRS-GTR

81 estmato, has bee elmated. Wth bootstrappg t should be smple to geerate varace estmates for ay umber of remeasuremets ad estmato schemes. All of the FIA uts the USA ow use complete remeasuremet samplg, although oe formerly used samplg wth partal replacemet. I geeral, SPR samplg s probably most effcet but becomes qute complex from a aalyss pot of vew, whch makes t hard to deal wth the umerous specal requests of estmates for specfc subpopulatos of the survey populato covered. 76 USDA Forest Servce RMRS-GTR

82 X. Buldg Models ad Cause-Effect The statemet by Box ad Draper (987) that all models are wrog; the practcal questo s how wrog do they have to be to ot be useful s geerally accepted the statstcal world, ad ca be paraphrased as all models are wrog ad some are useful. The utlty of models s ofte assessed by the degree of correlato betwee the varables of terest ad covarates, but ote that correlato does ot prove causato (Ksh 967). Much f ot all of research revolves aroud model buldg, ad the potetal msuse of models has bee greatly facltated by the ready avalablty of computers ad easy use of regresso programs. Ideally, a researcher observes the real world or carefully studes substatve scetfc theores. Models are the developed o the bass of the sghts accorded, recogzg the fact that besdes the explaatory varables, there are other sources of varato to be cosdered. Ksh (967) separates all sources of varato to four classes:. The explaatory or expermetal varables that are the objectves of the research explag or establshg a relatoshp betwee both the depedet (ofte called the respose varables ths cotext) ad the depedet (ofte called predctor varables ths cotext) varables.. Extraeous varables that ca be cotrolled ether sample selecto or estmato. 3. Extraeous (umeasured, ofte umeasurable) varables that may be cofouded wth the varables class above. 4. Extraeous, dffcult to cotrol or ucotrollable varables that have to be treated as radomzed errors. I deal expermets, they ca be radomzed, whereas surveys they ca oly be assumed to be radomzed. I all research oe wats to place as may extraeous varables as possble class. Sce ths usually caot be doe, we have expermets ad surveys. Expermets, the coduct of a systematc, cotrolled test or vestgato, tres to cotrol the varables class 3 as much as possble by tryg to place all of the thrd class of varato to the fourth through radomzato. I a deal expermet, there are o varables the thrd class. I a deal survey, all varables class 3 are separated from those class through regresso adjustmets, matchg of uts, ad stadardzato. If there was complete commad over the research stuato, oe could troduce the desred effects ad measuremets to cotrolled ad radomzed portos of the target populato wth frm expermetal cotrols ad buld a true model (Ksh 967). Such stuatos are rare so that we have expermets that are strog o cotrol through radomzato but weak o represetg actual populatos of terest ad frequetly weak o the atural settg of the model beg bult. Surveys ofte are feasble whe expermets caot be doe, the most obvous beg that we do ot expermet o humas. Surveys are strog o represetato but are ofte weak o cotrol. The latter s a ready explaato of why so may studes are publshed clamg ths or that chemcal s bad for you ad subsequet research does ot support such clams. Ofte survey data are used to buld models, to lead to a better uderstadg of what s gog o. May models appear to have poor predctve ablty; for example, ths s true for the staple of forest research: buldg growth ad mortalty models. A mssg gredet s key data that would help detfyg cause-effect relatoshps such as daly rafall, atmospherc deposto, sol mosture cotet, etc. Such data caot be collected yet a practcal maer cojucto wth atural resources surveys, but developmet of ew strumetato should make that feasble some day. Utl ths formato s routely avalable for the plots, predcto models for growth, mortalty, eroso, ad other key varables are ulkely to be relable. Large-scale surveys such as FIA ad the Natural Resources Ivetory (NRI) of the Natoal Resources Coservato Servce (NRCS) ca establsh treds chage for large areas, ca be used to suggest ad detfy potetal cause-effect relatoshps, ad ca suggest useful hypotheses to USDA Forest Servce RMRS-GTR

83 documet relatoshps (Olse ad Schreuder 997). Ifereces about possble cause-effect relatoshps have to be terpreted cautously because screeg of data makes t dffcult to defe the populato of ferece (see for example Schreuder ad Thomas 99). It s ufortuate the USA that there are two atural resources surveys, Forest Ivetory ad Aalyss of the USFS ad the NRI of the NRCS where both ageces are the USDA. Narrowly focused surveys seem to be the rule most other coutres too as evdeced from descrptos Europea Commsso (997). Complemetaress of the data collected would make t more lkely to detfy promsg causeeffect relatoshps for a wder rage of resource varables. For example, t would be desrable to have the relable formato o sols collected by the NRI also avalable o the FIA plots to develop better growth ad yeld models. Mosteller ad Tukey (977) detfy three crtera, of whch two have to be satsfed to fer cause-effect relatoshps: cosstecy, resposveess, ad mechasm. Cosstecy mples the presece ad magtude of the effect y, assocated wth a mmal level of the suspected causal aget x. Resposveess s establshed by a expermetal exposure of the suspected causal aget ad reproducg the symptoms. Mechasm demostrates the bologcal or ecologcal process causg the observed effect. To establsh all three crtera s dffcult. For example, the cosstecy betwee smokg ad cacer was establshed the 950s. The resposveess was well documeted the ad the 960s, but the actual causal mechasm was ot establshed utl the 990s (Pefer 997). Ad ths lkage was relatvely easy to establsh because the effect s dramatc; see for example Taubes (995). Feste (988) advocated the followg scetfc prcples for establshg cause-effect: stpulate a hypothess pror to aalyss, study a well-defed cohort havg a statstcal factor commo, collect hgh-qualty data, study possble explaatos, ad avod detecto bas. Hll (965) the epdemologcal lterature suggests a weght of evdece approach cosstg of e crtera for ferrg causalty: stregth, cosstecy, specfcty, temporalty, bologcal gradet, plausblty, coherece, expermetal evdece, ad aalogy. Stregth refers to havg a hgh magtude of a effect assocated wth exposure to the stressor; cosstecy to repeatedly observg the assocato of the observed effect ad stressor uder dfferet codtos; specfcty to the degree of the effect beg more lkely to be dagostc of the stressor ad the ease of assocatg t wth a effect; temporalty to the fact that the stressor always precedes the effect tme; bologcal gradet to the chage effect wth correspodg chages the stressor; plausblty that the assocato betwee effect ad stressor s cosstet wth physcal, chemcal, ad bologcal prcples; expermetal evdece that chages effect are documeted after expermetal mapulato or through recovery of the populato followg relef of the stressor; ad aalogy s havg smlar stressors assocated wth smlar effects. The more of these crtera that are satsfed, the more weght ca be gve to the evdece that there s probable cause. Survey data ca oly provde formato for detfyg possble cause-effect relatoshps. Establshg that there s a correlato betwee possble cause ad effect varables s a useful frst step ths detfcato For readers who wat more formato o how to maxmze the possbltes of such detfcato, we refer them to Olse ad Schreuder (997) ad Gadbury ad Schreuder (003). 78 USDA Forest Servce RMRS-GTR

84 XI. Forest Samplg Stuatos Ptfalls Now that you have studed samplg tesvely you may thk you kow what to do. Ths secto covers some major errors commtted by serous samplers over tme the USA.. Movg subplots. FIA used to select 0.4 ha ( acre) prmary sample uts subsampled by a seres of 5 or 0 VRP subplots. As s logcal, some cases some of the subplots mght fall a dfferet codto tha the ceter subplot. For example, subplots -3 mght be samplg a pe platato ad subplots 4-5 a hardwood stad. The decso was made as early as the 930s ad cotued to the 990s by several FIA uts to keep all subplots the same forest type. For example f subplot (the ceter oe) was a pe platato, all subplots ot fallg the pe platato would be moved by some systematc rule to the pe platato. Ths procedure bases the results (Wllams ad others 996).. Averagg codtos. Related to the above, aother ut dd ot move the subplots the above stuato but made the equally udesrable decso to average forest types,.e., they dd ot keep track of what type was beg sampled. They would call the plot descrbed () above a mxed pe-hardwood stad. Stuatos ad led to the terestg stuato where two states the USA that are qute smlar showed huge dffereces the area pe/hardwood stads as a percetage of the total forest. 3. VRP samplg to get 6-8 trees per pot. A reasoable recommedato was made a forest mesurato textbook that VRP samplg oe should select a prsm factor yeldg o average 6-8 trees/pot. Ths recommedato was followed up correctly several places the Wester USA. Feld crews would take varous prsms to the feld wth them ad the select oe at a locato that would yeld them betwee 6 ad 8 trees. Ths based approach surprsgly was supported by several promet bometrcas. I several expermets of ths method, lttle or o bas resulted but oe author got volved a stuato Calfora where such serous bas was oted (see Wesel ad others 980 ad Schreuder ad others 98). 4. Msuse of model predctos. A FIA ut developed growth ad mortalty models based o growth ad yeld studes ad used those models to update the formato o plots they could ot remeasure from a cost pot of vew. The predcted plot values were the used as real plot data for geeratg state-wde estmates. 5. Droppg subplots to meet producto targets. FIA program maagers put heavy emphass o meetg producto targets. Ths s why oe ut approved of the elmato of subplot 4 to meet the producto of 8 plots per weeks for crews f they felt they could ot meet ther producto targets. Ths bases the results, especally f crews decde to judcously drop subplots 4 such as dffcult samplg codtos. 6. Forgettg probabltes of selecto. A govermet agecy selected a tmber crusg sample usg stratfed samplg to obta volume estmates for dfferet strata. Te years later they decded they wated to revst the locatos for other purposes but had ot kept track of the probabltes of selecto. They wated to treat the exstg sample as a SRS for remeasuremet purposes. Schreuder ad Alegra (995) llustrate how ths may serously bas results. 7. Treatg subplots as plots because ther formato s cosdered ucorrelated. Clearly the subplots are ot depedet observatos ad hece should ot be treated as such. 8. Dfferet results by dfferet ageces. Two ageces the same departmet estmated vastly dfferet areas of forest several states. It tured out that ths was due to dffereces terpretg a commo defto of forest, defto of what s a tree, ad stadards measuremet ad estmato techques. Several of these dffereces are also ted to USDA Forest Servce RMRS-GTR

85 cosderg forest as a use class (Goebel ad others 998). A forester may prefer to see as much forest a state as possble, whle a rage maager may see the same lad as rage. Problem: Usg a tree growth model developed from growth ad yeld study data, how would the predctos for such a model compare to the actual growth of trees of the same speces o vetory plots? Aswer: Growth ad yeld studes typcally use plots wth 00 percet stockg levels wth a much more favorable evromet tha that of vetory plots whch are more lkely mpacted by sects ad dseases, by huma actvtes, etc. It s therefore more lkely that the predctos wll yeld overestmates of the actual growth of the vetory trees. Problem: A umotvated crew usg the stuato descrbed #5 decdes to always drop subplot 4 whe they kow t s dffcult to measure. What are the cosequeces? Aswer: Clearly ths wll bas estmates for the area vetored because t chages the probabltes of selecto of the subplots ad hece the plot that they are part of. It s really ot possble to aswer whch way the bas wll go. Some of the subplots may be areas dffcult to measure because they are a swamp whle others may be hghly productve areas where ves ad uderbrush make access to the subplots dffcult ad others may be o a very uproductve, steep clff. Problem: A well-tetoed crew usg the stuato descrbed #5 decdes to measure subplot 4 oly whe they kow t cotas ce tmber trees. What are the cosequeces? Aswer: Ths should bas certa estmates upward such as those for tmber volume. It mght ot have much effect o estmates for varables ot much correlated wth ceess of trees such as umber of trees or mortalty. Problem: A crew used multple prsms selectg the oe gvg 6-8 trees at each pot VRP samplg. Geeratg estmates for the 00 VRP plots they took geerates a estmate of 00,000 3 m for the area. They sell the tmber based o that amout of volume. The compay that buys the tmber fds 3 oly 60,000 m. They are ot happy ad fle a lawsut agast the compay that dd the vetory. Both sdes approach you, a rekow vetory expert, to testfy o ther behalf. Sce truth s more mportat to you tha moey, you ca pck ether sde. Whch oe would you pck? Aswer: It would be more sesble to take the sde of the compay that bought the tmber. Certaly the vetory method used was faulty as dcated #3 above. Suggestos Our experece leads us to beleve that oe ca do better tha what s avalable ow regards to surveys. It s hard to chage exstg surveys. Hece:. Be more flexble, less hdeboud.. Do research o what s avalable ad what ca be doe better. 3. Documet well what you are dog. 4. Observe the Keep t smple (KIS) prcple desg, less so estmato. 5. Use creatve, competet aalysts. 80 USDA Forest Servce RMRS-GTR

86 6. Focus o the objectves. For example, there may be eed to resolve potetal coflcts betwee what s wated for tmber surveys for whch there s ofte strog poltcal support vs. ecology parameters where such support may be less powerful. Atcpate what may be eeded the future too. 7. Keep up o the world lterature ad cotrbute to t. 8. Defe measurable varables (see Schreuder ad others 993, p. 9, specfcally the warg by Ies for example). Over tme, we have leared the followg lessos:. The objectves of a successful survey wll chage over tme ad wll become more ecompassg.. Do t lock yourself to exstg approaches. Allow for chage. A example s plot desg where the USA we have goe from rectagular plot samplg to varable radus plot or VRP samplg, to crcular plot samplg ad more tha lkely at a future date should move to samplg usg dfferet plots for dfferet varables cludg a log rectagular or square plot closely ted to remotely sesed formato. Over tme, more formato ca be collected by remote sesg. Large-scale surveys are gettg away from a pure tmber oretato to beg more ecologcally based so that we are terested also lear features such as rparara areas, uderstory vegetato, ad rare ad edagered plat speces. Because plots ca be more accurately colocated o both the remote sesg ad the actual groud plots by the use of geographc postog systems (GPS) over tme, ad because more ad more detal ca be dscered wth ewer remote sesg platforms, more effcet estmato wll be possble by combg groud ad remote sesg formato statstcal regresso estmato models. 3. The estmates/aalyses ca be ad should be as defesble as possble. A fudametal prcple FIA s to keep thgs smple: KIS (keep t smple). Our recommedato s to keep the desg smple but allow for more complexty the aalyses, sce dfferet people wat to use the data dfferet ways. We are lkely to have much cotroversy aalyzg the aualzed data sets before agreemet s reached. USDA Forest Servce RMRS-GTR

87 XII. Refereces Aldrch, R. C Remote sesg of wldlad resources: a state-of-the-art revew. Ge. Tech. Rep. RM- 7. Fort Colls, CO: Rocky Mouta Forest ad Rage Expermet Stato. 56 p. Arvats, L. G.; Rech, R. M Natural resources samplg. New York: Sage. Avery, T. E.; Burkhart, H. E Forest measuremets. 3d ed. New York: McGraw-Hll. 33 p. Bggs, P. H.; Pearce, C. J.; Wescott, T. J GPS avgato for large-scale photography. Photogrammetrc Egeerg ad Remote Sesg. 55: Bggs, P. H.; Specer, R. D New approaches to extesve forest vetory Wester Australa usg large-scale aeral photography. Australa Forestry. 53: Btterlch, W The agle cout method ( Germa). Allgemees Forst-ud Holzwrtschaftlche Zetug. 58: Bolstad, P. V.; Smth, J. L. 99. Errors GIS. Joural of Forestry. November: 9. Box, George; Draper, Norma Emprcal model buldg ad respose surfaces. Joh Wley & Sos: 74. Brewer, K. R. W.; Haf, M Samplg wth uequal probabltes (lecture otes statstcs). New York: Sprger-Verlag. 64 p. Bucklad, S. T.; Aderso, D. R.; Burham, K. P.; Laake, J. L.; Borchers, D. L.; Thomas, L. 00. Itroducto to dstace samplg. Oxford Uversty Press. 43 p. Buge, J.; Ftzpatrck, M Estmatg the umber of speces: a revew. Joural of the Amerca Statstcal Assocato. 88: Burham, K. P Is fte populato samplg always applcable to fte populatos? Ivted presetato to Amerca Statstcal Assocato atoal meetg; 980 August; Housto, TX. Carroll, R. J.; Rupert, D Trasformatos ad weghtg regresso. New York: Chapma ad Hall. 49 p. Cassel, C-M.; Sardal, C-E.; Wretma, J. H Foudatos of ferece survey samplg. New York: Joh Wley & Sos. 9 p. Chao, A.; Lee, S-M. 99. Estmatg the umber of classes va sample coverage. Joural of the Amerca Statstcal Assocato. 87: 0 7. Cochra, W. G Samplg techques. 3d ed. New York: Joh Wley & Sos. 48 p. Cogalto, R. G.; Gree, K. 99. The ABCs of GIS. Geographc Iformato Systems. Part. Joural of Forestry. 90(): 3 0. Cramer, H Mathematcal methods of statstcs. Prceto, NJ: Prceto Uversty Press. 575 p. Czaplewsk, R. C Multstage remote sesg. Towards a aual atoal vetory. Joural of Forestry. 97(): Czaplewsk, R. C Ca a sample of Ladsat sesor scees relably estmate the global extet of tropcal deforestato? Iteratoal Joural of Remote Sesg. 4: Dawd, A. P Iferece, statstcal: I. I: Kotz, S.; Johso, N. L. New York: Joh Wley & Sos. Ecyclopeda of statstcal scece. 4: De Vres, P. G Samplg theory for forest vetory. A teach-yourself course. New York: Sprger- Verlag. 399 p. Demg, W. E O probablty as a bass for acto. Amerca Statstca. 9: Ducey, M. J.; Gove, J. H.; Valete, H. T A walkthrough soluto to the boudary overlap problem. Forest Scece ( process). Duca, G. J.; Kalto, G.987. Issues of desg ad aalyss of surveys across tme. Iteratoal Statstcal Revue. 55: Europea Commsso. Study o Europea forestry formato ad commucato system reports o forestry vetory ad survey systems. Vol. ad. Luxembourg: Offce for Offcal Publcatos of the Europea Commutes: L-985. Feste, A. R Scetfc stadards epdemologcal studes of the meace of daly lfe. Scece. 4: Fraco-Lopez, H Updatg forest motorg systems estmates. EM Meapols: The Uversty of Mesota. 48 p. Dssertato. Fraser, D. A. S Iferece, statstcal: II. I Kotz, S.; Johso, N. L. New York: Joh Wley & Sos. Ecyclopeda of statstcal scece. 4: Freese, F. 96. Elemetary forest samplg. Agrculture Hadbook. 3. Washgto, DC: U.S. Departmet of Agrculture, Forest Servce. 9 p. Gadbury, G. L.; Schreuder, H. T Cause-effect relatoshps aalytcal surveys: a llustrato of statstcal ssues. Evrometal Motorg ad Assessmet. 83: USDA Forest Servce RMRS-GTR

88 Goebel, J. J.; Schreuder, H. T.; House, C. C.; Gesler, P. H.; Olse, A. R.; Wllams, W. W Itegratg surveys of terrestral atural resources: the Orego demostrato project. Tech. Rep.. Fort Cols, CO: U.S. Departmet of Agrculture, Forest Servce, Forest Ivetory ad Motorg Isttute. 0 p. Gree, K. 99. Spatal magery ad GIS. Joural of Forestry. November: Gregore, T. G Desg-based ad model-based ferece survey samplg: apprecatg the dfferece. Caada Joural of Forest Research. 8: Gregore, T. G.; Scott, C. T Samplg at the stad boudary: a comparso of the statstcal performace amog eght methods. I: Proceedgs XIX World Forestry Cogress IUFRO; 990 August 5 ; Motreal, Caada. Publ. FWS Blacksburg: Vrga Polytech Isttute ad Uversty: Gregore, T. G.; Valete, H. T Samplg techques for atural resources ad the evromet. New York: Chapma Hall CRC Press. I process. Grosebaugh, L. R Some suggestos for better sample-tree measuremet. I: Proceedgs Socety of Amerca Foresters; 963; Bosto, MA: Grosebaugh, L. R The gas from sample-tree selecto wth uequal probabltes. Joural of Forestry. 65: Haas, P. J.; Stokes, L Estmatg the umber of classes a fte populato. Joural of the Amerca Statstcal Assocato. 93: Hah, G. J.; Meeker, W. O Assumptos for statstcal ferece. Amerca Statstca. 47:. Hajek, J Some cotrbutos to the theory of probablty samplg. Bullet of the Iteratoal Statstcal Isttute. 36: Hase, M. H.; Hurwtz, W. N.; Madow, W. G Sample survey methods ad theory. Vol. I ad II. New York: Joh Wley & Sos. 638 p, 33 p. Hll, A. B The evromet ad dsease: assocato or causato? Proceedgs of the Royal Socety of Medce. 58: Holmgre, P.; Thuresso, T Satellte remote sesg for forestry plag: a revew. Scadava Joural of Forest Research. 3: Hush, B. 97. Plag a forest vetory. FAO Forest Products Studes No. 7. Rome, Italy. p. Iles, K A sampler of vetory topcs. Naamo, B.C., Caada: Km Iles Assocates. 869 p. Johso, E. W Forest samplg desk referece. New York: CRC Press. 985 p. Kalto, G.; Aderso, D. W Samplg rare populatos. Joural of the Royal Statstcal Socety A. 49: Ksh, L Survey samplg. d ed. New York: Joh Wley & Sos. 643 p. Koch, G. G.; Gllgs, D. B Iferece, desg based vs model based. I: Kotz, S.; Johso, N. L. New York: Joh Wley & Sos. Ecyclopeda of statstcal scece. 4: Kotz, S.; Johso, N. L. New York: Joh Wley & Sos. Ecyclopeda of statstcal scece p. Kruskal, W. H.; Mosteller, F Represetatve samplg. I: Kotz, S.; Johso, N. L. New York: Joh Wley & Sos. Ecyclopeda of statstcal scece. 8: Kuter, M.; Neter, J.; Nachtshem, C.; Wasserma, W Appled lear regresso models. 4th ed. New York: McGraw-Hll/Irw. 67 p. Lachowsk, H.; Maus, P.; Platt, B. 99. Itegratg remote sesg wth GIS. Joural of Forestry. : 6. Lefsky, M. A.; Cohe, W. B.; Parker, G. G.; Hardg, D. J. 00. Ldar remote sesg for ecosystem studes. Boscece. 5: Lllesad, T. M.; Kefer, R. W Remote sesg ad mage terpretato. d ed. New York: Joh Wley & Sos. 7 p. L, J-M Small area estmato. Fort Colls: Colorado State Uversty, Statstcs Departmet. 344 p. Dssertato. Max, T. A.; Schreuder, H. T.; Hazard, J. W.; Oswald, D. D.; Teply, J.; Alegra, J The Pacfc Northwest Rego Vegetato ad Ivetory Motorg System. Res. Pap. PNW-RP-493. Portlad, OR: U.S. Departmet of Agrculture, Forest Servce, Pacfc Northwest Research Stato. Mosteller, F.; Tukey, J. W Data aalyss ad regresso. Readg, MA: Addso-Wesley Publshg Co. 586 p. Murthy, M. N Samplg theory ad methods. Calcutta, Ida: Statstcal Publshg Co. 684 p. Olse, A. R.; Schreuder, H. T Perspectves o large-scale atural resource surveys whe cause-effect s a potetal ssue. Evrometal ad Ecologcal Statstcs. 4: Overto, W. S.; Stehma, S. V The Horvtz-Thompso Theorem as a ufyg perspectve for probablty samplg: wth examples from atural resource samplg. Amerca Statstca. 49: Pefer, M Cacer-beta-cate as ocogee: the smokg gu. Scece. 75: Pkham, R. S A effcet algorthm for drawg a smple radom sample. Appled Statstcs. 36: Rosefeld, G. H.; Ftzpatrck-Ls, K A coeffcet of agreemet as a measure of thematc classfcato accuracy. Photogrammetrc Egeerg ad Remote Sesg. 5: 3 7. USDA Forest Servce RMRS-GTR

89 Sardal, C-E A two-way classfcato of regresso estmato strateges probablty samplg. Caada Joural of Statstcs. 8: Sardal, C-E.; Swesso, B.; Wretma, J. 99. Model asssted survey samplg. New York: Sprger-Verlag. 694 p. Schreuder, H. T Smplcty versus effcecy samplg desgs ad estmato. Evrometal Motorg ad Assessmet. 33: Schreuder, H. T.; Alegra, J Stratfcato ad plot selecto rules, msuses ad cosequeces. Res. Note RM-RN-536. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Forest ad Rage Expermet Stato. 4 p. Schreuder, H. T.; Ba, S.; Czaplewsk, R. C Accuracy assessmet of percet caopy cover, cover type ad sze class. Ge. Tech. Rep. RMRS-GTR-08. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Research Stato. 0 p. Schreuder, H. T.; Czaplewsk, R. L. 99. Log-term strategy for the statstcal desg of a forest health motorg system. Evrometal Motorg ad Assessmet. 7: Schreuder, H. T.; Gessler, P. H Plot desgs for ecologcal motorg of forest ad rage. North Amerca Scece Symposum. Towards a ufed framework for vetoryg ad motorg forest ecosystem resources symposum; 998 November 4; Guadalajara, Mexco. Proc. RMRS-P-. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Forest ad Rage Expermet Stato: Schreuder, H. T.; Gregore, T. G. 00. For what applcatos ca probablty ad o-probablty samplg be used? Evrometal Motorg ad Assessmet. 66: 8 9. Schreuder, H. T.; Gregore, T. G.; Wood, G. B Samplg methods for multresource forest vetory. New York: Joh Wley & Sos. 446 p. Schreuder, H. T.; L, H. G.; Sadoogh-Alvad, S. M Suter s pps wthout replacemet samplg as a alteratve to Posso samplg. Res. Pap. RMRS-RP-90. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Forest ad Rage Expermet Stato. 6 p. Schreuder, H. T.; L, J-M. S.; Teply, J Estmatg the umber of tree speces forest populatos usg curret vegetato survey ad forest vetory ad aalyss approxmato plots ad grd testes. Res. Note RMRS-RN-8. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Research Stato. 7 p. Schreuder, H. T.; Schreer, D. A.; Max, T. E. 98. Esurg a adequate sample at each locato pot samplg. Forest Scece. 7: Schreuder, H. T.; Thomas, C. E. 99. Establshg cause-effect relatoshps usg forest survey data. Forest Scece. 37: Schreuder, H. T.; Wllams, M. S Relablty of cofdece tervals calculated by bootstrap ad classcal methods usg the FIA -ha plot desg. Ge. Tech. Rep. RMRS-GTR-57. Fort Colls, CO: U.S. Departmet of Agrculture, Forest Servce, Rocky Mouta Research Stato. 6 p. Schreuder, H. T.; Wllams, M. S.; Rech, R Estmatg the umber of tree speces a forest commuty usg survey data. Evrometal Motorg ad Assessmet. 56: Schwarz, C. J.; Seber, G. A. F.999. Estmatg amal abudace. Revew III. Statstcal Scece. 4: Shver, B. D.; Borders, B. E Samplg techques for forest resource vetory. New York: Joh Wley & Sos. 356 p. Smth, T. M. F Sample surveys: ; a age of recoclato? Iteratoal Statstcal Revew. 6: Stahl, G.; Rgvall, A.; Frdma, J. 00. Assessmet of coarse woody debrs a methodologcal overvew. Ecologcal Bullet. 49: Stuart, A Some remarks o samplg wth uequal probabltes. Bullet of the Iteratoal Statstcal Isttute. 40: Taubes, G Epdemology faces ts lmts. Specal ews report. Scece. 69: Thompso, W. L.; Whte, G. C.; Gowa, C Motorg vertebrate populatos. New York: Academc Press. 365 p. USDA Forest Servce Implemetato of remote sesg for ecosystem maagemet. Salt Lake Cty, Utah. U.S. Departmet of agrculture, Forest Servce, Egeerg Staff, Remote Sesg Applcatos Ceter. Wesel, L.; Levta, J.; Barber, K Selecto of basal area factor pot samplg. Joural of Forestry. 78: Wllams, M. S.; Gove, J. H Perpedcular dstace samplg: a alteratve method for samplg dowed coarse woody debrs. Caada Joural of Forest Research. 33: 6. Wood, G. B Geeratg the lst of radom umbers for 3P samples. Australa Forester. 50: Wood, G. B Groud samplg methods used to vetory tropcal mxed/most forest. Forest Ecology ad Maagemet. 35: USDA Forest Servce RMRS-GTR

90 XIII. Glossary Accuracy. Freedom from error or the closeess of a measuremet or estmate to the true value. More broadly, t s the degree to whch a statemet or quattatve result approaches the truth. Note that Accuracy = Precso + Bas usg these statstcal deftos. Thus, f bas s elmated, Accuracy = Precso. Asymptotcally ubased. Estmato bas goes to 0 as sample sze approaches populato sze. It s the same as cosstecy as used by Cochra (977). Attrbute. Uts classfed as havg or ot havg some specfc qualty. Basal area (per ste). The cross-sectoal area at breast heght of all trees o the ste. Basal area (per tree). The cross-sectoal area of a tree at breast heght. Bas. A systematc error troduced to samplg, measuremet, or estmato by selectg or favorg, possbly utetoally, oe outcome or aswer over others. Breast heght. The pot o a tree stem at.4 m (4'6") the USA, New Zealad, Burma, Ida, Malaysa, South Afrca, ad some other coutres ad.3 m (4'3") above groud cotetal Europe, Great Brta, Australa, Caada, ad Mexco. Cosstecy. The same as asymptotcally ubased as defed above. Cotuous varable. A varable expressed a umercal scale of measuremet, where ay terval of t ca be subdvded to a fte umber of values. Correlato coeffcet. A measure of the degree of lear assocato betwee two varables that s uaffected by the szes or scales of the varables. Covarace. A varace or measure of assocato betwee pared measuremets of two varables. Covarate. A quattatve, ofte explaatory varable a model such as a regresso model. Covarates are ofte mportat mprovg estmato. dbh. The dameter at breast heght of a tree. Dscrete varable. Qualtatve varables or those represeted by tegral values or ratos of tegral values. Double samplg. Two levels of samplg where the frst level provdes formato o covarates ad the secod o the varable of terest to estmate parameter(s). Effcet estmator. A estmator that predcts a parameter more relably tha competg estmators where relablty s usually measured by the rato of the mea square errors of the estmators. Estmate. The umercal value calculated from a estmator for a sample. Estmator. A fucto of the values a sample or a formula used for estmatg a parameter based o a sample. Estmator of populato mea. The formula used estmatg the populato mea from a sample. Estmator of populato varace. The formula used estmatg the populato varace from a sample. Expermet. The coduct of a systematc, cotrolled test or vestgato. Global postog system (GPS). A system usg satelltes to locate groud postos. Iferece. The drawg of coclusos based o data or observatos. Mea. The average value of a varable for all uts a populato or sample. Meda. The value of a varable so that half of the values are larger ad half are smaller tha ths value a populato or sample. Mode. The value of a varable that occurs most frequetly a populato or sample. USDA Forest Servce RMRS-GTR

91 Multlevel samplg. A samplg desg where more tha oe phase or stage of samplg s used. The frst levels are used to collect formato o covarates useful for more effcet estmato of the ultmate parameter(s) of terest for whch formato s usually collected at the last phase or stage. Parameter. A characterstc or fucto of the values of the uts a populato,.e., the populato characterstc of terest, such as average volume per ha or total volume of trees a forest. Populato. A aggregate of tems each wth a commo characterstc or commo set of characterstcs. I the statstcal sese, a populato s a assembly of dvdual uts formed order to descrbe the populato quattatvely. For example, t mght be all the trees a partcular forest stad or all the users of a recreato area. pps samplg. A samplg desg where sample uts are selected wth a probablty proportoal to a measure of sze, usually a covarate such as dbh or basal area the case of tree volume. Precso. Relatve freedom from radom varato. I samplg t s expressed as the stadard error of the estmate ad relates to the degree of clusterg of sample values about ther ow average or the reproducblty of a estmate repeated samplg. It s also used to dcate the resolvg power of a measurg devce. Probablstc samplg. Procedures whch samples are selected such that all uts ad each par of uts the populato have a postve probablty of selecto. Radomzato. A delberately haphazard arragemet of observatos to smulate selecto by chace. Sample surveys. The desg ad executo of surveys to provde estmates of characterstcs (parameters) of well-defed fte populatos. Sample ut. A ut from a populato,.e., a tree or all trees located wth a plot (.e., fxed-area, strp or pot sample). Sample. A subset of a populato used to obta estmates of oe or more of ts parameters. I ths book we focus o probablstc samples. For example, a sample ca be the dameters (dbh) of all trees o a sample of plots or the amout of tme spet pcckg by users of a recreato area o gve days. Samplg desg. A formalzed method of selectg a sample from the populato, for example smple radom samplg. Samplg frame. A lst of all sample uts used to represet a populato. Samplg strategy. Comprses both the samplg desg ad the estmator(s) used, for example smple radom samplg wth the estmator of the populato mea, say the sample mea. Sgle-level samplg. A samplg desg where uts are selected drectly from the samplg frame of the populato. Stadard devato. The square root of the varace defed below. Statstcal ferece. Expressg the coecto betwee the ukow state of ature ad observed formato probablstc terms. Statstcal survey. Ivolves the desg ad executo of surveys to provde estmates of characterstcs of well-defed fte populatos. Uequal probablty samplg. Samplg desgs where uts are selected wth dfferet probabltes. These probabltes eed to be kow for ubased estmato. Ut. The basc sample ut used; e.g., that used the last stage of multstage samplg. Varable. A characterstc that vares from ut to ut; for example, the age of a tree. Varace. The average of the devatos squared betwee the values of the varables ad the overall mea the case of a populato or betwee the values of the varables ad the sample mea the case of a sample; the frst case t s a populato parameter, the secod a sample statstc. 86 USDA Forest Servce RMRS-GTR

92 Appedx. Iferece Iductve logc, the drawg of coclusos from aalyss of observed data about uobserved parameters or uderlyg laws, s oe of the most cotroversal ssues phlosophy (Gregore 998, Schreuder ad Gregore 00). Although ferece, the drawg of coclusos based o data or observatos, s ot lmted to the arrow feld of scetfc ad statstcal ferece, the latter s mportat ths cotetous world, ad a proper uderstadg of t s crucal to dscussg the role of samplg the feretal process. Scetfc ferece becomes statstcal ferece whe the coecto betwee the ukow state of ature ad the observed formato s expressed probablstc terms (Dawd 983). Statstcal ferece comprses the whole feld of statstcs, ts focus beg what s logcally mpled by the formato avalable (Fraser 983). Cramer (946) summarzes the role of statstcal ferece as havg three fuctos: descrpto, aalyss, ad predcto. Descrpto s the reducto of data sets to as small a set of umbers as possble, such as the mea, varace, skewess of a dstrbuto, etc. Ths eables oe to descrbe a populato as cocsely ad brefly as possble ad ca allow for comparso betwee populatos. Aalyss s the summarzato of data for a partcular purpose or objectve. Examples are: What are the estmates of certa populato characterstcs? Dd the sample arse from a gve dstrbuto? Or gve two samples, dd they arse from the same populato or ot? Statstcs provdes methods of how to do such aalyses. Statstcal methods are used to predct ad expla pheomea, ofte a very challegg task. Ideally, statstcal ferece would always be based o Bayes theorem, whch combes pror formato wth formato from surveys or expermets ad would be acceptable to may statstcas f the pror belef s objectve. The problem s that usually pror formato s subjectve, where subjectve dcates that the formato avalable vares from perso to perso. Objectve pror formato dcates that people would ormally agree o t. As a example of subjectve pror formato, a forest dustry perso could beleve that there s plety of old growth dstrbuted cely over the forest for habtat for edagered speces whereas a evrometalst could equally strogly beleve that the old growth the forest s lmted ad badly dstrbuted. People wllg to accept pror subjectve formato are called Bayesas ad rely o Bayes theorem for ferece. No-Bayesas or frequetsts, a majorty, use classcal ferece procedures relyg oly o objectve data ofte based o ormalty assumptos ad large sample theory based o the cetral lmt theorem ad related statstcal propertes. It s our belef that Bayesa procedures should be used whe mmedate logcally defesble decsos eed to be made, ad classcal oes whe buldg a body of scetfc kowledge. A forest maager who has to make decsos about whether or ot to cut old growth ad where for maagemet purposes, may well choose to use all hs pror formato to costruct a (subjectve) pror dstrbuto to combe wth actual sample data order to use Bayes theorem to make such decsos. Such decsos ca be defeded at least o the bass of a systematc approach. Scetfc databases ca be used by dfferet users applyg dfferet prors to make ther decsos. Statstcal ferece from sample surveys ca be ether model-based or desg-based. I modelbased samplg, ferece reles o a statstcal model to descrbe how the probablty structure of the observed data depeds o ucotrollable chace varables ad, frequetly, o other ukow usace varables. Such models ca be based o a theoretcal uderstadg of the process by whch the data were geerated, expermetal techques used, or past experece wth smlar processes. For ferece desg-based samplg, relace s placed o probablstc samplg. It s the most wdely accepted approach ow. The followg s a bref summary of both approaches. I o-probablstc or model-based samplg, ferece s made by specfyg a uderlyg superpopulato model ξ for the values of the varable the actual populato beg sampled. The actual values are cosdered to be radom varables from ths superpopulato. It s the assumed that the actual populato or a sample from t s a sample from ths superpopulato of terest. The usg ξ, for estmator Ŷ of the quatty of terest Y, the dstrbuto of Ŷ-Y ca be derved for the specfc sample ad the model-based mea square error of Ŷ-Y ca be obtaed ad estmated, leadg to a model-based predcto terval for Y. Iferece exteds to parameters of the USDA Forest Servce RMRS-GTR

93 superpopulato model, so that the ferece space s broader tha for desg-based ferece. Sample uts do ot have to be chose at radom or wth kow probablty as log as they are ot selected based o ther values of terest y, =,...,N. Coclusos ad fereces rely heavly o the model assumed, whch ca be a serous lablty f the model s ot specfed correctly. But f a model s correctly specfed, a crease precso ca be expected over the desg-based approach. Our experece s that very few models are relable. The statemet by Box ad Draper (987) that all models are wrog; the practcal questo s how wrog do they have to be to ot be useful deserves cosderato. Noetheless, models are useful buldg a body of kowledge every subject ad may have to be reled o whe a quck decso eeds to be made. The Bayesa approach to ferece wth subjectve prors fts well to the model-based ferece approach, although may advocates of model-based samplg would ot cosder themselves Bayesas at all. As oted by Koch ad Gllgs (983), model-based ferece ecompasses Bayesa ad superpopulato ferece sce the valdty of the clamed geeralty s model depedet,.e., s sestve to model msspecfcato. The desg-based approach to ferece reles heavly o probablstc samplg, whch each sample ut of the populato has a postve probablty of beg selected ad the probablty of each sample ca be calculated. The statstcal behavor of estmators of a populato attrbute s based o these probabltes ad the probablty-weghted dstrbuto of all possble sample estmates. The dstrbuto of the varable, probablstc or otherwse, s ot cosdered here. A obvous weakess of ths approach s that samples that were ot draw are cosdered heavly evaluatg the propertes of the ferece procedure, yet should ot ferece about a populato parameter be based solely o the actual sample draw? Nevertheless, the approach s objectve ad the oly assumpto made s that observed uts are selected at radom so the valdty of the ferece oly requres that the targeted ad sampled populatos are the same. Ad, careful atteto to sample selecto wth the framework of probablstc samplg wll elmate some udesrable samples from cosderato ad gve others a low probablty of selecto. The whole dea behd probablstc samplg s to make the sample represetatve of the populato beg sampled. However, as llustrated especally well Kruskal ad Mosteller (979), represetatve s subject to a wde array of terpretatos. Smth (994, p.7), formerly a strog advocate of model-based ferece, states: My vew s that there s o sgle rght method of ferece. All fereces are the product of ma s magato ad there ca be o absolutely correct method of ductve reasog. Dfferet types of ferece are relevat for dfferet problems ad frequetly the approach recommeded reflects the statstca s backgroud such as scece, dustry, socal sceces or govermet I ow fd the case for hard-le radomzato ferece based o the ucodtoal dstrbuto to be acceptable Complete recoclato s ether possble or desrable. Vve la dfferece. A crucal dfferece betwee desg-based ad model-based approaches s that for the former ferece s made about the fte, usually large populato sampled, whereas model-based samplg makes fereces about superpopulatos by the compulsory use of models. The ferece the s about the actual populato that s represeted some sese by the exstg populato, assumg that the models used uderle the real populato. Demg (975) recommeds that a dstcto should be made betwee eumeratve (or descrptve) ad aalytcal (or comparatve) surveys. I eumeratve surveys terest s a fte, detfable, ad uchagg populato from whch samples are draw. Acto s take o the populato of uts studed (e.g., all forests the state of Motaa) at the tme of samplg to decde how much tmber to harvest. Ths s the type of survey coducted by Forest Ivetory ad Aalyss (FIA) program of the USDA FS. Here desg-based ferece s dcated. I cotrast, aalytcal surveys focus o populatos where acto s to be take o the process or cause system, the purpose beg to mprove codtos the future. For the Motaa forests, we are stll talkg about the same forests but dfferet codtos exst whe we apply treatmets to mprove codtos. For example, lad maagemet ageces such as the Natoal Forest System (NFS) of the US Forest Servce may be terested collectg formato o maagg rare ad edagered wldlfe speces so as to create or modfy exstg vegetato codtos to crease the umber of such amals the forests over tme. Although we stll wat to take a desgbased sample here from the exstg populato, the ferece clearly s for populatos of the future so 88 USDA Forest Servce RMRS-GTR

94 s model-based the sese that we are extrapolatg from the exstg to future populatos. Ths s clarfed further by the followg. Demg (975) suggests that eumeratve surveys a 00 percet sample of the populato provdes the complete aswer to the questos posed, whereas aalytcal surveys the aswer s stll coclusve. Hah ad Meeker (993) make the further dstcto that aalytcal studes requre the assumpto, usually ot verfable, that the process about whch oe wats to make fereces s statstcally detcal to that from whch the sample was selected. Fgure A- from Hah ad Meeker (993) llustrates the dffereces betwee aalytcal ad eumeratve surveys. A problem wth ths dstcto s that ofte surveys coducted by lad maagemet ageces, there s terest both types of ferece. For example: oe am may be to determe how much tmber to harvest ad from where or what areas to deleate for weed or eroso cotrol (requrg eumeratve surveys) ad aother to assess what eeds to be doe to mprove habtat for wldlfe or dmsh t for oxous weeds (requrg aalytcal surveys). Objectves Eumeratve survey Defe samplg frame Defe the target or process about whch to draw ferece Type of survey eeded () Aalytcal Survey Determe assessmet process Is frame detcal to target populato yes Defe samplg procedure o Assess relevace of frame () Defe samplg procedure Determe relevace of sampled Process (6) Select SRS (3) Other Probablstc Sample (4) Select oradom sample (5) Select desred sample Fgure A-. A Comparso of Eumeratve ad Aalytcal Surveys. Reprted wth permsso from The Amerca Statstca. Copyrght 993 by the Amerca Statstcal Assocato. All rghts reserved. The umbers refer to the followg commets: () Is the purpose of the study to draw coclusos about a exstg fte populato (eumeratve study), or s t to act o ad/or predct the performace of a (frequetly future) process (aalytc study)? () Statstcal tervals apply to the frame from whch the sample s take. Whe the frame does ot correspod to the target populato, fereces about the target populato could be based, ad a statstcal terval provdes oly a lower boud o the total ucertaty. (3) Most statstcal tervals assume a smple radom sample from the frame. (4) More complex statstcal tervals tha those for smple radom samples apply; see Cochra (977). (5) Statstcal tervals do ot apply. If they are calculated, they geerally provde oly a lower boud o the total ucertaty. (6) Statstcal tervals apply to the sampled process, ad ot ecessarly to the process of terest. Thus, ay statstcal terval geerally provdes oly a lower boud o the total ucertaty wth regard to the process of terest. USDA Forest Servce RMRS-GTR

95 Appedx. Dstrbutos Populatos are dscrete or fte ( N < ) but ofte are assumed to be ftely large ( N = ) sce cotuous dstrbutos are more lkely to approxmate the real populato. Ifte populatos have propertes that are crucal to statstcal ferece. A few of the more mportat dstrbutos are preseted below as well as some key results from statstcal theory based o cotuous dstrbutos. The materal s a codesato of materal dscussed Schreuder ad others (993). Dstrbutos are ofte characterzed by ther momet geeratg fucto (mgf). Defto. If Y s a radom varable wth probablty desty f(y), the the expected value, E, of ty e s called the mgf of Y f t exsts for every value of t some terval h < t < h. Ths s deoted by () ( ty ty m t E e ) e f( y) dy = = where for dscrete dstrbutos, f(y) s the probablty mass fucto, ad () ( ty ty m t = E e ) = e f( y) dy where for cotuous dstrbutos, f(y) s the probablty desty fucto (Mood ad others 974). The logarthm of the mgf, called the cumulat geeratg fucto, s ofte used. The momets of ths fucto are called the cumulats. m(t) geerates the momets of dstrbutos. For example, survey samplg we are ofte terested estmatg the frst two momets of the ormal dstrbuto. The frst momet s the mea ad s obtaed from the mgf by dfferetatg wth respect to t oce ad settg t = 0. Smlarly the secod momet, the varace, s obtaed by dfferetatg the mgf wth respect to t twce ad settg t = 0 ad the subtractg the frst momet squared at t = 0. Cotuous Dstrbutos Three mportat cotuous dstrbutos are the ormal, gamma, ad multvarate. Normal Dstrbuto The cumulatve dstrbuto, usually smply called dstrbuto, s defed as y y ( ) = { } = (/ πσ exp[ / {( µ ) / σ } ] = ( ; µ, σ ) F y P Y y y dy f y dy wth parameters µ ad σ ( σ > 0), ad < Y <. The parameters µ ad σ are called the mea ad varace of the dstrbuto, respectvely. The mgf of the ormal dstrbuto s m t t t ( ) = exp( µ + σ / ) ' wth mea m '(0) = µ = µ ad varace m''(0) [ m '(0)] = µ = σ so fact the mea ad varace are the two parameters of the dstrbuto. Although there are umerous stuatos where the ormal dstrbuto approxmates the dstrbuto of uts a populato (such as the heghts of all trees a large platato), t s most commoly used as a coveet approxmato to other dstrbutos. The ormal dstrbuto s mportat probablty theory because t s the lmtg dstrbuto of almost ay stadardzed sums (or meas) of radom varables as the umber of varables the sum creases. y 90 USDA Forest Servce RMRS-GTR

96 If a statstc θ s a ubased estmator of a parameter θ, wth estmated varace v( θ ˆ), ad s approxmately ormally dstrbuted, the the statstc t = ( θˆ θ)/ v( θˆ ) follows Studet s ( )/ t-dstrbuto wth desty ( ) {( ) / )}(/ )( / ) v + f t = τ v+ v + t v /{ τ( v/ ) τ(/ )} where v s the umber of degrees of freedom o whch the estmate of the stadard error s based v/ t adτ ( v/) = t e dt. The t-dstrbuto s fudametal to costructg cofdece tervals, 0 ad tables for ths dstrbuto are wdely avalable (see Appedx 3, Table ). Gamma Dstrbuto Ths dstrbuto appears aturally as the dstrbuto of the sum of squares of depedet, stadard, ormally dstrbuted radom varables, Z, Z,..., Z. The Z has a χ dstrbuto = wth parameter where χ s a specal case of the gamma dstrbuto ad s the umber of degrees of freedom. The gamma dstrbuto s ofte used survey samplg to eable comparsos of samplg strateges. The dstrbuto s: y y ( ) = [ α α ] = exp( / β) /{ β τ( α )} = ( ) 0 y 0 ( α ) t F y P Y y y y dy f y dy wth parameters α ad β > 0, y>0, ad The gamma dstrbuto has the mgf wth mea µ = αβ ad varace Multvarate Dstrbutos µ αβ =. τα ( ) = t e dt. m() t = ( βt) α The multvarate ormal dstrbuto has bee studed much more extesvely tha other multvarate dstrbutos ad s used more frequetly for ferece amog multvarate cotuous dstrbutos tha s the ormal dstrbuto amog uvarate cotuous dstrbutos. The bvarate ormal dstrbuto s defed as F( x, y) = P( X x, Y y) = (πσ xσ y ρ ) exp{[ /{( ρ )}]{( x µ x) / σ x ρ {( x µ x) / σ x} {( y µ y)/ σ y + ( y µ y) / σ y}] wth < x<, < y<, where µ = E( X), u = E( Y), σ = V( X), σ = V( Y), 0 x y x y ad ρ = EX ( µ x)( Y µ y)/( σ xσ y) = σ xy /( σ xσ y),( < ρ < ) s called the correlato betwee X ad Y, ad σ xy s the covarace betwee X ad Y. Dscrete Dstrbutos Importat examples of dscrete dstrbutos are the bomal, hypergeometrc, Posso, ad multomal dstrbutos. Bomal Dstrbuto If depedet trals are made (such as whether a load of logs should be sampled or ot) ad each tral has the probablty p of outcome occurrg, the the umber of tmes whch occurs USDA Forest Servce RMRS-GTR

97 may be represeted as a radom varable Y followg the bomal dstrbuto wth parameters ad p. Ths dstrbuto s defed as the dstrbuto of a radom varable Y (= umber of occurreces of ) for whch [ ] [!/{!( )!}] y y PY= y = y y p ( p) (y = 0,,,,). The mgf of the bomal dstrbuto s m() t = ( p+ pe t ). From the mgf, the mea ad varace are derved as ' ' ' µ = p, µ = µ ( µ ) = p( p) where µ = p + ( ) p. There are may approxmatos to the bomal dstrbuto. Such approxmatos ofte volve lmtg dstrbutos that arse whe oe or both parameters coverge to a specfc value. Lmtg dstrbutos are the (dscrete) Posso dstrbuto dscussed below (, p 0) wth p = a costat θ, ad the ormal dstrbuto, whch s a specal case of the stadardzed bomal varable ( Y p)/ p( p) as. Hypergeometrc Dstrbuto The classc stuato whch t arses forestry s as follows. Suppose we have a populato wth N trees, M of whch are dead, ad N - M of whch are alve. If trees are draw at radom wthout replacemet, the the probablty of selectg y dead trees s PY [ = y] = { M!/[ y!( M y)!]}{( N M)!/[( N M + y)!( y)!]}/[ N!/{!( N )!}] for max(0, N + M) y m(, M) wth parameters M, N, ad ad! = (-)(-) ad 0! =. The mgf for the hypergeometrc dstrbuto s m( t) = [( N )!( N M)!/ N!] H(, M; N M + ; e t ) where H( α, β; γ; z) = + ( αβ / γ)( z/!) + [ α( α + ) β( β + )] /{ γ( γ + )} z /! +... s a hypergeometrc fucto whch coverges for absolute value of z<. The mea s µ = M / N ad varace µ = [( N ) /( N )] ( M / N)( M / N). Several approxmatos to the hypergeometrc dstrbuto exst, a smple oe beg the bomal dstrbuto PY [ = y] = [!/{ y!( y)!}] M/ N) ( M/ N) whch s usually adequate whe y y < 0.N. Cofdece tervals ca be costructed as descrbed uder the bomal dstrbuto assumg ether the bomal or ormal approxmato to the hypergeometrc dstrbuto. Posso Dstrbuto If the future lfetme of a tem of equpmet (say a chasaw) s depedet of ts preset age, the the lfetme ca be represeted by a radom varable Y wth dstrbuto y PY [ = y] = e θ θ / y!, y= 0,,,...; θ > 0 where θ s the oly parameter (= p the bomal dstrbuto wth as p 0 ). A wdely quoted applcato of ths dstrbuto cocers the umber of solders aually kcked to death by mules a army at the mddle of the 9 th cetury. (A aalogous stuato mght be the umber of loggers klled by fallg trees a forest.) The probablty of death was small ad the umber of solders exposed was large. It s doubtful that the codtos of depedece ad costat probablty (p) were satsfed but the data avalable were satsfactorly ftted by ths dstrbuto. t The momet geeratg dstrbuto s m( t) = exp[ θ ( e )] wth mea µ = θ ad varace µ = θ. 9 USDA Forest Servce RMRS-GTR

98 Multomal Dstrbuto Multvarate dscrete dstrbutos are ofte closely related to uvarate oes. For example, the margal dstrbuto of dvdual varables s geerally a smple bomal, Posso, or hypergeometrc dstrbuto or a dstrbuto obtaed by modfyg or compoudg oe of the uvarate dstrbutos. For example, the jot dstrbuto of the radom varables,,..., k represetg the umber of occurreces of evets O, O,..., O k trals s the multomal dstrbuto k P (,,..., )! ( j k = Π j = pj / j!) wth 0 for all j =,..., k ad j k j= j =. Ths dstrbuto s a atural exteso of the bomal dstrbuto, whch s a specal case f k =. The jot dstrbuto of ay subset s < k s also a multomal, hece the mportat subset where we have two classes, that s, class ad all others, s also a bomal. t t The mgf of the multomal s (,..., ) (... k m t tk = pe + + pke ) where the momets of the ( =,..., k) are smply ad Note that the covarace of ad ad Cor(, ) = p p /{( p )( p ) j j j ' µ = = ( t) p,,..., k µ ( ) ( ) t = p p. j ad the correlato betwee them are respectvely Cov(, ) = p p j j Cofdece tervals for ay p or pare costructed by treatg the multomal as a bomal wth probablty of selectg class, p, ad of all other classes, p. The the dscusso for costructg cofdece tervals uder the bomal s approprate. Multvarate Hypergeometrc Dstrbuto A geeralzato of the hypergeometrc dstrbuto s the multvarate hypergeometrc dstrbuto defed as follows: If there s a populato of N uts, N of whch are of type ( =,..., k) k so that N = N ad a sample of sze s take wthout replacemet from the N uts, the = P (,,..., ) = { N!/[( N )!!]}/[ N!/{( N )!!}] k k = s the multvarate hypergeometrc dstrbuto wth k = = ;0 N, =,..., k The momets of the multvarate hypergeometrc dstrbuto are aalogous to those for the hypergeometrc dstrbuto ad the correlato betwee ad j s Corr(, ) = N N /{( N N )( N N )}. Laws of Large Numbers j j j I ductve ferece we determe somethg about a populato of terest, say ts mea, by examg a sample from the populato. The followg theorems assumg radom samplg are crtcal to survey samplg ferece. A fte umber of values of Y ca be used to make relable USDA Forest Servce RMRS-GTR

99 fereces about E(Y), the average of a fte (or very large fte) umber of values of Y (.e., N y / N the average for the whole populato). For smple radom samples, the followg three = theorems apply: () Theorem (Tchebysheff s Iequalty) For a dstrbuto F(y) wth mea µ ad fte varace σ, ad f y s the mea of a radom sample of sze from ths dstrbuto ad α ay postve umber, the P[ / y / ] / () Theorem (Weak Law of Large Numbers) ασ µ ασ α. Let F(y) be a dstrbuto wth mea µ ad fte varace σ ad letε ad δ be two specfed small umbers where ε > 0 ad 0< δ <. If s ay teger greater tha σ /( ε δ ) ad y s the mea of a radom sample of sze from F(y), the P[ ε < y µ < ε] δ. Thus ths theorem states that for ay two small umbers ε ad δ, where ε > 0 ad 0< δ <, there s a teger such that for a radom sample of sze or larger from the dstrbuto of the populato of y-values F(y), the probablty that the mea of the sample of y-values y s arbtrarly close to the populato mea µ ca be made as close to as desred. The weak law of large umbers s a example of covergece probablty. The followg theorem exemplfes the oto of covergece dstrbuto. It uderles the wde applcato of the t-dstrbuto costructg cofdece tervals aroud estmates of parameters of terest ad hghlghts the crtcal mportace of the ormal dstrbuto. (3) Theorem 3 (Cetral Lmt Theorem) Let F(y) be a dstrbuto wth mea µ ad fte varace σ ad let y be the mea of a radom sample of sze from F(y). If z = ( y µ ) / σ, the the dstrbuto of z approaches the stadard ormal dstrbuto as creases wthout boud. Ths theorem states that the mea y of a radom sample from ay dstrbuto wth fte mea µ ad varace σ s approxmately dstrbuted as a ormal varable wth mea µ ad fte varace σ /. Sce we usually deal wth populatos of sze N wth N cosderably less tha, caot crease wthout boud, so the applcablty of ths theorem to fte populatos s arguable. I may cases, whe s ot too small ad the dstrbuto of y s ot too far from symmetry, the dstrbuto of z wll be approxmately ormal for ferece. 94 USDA Forest Servce RMRS-GTR

100 Appedx 3. Tables Tabulated values are frequetly used data aalyss ad hypothess testg. Amog the more commo tables (cluded here) are: Table of radom umbers, Dstrbuto of Studet s t, cofdece tervals (95 percet) for the bomal dstrbuto, ArcSe trasformato, ad two-taled sgfcace levels of correlato coeffcet r. The tables preseted here have bee geerated usg the ope source package R. The code used to create these tables ca be dowloaded from ad ca be modfed ad used for more detaled tables to meet specfc eeds. Table. Table of radom umbers USDA Forest Servce RMRS-GTR

101 Table. Cotued USDA Forest Servce RMRS-GTR

102 Table. Dstrbuto of Studet s t. Probablty of a larger value of t, sg gored df USDA Forest Servce RMRS-GTR

103 Table 3. Cofdece tervals (95 percet) for the bomal dstrbuto. Observed = 0 = 5 = 0 = 30 = 50 = USDA Forest Servce RMRS-GTR

104 Table 4. The ArcSe Percetage trasformato. Trasformato of bomal percetages the margs to agles equals formato degrees. % USDA Forest Servce RMRS-GTR

105 Table 4. Cotued. % USDA Forest Servce RMRS-GTR

106 Table 5. Two-taled sgfcace levels of the correlato coeffcet r. Sgfcace level df USDA Forest Servce RMRS-GTR

107 Appedx 4. Statstcal Aalyss Worked Examples The best way to uderstad statstcs s to work through may examples. Ufortuately, most examples are computatoally tesve. But wth the computg power avalable o the desktop, there are may good statstcal computg packages avalable. I ths Appedx, we show the commads as well as output for may of the examples preseted throughout. We choose the R laguage because t s freely avalable ad because t provdes a extesve array of statstcal aalyss procedures. The R package ca be dowloaded from the R project web ste at The applcato of the methods dscussed throughout ths book s computatoally tesve. We preset examples of these aalyses ths secto; these worked examples may be used as startg pots or templates for other aalyses. Icluded ths Appedx are descrptos of the data sets as well as a varety of samplg methods wth results. The data sets as well as the programs ca be dowloaded from RMRS Fle Name schreuderworkedexamples.xls suram.csv macros.r schreuder.r schreudertables.r workedexamples.r Descrpto Orgal large data set wth descrpto ad summary of results Text export of large data set that s read by R Mscellaeous fucto deftos used by the R programs Calculatos used the text body Developmet of tables used appedces Worked examples usg the large data set Aalyss Software The choce of software rug o Wdows, Lux, or other platforms s very broad. The commercally avalable packages such as SAS, SPSS, or S/S-Plus ru o the full rage of platforms, from PC to maframe. Because of the uqueess ad selectve avalablty of each of these packages, we do ot attempt to work these examples terms of these systems. Istead, we llustrate the aalyses wth the readly avalable ope source package, R. The R data hadlg package s extremely robust ad powerful, ad t offers a wde array of statstcal aalyss procedures; most programs wrtte for the wdely avalable commercal packages S ad S-Plus wll ru uder the R system. Lks from the R home page wll take you to the dowloads for the package tself (the complete stallato cotas the executables alog wth complete documetato), as well as cotrbuted packages ad varous electroc publcatos Eglsh, Spash, Frech, ad Germa. Data Sets The frst data set s a small data set of 0 trees that s preseted Table the text body. Although cotrved, t s a easy data set to aalyze by had. The secod data set cossts of a 60 ha stem-mapped populato of trees from a tropcal forest Suram. These data were used ad descrbed by Schreuder ad others (997). The tree heghts ad volumes were added by usg trees of the same sze from FIA data for very dfferet speces by ecessty. Ths populato of 6,806 trees has the relatve spatal locato of the trees ad s used to llustrate the effcecy of several samplg strateges. The populato stem map s dsplayed Fgure A- below. 0 USDA Forest Servce RMRS-GTR

108 Locato of trees wth crcles proportoal to dameters Locato oftrees wth crcles proportoalto dam eters Sample ple plots plots as as thck ck squares ad sam sample trees trees as th as ck thck crcles crcles Dstace (m) alog Y Dstace (m) alog X Fgure A-. Stem map for the Suram populato wth sample locatos for a SRS ad a cluster sample. The attrbutes recorded for each tree were: Colum ames Descrpto Dameter_cm Dameter of tree measured cetmeters Logtude X-offset of tree measured 0.m Latttude Y-offset of tree measured 0.m Heght_m Heght of the tree m Volume_cum Volume of the tree m 3 Subplot DBHClass Dameter_ Heght_ft Volume_cuft CC Subplot detfcato based o grd labeled wth letters for oe dmeso ad umbers for the other Dameter class Dameter of tree measured ches - hard coverso Heght of the tree feet - hard coverso Volume of the tree cubc ft - hard coverso Crow class of (D)omat or (S)ubdomat derved from heght The tree locatos are dcated wth crcles that are proportoal to the dameter of the tree. Te trees were selected at radom from the populato to llustrate smple radom samplg; these trees are dcated wth thck crcles. Te 30-m by 30-m plots were also radomly selected to llustrate cluster samplg; these plots are dcated wth thck squares. Stratfed samplg s llustrated by categorzg the trees as ether domats or subdomats o the bass of heght. Results The results from aalyzg the small data set are tabulated the ma body text. The worked example calculatos ca be created by rug the R program the fle schreuder.r. The Suram data set ca be used for realstc exercses applyg the methods dscussed ths book. Oe of the most useful steps ay aalyss s to produce some descrptve statstcs, ether tabular or graphcal. Some useful graphcs clude the boxplot. Examples of the boxplot for the volume ( m 3 ) for the etre populato ad the boxplot for the stratfed populato are show Fgure A-3. USDA Forest Servce RMRS-GTR

109 Fgure A-3. Sample boxplots. Clearly, the mea volume per tree s related to the crow posto of the tree, ad thus would be useful stratfyg the populato. The samplg methods dscussed ths paper all have stregths ad weakesses. As a exercse, the orgal populato ad fve samples were used to demostrate actual calculatos performed the R aalyss program. The commads to duplcate these results are the R commad fle amed workedexamples.r Appedx 3, Table 6 summarzes some of the results. The frst row of the table cotas the populato parameters calculated from all 6,806 trees; ths s the bechmark agast whch all estmates are judged. The ext two les of the table are for two cotrved samples that pcked three small trees ad the three large trees. Eve though ths s a woefully adequate sample sze, ether could actually result from a radom tral. Both result poor estmates of the populato parameter. The ature of estmates resultg from a radom draw do ot guaratee relable estmates. The ext tral was a SRS of sze 0 selected from ths populato. The estmate, aga sample based, s better, but stll ot relable. A radom sample of 0 clusters resulted the measuremet of 73 trees ad yelded a relable result. A stratfed sample measured 0 trees, but ths tme, fve from the domat ad fve from the subdomat classes. Ths partcular tral resulted aother relable estmate of the populato parameter. Table 6. Summary results for populato parameters of Suram populato ad results for some samples. Method Sze Mea Varace Orgal populato Sample of small trees Sample of large trees Smple radom sample Cluster sample 0 plots 73 trees Stratfed sample USDA Forest Servce RMRS-GTR

110 The output resultg from rug the large data set aalyss fle follow: *********************************************************** Worked examples: Suram data set. Summary formato for the populato. The followg varables are avalable: Dameter.cm Logtude Latttude Heght.m Volume.cum SubPolt x x3 DBHClass Dameter. Heght.ft Volume.cuft CC The basc pop statstcs for these varables are: Statstcal summary: Dameter.cm Heght.m Volume.cum Mea Varace N The statstcs are saved to fle:..//data/allsuramresults.csv Wth the dstrbuto summary: Dameter.cm Heght.m Volume.cum M. : 5.00 M. : 9.0 M. : 0.76 st Qu.: 9.00 st Qu.:9.50 st Qu.: Meda : Meda :.60 Meda : 0.96 Mea : 4.79 Mea :4.8 Mea :.886 3rd Qu.: rd Qu.:5.90 3rd Qu.:.903 Max. :65.00 Max. :70.90 Max. : NA NA NA *********************************************************** See plot ( aother wdow) for spatal arragemet of trees. I addto, type: detfy(suram$logtude, suram$latttude,suram$dameter.cm) to teractvely clck o pots to detfy dameter. NOTE: be sure to rt-clck-stop f you do ths. See plot ( aother wdow) example of boxplot for dameter. *********************************************************** Suppose we select a SRS of sze three from the populato, say observatos: The basc sample statstcs for ths sample are: Statstcal summary: Dameter.cm Heght.m Volume.cum Mea Varace The statstcs wll be saved to fle:..//data/suramsampleresults.csv *********************************************************** Suppose we select aother SRS of sze three from the populato, say observatos: The basc sample statstcs for ths sample are: Statstcal summary: Dameter.cm Heght.m Volume.cum Mea Varace The statstcs wll be saved to fle://data/suramsampleresults.csv *********************************************************** Compare the estmates from these two samples wth the actual populato parameters. *********************************************************** Now let us select a true radom sample of sze 0, say observatos: The basc sample statstcs for ths sample are: Statstcal summary: Dameter.cm Heght.m Volume.cum USDA Forest Servce RMRS-GTR

111 Mea Varace *********************************************************** How dd we do estmatg the populato parameters? *********************************************************** *********************************************************** We could also use plots or clusters to sample. Suppose we select a true radom sample of 0 plots The plot samples selected are: Dameter.cm Heght.m Volume.cum IsOPlot USDA Forest Servce RMRS-GTR

112 Cluster Statstcs for the volume (CuM): [] [] [3] [4] [5] [6] [7] m ybar var [8] [9] [0] m ybar var Thus the estmates of the volume for the total populato are: Mea: ad varace: *********************************************************** *********************************************************** Assume we stratfy the populato by crow class. Stratfed Suram data set. Summary fo for the stratfed populato. The followg varables are avalable: Dameter.cm Logtude Latttude Heght.m Volume.cum SubPolt x x3 DBHClass Dameter. Heght.ft Volume.cuft CC IsSRS IsOPlot IsStratfedSample The basc populato statstcs for the domats are: Statstcal summary: Dameter.cm Heght.m Volume.cum Mea Varace N The statstcs wll be saved to fle://data/domsuramresults.csv Whle the statstcs for the suppressed trees are: Statstcal summary: Dameter.cm Heght.m Volume.cum Mea Varace N The statstcs wll be saved to fle://data/supsuramresults.csv *********************************************************** Compare the parameters from these two strata wth the sgle populato parameters. *********************************************************** We ca sample from these two strata wth the results: Stratfed Stats for the volume (CuM): [] [] ID D S h 5 5 ybarh varh Resultg populato estmates of: Mea:.3539 ad varace: See plot ( aother wdow) example of boxplot for dameter, by domace. USDA Forest Servce RMRS-GTR

113 Idex A adaptve 67 aeral 4, 9, 44, 45, 60, 6, 64, 7, 8 arthmetc mea 8, 3 AVHRR 59, 60, 6 B basal area 4, 5, 8, 8, 6, 9, 30, 33, 40, 4, 44, 46, 47, 66, 7, Bayesa 87, 88 beta 83 bomal 0,, 53, 57, 9-93, 95 bomal samplg 4, 53 Btterlch samplg 40 bvarate ormal 9 bootstrappg 33, 34, 63, 7, 76 C cesus, 3 CIR photography 60 cluster 6, 7,, 7-9, 3, 33, 44, 45, 53-55, 57, 68, 69, 03, 07 coeffcet of varato 6 complete remeasuremet 73, 76 complete remeasuremet samplg 73, 76 cofdece terval 34, 4, 43, 5, 5, 54 cosstet 8, 33, 4, 43, 75, 78 cotuous 9, 0, 55, 57, 63, 65, 85, 90, 9 correlato coeffcet 7, 8, 85, 95 cout method 8 D descrptve 88, 03 desg-based 83, 87, 88 dscrete 9, 0, 5, 6, 64, 85, dstrbutos 0, 57, DNA 50, 67 double samplg 36, 70, 7, 7, 85 E ease of mplemetato 9 Edge Effect 46 effectve 6, 9, 3, 33, 48, 65 eumeratve 88, 89 estmato bas 8, 85 F FIA. See also Forest Ivetory ad Aalyss Program fte, 9,, 50, 5, 54, 55, 57, 68, 8, 83, 85, 86, 88, 90, 93, 94 fxed 5, 9, 0, 5, 9, 40, 4, 44, 47, 49, 53, 57, 86 forest vetory ad aalyss 3, 78, 84, 88 Forest Ivetory ad Aalyss Program 08 USDA Forest Servce RMRS-GTR

114 G gamma 90, 9 geographc formato system 59, 64, 66, 8 Global Postog system 85 H heght, 7-0, 3, 4, 6, 8, 9, 40, 48, 53, 59, 68, 85, 03, Horvtz-Thompso estmator, 5, 9, 34-36, 39, 49 hypergeometrc 9-93 hypsometer 48 I Iferece 78, 8, 83, 85, 87 Istrumets Hypsometer 48 Relaskop 40 vetory, 3, 4, 48, 49, 53, 59-6, 64, 65, 80, 8-84 J jackkfe 36 L Ladsat 59, 60 le tercept 45, 49 M mappg 64 mea 3, 5, 7-9, 3-39, 43, 53-58, 68, 70, 7, 85-87, 90-94, mea-of-ratos 36 meda, 3, 85, 05 methodology Mcrowave 59, 6 mrage method 46, 47 mssg data 5, 47 mode, 3, 3, 85 model-based 6, 83, motorg,, 4, 59, 6, 64, 66, 73, 8-84 multomal 9, 93 multphase 68, 69, 7 multphase samplg 69 multplcty samplg 67 multstage 68, 69, 7, 8, 86 multstage samplg 68, 69, 7, 86 multvarate ormal 9 N egatve bomal 57 ormal, 0,, 33, 43, 5, 54, 57, 90-9, 94 O optmal 7, 33, 43, 7 USDA Forest Servce RMRS-GTR

115 P parameter 7, 9, -4, 6, 8, 5, 39, 4, 43, 85, 86, 88, 9, 9, 04 permaet 4, 59, 73 Ptfalls 79 pot 4, 30, 40, 44, 46, 48, 59, 73, 76, 79, 80, Posso 0,, 40-4, 57, 7, 84, 9-93 precso 8, 33, 34, 43, 45, 5, 66, 73, 85, 86, 88 pror 44, 53, 60, 78, 87 probablstc, 4, 6,, 4, 5, 33, 67, probablstc samplg R radom sample 6, 3, 4, 3, 39, 4, 44, 5, 53, 6, 68, 7, 83, 94, radomzato 6, 77, 86, 88 rato-of-meas 34, 36, 37, 39, regresso 6-8, 34-39, 45, 65, 69, 70-77, 8-85 regresso estmators 36, 37, 7 Relaskop 48 relatve 0, 6,, 3, 34, 4, 45, 5, 53, 6, 65, 73, 86, 0 remote sesg 4, 7, 45, 50, 59, 60-66, 70-7, 8-84 S sample plots 4, 5, 8, 37, 47, 53, 63, 73 sample sze 8, 0, 3, 4, 7, -5, 9, 39, 40-43, 5-53, 64, 67, 70, 7, 85, 04 sample survey, 7, 6, 0, 4, 83, 84, 86, 87 samplg -0, 4-9, -76, 79-93, 0-04 screeg 67, 78 selecto 6-8,, -34, 39, 40-4, 44, 49, 50, 53, 7, 77, 79, 80, 83, 84, 86, 88 sequetal samplg 67 smple lear regresso 39 smple radom 6, 3, 6,, 5, 7, 30, 33-35, 4, 5, 54, 57, 6, 83, 86, 94, 03 smple radom samplg (SRS), 30, 5 sze of 5, 6, 7, 4, 37, 39, 40, 44, 47, 57, 59, 60, 69 small area estmato 3, 65, 66, 83 sowball samplg 67 SPOT 60, 6 stadard devato -6, 9, 35, 36, 86 stadard error 4, 5, 9, 35, 36, 37, 43, 5, 54, 55-58, 86, 9 statstcal, 0,, 33, 4, 50, 65, 77, 78, 8-84, 87-90, 0 statstcal ferece 83, 86, 87, 90 stratfcato 6, 7, 3, 34, 39, 69, 70, 7, 84 stratfed 7,, 6, 7, 9, 30, 3, 33, 35, 39, 70-7, 79, 03, 04, 07 strp 44, 49, 86 suggestos, 80, 83 survey samplg, 3, 8-84, 90, 9, 93 systematc 8,, 3, 33, 4, 44, 53, 77, 79, 85, 87 systematc samplg 3, 3 T t-dstrbuto 9, 94 trasects 46 0 USDA Forest Servce RMRS-GTR

116 U uequal probablty, 5, 6, 3, 33, 34, 44, 55 uequal probablty samplg 5, 86 V varable probablty 7 varable radus plots 5 varace 6, 9, 0, 86, varace estmato 9, 7, 75 vertcal 4, 48, 6 volume 8, 03, VRP samplg 40, 4, 44, 46, 7, 79-8 W walkthrough 47, 48, 8 weghted regresso 35 wldlfe samplg 50 USDA Forest Servce RMRS-GTR

117 The use of trade ad compay ames s for the beeft of the reader; such use does ot costtute a offcal edorsemet or approval of ay servce or product by the U.S. Departmet of Agrculture to the excluso of others that may be sutable.

118 The Rocky Mouta Research Stato develops scetfc formato ad techology to mprove maagemet, protecto, ad use of the forests ad ragelads. Research s desged to meet the eeds of the Natoal Forest maagers, Federal ad State ageces, publc ad prvate orgazatos, academc sttutos, dustry, ad dvduals. Studes accelerate solutos to problems volvg ecosystems, rage, forests, water, recreato, fre, resource vetory, lad reclamato, commuty sustaablty, forest egeerg techology, multple use ecoomcs, wldlfe ad fsh habtat, ad forest sects ad dseases. Studes are coducted cooperatvely, ad applcatos may be foud worldwde. Research Locatos Flagstaff, Arzoa Fort Colls, Colorado* Bose, Idaho Moscow, Idaho Bozema, Motaa Mssoula, Motaa Lcol, Nebraska Reo, Nevada Albuquerque, New Mexco Rapd Cty, South Dakota Loga, Utah Ogde, Utah Provo, Utah Larame, Wyomg *Stato Headquarters, Natural Resources Research Ceter, 50 Cetre Aveue, Buldg A, Fort Colls, CO The U.S. Departmet of Agrculture (USDA) prohbts dscrmato all ts programs ad actvtes o the bass of race, color, atoal org, sex, relgo, age, dsablty, poltcal belefs, sexual oretato, or martal or famly status. (Not all prohbted bases apply to all programs.) Persos wth dsabltes who requre alteratve meas for commucato of program formato (Bralle, large prt, audo-tape, etc.) should cotact USDA s TARGET Ceter at (0) (voce ad TDD). To fle a complat of dscrmato, wrte USDA, Drector, Offce of Cvl Rghts, Room 36 W, Whtte Buldg, 400 Idepedece Aveue, SW, Washgto, D.C or call (0) (voce ad TDD). USDA s a equal opportuty provder ad employer. Federal Recyclg Program Prted o Recycled Paper