Model Quality Report in Business Statistics

Transcription

1 Moel Quali Repor in Buine Saiic Ma Bergal, Ole Blac, Ruell Boaer, Ra Camber, Pam Davie, Davi Draper, Eva Elver, Suan Full, Davi Holme, Pär Lunqvi, Sien Lunröm, Lennar Norberg, Jon Perr, Mar Pon, Mie Preoo, Ian Ricaron, Cri Sinner, Paul Smi, Ceri Uneroo, Mar William General Eior: Pam Davie, Paul Smi Volume I Teor an Meo for Quali Evaluaion

2 Preface Te Moel Quali Repor in Buine Saiic proec a e up o evelop a eaile ecripion of e meo for aeing e quali of urve, i paricular applicaion in e cone of buine urve, an en o appl ee meo in ome eample urve o evaluae eir quali. Te or a pecifie an iniiae b Euroa folloing on from e Woring Group on Quali of Buine Saiic. I a fune b Euroa uner SUP-COM 997, lo 6, an a been uneraen b a conorium of e UK Office for Naional Saiic, Saiic Seen, e Univeri of Souampon an e Univeri of Ba, i e Office for Naional Saiic managing e conrac. Te repor i ivie ino four volume, of ic i i e fir. Ti volume eal i e eor an meo for aeing quali in buine urve in nine caper folloing e urve proce roug i variou age in orer. Tee fall ino ree par, one ealing i ampling error, one i a varie of non-ampling error, an one covering coerence an comparabili of aiic. Oer volume of e repor conain: a comparion of e ofare meo an pacage available for variance eimaion in ample urve (volume II); eample aemen of quali for an annual an a monl buine urve from Seen an e UK (volume III); guieline for an eperience of implemening e meo (volume IV). An ouline of e caper in e repor i given on e folloing page. Acnolegemen Apar from e auor, everal oer people ave mae large conribuion iou ic i repor oul no ave reace i curren form. In paricular e oul lie o menion Tim Jone, Ania Ullberg, Jeff Evan, Trevor Fenon, Jonaan Goug, Dan Helin, Sue Hibbi an Seve Jame, an e oul alo lie o an all e oer people o ave been o elpful an uneraning ile our aenion a been focue on i proec! Ouline of Moel Quali Repor Volume Volume I. Meoolog overvie an inroucion Par : Sampling error. Probabili ampling: baic meo 3. Probabili ampling: eenion 4. Sampling error uner non-probabili ampling Par : Non-ampling error

3 5. Frame error 6. Meauremen error 7. Proceing error 8. Non-repone error 9. Moel aumpion error Par 3: Oer apec of quali 0. Comparabili an coerence Par 4: Concluion an Reference. Concluion. Reference Volume II. Inroucion. Evaluaion of variance eimaion ofare 3. Simulaion u of alernaive variance eimaion meo 4. Variance in STATA/SUDAAN compare i analic variance 5. Reference Volume III. Inroucion Par : Te rucural urve. Quali aemen of e 995 Sei Annual Proucion Volume Ine 3. Quali aemen of e 996 UK Annual Proucion an Conrucion Inquirie Par : Sor-erm aiic 4. Quali aemen of e Sei Sor-erm Proucion Volume Ine 5. Quali aemen of e UK Ine of Proucion 6. Quali aemen of e UK Monl Proucion Inquir Par 3: Te UK Sampling Frame 7. Sampling frame for e UK Volume IV. Inroucion. Guieline on implemenaion 3. Implemenaion repor for Seen 4. Implemenaion repor for e UK ii

4 Conen Meoolog overvie an inroucion.... General rucure.... A guie o e conen..... Toal urve error..... Sampling error Non-ampling error Comparabili an coerence Concluing remar...6 Par : Sampling Error Probabili ampling: baic meo...7. Baic concep Targe populaion an ample populaion Sample frame an auiliar informaion Probabili ampling...8. Saiical founaion Y an X variable Finie populaion parameer Populaion moel Sample error an ample error iribuion Te repeae ampling iribuion v. e uperpopulaion iribuion Bia, variance an mean quare error....3 Eimae relae o populaion oal Te eign-bae approac Sample incluion probabiliie Te Horviz-Tompon eimae Deign-bae eor for e Horviz-Tompon eimae Deign-bae eor for fie ample ize eign Approimaing econ orer incluion probabiliie Problem i e eign-bae approac Te ue of moel for eimaing a populaion oal Te uperpopulaion moel Te omogeneou raa moel Te imple linear regreion moel Te general linear regreion moel Te cluer moel Ignorable ampling Bia, variance an mean quare error uner e moel-bae approac Weanee of e moel-bae approac Linear preicion Robu preicion variance eimaion Te moel-aie approac Te GREG an GRAT eimae for a populaion oal Variance eimae for e GREG an GRAT Calibraion eiging Meo for nonlinear funcion of e populaion value Variance eimaion via Talor erie lineariaion Differeniable funcion of populaion oal Funcion efine a oluion of eimaing equaion Replicaion-bae meo for variance eimaion Ranom group eimae of variance Jacnife eimae of variance Te linearie acnife Boorapping Concluion Probabili ampling: eenion Domain eimaion Deign-bae inference for omain Deign-bae inference uner SRSWOR Moel-bae inference en N i unnon...4 iii

5 3..4 Moel-bae inference en N i non Moel-bae inference uiliing auiliar informaion An eample Domain eimaion uing a linear eige eimae Moel-aie omain inference Eimaion of cange Linear eimaion Eimae of cange for funcion of populaion oal Eimae of cange in omain quaniie Oulier robu eimaion Oulier robu moel-bae eimaion Winoriaion-bae eimaion Variance eimaion for inice Concluion Sampling error uner non-probabili ampling Inroucion Volunar ampling Quoa ampling Jugemenal ampling Proucer price ine conrucion in e EU Te UK eperience Cu-off ampling Variaion : Ignore e cu-off uni Variaion : Moel e cu-off uni Concluion...80 Par : Non-ampling error 5 Frame error Inroucion A Buine Regier an i ue a a frame Uni, elineaion, an variable Upaing e BR uing everal ource Te BR a a frame uni, variable an reference ime Frame an arge populaion Targe populaion Frame, an frame populaion Difference beeen e frame populaion an e arge populaion Uner- an over-coverage of e populaion Difference iin e populaion Some commen on frame error Defining a Buine Regier covering a ime perio Te arge populaion: eimaion an inaccurac Eimaion proceure an informaion neee Uing e frame populaion onl Upaing e ample onl Uiliing laer BR informaion on e populaion Uiliing a BR covering e reference perio Some commen on e BR an effec of coverage eficiencie Illuraion aminiraive aa an buine emograp Illuraion ime ela an aing frame Te UK Buine Regier Te Sei Buine Regier Some comparion beeen UK an Seen Illuraion cange beeen frame an eir effec Difference beeen UK curren an frozen claificaion Difference iin e Sei populaion one ear apar Difference for e populaion a a ole; Seen A fe ummariing concluion Meauremen error Naure of meauremen error True value...04 iv

6 6.. Source of meauremen error Tpe an moel of meauremen error Te conribuion of meauremen error o oal urve error Toal urve error Bia Variance inflaion Diorion of eimae b gro error Deecing meauremen error Comparion a aggregae level i eernal aa ource Comparion a uni level i eernal aa ource Inernal comparion an eiing Follo-up Embee eperimen an obervaional aa Quali meauremen Quali inicaor Aeing e bia impac of meauremen error Aeing e variance impac of meauremen error Proceing error Inroucion o proceing error Sem error Meauring em error Sem error: o eample Sampling in e ONS Variable forma in compuer program Minimiing em error Daa anling error Daa ranmiion Daa capure Daa eing from pencil an paper queionnaire Meauring error occurring uring aa eing Minimiing error occurring uring aa eing Daa capure uing canning an auomae aa recogniion Meauring error aociae i canning an auomae aa recogniion Minimiing error aociae i canning an auomae aa recogniion Coing error Meauring coing error Conienc Accurac Te impac of coer error on e variance of urve eimae Te ri of coer error inroucing bia in urve eimae Minimiing coing error Daa eiing Meauring e impac of eiing on aa quali Minimiing error inrouce b eiing An eample of error a e publicaion age Nonrepone error Inroucion Tpe of nonrepone Paern of miing aa Miing aa mecanim Problem caue b nonrepone A baic eing Bia Variance inflaion Effec of confuing uni ouie e populaion i nonrepone Effec of nonrepone on coerence Quali meauremen Repone rae Meaure bae on follo-up aa Comparion i eernal aa ource an bencmar...3 v

7 8.4.4 Comparion of alernaive aue poin eimae Weiging aumen Te baic meo Ue of auiliar informaion Poraificaion Regreion eimaion an calibraion Weiging an nonrepone error Variance eimaion Impuaion Ue Deucive impuaion an eiing La value impuaion Raio an regreion impuaion Donor meo Socaic meo Impuaion an nonrepone error Variance eimaion Moel Aumpion Error Inroucion Ine number Bencmaring Seaonal aumen Cu-off ampling Small omain of eimaion Non-ignorable nonrepone Selecion moel for coninuou oucome Paern-miure moel for caegorical oucome Concluion...60 Par 3: Oer Apec of Quali 0 Comparabili an coerence Inroucion Coerence empaiing e uer perpecive Definiion in eor Definiion in pracice Accurac an conien eimae Comparabili over ime Inernaional comparabili Some uer-bae concluion Proucer apec on coerence, incluing comparabili Definiion in eor Definiion in pracice Accurac an conien eimae Some commen on meoolog, epeciall bencmaring Comparabili over ime Inernaional comparabili Some proucer-bae concluing commen Some illuraion of coerence an co-orinaion...74 Par 4: Concluion an Reference Concluing remar Meoolog for quali aemen Recommenaion for quali aemen...79 Reference Ine...89 vi

8 Meoolog overvie an inroucion. General rucure Paul Smi, Office for Naional Saiic Ti volume cover e eor an meo for aeing quali in buine urve uner eig main eaing. Te main bo of e repor i ivie ino nine caper, i e probabili ampling main eaing pli ino o caper. Te non-ampling error ecion follo e claificaion of e Euroa oring group on Quali of Buine Saiic. Te caper are. Probabili ampling: baic meo 3. Probabili ampling: eenion 4. Sampling error uner non-probabili ampling 5. Frame error 6. Meauremen error 7. Proceing error 8. Nonrepone error 9. Moel aumpion error 0. Comparabili an coerence Tee fall ino ree par, i caper -4 ealing i ampling error (par ), caper 5-9 i variou apec of non-ampling error (par ) an caper 0 forming a par on i on (par 3). Te coverage of eac caper i ecribe in ummar in ecion., an e iea are neie an line o e Moel Quali Repor in e final caper, caper. Reference o oer or menione in i volume appear a e en, an e noaion generall follo Särnal, Senon & Wreman (99) ecep ere furer noaion i require, in ic cae i i efine.. A guie o e conen.. Toal urve error I i enible o r o lin e meo in ee ampling an non-ampling error caper ino a common frameor (a) a a guie o a i of mo inere an relevance an ic ource of error i liel o be mo imporan in a given cone, an (b) o elp in navigaion roug e opic conaine in e variou caper. Ti i epeciall imporan in ome of e non-ampling error caper ere opic ill ofen fi comforabl uner more an one eaing, an i ma no be immeiael obviou ere o loo for informaion on a paricular opic. Te be concep for proviing a unifing frameor i e concep of oal urve error (Grove 989), ic emboie e ifference beeen e urve eimae an e concepual real or rue value. In buine urve e real value (oal ale b manufacuring inurie, for eample) mol a a founaion in reali if i ere poible o loo a ever manufacuring buine ale an recor em accurael, e coul arrive a

9 e real value. For oer aiic uc a e average price movemen e rue value i no ell-efine an i conruc brea on. So, auming a e real value i ell-efine, e can imagine a e an o meaure e ifference beeen our urve eimae an e rue value. Conier e problem of eimaing a oal pical eimaor ae e form i i U i of a variable acro a populaion U. Te, ere i i e urve eig, i i e repore value of i an e um i over e ample. Te oal urve error i en oal urve error i i U i an i ma be broen on ino o componen (ee Grove, 989, p.): error from obervaion error from non - obervaion i i i i U i i i ( ) Te fir (obervaion error) componen reflec meauremen error, a ell a proceing, coing an impuaion error an oul iappear if e recore value i ere equal o e rue value i. Te econ (non-obervaion error) componen reflec ampling error, frame error an nonrepone error an oul iappear if e uni upon ic e eimae i bae comprie preciel e arge populaion U. Te oal urve error provie an overall meaure of quali. Te problem i o o ae i magniue. To meaure e ampling error i i uual o e up a moel for e iribuion of e ampling error an en o eimae e caraceriic of i iribuion. Uuall, i i aume (e aumpion being bae on ampoic eor) a i ampling iribuion i approimael normal an cenre a zero o a e onl a i o eimae e variance of e iribuion. To een i iea o oal urve error i i necear o e up a moel for e iribuion of e oer componen of error. Toal urve error can be coniere in a ifferen a oo broen on ino o componen, a ifference ic i approimael invarian over repeiion of e urve, e bia, an a ifference ic varie i ifferen repeiion of e urve, e variance. Te repeiion ue in i efiniion are ofen poeical, a i e urve i no acuall repeae. We eplore ee o pe of error in more eail belo. Te bia an variance ogeer conribue o a meaure of e oal urve error, calle e mean quare error (me), uc a me bia variance alo omeime epree a i quare roo, e roo mean quare error (rme). Bo e bia an e variance are mae up of everal componen erm correponing o paricular pe of error. In e cae of e bia ome of ee componen ill almo cerainl cancel eac oer ou (e a a ere are poiive an negaive biae), o a e overall bia ill be i i i

10 e ne of ee effec. Variance are ala non-negaive an o ill cumulae over componen. If all e relevan biae an variance are inclue in calculaing e me, i ill be a goo eimaor of e oal urve error. Ti give u o broa approace o man error. We can rea e repone of a given uni a fie for an occaion en i i inclue in e ample (a in of eerminiic approac). Ta i, if a buine i inclue in e ample, e aume a i ala mae e ame repone/nonrepone eciion, ala give e ame aner on e queionnaire, an o on. Ti almo ala lea u o eimae biae. Alernaivel e can conier a a buine repone/nonrepone eciion arie from ome probabili iribuion, an a i aner alo come from ome iribuion, in ic cae mo of e error ill aiionall ave a variance componen. Ti laer approac i ain o e moel-bae ampling approac (ecion.3.), a e aume a uperpopulaion of poible oucome i e ampling forming onl one componen of eermining ic oucome e acuall oberve in e urve. We ill ue i iincion in approac beeen eerminiic an uperpopulaion moel in icuing e error ic mae up oal urve error... Sampling error Cerain aumpion an moel are require o eimae e componen of oal urve error, an e begin b coniering ranom ampling mecanim; in i ecion e aume a all urve age afer ampling are error-free. Wen a urve i o be conuce, e ample can be elece accoring o ome probabili mecanim. A lea concepuall e can elec more en one ample uing e ame probabili mecanim (b running e elecion proce everal ime), an eac ample oul reul in a ifferen eimae if e urve ere acuall run, impl becaue ifferen uni oul be inclue in e ample. Eac of ee poenial eimae oul in general be ifferen from e rue oal. We ave ere e iuaion a e urve eimae are ifferen b repeiion over ifferen ample, an e can meaure o muc ee eimae iffer from eir mean on average, uing e average iance of e ample elemen from eir mean o eimae e average iance of populaion elemen from e mean. Ti give u a variance, e ampling variance. Over all poible ifferen ample, e mean of e eimae i e ame a e rue value (ill auming no oer error); in pracice e normall ave onl one ample, an ave o ue e mean of a ample o approimae e rue populaion mean. Effecivel, a menione in ecion.., e aume a e ampling error i cenre aroun e eimae e o ave. Caper cover e eor an meo ic give rie o ampling error an ampling error eimae uing firl e eign-bae an moel-aie approace, uner ic ifferen moel of e relaionip beeen a urve repone an non auiliar value are ue o improve e eimaion. Tee approace baicall involve accouning for e elecion probabiliie from e ampling in all e eimaion an variance calculaion in an appropriae a. Ti caper alo inrouce e moel-bae approac, ic aume a Unle eimae b a variance componen moel; if a negaive variance i obaine i probabl inicae a e moel i inappropriae. 3

11 e urve repone are realiaion from an poeical infinie populaion of poible oucome. In i cae, i an appropriae moel e elecion probabiliie are ignorable, a i e ave no effec on e eimaion or variance eimaion an o no nee o be inclue eplicil. Caper 3 ae ee o approace an een em from raigforar eimaion meo o more complicae aiic, incluing eimaion of cange, eimaion for omain (ube of e populaion) an eimaion in e preence of oulier. Tere i alo a ummar of ome or on e variabili of a muliource inicaor, ic conier e effec of e variabili of ifferen erie ic go o mae up an ine on i oal variance. Conier no ample elecion mecanim ic are no bae on probabili. In ee cae e pe of error e obain epen on e acual mecanim of elecion. If repeiion a no effec on e ample compoiion (a i, e ame ample elemen are coen ever ime), en e ifference beeen e urve eimae an e rue value i conan over repeiion: i i a (pure) bia. If e ample can be ifferen over repeiion, en ere ill be a range of poenial eimae, an ere ill be a variance componen an a bia. In pracice e o effec ma no be eparael eimable, or even eimable a all if e rue value i unnon (ic i picall e cae). Ti ubec i aree in caper 4 (nonprobabili ampling), concenraing paricularl on cu-off ampling an volunar ampling (ample obaine from volunar urve), bu alo menioning quoa ampling an ugemenal ampling...3 Non-ampling error No relaing e aumpion from ecion.. a evering ele apar from ampling i perfec, le u conier e oer poible error. Tee are arrange o follo approimael e orer of proceing in a buine urve. Frame error conribuing mainl o e bia componen of e oal error are icue in caper 5. Tee error generall em from ifference beeen frame- an arge populaion. Hence problem of uner- an over-coverage are imporan. Since buine populaion uuall cange rapil, e upaing of uni an of variable aace o ee uni become imporan. Delineaion of buinee ino ifferen pe of uni (local uni, in-of-acivi uni ec) i anoer acivi i a large impac on frame quali. All of ee iue are eal i in caper 5. Meauremen error are error ic are inrouce en ring o ge e eire informaion from conribuor. In caper 6, e loo a a meauremen error moel for o aner var over ifferen (concepual) repeae queioning, an i conribue o e variabili of e eimae b giving a variable meauremen for a ingle reponen. Meauremen error are liel o conribue o bo componen bia an variance of e oal error bu e are ofen ifficul or epenive o ae, epeciall in cae ere folloup uie become necear. Ye meauremen error ma ofen ave a large influence on accurac in buine urve. Approace o eecion an aemen of meauremen error are icue in caper 6. 4

12 Proceing error are icue in caper 7. Tee are error connece i aa anling aciviie eer manual or auomae uc a aa ranmiion, aa capure, coing an aa eiing. A paricular form of proceing error, calle em error in caper 7, are error ariing from ofare an arare. I i ifficul o enviage a probabili mecanim i a real inerpreaion for em error, an in fac e are ver ifficul o meaure a all. Proceing error in general ma conribue o bo componen e bia an e variance of e oal error aloug e bia i liel o be e more imporan one. Nonrepone, reae in caper 8, arie en a ample uni fail o provie complee repone o all queion ae in a urve. Tere are o a of coniering nonrepone in a fie ample. Te eerminiic approac aume a fie bu unnon repone inicaor value ( if value i recore, 0 if value i miing) for ever uni in e ample. Te ocaic approac rea e repone inicaor variable a oucome of ranom variable. Te naure of error ariing from nonrepone en epen on aumpion abou i ranom mecanim. Te ocaic approac i e one folloe in caper 8. Meo o meaure or inicae e impac of nonrepone on accurac are reae. Ti caper alo rea implicaion of nonrepone uc a bia, variance inflaion an effec of confuing nonrepone i over-coverage. Re-eiging an impuaion meo o compenae for bia caue b nonrepone are icue. Caper 9 icue error an inaccurac caue b uing moel aumpion concenraing on eimaion problem an pe of moel ic are no menione eleere. Te aim of inroucing a moel ma be o reuce variance an/or o reuce bia, bu ere i alo a ri of inroucing bia if e moel i no ell coen. Small area eimaion i one par of e urve proce ere moel are imporan, bencmaring anoer (noe a calibraion belong o ampling error; e iea i imilar bu e ecnique ifferen). Non-ignorable nonrepone i icue ere, aloug i a rong lin o e non-repone meo in caper 8. Te icuion of cu-off ampling a are in caper 4, non-probabili ampling, an i i coninue ere, empaiing e ue of moel o eimae for e par of e populaion a a cu off. Anoer reaon for uing moel i o elp o compenae for a lac of up-o-ae informaion, for eample on eig in caine price inice, a problem ic i inrouce in i caper. Seaonal aumen i alo ecribe, incluing commen on e ofare in ue; aemen of e reuling accurac i a ifficul maer...4 Comparabili an coerence Ti i an area ic oe no fi uner e uual efiniion of oal urve error, becaue i oe no eal i e error in a ingle urve, bu inea conier o ell o or more e of aiic can be ue ogeer. Ti caper cover efiniion in eor an in pracice, accurac, ifferen co-orinaion aciviie, an comparabili of urve over ime an naional bounarie. Bo uer an proucer perpecive are coniere, an illuraion are given. 5

13 ..5 Concluing remar Te final caper in i volume, caper, lin e concep ecribe in i inroucion an ra ou e imporan eme for aeing oal urve error in ome given cone. I alo correpon i caper of e Implemenaion Guieline (volume IV), ic provie a ummar of e meo ecribe in i volume a e are applie in e Moel Quali Repor. Tere i an eample running roug e ampling error caper ( an 3), an ic alo appear in caper 4, 8 an 9, ic correpon rongl i e Annual Buine Inquir in e UK, ic i e annual rucural urve eample from e UK in e Moel Quali Repor (volume III, caper 3). 6

14 Par : Sampling Error Probabili ampling: baic meo. Baic concep Ra Camber, Univeri of Souampon Man cienific an ocial iue revolve aroun e iribuion of ome pe of caraceriic over a populaion of inere. Tu e number of unemploe people in a counr labour force an e average annual profi mae b buinee in e privae ecor of a counr econom are o e inicaor of a counr economic ell-being. Te fir of ee number epen on e iribuion of labour force ae among e iniviual maing up e counr labour force ile e econ i eermine b e iribuion of annual profi acieve b e counr buinee. Bo ee number are picall meaure b ample urve. Ta i, a ample of iniviual belonging o e counr labour force i urvee an eir emplomen/unemplomen aue eermine. Similarl a ample of privae ecor buinee i urvee an eir annual profi meaure. In bo cae e informaion obaine from e urve can be ue o infer e unnon correponing value (unemplomen oal or average profi) for e counr... Targe populaion an ample populaion Since in general i i meaningle o al abou a ample iou referring o a i i a ample of, e concep of a populaion from ic a ample i aen i baic o ample urve eor. In e eample above ere are o populaion e populaion of iniviual maing up e labour force of e counr, an e populaion of buinee maing up e privae ecor econom of e counr. In general, oever, e populaion from ic a ample i aen, an e populaion of inere can an o iffer. Te arge populaion of a urve i e populaion a ic e urve i aime, a i e populaion of inere. Hoever, a arge populaion i no necearil a populaion a can be urvee. Te acual populaion from ic e urve ample i ran i calle e urve populaion. A baic meaure of e overall quali of a ample urve i e coverage of e urve populaion, or e egree o ic arge an ample populaion overlap. Aemen of i quali i coniere in Caper 5. Here e all aume ere i no ifference beeen e arge an urve populaion. Ta i, e ave complee coverage. From no on e ill u refer o e populaion... Sample frame an auiliar informaion A anar meo of ampling i o elec e ample from a li (or erie of li) ic enumerae e uni (iniviual, buinee, ec) maing up e ample populaion. Ti li i calle e (ample) frame for e urve. Eience of a ample frame i necear for e ue of man ampling meo. Furermore, applicaion of ee meo ofen require a 7

15 a frame conain more an u ienifier (for eample, name an aree) for e uni maing up a ample populaion. For eample, raifie ampling require e frame o conain enoug ienifing informaion abou eac populaion uni for i raum memberip o be eermine. In general, e refer o i informaion a auiliar informaion. Tpicall, i auiliar informaion inclue caraceriic of e urve populaion a are relae o e variable meaure in e urve. Tee inclue raum ienifier an meaure of ize. For economic populaion, e laer correpon o value for eac uni in e populaion ic caracerie e level of economic acivi b e uni. Te een o ic e ample frame enumerae e ample populaion i anoer e meaure of ample urve quali. Ti iue i coniere in Caper 5. In a follo oever e all aume a ample frame ei an i perfec. Ta i, i li ever uni in e populaion once an onl once, an ere i a non number N of uc uni...3 Probabili ampling A probabili ampling meo i one a ue a ranomiaion evice o ecie ic uni on e ample frame are in e ample. Wi i pe of elecion meo, i i no poible o pecif in avance preciel ic uni on e frame mae up e ample. Conequenl uc ample are free of e (ofen ien) biae a can occur i ampling meo a are no probabili-bae. In a follo e mae e baic aumpion a e probabili ampling meo ue i uc a ever uni on e frame a a non-zero probabili of elecion ino e ample. Ti aumpion i necear for valii of e eign-bae approac o urve eimaion an inference ecribe in ecion.3. belo. Some relevan eor for e cae ere a non-probabili ampling meo i ue i e ou in Caper 4.. Saiical founaion A noe earlier, e baic aim of a ample urve i o allo inference abou one or more caraceriic of e populaion. Suc caraceriic are picall efine b e value of one or more populaion variable. A populaion variable i a quani a i efine for ever uni in e populaion, an i obervable en a uni i inclue in e ample. In general, urve are concerne i man populaion variable. Hoever, mo of e eor for ample urve a been evelope for e cae of a mall number of variable, picall one or o. In a follo e aop e ame implificaion. Iue ariing ou of e nee o meaure man variable imulaneoul in a ample urve are coniere in ecion Y an X variable Aociae i eac uni in e populaion i a e of value for e populaion variable. Some of ee are recore on e frame, an o are non for ever uni in e populaion. We refer o ee auiliar variable a X-variable. Te oer coniue e variable of inere (e u variable) for e urve. Tee are no non. Hoever e aume a eir value are meaure for e ample uni, or can be erive from ample aa. We uuall refer o ee variable a Y-variable. 8

16 For eample, e quarerl urve of capial epeniure (CAPEX) carrie ou b e U.K. Office for Naional Saiic (ONS) a everal u (Y) variable, e mo imporan being acquiiion, ipoal an e ifference beeen acquiiion an ipoal, e ne capial epeniure. Te frame for i urve i erive from e Iner-Deparmenal Buine Regier (IDBR) of e ONS. Tere are a number of X-variable on e urve frame, e mo imporan of ic are e inur claificaion of a buine (Sanar Inur Claificaion), e number of emploee of e buine an e oal VAT urnover of e buine in e preceing ear... Finie populaion parameer Te populaion caraceriic a are e focu of ample urve are omeime referre o a i arge of inference. In general, ee arge are ell-efine funcion of e populaion value of Y-variable, picall referre o a parameer of e populaion. An populaion covere b a frame-bae urve i necearil finie in erm of e number of uni i conain. Suc a parameer ill be referre o a a finie populaion parameer (FPP) in a follo in orer o iingui i from e parameer a caracerie e infinie populaion ue in anar aiical moelling. Some common eample of FPP are: - e populaion oal an average of a Y-variable; - e raio of e populaion average of o Y-variable; - e populaion variance of a Y-variable; - e populaion meian of a Y-variable...3 Populaion moel A populaion of Y-value a an one poin in ime repreen e oucome of man cance occurrence. Hoever, i oe no mean a ee value are compleel arbirar. Tere i picall a rucure ineren in a e of populaion value a can be caracerie in erm of a moel. Suc moel are uuall bae on pa epoure o aa from oer populaion ver muc lie e one of inere, or ubec maer nolege abou o e populaion value oug o be iribue. Conequenl i moel i no caual i oe no a o ee Y-value came o be bu ecripive, in e ene a i i a maemaical ecripion of eir iribuion. In man cae i moel i ielf efine in erm of parameer ic capure ee iribuional caraceriic. A anar a of pecifing uc a aiical moel i in erm of an unerling ocaic proce. Ta i, e N value coniuing e finie populaion of inere are aume o be realiaion of N ranom variable oe oin iribuion i ecribe b e moel. If i approac i aen, en e moel ielf i referre o a a uperpopulaion moel for e finie populaion of inere. Te parameer a caracerie i moel are picall unnon, an are referre o a e uperpopulaion parameer for e populaion. Unlie FPP, uperpopulaion parameer are no real e can never be non preciel, even if e uperpopulaion moel i an accurae epicion of o e finie populaion value are iribue an ever populaion value i non. Some eample of uc uperpopulaion parameer are momen (mean, variance, covariance) of e oin iribuion of e Y- 9

17 variable efining e populaion value an relae quaniie (for eample regreion coefficien)...4 Sample error an ample error iribuion Once a ample a been elece, an ample value of Y-variable obaine, e are in a poiion o calculae e value of variou quaniie bae on ee aa. Tee quaniie are picall referre o a aiic. Te aim of ample urve eor i o efine o pe of aiic: (i) eimae of e FPP of inere; (ii) quali meaure for e eimae in (i). In i repor e ill be mainl concerne i e econ pe of aiic above, a i aiic meauring e quali of e eimae. Hoever, before e can ecribe o uc aiic can be erive, e nee o icu e concep of ample error an ample error iribuion. Te ample error of a urve eimae i u e ifference beeen i oberve value an e unnon value of e FPP of ic i i an eimae. Clearl one oul epec a ig quali urve eimae o ave a mall ample error. Hoever, ince e acual value of e FPP being eimae i unnon, e ample error of i eimae i alo unnon. Bu i oe no mean a ere i noing e can a abou i error. Te meo b ic e ample i coen, an e uperpopulaion moel for e populaion, allo u o pecif a varie of iribuion for e ample error. In urn, i allo u o ue aiical meo o meaure e quali of e urve eimae in erm of e caraceriic of ee iribuion. Before going on o ecribe o ee iribuion are erive an inerpree, i i imporan o noe a i quali meauremen relae o a quani (e ample error) ic aume a ere are no oer ource of error in e urve. In reali, ere are man oer ource of error (frame error, nonrepone error, meauremen error, moel pecificaion error, proceing error) in a urve. Meo for aeing ee are icue in Par of i repor...5 Te repeae ampling iribuion v. e uperpopulaion iribuion Tere are o anar a of efining a iribuion for a ample error. One i i repeae ampling iribuion. Ti i e iribuion of poible value i error can ae uner repeiion of e ampling meo. Concepuall, i correpon o repeaing e ampling proce, elecing ample afer ample from e populaion, calculaing e value of e eimae for eac ample, generaing a (poeniall) ifferen ample error eac ime an ence a iribuion for ee error. Te oer a of efining a iribuion for a ample error i in erm of e uperpopulaion iribuion. Uner i iribuion e ample eimae a ell a e FPP are bo bae on realiaion of e Y-variable a efine e populaion value. Conequenl e ample error i alo a ranom variable i a iribuion efine b e uperpopulaion moel. 0

18 Operaionall i iribuion correpon o e range of poenial value e ample error can ae given e range of poenial value for e populaion Y-variable uner i moel. Tere are funamenal ifference beeen ee iribuion. Te repeae ampling iribuion rea e populaion value a fie. Conequenl e ource of variabili unerling i iribuion i e ample elecion meo. Sample elecion meo a are no probabili bae are erefore no uie o evaluaion uner i iribuion. In conra, e uperpopulaion iribuion rea e ample a fie. Ta i, e unerling variabili in i cae arie from e uncerain abou e iribuion of Y-value for e ample uni an non-ample uni, bu e ample/non-ample iincion i fie accoring o a acuall oberve. To iingui beeen ee o iribuion, e ue a ubcrip of p in a follo o enoe epecaion, variance, ec, aen i repec o e repeae ampling iribuion, an a ubcrip of o enoe correponing quaniie aen i repec o e uperpopulaion iribuion. Tere are aiical argumen for an again e ue of ee o iribuion for e ample error en e an o caracerie e quali of e acual ample eimae. Baicall, e repeae ampling (or ranomiaion) iribuion of e ample error i viee a appropriae for meauring e quali of a urve eign, a i e meo ue o elec e ample. Ti i becaue i reflec our uncerain abou ic ample ill be coen prior o e acual coice of ample. Hoever, bo meo ave been ue o caracerie uncerain abou e ize of e ample error afer e ample aa are obaine. Te argumen for uing e ranomiaion iribuion involve e aumpion a ee aa o noing o cange e ource of our uncerain, e u provie u i a mean o meaure i. We ill caracerie uncerain b e iribuion of ample error aociae i ample a mig ave been coen bu ere no. In conra, ue of e uperpopulaion iribuion eeniall come on o aing a e populaion Y-value, being unnon, repreen e rue ource of uncerain a far a urve inference i concerne. In paricular, afer e ample aa are obaine e ave no uncerain abou ic ample a elece, bu e ill ave uncerain abou e populaion Y-value efining e FPP of inere. In i repor e ill evelop meaure bae on bo iribuion, inicaing eir reng an eanee ere appropriae...6 Bia, variance an mean quare error In orer o ue a iribuion for e ample error o meaure e quali aociae i e acual ample eimae, e nee o pecif e caraceriic of i iribuion a are appropriae for i purpoe. Saiical pracice eeniall focue on o uc caraceriic e cenral locaion of e iribuion, a efine b i mean or epecaion, an e prea of i iribuion aroun i mean, a efine b i variance. Ofen bo are combine in e mean quare error, ic i e variance plu e quare mean. Te mean of e ample error iribuion i picall referre o a e bia of e eimaion meo, o e mean quare error become variance plu quare bia.

19 A ig quali eimae ill be aociae i a ample error iribuion a a bia cloe o or equal o zero an lo variance. In i cae e can be ure a e oberve value of e eimae ill, i ig probabili, be cloe o e unnon FPP being eimae. Conequenl e focu on e bia an variance of e ample error iribuion a e e quali meaure of a ample eimaion meo. In e ne ecion e evelop epreion for ee quaniie, ogeer i relevan meo for eimaing em from e ample aa. In oing o e focu on one FPP a i of paricular inere in man urve ampling iuaion. Ti i e FPP efine b e oal of e value aen b a ingle Y-variable..3 Eimae relae o populaion oal Le U enoe e finie populaion of inere, an le U enoe e N uni maing up i populaion. For eac uni e aume a a Y-variable i efine, i e realie (bu unnon) value of i variable for e uni enoe b. Te oal of e N value of i Y-variable in e populaion ill be enoe. Folloing common pracice e o no iingui beeen a a realiaion (a i a number) an a e ranom variable a le o a realiaion. I ill be clear from e cone a paricular inerpreaion oul be place on i quani. Similarl, e ill no iingui beeen an eimae (a realie value) an an eimaor (e proceure a le o e realie value)..3. Te eign-bae approac Ti approac, ofen referre o a eign-bae eor, evaluae an eimae of in erm of e repeae ampling iribuion of i ample error. Ta i, a goo eimae for i efine a one for ic e aociae ample error i non o be a ra from a repeae ampling iribuion a a eier zero bia or bia a i approimael zero an a mall variance. A ill become clear belo, e uefulne of i approac epen on eer or no a ranom meo i non ample incluion probabiliie i emploe for ample elecion..3.. Sample incluion probabiliie In orer o generae i repeae ampling iribuion e nee o inrouce e concep of a ample incluion inicaor. Ti i a binar value ranom variable a ae e value if a uni i inclue in ample an i zero oerie. We enoe i b I in a follo. Clearl e iribuion of I epen on e proce ue o cooe e ample. Suppoe no a i proce i ranom in ome a. Ten e can pu π Pr p (I ) Pr(uni i inclue in ample given fie populaion value for Y an e auiliar variable X). Since e aume a ever uni in e populaion a a non-zero probabili of incluion in e ample, e mu ave π > 0 for all U. Noe a e o NOT aume a e I are inepenen ranom variable. Te properie of e oin iribuion of an ube of ee ranom variable ill epen on e acual ampling meo emploe. Te imple oin iribuion i of o incluion variable, I an I, ere. In i cae e pu π Pr p (I, I ) Pr(uni an are bo

20 inclue in ample given fie populaion value for Y an X). I i anar o refer o π a e incluion probabili for uni, an π a e oin incluion probabili for uni an..3.. Te Horviz-Tompon eimae Suppoe no a e value π are non for eac uni in e populaion. Ten, irrepecive of ic ample i acuall coen, e can efine an eimae of of e form HT π. Te noaion mean a e ummaion above i rerice o e ample uni, ile e ubcrip HT refer o e fac a i eimae a fir pu forar in Horviz & Tompon (95) Deign-bae eor for e Horviz-Tompon eimae I i raigforar o o a e repeae ampling iribuion of e ample error of e HTE (Horviz-Tompon eimae) a mean zero. An equivalen a of aing i i o a a HT i unbiae uner repeae ampling, or, more commonl, a i i eign unbiae a i, unbiae i repec o repeae ampling uner e probabili ampling eign. Te mean an variance of e repeae ampling iribuion of (e ample error efine b) HT are eail obaine. I u require one o noice a e onl ranom variable conribuing o i iribuion are e ample incluion variable I efine above. All oer quaniie (an in paricular e value of Y) are el fie a eir populaion value. Conequenl, ince E p (I ) π, E p ( ) HT E E p p U E U p π I ( I π π ) U U U 0. Ta i, e ample error iribuion of e HTE a zero bia uner repeae ampling. Noe a i proof i epenen on ever uni in e populaion aving a non-zero probabili of incluion in ample. Te eign variance of HT (a i e variance of e repeae ampling iribuion of e ample error efine b HT ) i obaine roug a ver imilar argumen. Since i coniere fie in i cae, i variance i given b 3

21 4 ( ) ( ). ) ( ), ( C ) ( V V V V U U U U U p U p U p J p HT p I I I I π π π π π π π π π π π π Wiou lo of generali e efine π π. Ten e above variance i ( ) ( ) U U HT p π π π π π V. Noe a i variance i a FPP. Conequenl e can ue e argumen a o HT i eign unbiae o obain an eimae of i variance a i alo eign unbiae. Ti i e o-calle HT eimae of variance ( ) ( ) HT HT p π π π π π π V Deign-bae eor for fie ample ize eign An imporan cla of ample eign ave fie ample ize. For uc eign e um of an realiaion of e N ample incluion inicaor equal a fie number n (e ample ize). I immeiael follo a for fie ample ize eign e um of e populaion value of π mu alo equal n. Furermore, en U U U n I n I I I I I π π ) ( ) (. Tee equaliie allo u o epre e eign variance of HT a lile ifferenl. Ta i, en a fie ample ize eign i ue i variance i ( ) ( ) U U HT p V π π π π π. A eign unbiae eimae of i variance i eail een o be ( ) HT SYG p V π π π π π π. Te upercrip SYG above an for Sen-Yae-Grun, e original eveloper of i paricular variance eimae (Yae & Grun, 953; Sen, 953).

22 Te HT variance eimae can ae negaive value en ample uni ave ig incluion probabiliie. Similarl, e SYG variance eimae can be negaive if π π < π for ome. Since in mo pracical cae i coniion oe no ol, e SYG eimae i uuall preferre for eimaing e eign variance of e HTE Approimaing econ orer incluion probabiliie An imporan pracical problem unerling bo variance eimae above i a e require e urve anal o no e oin incluion probabiliie π. In e cae of imple ranom ampling, or raifie ranom ampling, ee probabiliie are non. For eample, uner raifie ranom ampling π n ( n N ( N nng N N g ) ) if, are bo in raum ; if i in raum an i in raum g. For oer meo of ampling, oever, e oin incluion probabiliie are rarel non. In uc cae, one can approimae ee probabiliie o a, iin raa, e are a lea correc for imple ranom ampling. Ta i, e pu π N n ( n ) ( N ) π π en an are in e ame raum. Obvioul, en an are in ifferen raa e ave π π π. In e pecial cae of probabili proporional o ize (PPS) ampling Berger (998) a propoe an alernaive approimaion. Ti i bae on e folloing approimaion o e variance of e HTE (Hae, 964): ( ) V ~ π H ( N p HT ) ( π ) G ( π ) N U π ere an ( π ) π ( π ) U G( π ) ( π ) U ( π ). Berger variance eimae replace e populaion quaniie in Hae approimaion b eign-unbiae eimae, leaing o e variance eimae 5

23 V B p ( ) HT n ( π ) ( n ) ( π ) ( π ) G ( π ) π ere ( π ) ( ) π an ( π ) ( π ) G ( π ) U π. I oul be empaie a i variance eimaor i onl uiable for PPS eign. I can give erioul mileaing reul if ue i general unequal probabili eign. For eample, if ue i raifie ranom ampling i a a large poiive bia. Coniion for B V are e ou in Berger (998). applicabili of ( ) p HT.3..6 Problem i e eign-bae approac Te main reng of eign-bae eor i a i mae no aumpion abou e populaion value being ample. Hoever i i alo i eane, ince ere i noing in e approac o inicae o o mae efficien inference. In paricular, e HTE can be quie inefficien. Uner e eign-bae approac o ample urve inference, eign unbiaene i a e meaure of quali for a urve eimae. A ill be clear from e evelopmen above, i proper a noing o o i e acual value of e ample error of i eimae. I i a proper of e probabili ampling meo. On average, over repeae ampling from e fie finie populaion of Y-value acuall ou ere, i error i zero. Bu e ize of e acual error ma be far from zero. If e variance of e repeae ampling iribuion i alo mall, en i error ill be mall i ig probabili. Sanar probabili eor aure u a i ill be e cae provie e ample ize i large. Hoever, ere i lile o guie one on a large mean ere, ince e coniion require for i eor o ol epen on e (unnon) caraceriic of e populaion. Furermore, in man pracical iuaion ample ize are NOT large, an eign-unbiaene i of limie uefulne. Tee commen appl equall ell o a eign-unbiae eimae of e eign variance of an eimae. Wen a ample i no large e accurac of i eimae of variance (a i e ifference beeen i an e rue ampling variance of e eimae) uffer from e ame problem a e acual ample error ielf e canno a o mall (or o large) i acuall i. All e can a i a e proceure ue o calculae i eimae ill on average prouce an eimae a i e rig value. A furer problem relae o e ue of e eign variance a e meaure of e error of a paricular ample eimae. Ti quani i no e acual value of i error. In fac, e eign variance remain e ame irrepecive of e ize of i error. Ti invariance a 6

24 been criicie (Roall, 98). Furermore, e anar eimae of i eign variance (ic, ince e var from ample o ample, DO var i e acual error) ave been criicie a being mileaing. In paricular, in ome circumance ee variance eimae can be negaivel correlae i e acual error, leaing o mileaing quali aemen for e urve eimae. See Roall & Cumberlan (98). Bo e above problem (efficien eimae an meaningful variance eimae) can be reolve if one aop a moel-bae approac o ample urve inference. Hoever, i i no free of co. One en a o rel on e aequac of one moel for e uperpopulaion iribuion of e Y-variable of inere. Since all moel are, o a greaer or leer een, incorrec i mean a one oul aop robu moel-bae meo, a i meo a o no erioul loe efficienc uner moo eviaion from aumpion. Ti iue i aen up in more eail in Belo e evelop e baic eor unerlining e moelbae approac..3. Te ue of moel for eimaing a populaion oal A on above, e eign variance of e HTE epen on e acual populaion value of Y. Conequenl, iou ome a of moelling e iribuion of ee populaion Y- value, ere i lile one can a abou e properie of e HTE. Over e la 5 ear a conierable bo of eor a erefore evelope ic aemp o uilie nolege abou e probable iribuion of populaion value for Y in orer o improve eimaion of a FPP. Tpicall, i informaion i caracerie in erm of a ocaic moel for i iribuion. Tere are o baic a uc a moel can be ue. Te moel-aie approac eeniall ue i o improve eimaion of e FPP iin e eign-bae frameor. Ta i, e moel i ue o moivae an eimae i goo moel-bae properie. Hoever, i eimae i ill aee in erm of eirable eign-bae properie lie eign unbiaene an lo eign variance. Furermore, e e quali meaure of an eimae uner i approac remain i eimae eign variance. Te oer baic approac i full moel-bae. Here e rericion of eign unbiaene an lo eign variance are ipene i, being replace b moel unbiaene an lo moel variance. Belo e ecribe e baic of e moel-bae approac. Correponing evelopmen of e moel-aie approac i e ou in ecion Te uperpopulaion moel In orer o ecribe i approac, e inrouce e iea of a uperpopulaion moel. Ti i a moel for e oin iribuion of e N ranom variable Y, U oe realiaion correpon o e populaion Y-value, given e value of e auiliar variable X. Tpicall uc a moel pecifie e fir an econ orer momen of i oin iribuion raer an e complee iribuion. Tu e can rie 7

25 C E ( ) µ ( ; ω ) V ( ) σ ( ; ω ) (, ) 0 for ere µ an σ are pecifie funcion of oe value epen on ω, a picall unnon parameer. Noe a e aumpion a iinc populaion uni are uncorrelae given X ma eem rericive, bu i anar for urve of economic uni ere X can be quie informaive abou Y. In oueol urve X ma provie ver lile informaion abou Y, in ic cae i i anar o allo uni a group ogeer (for eample iniviual in oueol) o be correlae. See ecion.3..5 belo..3.. Te omogeneou raa moel Ti moel i iel ue in buine urve pracice. Here, e populaion i pli ino raa an i i aume a e mean an variance of e populaion Y-variable are e ame for all uni iin a raum, bu ifferen acro raa. In i cae X i a raum inicaor. Auming e raa are inee b,,, H, en for in raum e ave ( ; ω ) µ an σ ( ω ) σ µ beeen e raum mean an variance Te imple linear regreion moel ;. Noe a i moel oe no aume an relaionip Anoer commonl ue moel i ere i a meaure of e ize of e populaion uni, an i i reaonable o aume a linear relaionip beeen Y an X. Tpicall i linear relaionip i couple i eeroeaici in X, in e ene a e variabili in Y en o increae i increaing X. A pecificaion a allo for i beaviour for poiive γ value X i µ ( ; ω ) α β an ( ) σ ; ω ψ φ. In man economic populaion e regreion of Y on X goe roug e origin, an i moel reuce o e imple raio form efine b α ψ Te general linear regreion moel Bo e omogeneou raa moel an e imple linear regreion moel are pecial cae of a moel ere e auiliar informaion correponing o X conain a mi of raum ienifier an ize variable. We enoe i mulivariae auiliar variable b X. Ten T ( ω ) β µ. I i anar in i cae o epre e eeroeaici in Y in erm of a ; ingle auiliar variable Z, ic can be one of e auiliar ize variable in X, or ome poiive value funcion of e componen of i vecor (for eample a poer ranformaion lie γ above). In eier cae e pu σ ( ; ω ) σ z. I i imporan o noe a e pecificaion of X i quie general. In mo applicaion i vecor conain onl main effec, bu concepuall ere i noing o op i conaining an funcion (incluing ineracion erm) efine b e auiliar informaion on e ample frame. 8

26 .3..5 Te cluer moel A common feaure of e moel e ou above i a e aume iniviual populaion uni are uncorrelae, irrepecive of eir iance from oer populaion uni. Ta i, afer coniioning on e auiliar informaion in X, ere i no reaon o epec populaion uni a are coniguou in ome ene o be more alie i repec o eir value of Y an uni a are no coniguou. Anoer a of epreing i i a ee moel aume e oberve imilari in Y value for coniguou uni i compleel eplaine b eir imilar value of X. Wen e eplanaor poer of X i ea, a i e cae in mo uman populaion, i aumpion of lac of correlaion canno be uaine. In uc cae i i uual o epan e moel in.3.. o allo correlaion beeen coniguou uni. In paricular, a ierarcical rucure for e populaion i ofen aume, i iniviual groupe ogeer ino mall non-overlapping cluer (for eample oueol). All cluer are aume o be more or le imilar in ize, an eeniall imilar in erm of e range of Y value e conain. Hoever, iniviual from e ame cluer are aume o be more alie an iniviual from ifferen cluer. Tpicall i i moelle b an unobervable cluer effec variable ic a a iribuion acro e cluer maing up e populaion. Te effec of i variable i o inuce a iin cluer correlaion for Y. Since e focu of i repor i quali meaure for buine urve, an cluer pe moel are rarel ue o moel buine populaion, e ill no purue i iue an furer. See Roall (986) for furer icuion of moel-bae eimaion uner a cluer pecificaion Ignorable ampling An imporan aumpion a i picall mae a i age i a e oin iribuion of e ample value of Y can be euce from e aume uperpopulaion moel. In paricular, i i ofen aume a if uni i in ample, en e mean an variance of are e ame a pecifie b e moel. Ta i, e fac a a uni i elece in e ample a no impac on our uncerain abou e iribuion of poenial value aociae i i correponing Y- value. Ti i e o-calle ignorable ampling aumpion. I i aifie b an meo of probabili ampling a epen a mo on non populaion auiliar informaion. We all aume ignorable ampling in a follo, ince i i a i one in pracice. An inveigaion of non-ignorable ampling i e ou in Caper Bia, variance an mean quare error uner e moel-bae approac Uner e moel-bae approac e oal of e populaion value of Y i a ranom variable, o e problem of eimaing i FPP i acuall a preicion problem. An eimae of e populaion oal of Y i a funcion of e ample Y-value, eac one of ic i a realiaion of a ranom variable uner e aume uperpopulaion moel. Conequenl i alo e realiaion of a ranom variable. Te ample error i a preicion error uner i approac. Te moel bia of an eimae of i en e epece value of i ample 9