SAMPLING DESIGNS - RANDOM SAMPLING, ADAPTIVE AND SYSTEMATIC SAMPLING

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1 (Versio 4, 14 Marc 013) CHAPTER 8 SAMPLING DESIGNS - RANDOM SAMPLING, ADAPTIVE AND SYSTEMATIC SAMPLING Page 8.1 SIMPLE RANDOM SAMPLING Estimatio of Parameters Estimatio of a Ratio Proportios ad Percetages STRATIFIED RANDOM SAMPLING Estimates of Parameters Allocatio of Sample Size Proportioal Allocatio Optimal Allocatio Costructio of Strata Proportios ad Percetages ADAPTIVE SAMPLING Adaptive cluster samplig Stratified Adaptive Cluster Samplig SYSTEMATIC SAMPLING MULTISTAGE SAMPLING Samplig Uits of Equal Size Samplig Uits of Uequal Size SUMMARY SELECTED REFERENCES QUESTIONS AND PROBLEMS Ecologists sample weever tey caot do a complete eumeratio of te populatio. Very few plat ad aimal populatios ca be completely eumerated, ad so most of our ecological iformatio comes from samples. Good samplig metods are critically

2 Capter 8 Page 35 importat i ecology because we wat to ave our samples be represetative of te populatio uder study. How do we sample represetatively? Tis capter will attempt to aswer tis questio by summarizig te most commo samplig desigs tat statisticias ave developed over te last 80 years. Samplig is a practical busiess ad tere are two parts of its practicality. First, te gear used to gater te samples must be desiged to work well uder field coditios. I all areas of ecology tere as bee tremedous progress i te past 40 years to improve samplig teciques. I will ot describe tese improvemets i tis book - tey are te subject of may more detailed adbooks. So if you eed to kow wat plakto sampler is best for oligotropic lakes, or wat ligt trap is best for octural mots, you sould cosult te specialist literature i your subject area. Secod, te metod of placemet ad te umber of samples must be decided, ad tis is wat statisticias call samplig desig. Sould samples be placed radomly or systematically? Sould differet abitats be sampled separately or all togeter? Tese are te geeral statistical questios I will address i tis ad te ext capter. I will develop a series of guidelies tat will be useful i samplig plakto wit ets, mots wit ligt traps, ad trees wit distace metods. Te metods discussed ere are addressed i more detail by Cocra (1977), Jesse (1978), ad Tompso (199). 8.1 SIMPLE RANDOM SAMPLING Te most coveiet startig poit for discussig samplig desigs is simple radom samplig. Like may statistical cocepts, "radom samplig" is easier to explai o paper ta it is to apply i te field. Some backgroud is essetial before we ca discuss radom samplig. First, you must specify very clearly wat te statistical populatio is tat you are tryig to study. Te statistical populatio may or may ot be a biological populatio, ad te two ideas sould ot be cofused. I may cases te statistical populatio is clearly specified: te wite-tailed deer populatio of te Savaa River Ecological Area, or te witefis populatio of Brooks Lake, or te black oaks of Warre Dues State Park. But i oter cases te statistical populatio as poorly defied boudaries: te mice tat will eter live traps, te addock populatio of George s Bak, te aerial apid populatio over souter Eglad, te seed bak of Erigero caadesis. Part of tis vagueess i ecology depeds o

3 Capter 8 Page 36 spatial scale ad as o easy resolutio. Part of tis vagueess also flows from te fact tat biological populatios also cage over time (Va Vale 198). Oe strategy for dealig wit tis vagueess is to defie te statistical populatio very sarply o a local scale, ad te draw statistical ifereces about it. But te biological populatio of iterest is usually muc larger ta a local populatio, ad oe must te extrapolate to draw some geeral coclusios. You must tik carefully about tis problem. If you wis to draw statistical ifereces about a widespread biological populatio, you sould sample te widespread populatio. Oly tis way ca you avoid extrapolatios of ukow validity. Te statistical populatio you wis to study is a fuctio of te questio you are askig. Tis problem of defiig te statistical populatio ad te relatig it to te biological populatio of iterest is eormous i field ecology, ad almost o oe discusses it. It is te first tig you sould tik about we desigig your samplig sceme. Secod, you must decide wat te samplig uit, is i your populatio. Te samplig uit could be simple, like a idividual oak tree or a idividual deer, or it ca be more complex like a plakto sample, or a 4 m quadrat, or a brac of a apple tree. Te sample uits must potetially cover te wole of te populatio ad tey must ot overlap. I most areas of ecology tere is cosiderable practical experiece available to elp decide wat samplig uit to use. Te tird step is to select a sample, ad a variety of samplig plas ca be adopted. Te aim of te samplig pla is to maximize efficiecy - to provide te best statistical estimates wit te smallest possible cofidece limits at te lowest cost. To acieve tis aim we eed some elp from teoretical statistics so tat we ca estimate te precisio ad te cost of te particular samplig desig we adopt. Statisticias always assume tat a sample is take accordig to te priciples of probability samplig, as follows: 1. Defie a set of distict samples S 1, S, S 3... i wic certai specific samplig uits are assiged to S 1, some to S, ad so o.. Eac possible sample is assiged a probability of selectio.

4 Capter 8 Page Select oe of te S i samples by te appropriate probability ad a radom umber table. If you collect a sample accordig to tese priciples of probability samplig, a statisticia ca determie te appropriate samplig teory to apply to te data you gater. Some types of probability samplig are more coveiet ta oters, ad simple radom samplig is oe. Simple radom samplig is defied as follows: 1. A statistical populatio is defied tat cosists of N samplig uits;. uits are selected from te possible samples i suc a way tat every uit as a equal cace of beig cose. Te usual way of acievig simple radom samplig is tat eac possible sample uit is umbered from 1 to N. A series of radom umbers betwee 1 ad N is te draw eiter from a table of radom umbers or from a set of umbers i a at. Te sample uits wic appe to ave tese radom umbers are measured ad costitute te sample to be aalyzed. It is ard i ecology to follow tese simple rules of radom samplig if you wis te statistical populatio to be muc larger ta te local area you are actually studyig. Usually, oce a umber is draw i simple radom samplig it is ot replaced, so we ave samplig witout replacemet. Tus, if you are usig a table of radom umbers ad you get te same umber twice, you igore it te secod time. It is possible to replace eac uit after measurig so tat you ca sample wit replacemet but tis is less ofte used i ecology. Samplig witout replacemet is more precise ta samplig wit replacemet (Caugley 1977). Simple radom samplig is sometimes cofused wit oter types of samplig tat are ot based o probability samplig. Examples aboud i ecology: 1. Accessibility samplig: te sample is restricted to tose uits tat are readily accessible. Samples of forest stads may be take oly alog roads, or deer may be couted oly alog trails.

5 Capter 8 Page 38. Hapazard samplig: te sample is selected apazardly. A bottom sample may be collected weever te ivestigator is ready, or te dead fis may be picked up for cemical aalysis from a large fis kill. 3. Judgmetal samplig: te ivestigator selects o te basis of is or er experiece a series of "typical" sample uits. A botaist may select 'climax' stads of grasslad to measure. 4. Voluteer samplig: te sample is self-selected by voluteers wo will complete some questioaire or be used i some pysiological test. Huters may complete survey forms to obtai data o kill statistics. Te importat poit to remember is tat all of tese metods of samplig may give te correct results uder te rigt coditios. Statisticias, owever, reject all of tese types of samplig because tey ca ot be evaluated by te teorems of probability teory. Tus te uiversal recommedatio: radom sample! But i te real world it is ot always possible to use radom samplig, ad te ecologist is ofte forced to use o-probability samplig if e or se wises to get ay iformatio at all. I some cases it is possible to compare te results obtaied wit tese metods to tose obtaied wit simple radom samplig (or wit kow parameters) so tat you could decide empirically if te results were represetative ad accurate. But remember tat you are always o saky groud if you must use o-probability samplig, so tat te meas ad stadard deviatios you calculate may ot be close to te true values. Weever possible use some covetioal form of radom samplig Estimatio of Parameters I simple radom samplig, oe or more caracteristics are measured o eac experimetal uit. For example, i quadrat samplig you migt cout te umber of idividuals of Solidago spp. ad te umber of idividuals of Aster spp. I samplig deer, you migt record for eac idividual its sex, age, weigt, ad fat idex. I samplig starlig ests you migt cout te umber of eggs ad measure teir legt.

6 Capter 8 Page 39 I all tese cases ad i may more, ecological iterest is focused o four caracteristics of te populatio 1 : 1. Total = X ; for example, te total umber of Solidago idividuals i te etire 100 a study field.. Mea = x ; for example, te average umber of Solidago per m. 3. Ratio of two totals = R = xy; for example, te umber of Solidago per Aster i te study area. 4. Proportio of uits i some defied class; for example, te proportio of male deer i te populatio. We ave see umerous examples of caracteristics of tese types i te previous seve capters, ad teir estimatio is covered i most itroductory statistics books. We cover tem agai ere briefly because we eed to add to tem a idea tat is ot usually cosidered i itroductory books - te fiite populatio correctio. For ay statistical populatio cosistig of N uits, we defie te fiite populatio correctio (fpc) as: were: N fpc 1- (8.1) N N fpc N Fiite populatio correctio Total populatio size Sample size ad te fractio of te populatio sampled (/N) is sometimes referred to as f. I a very large populatio te fiite populatio correctio will be 1.0, ad we te wole populatio is measured, te fpc will be 0.0. Ecologists workig wit abudace data suc as quadrat couts of plat desity immediately ru ito a statistical problem at tis poit. Stadard statistical procedures 1 We may also be iterested i te variace of te populatio, ad te same geeral procedures apply.

7 Capter 8 Page 330 deal wit ormally distributed raw data i wic te stadard deviatio is idepedet of te mea. But ecological abudace data typically sow a positive skew (e.g. Figure 4.6 page 000) wit te stadard deviatio icreasig wit te mea. Tese complicatios violate te assumptios of parametric statistics, ad te simplest way of correctig ecological abudace data is to trasform all couts by a log trasformatio: Y log X (8.) were Y = trasformed data X = origial data All aalysis is ow doe o tese Y-values wic ted to satisfy te assumptios of parametric statistics 1. I te aalyses tat follow wit abudace data we effectively replace all te observed X values wit teir Y trasformed couterpart ad use te covetioal statistical formulas foud i most textbooks tat defie te variable of iterest as X. Cocra (1977) demostrates tat ubiased estimates of te populatio mea ad total for ormally distributed data are give by te followig formulas. For te mea: x x i (8.3) were: x = Populatio mea xi = Observed value of x i sample i = Sample size For te variace of te measuremets we ave te usual formula: s x i x 1 (8.4) 1 Tis trasformatio will be discussed i detail i Capter 15, page 000. If te data are compiled i a frequecy distributio, te appropriate formulas are supplied i Appedix 1, page 000.

8 Capter 8 Page 331 ad te stadard error of te populatio mea x is give by: were: s s x f s x s 1 f (8.5) Stadard error of te mea x Variace of te measuremets as defied above (8.4) Sample size Samplig fractio / N Tese formulas are similar to tose you ave always used, except for te itroductio of te fiite populatio correctio. Note tat we te samplig fractio (/N) is low, te fiite populatio correctio is early 1.0 ad so te size of te populatio as o effect o te size of te stadard error. For example, if you take a sample of 500 measuremets from two populatios wit te same variace ( s = 484.0) ad populatio A is small (N = 10,000) ad populatio B is a tousad times larger (N = 10,000,000), te stadard errors of te mea differ by oly.5% because of te fiite populatio correctio. For tis reaso, te fiite populatio correctio is usually igored weever te samplig fractio (/N) is less ta 5% (Cocra 1977). Estimates of te populatio total are closely related to tese formulas for te populatio mea. For te populatio total: ˆX N x (8.6) were: Xˆ Estimated populatio total N x Total size of populatio Mea value of populatio Te stadard error of tis estimate is give by: were: s X sx N s x N s (8.7) x = Stadard error of te populatio total = Total size of te populatio = Stadard error of te mea from equatio (8.4) Cofidece limits for bot te populatio mea ad te populatio total are usually derived by te ormal approximatio:

9 Capter 8 Page 33 were: x t s x (8.8) t = Studet's t value for - 1 degrees of freedom for te 1- level of cofidece Tis formula is used all te time i statistics books wit te warig ot to use it except wit "large" sample sizes. Cocra (1977, p.41) gives a rule-of-tumb tat is useful for data tat as a strog positive skew (like te egative biomial curves i Figure 4.8, page 000). First, defie Fiser s measure of skewess: g1 Fiser's measure of skewess xx s (8.9) Cocra s rule is tat you ave a large eoug sample to use te ormal approximatio (equatio 8.8) if were: 5g (8.10) 1 = Sample size s = Stadard deviatio Sokal ad Rolf (1995, p. 116) sow ow to calculate g 1 from sample data ad may statistical packages provide computer programs to do tese calculatios. samplig. Box 8.1 (page 000) illustrates te use of tese formulae for simple radom Box 8.1 ESTIMATION OF POPULATION MEAN AND POPULATION TOTAL FROM SIMPLE RANDOM SAMPLING OF A FINITE POPULATION A biologist obtaied body weigts of male reideer calves from a erd durig te seasoal roudup. He obtaied weigts o 315 calves out of a total of 16 i te erd, cecked te assumptio of a ormal distributio, ad summarized te data: Body weigt class (kg) Midpoit, x Observed frequecy, f x

10 Capter 8 Page Te observed mea is, from te grouped versio of equatio (8.3), x fx x 17,55 = = kg 315 (4)(3) + (13)(37) + (0)(4) Te observed variace is, from te grouped versio of equatio (8.4), s x fxx f x / ,685 - (17,55) / Te stadard error of te mea weigt, from equatio (8.5): s x s 1 f From equatio (8.8) we calculate g 1 = 3 equatio (8.10) 5g ( 0.164) / s x x = Hece, applyig so tat we ave a large sample accordig to Cocra s rule of tumb. Hece we ca compute ormal cofidece limits from equatio (8.8): for =.05 te t value for 315 d.f. is 1.97 x t s x (0.431) or 95% cofidece limits of to kg. To calculate te populatio total biomass, we ave from equatio (8.6):

11 Capter 8 Page 334 Xˆ N x 16(54.78) 69,19.6 kg ad te stadard error of tis total biomass estimate for te erd is, from equatio (8.7), s ˆ Ns 16(0.4311) X x ad ormal 95% cofidece limits ca be calculated as above to give for te populatio total: 69, kg It is importat to remember tat if you ave carried out a log trasform o abudace data, all of tese estimates are i te log-scale. You may wis to trasform tem back to te origial scale of measuremet, ad tis procedure is ot immediately obvious (i.e. do ot simply take ati-logs) ad is explaied i Capter 15, Sectio (page 000) Estimatio of a Ratio Ratios are ot as commoly used i ecological work as tey are i taxoomy, but sometimes ecologists wis to estimate from a simple radom sample a ratio of two variables, bot of wic vary from samplig uit to samplig uit. For example, wildlife maagers may wis to estimate te wolf/moose ratio for several game maagemet zoes, or beavioral ecologists may wis to measure te ratio of breedig females to breedig males i a bird populatio. Ratios are peculiar statistical variables wit strage properties tat few biologists appreciate (Atcley et al. 1976). Ratios of two variables are ot just like ordiary measuremets ad to estimate meas, stadard errors, ad cofidece itervals for ecological ratios, you sould use te followig formulae from Cocra (1977): For te mea ratio: Rˆ x (8.11) y were:

12 Capter 8 Page 335 Rˆ Estimated mea ratio of x to y x Observed mea value of x y Observed mea value of y Te stadard error of tis estimated ratio is: s Rˆ 1 f y x Rˆ xy Rˆ y (8.1) 1 were: sr ˆ f y Estimated stadard error of te ratio R Samplig fractio / N Sample size Observed mea of Y measuremet (deomiator of ratio) ad te summatio terms are te usual oes defied i Appedix 1 (page 000). Te estimatio of cofidece itervals for ratios from te usual ormal approximatio (eq. 8.8) is ot valid uless sample size is large as defied above (page 33) (Sukatme ad Sukatme 1970). Ratio variables are ofte skewed to te rigt ad ot ormally distributed, particularly we te coefficiet of variatio of te deomiator is relatively ig (Atcley et al. 1976). Te message is tat you sould treat te computed cofidece itervals of a ratio as oly a approximatio uless sample size is large. Box 8. illustrates te use of tese formulas for calculatig a ratio estimate. Box 8. ESTIMATION OF A RATIO OF TWO VARIABLES FROM SIMPLE RANDOM SAMPLING OF A FINITE POPULATION Wildlife ecologists iterested i measurig te impact of wolf predatio o moose populatios i Britis Columbia obtaied estimates by aerial coutig of te populatio size of wolves ad moose ad 11 subregios wic costituted 45% of te total game maagemet zoe. Subregio No. of wolves No. of moose Wolves/moose A B C D E F G

13 Capter 8 Page 336 H I J K Te mea umbers of wolves ad moose are x = SE = % CL = to x y x wolves y moose Te mea ratio of wolves to moose is estimated from equatio (8.11): ˆ x 1.77 R wolves/moose y Te stadard error of tis estimate (equatio 8.1) requires tree sums of te data: x y ,09,844 xy From equatio (8.1): (8)(190) + (15)(370) + (9)(460) + 66,391 s Rˆ 1 f y x Rˆ xy Rˆ y (0.03)(66,391) + (0.03 )(,09,844) 11 (395.64) te 95% cofidece limits for tis ratio estimate are tus ( t for 10 d.f. for =.05 is.8): Rˆ t s R or ( )

14 Capter 8 Page 337 or to wolves per moose Proportios ad Percetages Te use of proportios ad percetages is commo i ecological work. Estimates of te sex ratio i a populatio, te percetage of successful ests, te icidece of disease, ad a variety of oter measures are all examples of proportios. I all tese cases we assume tere are classes i te populatio, ad all idividuals fall ito oe class or te oter. We may be iterested i eiter te umber or te proportio of type X idividuals from a simple radom sample: Populatio Sample No. of total idividuals N No. of idividuals of type X A a Proportio of type X idividuals P A N pˆ a I statistical work te biomial distributio is usually applied to samples of tis type, but we te populatio is fiite te more proper distributio to use is te ypergeometric distributio 1 (Cocra 1977). Fortuately, te biomial distributio is a adequate approximatio for te ypergeometric except we sample size is very small. For proportios, te sample estimate of te proportio P is simply: pˆ a (8.13) were: pˆ Proportio of type X idividuals a Number of type X idividuals i sample Sample size Te stadard error of te estimated proportio ˆp is from Cocra (1977), s Pˆ 1 f pq ˆˆ 1 (8.14) 1 Zar (1996, pg. 50) as a good brief descriptio of te ypergeometric distributio.

15 Capter 8 Page 338 were: s ˆ = Stadard error of te estimated populatio p P f Samplig fractio / N pˆ Estimated proportio of X types qˆ 1 - pˆ Sample size For example, if i a populatio of 3500 deer, you observe a sample of 850 of wic 400 are males: pˆ 400/ (0.4706)( ) s ˆ P To obtai cofidece limits for te proportio of x-types i te populatio several metods are available (as we ave already see i Capter, page 000). Cofidece limits ca be read directly from graps suc as Figure. (page 000), or obtaied more accurately from tables suc as Burstei (1971) or from Program EXTRAS (Appedix, page 000). For small sample sizes te exact cofidece limits ca be read from tables of te ypergeometric distributio i Lieberma ad Owe (1961). If sample size is large, cofidece limits ca be approximated from te ormal distributio. Table 8.1 lists sample sizes tat qualify as "large". Te ormal approximatio to te biomial gives cofidece limits of: or pq ˆˆ 1 pˆ z 1 f 1 ˆ p z s P ˆ 1 (8.15) were:

16 Capter 8 Page 339 pˆ = Estimated proportio of X types z = Stadard ormal deviate (1.96 for 95% cofidece itervals,.576 for 99% cofidece itervals) s ˆ Stadard error of te estimated proportio (equatio 8.13) P f Samplig fractio / N qˆ 1 - pˆ proportio of Y types i sample Sample size TABLE 8.1 SAMPLE SIZES NEEDED TO USE THE NORMAL APPROXIMATION (EQUATION 8.15) FOR CALCULATING CONFIDENCE INTERVALS FOR PROPORTIONS a a Proportio, p Number of idividuals i te smaller class, p Total sample size, For a give value of p do ot use te ormal approximatio uless you ave a sample size tis large or larger. Source: Cocra, Te fractio (1/) is a correctio for cotiuity, wic attempts to correct partly for te fact tat idividuals come i uits of oe, so it is possible, for example, to observe 16 male deer or 17, but ot Witout tis correctio te ormal approximatio usually gives a cofidece belt tat is too arrow. For te example above, te 95% cofidece iterval would be: (0.0149) (850) or (0.44 to 0.50 males) Note tat te correctio for cotiuity i tis case is very small ad if igored would ot cage te cofidece limits except i te fourt decimal place. N

17 Capter 8 Page 340 Not all biological attributes come as two classes like males ad females of course, ad we may wis to estimate te proportio of orgaisms i tree or four classes (istar I, II, III ad IV i isects for example). Tese data ca be treated most simply by collapsig tem ito two-classes, istar II vs. all oter istars for example, ad usig te metods described above. A better approac is described i Sectio for multiomial data. Oe practical illustratio of te problem of estimatig proportios comes from studies of disease icidece (Ossiader ad Wedemeyer 1973). I may atcery populatios of fis, samples eed to be take periodically ad aalyzed for disease. Because of te cost ad time associated wit disease aalysis, idividual fis are ot always te samplig uit. Istead, groups of 5, 10 or more fis may be pooled ad te resultig pool aalyzed for disease. Oe diseased fis i a group of 10 will cause tat wole group to be assessed as disease-positive. Worlud ad Taylor (1983) developed a metod for estimatig disease icidece i populatios we samples are pooled. Te samplig problem is acute ere because disease icidece will ofte be oly 1-%, ad at low icideces of disease, larger group sizes are more efficiet i estimatig te proportio diseased. Table 8. gives te cofidece itervals expected for various sizes of groups ad umber of groups we te expected disease icidece varies from 1-10%. For group size = 1, tese limits are te same as tose derived above (equatio 8.14). But Table 8. sows clearly tat, at low icidece, larger group sizes are muc more precise ta smaller group sizes. Worlud ad Taylor (1983) provide more details o optimal samplig desig for suc disease studies. Oe problem wit disease studies is tat diseased aimals migt be muc easier to catc ta ealty aimals, ad oe must be particularly cocered wit obtaiig a radom sample of te populatio. TABLE 8. WIDTH OF 90% CONFIDENCE INTERVALS FOR DISEASE INCIDENCE a Percet disease icidece No. 1.0%.0% 5.0% 10% of Group size, k Group size, k Group size,k Group size, k

18 Capter 8 Page 341 groups, a A umber of groups () of group size k are tested for disease. If oe idividual i a group as a disease, te wole group is diagosed as disease positive. Te umber i te table sould be read as d %, tat is, as oe-alf of te widt of te cofidece iterval. Source: Worlud ad Taylor, STRATIFIED RANDOM SAMPLING Oe of te most powerful tools you ca use i samplig desig is to stratify your populatio. Ecologists do tis all te time ituitively. Figure 8.1 gives a simple example. Populatio desity is oe of te most commo bases of stratificatio i ecological work. We a ecologist recogizes good ad poor abitats, e or se is implicitly stratifyig te study area. I stratified samplig te statistical populatio of N uits is divided ito subpopulatios wic do ot overlap ad wic togeter comprise te etire populatio. Tus: N = N 1 + N + N N L were L = total umber of subpopulatios

19 Capter 8 Page 34 Stratum A Stratum B Stratum C Figure 8.1 Te idea of stratificatio i estimatig te size of a plat or aimal populatio. Stratificatio is made o te basis of populatio desity. Stratum A as about te times te desity of Stratum C. Te subpopulatios are called strata by statisticias. Clearly if tere is oly oe stratum, we are back to te kid of samplig we ave discussed earlier i tis capter. To obtai te full beefit from stratificatio you must kow te sizes of all te strata (N 1, N,...). I may ecological examples, stratificatio is doe o te basis of geograpical area, ad te sizes of te strata are easily foud i m or km, for example. Tere is o eed for te strata to be of te same size. Oce you ave determied wat te strata are, you sample eac stratum separately. Te sample sizes for eac stratum are deoted by subscripts: 1 = sample size i stratum 1 = sample size i stratum ad so o. If witi eac stratum you sample usig te priciples of simple radom samplig outlied above (page 37), te wole procedure is called stratified radom samplig. It is ot ecessary to sample eac stratum radomly, ad you could, for example, sample systematically witi a stratum. But te problems outlied above would te mea tat it would be difficult to estimate ow reliable suc samplig is. So it is recommeded to sample radomly witi eac stratum.

20 Capter 8 Page 343 Ecologists ave may differet reasos for wisig to stratify teir samplig. Four geeral reasos are commo (Cocra 1977): 1. Estimates of meas ad cofidece itervals may be required separately for eac subpopulatio.. Samplig problems may differ greatly i differet areas. Aimals may be easier or arder to cout i some abitats ta tey are i oters. Offsore samples may require larger boats ad be more expesive to get ta earsore samples. 3. Stratificatio may result i a gai i precisio i te estimates of te parameters of te wole populatio. Cofidece itervals ca be arrowed appreciably we strata are cose well. 4. Admiistrative coveiece may require stratificatio if differet field laboratories are doig differet parts of te samplig. Poit 3 is peraps te most critical oe o tis list, ad I will ow discuss ow estimates are made from stratified samplig ad illustrate te gais oe ca acieve Estimates of Parameters For eac of te subpopulatios (N 1, N,...) all of te priciples ad procedures of estimatio outlied above ca be used. Tus, for example, te mea for stratum 1 ca be estimated from equatio (8.) ad te variace from equatio (8.3). New formulas are owever required to estimate te mea for te wole populatio N. It will be coveiet, before I preset tese formulae, to outlie oe example of stratified samplig so tat te equatios ca be related more easily to te ecological framework. Table 8.3 gives iformatio o a stratified radom sample take o a caribou erd i cetral Alaska by Siiff ad Skoog (1964). Tey used as teir samplig uit a quadrat of 4 sq. miles, ad tey stratified te wole study zoe ito six strata, based o a pilot survey of caribou desities i differet regios. Table 8.3 sows tat te 699 total samplig uits were divided very uequally ito te six strata, so tat te largest stratum (A) was times te size of te smallest stratum (D).

21 Capter 8 Page 344 TABLE 8.3 STRATIFIED RANDOM SAMPLING OF THE NELCHINA CARIBOU HERD IN ALASKA BY SINIFF AND SKOOG (1964) a Stratum Stratum size, N Stratum weigt, W Sample size, Mea o. of caribou couted per samplig uit, x Variace of Caribou couts, A B C ,556 D ,798 E ,578 F Total a Six strata were delimited i prelimiary surveys based o te relative caribou desity. Eac samplig uit was 4 square miles. A radom sample was selected i eac stratum ad couts were made from airplaes. Source: Siiff ad Skoog, We defie te followig otatio for use wit stratified samplig: s N Stratum weigt = W = N (8.16) were: N = Size of stratum (umber of possible sample uits i stratum ) N = Size of etire statistical populatio Te stratum weigts are proportios ad must add up to 1.0 (Table 8.3). Note tat te N must be expressed i "sample uits". If te sample uit is 0.5 m, te sizes of te strata must be expressed i uits of 0.5 m (ot as ectares, or km ). Simple radom samplig is ow applied to eac stratum separately ad te meas ad variaces calculated for eac stratum from equatios (8.) ad (8.3). We will defer util te ext sectio a discussio o ow to decide sample size i eac stratum. Table 8.3 gives sample data for a caribou populatio.

22 Capter 8 Page 345 Te overall mea per samplig uit for te etire populatio is estimated as follows (Cocra 1977): x ST = L 1 Nx N (8.17) were: xst = Stratified populatio mea per samplig uit N = Size of stratum = Stratum umber (1,, 3,, L) x = Observed mea for stratum N = Total populatio size = N Note tat x ST is a weigted mea i wic te stratum sizes are used as weigts. For te data i Table 8.3, we ave: x ST (400)(4.1) + (30)(5.6) + (61)( = 699 = caribou/sample uit Give te desity of caribou per samplig uit, we ca calculate te size of te etire caribou populatio from te equatio: were: Xˆ = N x (8.18) ST ST Xˆ ST Populatio total N Number of sample uits i etire populatio x Stratified mea per samplig uit (equatio 8.17) ST For te caribou example: X ˆ ST = 699(77.96) = 54,497 caribou so te etire caribou erd is estimated to be aroud 55 tousad aimals at te time of te study. Te variace of te stratified mea is give by Cocra (1977, page 9) as:

23 Capter 8 Page 346 were: L ws Variace of xst 1 f 1 (8.19) W Stratum weigt (equatio 8.16) s Observed variace of stratum (equatio 8.4) Sample size i stratum f Samplig fractio i stratum / N / Te last term i tis summatio is te fiite populatio correctio ad it ca be igored if you are samplig less tat 5% of te sample uits i eac stratum. Note tat te variace of te stratified meas depeds oly o te size of te variaces witi eac stratum. If you could divide a igly variable populatio ito omogeeous strata suc tat all measuremets witi a stratum were equal, te variace of te stratified mea would be zero, wic meas tat te stratified mea would be witout ay error! I practice of course you caot acieve tis but te geeral priciple still pertais: pick omogeeous strata ad you gai precisio. For te caribou data i Table 8.3 we ave: (0.57) (5575) 98 Variace of xst = (0.043) = Te stadard error of te stratified mea is te square root of its variace: ST Stadard error of x = Variace of x = = Note tat te variace of te stratified mea caot be calculated uless tere are at least samples i eac stratum. Te variace of te populatio total is simply: ST ˆ ST ST Variace of X = N variace of x (8.0) For te caribou te variace of te total populatio estimate is: ˆ ST Variace of X = 699 (69.803) 34,105,734

24 Capter 8 Page 347 ad te stadard error of te total is te square root of tis variace, or Te cofidece limits for te stratified mea ad te stratified populatio total are obtaied i te usual way: x t (stadard error of x ) (8.1) ST ST ST Xˆ t (stadard error of Xˆ ) (8.) ST Te oly problem is wat value of Studet's t to use. Te appropriate umber of degrees of freedom lies somewere betwee te lowest of te values ( - 1) ad te total sample size ( 1). Cocra (1977, p.95) recommeds calculatig a effective umber of degrees of freedom from te approximate formula: d.f. L 1 L gs 1 4 gs 1 (8.3) were: d.f. Effective umber of degrees of freedom for te cofidece limits i equatios (8.1) ad (8.) g N N / s Observed variace i stratum Sample size i stratum N Size of stratum For example, from te data i Table 8.3 we obtai Stratum g A B C D E F

25 Capter 8 Page 348 ad from equatio (8.): (34,106,39) d.f. = X 10 ad tus for 95% cofidece itervals for tis example t = Tus for te populatio mea from equatio (8.0) te 95% cofidece limits are: (8.35) or from 61.4 to 94.5 caribou per 4 sq. miles. For te populatio total from equatio (8.1) te 95% cofidece limits are: 54, (5840) or from 4,933 to 66,060 caribou i te etire erd. 8. Allocatio of Sample Size I plaig a stratified samplig program you eed to decide ow may sample uits you sould measure i eac stratum. Two alterate strategies are available for allocatig samples to strata - proportioal allocatio or optimal allocatio Proportioal Allocatio Te simplest approac to stratified samplig is to allocate samples to strata o te basis of a costat samplig fractio i eac stratum. For example, you migt decide to sample 10% of all te sample uits i eac stratum. I te termiology defied above: N N (8.4) For example, i te caribou populatio of Table 8.3, if you wised to sample 10% of te uits, you would cout 40 uits i stratum A, 3 i stratum B, 6 i stratum C, i stratum D, 7 i E ad 1 i F. Note tat you sould always costrai tis rule so tat at least uits are sampled i eac stratum so tat variaces ca be estimated. Equatio (8.3) tells us wat fractio of samples to assig to eac stratum but we still do ot kow ow may samples we eed to take i total (). I some situatios tis

26 Capter 8 Page 349 is fixed ad beyod cotrol. But if you are able to pla aead you ca determie te sample size you require as follows. were: Decide o te absolute size of te cofidece iterval you require i te fial estimate. For example, i te caribou case you may wis to kow te desity to ±10 caribou/4 sq. miles wit 95% cofidece. Calculate te estimated total umber of samples eeded for a ifiite populatio from te approximate formula (Cocra 1977 p. 104): 4 Ws (8.5) d W s d Total sample size required (for large populatio) Stratum weigt Observed variace of stratum Desired absolute precisio of stratified mea (widt of cofidece iterval is d ) Tis formula is used we 95% cofidece itervals are specified i d. If 99% cofidece itervals are specified, replace 4 i equatio (8.4) wit 7.08, ad for 90% cofidece itervals, use.79 istead of 4. For a fiite populatio correct tis estimated by equatio (7.6), page 000: were: * = 1 N * = Total sample size eeded i fiite populatio of size For te caribou data i Table 8.3, if a absolute precisio of 10 caribou / 4 square miles is eeded: N 4 (0.57)(5575) + (0.043)(4064) + 1 = Note tat tis recommeded sample size is more ta te total sample uits available! For a fiite populatio of 699 sample uits:

27 Capter 8 Page sample uits Tese 514 sample uits would te be distributed to te six strata i proportio to te stratum weigt. Tus, for example, stratum A would be give (0.57)(514) or 94 sample uits. Note agai te message tat if you wis to ave ig precisio i your estimates, you will ave to take a large sample size Optimal Allocatio We decidig o sample sizes to be obtaied i eac stratum, you will fid tat proportioal allocatio is te simplest procedure. But it is ot te most efficiet, ad if you ave prior iformatio o te samplig metods, more powerful allocatio plas ca be specified. I particular, you ca miimize te cost of samplig wit te followig geeral approac developed by Cocra (1977). fuctio: were: Assume tat you ca specify te cost of samplig accordig to a simple cost C co c (8.6) C co c Total cost of samplig Overead cost Cost of takig oe sample i stratum Number of samples take i stratum Of course te cost of takig oe sample migt be equal i all strata but tis is ot always true. Costs ca be expressed i moey or i time uits. Ecoomists ave developed muc more complex cost models but we sall stick to tis simple model ere. Cocra (1977 p. 95) demostrates tat, give te cost fuctio above, te stadard error of te stratified mea is at a miimum we: Ns is proportioal to c

28 Capter 8 Page 351 Tis meas tat we sould apportio samples amog te strata by te ratio: Ns / c Ns / c (8.7) Tis formula leads to tree useful rules-of-tumb i stratified samplig: i a give stratum, take a larger sample if 1. Te stratum is larger. Te stratum is more variable iterally 3. Samplig is ceaper i te stratum Oce we ave doe tis we ca ow go i oe of two ways: (1) miimize te stadard error of te stratified mea for a fixed total cost. If te cost is fixed, te total sample size is dictated by: were: C c O N s c C co Ns / c Ns c (8.8) Total sample size to be used i stratified samplig for all strata combied Total cost (fixed i advace) Overead cost Size of stratum stadard deviatio of stratum cost to take oe sample i stratum Box 8.3 illustrates te use of tese equatios for optimal allocatio. Box 8.3 OPTIMAL AND PROPORTIONAL ALLOCATION IN STRATIFIED RANDOM SAMPLING Russell (197) sampled a clam populatio usig stratified radom samplig ad obtaied te followig data: Stratum Size of stratum, N Stratum weigt, W Sample size, Mea (busels), x Variace, A B C s

29 Capter 8 Page 35 D N = 13, Stratum weigts are calculated as i equatio (8.16). I use tese data to illustrate ypotetically ow to desig proportioal ad optimal allocatio samplig plas. Proportioal Allocatio If you were plaig tis samplig program based o proportioal allocatio, you would allocate te samples i proportio to stratum weigt (equatio 8.4): Stratum Fractio of samples to be allocated to tis stratum A 0.43 B 0.10 C 0.10 D 0.38 Tus, if samplig was costraied to take oly 18 samples (as i te actual data), you would allocate tese as 7,,, ad 7 to te four strata. Note tat proportioal allocatio ca ever be exact i te real world because you must always ave two samples i eac stratum ad you must roud off te sample sizes. If you wis to specify a level of precisio to be attaied by proportioal allocatio, you proceed as follows. For example, assume you desire a absolute precisio of te stratified mea of d = 0.1 busels at 95% cofidece. From equatio (8.5): 4 Ws 4 (0.481)(0.068) (0.0953)(0.04) + d 17 samples (0.1) (assumig te samplig fractio is egligible i all strata). Tese 17 samples would be distributed to te four strata accordig to te fractios give above - 43% to stratum A, 10% to stratum B, etc. Optimal Allocatio I tis example proportioal allocatio is very iefficiet because te variaces are very differet i te four strata, as well as te meas. Optimal allocatio is tus to be preferred. To illustrate te calculatios, we cosider a ypotetical case i wic te cost per sample varies i te differet strata. Assume tat te overead cost i equatio (8.5) is $100 ad te coasts per sample are c 1 = $10 c = $0 c 3 = $30 c 4 = $40 Apply equatio (8.7) to determie te fractio of samples i eac stratum:

30 Capter 8 Page 353 Ns / c N s / c Tese fractios are calculated as follows: Stratum N s c N s / c Estimated fractio, / A B C D Total = We ca ow proceed to calculate te total sample size eeded for optimal allocatio uder two possible assumptios: Miimize te Stadard Error of te Stratified Mea I tis case cost is fixed. Assume for tis example tat $00 is available. Te, from equatio (8.8), C c N s c N s / c O / ( )( ) 44, (rouded to 71 samples) Note tat oly te deomiator eeds to be calculated, sice we ave already computed te umerator sum. We allocate tese 71 samples accordig to te fractios just establised: Stratum Fractio of samples Total o. samples allocatio of 68 total A (1) B (3) C (15) D (3) Miimize te Total cost for a Specified Stadard Error I tis case you must decide i advace wat level of precisio you require. I tis ypotetical calculatio, use te same value as above, d = 0.1 busels (95% cofidece limit). I tis case te desired variace (V) of te stratified mea is

31 Capter 8 Page 354 V d 0.1 = = =0.005 t Applyig formula (8.9): = Ws c Ws / c V N W s (1/ )( ) We eed to compute tree sums: Tus: W s W s c (0.481)(0.608)(3.16) + (0.0953)(0.049)(4.47) (0.481)(0.068) (0.0953)(0.049) / c Ws (0.481)(0.068) + (0.0953)(0.04) (3.3549)(0.1191) = = (rouded to 157 samples) ( / 13,34.) We allocate tese 157 samples accordig to te fractios establised for optimal allocatio Stratum Fractio of samples Total o. of samples allocated of 157 total A (47) B (6) C (34) D (71) Note tat i tis ypotetical example, may fewer samples are required uder optimal allocatio ( = 157) ta uder proportioal allocatio ( = 17) to acieve te same cofidece level ( d = 0.1 busels). Program SAMPLE (Appedix, page 000) does tese calculatios. () miimize te total cost for a specified value of te stadard error of te stratified mea. If you specify i advace te level of precisio you eed i te stratified mea, you ca estimate te total sample size by te formula: Ws c Ws / c V (1/ N) W s (8.9)

32 Capter 8 Page 355 were: Total sample size to be used i stratified samplig W Stratum weigt s Stadard deviatio i stratum c Cost to take oe sample i stratum N Total umber of sample uits i etire populatio V Desired variace of te stratified mea d / t d Desired absolute widt of te cofidece iterval for 1- t Studet's t value for 1- cofidece limits ( t for 95% cofidece limits, t.66 for 99% cofidece limits, t 1.67 for 90% of cofidece limits) Box 8.3 illustrates te applicatio of tese formulas. If you do ot kow aytig about te cost of samplig, you ca estimate te sample sizes required for optimal allocatio from te two formulae: 1. To estimate te total sample size eeded (): Ws V (1/ N) W s (8.30) were : V Desired variace of te stratified mea ad te oter terms are defied above.. To estimate te sample size i eac stratum: Ns Ns (8.31) were: Total sample size estimated i equatio (8.9) ad te oter terms are defied above. Tese two formulae are just variatios of te oes give above i wic samplig costs are presumed to be equal i all strata. Proportioal allocatio ca be applied to ay ecological situatio. Optimal allocatio sould always be preferred, if you ave te ecessary backgroud

33 Capter 8 Page 356 iformatio to estimate te costs ad te relative variability of te differet strata. A pilot survey ca give muc of tis iformatio ad elp to fie tue te stratificatio. Stratified radom samplig is almost always more precise ta simple radom samplig. If used itelligetly, stratificatio ca result i a large gai i precisio, tat is, i a smaller cofidece iterval for te same amout of work (Cocra 1977). Te critical factor is always to cose strata tat are relatively omogeeous. Cocra (1977 p. 98) as sow tat wit optimal allocatio, te teoretical expectatio is tat: were: S.E.(optimal) S.E.(proportioal) S.E.(radom) S.E.(optimal) = Stadard error of te stratified mea obtaied wit optimal allocatio of sample sizes S.E.(proportioal) = Stadard error of te stratified mea obtaied wit proportioal allocatio S.E.(radom) = Stadard error of te mea obtaied for te wole populatio usig simple radom samplig Tus comes te simple recommedatio: always stratify your samples! Uless you are perverse or very ulucky ad coose strata tat are very eterogeeous, you will always gai by usig stratified samplig Costructio of Strata How may strata sould you use, if you are goig to use stratified radom samplig? Te aswer to tis simple questio is ot easy. It is clear i te real world tat a poit of dimiisig returs is quickly reaced, so tat te umber of strata sould ormally ot exceed 6 (Cocra 1977, p. 134). Ofte eve fewer strata are desirable (Iaca 1985), but tis will deped o te stregt of te gradiet. Note tat i some cases estimates of meas are eeded for differet geograpical regios ad a larger umber of strata ca be used. Duck populatios i Caada ad te USA are estimated usig stratified samplig wit 49 strata (Joso ad Grier 1988) i order to ave regioal estimates of productio. But i geeral you sould ot expect to gai muc i precisio by icreasig te umber of strata beyod about 6.

34 Capter 8 Page 357 Give tat you wis to set up -6 strata, ow ca you best decide o te boudaries of te strata? Stratificatio may be decided a priori from your ecological kowledge of te samplig situatio i differet microabitats. If tis is te case, you do ot eed ay statistical elp. But sometimes you may wis to stratify o te basis of te variable beig measured (x) or some auxiliary variable (y) tat is correlated wit x. For example, you may be measurig populatio desity of clams (x) ad you may use water dept (y) as a stratificatio variable. Several rules are available for decidig boudaries to strata (Iaca 1985) ad oly oe is preseted ere, te cum f rule. Tis is defied as: cum f cumulative square-root of frequecy of quadrats Tis rule is applied as follows: 1. Tabulate te available data i a frequecy distributio based o te stratificatio variable. Table 8.4 gives some data for illustratio.. Calculate te square root of te observed frequecy ad accumulate tese square roots dow te table. 3. Obtai te upper stratum boudaries for L strata from te equally spaced poits: Maximum cumulative Boudary of stratum i = L f i (8.3) For example, i Table 8.4 if you wised to use five strata te upper boudaries of strata 1 ad would be: Boudary of stratum 1 = (1) = Boudary of stratum () Tese boudaries are i uits of cum f. I tis example, 8.3 is betwee depts 0 ad 1, ad te boudary 0.5 meters ca be used to separate samples belogig to stratum 1 from tose i stratum. Similarly, te lower boudary of te secod stratum is cum f uits wic falls betwee depts 5 ad 6 meters i Table 8.4.

35 Capter 8 Page 358 Usig te cum f rule, you ca stratify your samples after tey are collected, a importat practical advatage i samplig. You eed to ave measuremets o a stratificatio variable (like dept i tis example) i order to use te cum f rule. TABLE 8.4 DATA ON THE ABUNDANCE OF SURF CLAMS OFF THE COAST OF NEW JERSEY IN 1981 ARRANGED IN ORDER BY DEPTH OF SAMPLES a Class Dept, y (m) No,. of samples, f f cum f Observed o. of clams, x , 18, 13, , 4 Stratum , 5, , 3, 1, , , 11, 55, 3, 65 Stratum , 10, 0, , , , 30 Stratum , 5, , 0, 4, , , 10, 97 Stratum , , 1, , , 1, ,,, Stratum 5

36 Capter 8 Page , a Stratificatio is carried out o te basis of te auxiliary variable dept i order to icrease te precisio of te estimate of clam abudace for tis regio. Source: Iaca, Proportios ad Percetages Stratified radom samplig ca also be applied to te estimatio of a proportio like te sex ratio of a populatio. Agai te rule-of-tumb is to costruct strata tat are relatively omogeeous, if you are to acieve te maximum beefit from stratificatio. Sice te geeral procedures for proportios are similar to tose outlied above for cotiuous ad discrete variables, I will just preset te formulae ere tat are specific for proportios. Cocra (1977 p. 106) summarizes tese ad gives more details. We estimate te proportio of x-types i eac of te strata from equatio (8.1) (page 000). Te, we ave for te stratified mea proportio: pˆ = ST Np ˆ (8.33) N were: pˆ ST Stratified mea proportio N Size of stratum pˆ Estimated proportio for stratum (from equatio 8.13) N Total populatio size (total umber of sample uits) Te stadard error of tis stratified mea proportio is: ST S.E. pˆ = 1 N N pq ˆ ˆ (8.34) N N 1 1 were: S.E. p ˆST Stadard error of te stratified mea proportio qˆ 1 - ˆ p Sample size i stratum ad all oter terms are as defied above.

37 Capter 8 Page 360 Cofidece limits for te stratified mea proportio are obtaied usig te t- distributio as outlied above for equatio 8.0 (page 347). Optimal allocatio ca be acieved we desigig a stratified samplig pla for proportios usig all of te equatios give above ( ) ad replacig te estimated stadard deviatio by: s pˆ ˆ q 1 (8.35) were: s Stadard deviatio of te proportio p i stratum pˆ Fractio of x types i stratum qˆ 1- ˆ p Sample size i stratum Program SAMPLE i Appedix (page 000) does all tese calculatios for stratified radom samplig, ad will compute proportioal ad optimal allocatios from specified iput to assist i plaig a stratified samplig program. 8.3 Adaptive Samplig Most of te metods discussed i samplig teory are limited to samplig desigs i wic te selectio of te samples ca be doe before te survey, so tat oe of te decisios about samplig deped i ay way o wat is observed as oe gaters te data. A ew metod of samplig tat makes use of te data gatered is called adaptive samplig. For example, i doig a survey of a rare plat, a botaist may feel iclied to sample more itesively i a area were oe idividual is located to see if oters occur i a clump. Te primary purpose of adaptive samplig desigs is to take advatage of spatial patter i te populatio to obtai more precise measures of populatio abudace. I may situatios adaptive samplig is muc more efficiet for a give amout of effort ta te covetioal radom samplig desigs discussed above. Tompso (199) presets a summary of tese metods.