Chapter 4 Stratified Sampling

Transcription

1 Chapter 4 Stratfed Samplg A mportat obectve ay emato problem s to obta a emator of a populato parameter whch ca tae care of the salet features of the populato If the populato s homogeeous wth respect to the characterc uder udy, the the method of smple radom samplg wll yeld a homogeeous sample ad tur, the sample mea wll serve as a good emator of populato mea Thus, f the populato s homogeeous wth respect to the characterc uder udy, the the sample draw through smple radom samplg s expected to provde a represetatve sample Moreover, the varace of sample mea ot oly depeds o the sample sze ad samplg fracto but also o the populato varace I order to crease the precso of a emator, we eed to use a samplg scheme whch ca reduces the heterogeety the populato If the populato s heterogeeous wth respect to the characterc uder udy, the oe such samplg procedure s ratfed samplg The basc dea behd the ratfed samplg s to dvde the whole heterogeeous populato to smaller groups or subpopulatos, such that the samplg uts are homogeeous wth respect to the characterc uder udy wth the subpopulato ad heterogeeous wth respect to the characterc uder udy betwee/amog the subpopulatos Such subpopulatos are termed as rata Treat each subpopulato as separate populato ad draw a sample by SRS from each ratum [ote: Stratum s sgular ad rata s plural] Example: I order to fd the average heght of the udets a school of class to class, the heght vares a lot as the udets class are of age aroud 6 years ad udets class 0 are of age aroud 6 years So oe ca dvde all the udets to dfferet subpopulatos or rata such as Studets of class, ad 3: Stratum Studets of class 4, 5 ad 6: Stratum Studets of class 7, 8 ad 9: Stratum 3 Studets of class 0, ad : Stratum 4 Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

2 ow draw the samples by SRS from each of the rata,, 3 ad 4 All the draw samples combed together wll cotute the fal ratfed sample for further aalyss otatos: We use the followg symbols ad otatos: : Populato sze : umber of rata : umber of samplg uts th rata : umber of samplg uts to be draw from th ratum : Total sample sze Populato ( uts) Stratum uts Stratum uts Stratum uts Sample uts Sample uts Sample t Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

3 Procedure of ratfed samplg Dvde the populato of uts to rata Let the th ratum has,,,, umber of uts Strata are coructed such that they are o-overlappg ad homogeeous wth respect to the characterc uder udy such that Draw a sample of sze from th (,,, ) ratum usg SRS (preferably WOR) depedetly from each ratum All the samplg uts draw from each ratum wll cotute a ratfed sample of sze Dfferece betwee ratfed ad cluer samplg schemes I ratfed samplg, the rata are coructed such that they are wth homogeeous ad amog heterogeeous I cluer samplg, the cluers are coructed such that they are wth heterogeeous ad amog homogeeous [ote: We dscuss the cluer samplg later] Issue the emato of parameters ratfed samplg Dvde the populato of uts rata Let the th ratum has,,,, umber of uts ote that there are depedet samples draw through SRS of szes,,, from each of the rata So, oe ca have emators of a parameter based o the szes,,, respectvely Our tere s ot to have dfferet emators of the parameters but the ultmate goal s to have a sgle emator I ths case, a mportat ssue s how to combe the dfferet sample formato together to oe emator whch s good eough to provde the formato about the parameter We ow cosder the emato of populato mea ad populato varace from a ratfed sample Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 3

4 Emato of populato mea ad ts varace Let Y : characterc uder udy, y : value of th ut th ratum,,,,,,,, Y y y : populato mea of th ratum y : sample mea from th ratum Y Y wy : populato mea where w Emato of populato mea: Fr we dscuss the emato of populato mea ote that the populato mea s defed as the weghted arthmetc mea of ratum meas the case of ratfed samplg where the weghts are provded terms of rata szes Based o the expresso Y y y as a possble emator of Y Y, oe may choose the sample mea Sce the sample each ratum s draw by SRS, so E( y ) Y, thus E( y) E ( y) Y Y Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 4

5 ad y turs out to be a based emator of Y Based o ths, oe ca modfy y so as to obta a ubased emator of Y Cosder the ratum mea whch s defed as the weghted arthmetc mea of rata sample meas wth rata szes as weghts gve by y y ow E( y ) E( y ) Y Y Thus y s a ubased emator of Y Varace of y ( ) ( ) + (, ) ( ) Var y w Var y w w Cov y y Sce all the samples have bee draw depedetly from each of the rata by SRSWOR so Cov( y, y ) 0, Var( y) S where S ( Y Y ) Thus Var( y ) w S S w Observe that Var( y ) s small whe S s small Ths observato sugges how to coruct the rata If S s small for all,,,, the Var( y ) wll also be small That s why t was Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 5

6 metoed earler that the rata are to be coructed such that they are wth homogeeous, e, s small ad amog heterogeeous S For example, the uts geographcal proxmty wll ted to be more closer The cosumpto patter the households wll be smlar wth a lower come group housg socety ad wth a hgher come group housg socety whereas they wll dffer a lot betwee the two housg socetes based o come Emate of Varace Sce the samples have bee draw by SRSWOR, so Es ( ) S where s ( y y ) ad Var( y) s so Var( y ) w Var ( y ) w s ote: If SRSWR s used ead of SRSWOR for drawg the samples from each ratum, the ths case y E( y ) Y Var ( y ) wy σ Var( y ) w S w ws where σ ( y y ) Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 6

7 Advatages of ratfed samplg Data of ow precso may be requred for certa parts of the populato Ths ca be accomplshed wth a more careful vegato to few rata Example: I order to ow the drect mpact of he petrol prces, the populato ca be dvded to rata le lower come group, mddle come group ad hgher come group Obvously, the hgher come group s more affected tha the lower come group So more careful vegato ca be made the hgher come group rata Samplg problems may dffer dfferet parts of the populato Example: To udy the cosumpto patter of households, the people lvg houses, hotels, hosptals, prso etc are to be treated dfferetly 3 Admratve coveece ca be exercsed ratfed samplg Example: I tag a sample of vllages from a bg ate, t s more admratvely coveet to cosder the drcts as rata so that the admratve setup at drct level may be used for ths purpose Such admratve coveece ad the coveece orgazato of feld wor are mportat aspects atoal level surveys 4 Full cross-secto of populato ca be obtaed through ratfed samplg It may be possble SRS that some large part of the populato may rema urepreseted Stratfed samplg eables oe to draw a sample represetg dfferet segmets of the populato to ay desred extet The desred degree of represetato of some specfed parts of populato s also possble 5 Subatal ga the effcecy s acheved f the rata are formed tellgetly 6 I case of sewed populato, use of ratfcato s of mportace sce larger weght may have to be gve for the few extremely large uts whch tur reduces the samplg varablty 7 Whe emates are requred ot oly for the populato but also for the subpopulatos, the the ratfed samplg s helpful 8 Whe the samplg frame for subpopulatos s more easly avalable tha the samplg frame for whole populato, the ratfed samplg s helpful 9 If populato s large, the t s coveet to sample separately from the rata rather tha the etre populato 0 The populato mea or populato total ca be emated wth hgher precso by sutably provdg the weghts to the emates obtaed from each ratum Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 7

8 Allocato problem ad choce of sample szes s dfferet rata Queo: How to choose the sample szes,,, effectve way? There are two aspects of choosg the sample szes: () Mmze the co of survey for a specfed precso () Maxmze the precso for a gve co so that the avalable resources are used a ote: The sample sze caot be determed by mmzg both the co ad varablty smultaeously The co fucto s drectly proportoal to the sample sze whereas varablty s versely proportoal to the sample sze Based o dfferet deas, some allocato procedures are as follows: Equal allocato Choose the sample sze to be the same for all the rata Draw samples of equal sze from each rata Let be the sample sze ad be the umber of rata, the for all,,, Proportoal allocato For fxed, select such that t s proportoal to ratum sze or C where C s the coat of proportoalty or C C Thus C, e, Such allocato arses from the cosderatos le operatoal coveece Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 8

9 3 eyma or optmum allocato Ths allocato cosders the sze of rata as well as varablty S * C S where C* s the coat of proportoalty or or C * C S * C S * S S Thus S Ths allocato arses whe the Var ( y ) s mmzed subect to the corat (prespecfed) There are some lmtatos of the optmum allocato The owledge of S (,,, ) s eeded to ow If there are more tha oe charactercs, the they may lead to coflctg allocato Choce of sample sze based o co of survey ad varablty The co of survey depeds upo the ature of survey A smple choce of the co fucto s where C : C : 0 C 0 + C C C total co overhead co, eg, settg up of offce, trag people etc : co per ut the th ratum C : total co wth sample To fd multpler λ as uder ths co fucto, cosder the Lagraga fucto wth Lagraga Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 9

10 φ Var y + C C ( ) λ ( 0) + w S λ C + λ C λ C + terms depedet of Thus φ s mmum whe λ C or λ C for all How to determe λ? There are two ways to determe λ () Mmze varablty for fxed co () Mmze co for gve varablty We cosder both the cases () Let Mmze varablty for fxed co * C C 0 be the pre-specfed co whch s fxed So or C C C * 0 * 0 or λ * C0 λ C C C Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 0

11 Subtutg λ the expresso for C * * 0 C C, the optmum s obtaed as λ C The requred sample sze to emate Y such that the varace s mmum for gve co * * C C 0 s () Mmze co for gve varablty Let V V0 be the pre-specfed varace ow determe such that or or V V0 + λ C V0 + 0 V0 + or λ (after subtutg ) λ C C Thus the optmum s C C V 0 + So the requred sample sze to emate Y such that co C s mmum for a prespecfed varace V 0 s Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

12 Sample sze uder proportoal allocato for fxed co ad for fxed varace () If co C C0 s fxed the C0 C Uder proportoal allocato, w So C0 wc or C 0 wc Cw o Thus wc The requred sample sze to emate Y ths case s () If varace V 0 s fxed, the V0 or V0 + 0 w or V0 + or w V0 + or V + (usg w ) Ths s ow Bowley s allocato Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

13 Varaces uder dfferet allocatos ow we derve the varace of () Proportoal allocato Uder proportoal allocato ad Var( y) w S Var y S prop ( ) S () Optmum allocato Uder optmum allocato S S V y opt ( ) S S S S S y uder proportoal ad optmum allocatos Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 3

14 Comparso of varaces of sample mea uder SRS wth ratfed mea uder proportoal ad optmal allocato: (a) Proportoal allocato: VSRS ( y) S S Vp r op ( y ) I order to compare VSRS ( y) ad Vprop ( y ), fr we attempt to express Cosder ( ) S ( Y Y) ( Y Y) + ( Y Y) ( Y Y ) + ( Y Y) ( ) S + ( Y Y) + ( ) S S Y Y S as a fucto of S For smplfcato, we assume that Thus ad S S Y Y + ( ) s large eough to permt the approxmato - or S S + ( Y Y) (Premultply by o both sdes) Var ( Y ) V ( y ) + w ( Y Y ) SRS prop Sce w( Y Y) 0, Var ( y ) Var prop SRS ( y) Larger ga the dfferece s acheved whe Y dffers from Y more Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 4

15 (b) Optmum allocato opt ( ) V y Cosder Vprop ( y ) Vopt ( y ) where S S w( S S) Var ( y ) Var ( y ) 0 prop opt or Var ( y ) Var ( y ) opt prop Larger ga effcecy s acheved whe Combg the results (a) ad (b), we have Var ( y ) Var ( y ) Var ( y) opt prop SRS S dffer from S more Emate of varace ad cofdece tervals Uder SRSWOR, a ubased emate of S for the th ratum (,,,) s s ( y y ) I ratfed samplg, Var y w S ( ) Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 5

16 So a ubased emate of Var( y ) s Var( y ) w s ws ws ws ws The secod term ths expresso represets the reducto due to fte populato correcto The cofdece lmts of Y ca be obtaed as y ± t Var ( y ) assumg y s ormally drbuted ad Var ( y ) s well determed so that t ca be read from ormal drbuto tables If oly few degrees of freedom are provded by each ratum, the t values are obtaed from the table of udet s t-drbuto The drbuto of Var ( y ) s geerally complex A approxmate method of assgg a effectve umber of degrees of freedom ( e) to Var ( y ) s e gs 4 gs where g ( ) adm( ) ( ) e assumg y are ormally drbuted Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 6

17 Modfcato of optmal allocato Sometmes the optmal allocato, the sze of subsample exceeds the ratum sze I such a case, replace by ad recompute the re of ' s by the revsed allocato For example, f, > the tae the revsed ' s as ad ( ) ;,3,, provded for all,3,, Suppose revsed allocato, we fd that > the the revsed allocato would be ( ) ; 3, 4,, 3 provded < for all 3, 4,, We cotue ths process utl every < I such cases, the formula for mmum varace of where ( ) M Var( y ) * * * * deotes the summato over the rata whch sze the rata y eed to be modfed as ad * s the revsed total sample Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 7

18 Stratfed samplg for proportos If the characterc uder udy s qualtatve s ature, the ts values wll fall to oe of the two mutually exclusve complemetary classes C ad C Ideally, oly two rata are eeded whch all the uts ca be dvded depedg o whether they belog to C or ts complemet C Thus s dffcult to acheve practce So the rata are coructed such that the proporto C vares as much as possble amog rata Let A P : Proporto of uts C the th ratum a p : Proporto of uts C the sample from the th ratum A emate of populato proporto based o the ratfed samplg s p p whch s based o the dcator varable ad y p Here Y S th th whe ut belogs to the ratum s C 0 otherwse PQ where Q P Also Var y ( ) w S So Var( p ) ( ) PQ If the fte populato correcto ca be gored, the PQ Var( p ) w Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 8

19 If proportoal allocato s used for, the the varace of p s Var prop ad ts emate s PQ ( p ) w PQ pq Var prop ( p ) w The be choce of such that t mmzes the varace for fxed total sample sze s PQ PQ Thus PQ PQ Smlarly, the be choce of such that the varace s mmum for fxed co C C0 + C s PQ C PQ C Emato of the ga precso due to ratfcato A obvous queo crops up that what s the advatage of ratfyg a populato the sese that ead of usg SRS, the populato s dvded to varous rata? Ths s aswered by ematg the varace of emators of populato mea uder SRS (wthout ratfcato) ad ratfed samplg by evaluatg Var ( ) SRS y Var( y ) Var ( y ) Ths gves a dea about the ga effcecy due to ratfcato Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 9

20 Sce VarSRS ( y) S, so there s a eed to express based o a ratfed sample? Cosder ( ) S ( Y Y) ( Y Y ) + ( Y Y) ( Y Y) ( Y Y) + ( ) S ( Y Y) + + ( ) S wy Y S terms of S How to emate S I order to emate S, we eed to emates of oe S, Y ad Y We cosder ther emato oe by (I) For emate of S, we have Es ( ) S So ˆ s S (II) For emate of Y, we ow Var( y ) E( y ) [ E( y )] E( y ) Y or Y E( y ) Var ( y ) A ubased emate of Y s Y ˆ y Var( y) y s Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 0

21 (III) For the emato of Y, we ow Var( y ) E( y ) [ E( y )] E( y ) Y Y E( y ) Var( y ) So a emate of Y s Y ˆ y Var ( y ) y ws Subtutg these emates the expresso ( ) S as follows, the emate of S s obtaed as ( ) S ( ) S + wy Y Var( y ) w s ˆ ˆ ˆ ˆ as S ( ) S + w Y Y ( ) s w y s y ws + ( ) ( ) ( ) s w y y w w s + Thus Var ˆ SRS ( y) S ( ) ( ) s w ( y y ) w ( w ) s ( ) + ( ) ad Subtutg these expressos Var ( ) SRS y Var( y ), Var ( y ) the ga effcecy due to ratfcato ca be obtaed If ay other partcular allocato s used, the subtutg the approprate such ga ca be emated uder that allocato, Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

22 Iterpeetratg subsamplg Suppose a sample coss of two or more subsamples whch are draw accordg to the same samplg scheme The samples are such that each subsample yelds a emate of parameter Such subsamples are called terpeetratg subsamples The subsamples eed ot ecessarly be depedet The assumpto of depedet subsamples helps obtag a ubased emate of the varace of the compoe emator Ths s eve helpful f the sample desg s complcated ad the expresso for varace of the compoe emator s complex Let there be g depedet terpeetratg subsamples ad t, t,, t g be g ubased emators of parameter θ where t (,,, g) s based o th terpeetratg subsample The a ubased emator of θ s gve by g ˆ θ t t, say g The ad E( ˆ θ) Et ( ) θ g ˆ Var( θ ) Var( t ) ( t t ) gg ( ) ote that g E Var( t ) E ( t θ) g( t θ) gg ( ) g Var( t ) gvar( t ) gg ( ) g g Var t Var t gg ( ) ( ) ( ) ( ) If the drbuto of each emator t s symmetrc about θ, the the cofdece terval of θ ca be obtaed by Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page

23 g P M( t, t,, tg) < θ < Max( t, t,, tg) Implemetato of terpeetratg subsamples ratfed samplg Cosder the set up of ratfed samplg Suppose that each ratum provdes a depedet terpeetratg subsample So based o each ratum, there are L depedet terpeetratg subsamples draw accordg to the same samplg scheme Let Y ˆ ( tot ) be a ubased emator of the total of th ratum based o the th subsample,,,,l;,,, A ubased emator of the th ratum total s gve by Yˆ J Yˆ ( tot) ( tot) L ad a ubased emator of the varace of Y ˆ ( tot ) s gve by L Var( Yˆ ˆ ˆ ( tot) ) ( Y ( tot) Y( tot) ) LL ( ) Thus a ubased emator of populato total Y tot s L ˆ ˆ Y Y Yˆ tot ( tot) ( tot) Ad a ubased emator of ts varace s gve by Var ( Yˆ ) Var ( Yˆ ) tot ( tot) L Yˆ Yˆ LL ( ) ( ( tot) ( tot) ) Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 3

24 Po Stratfcatos Sometmes the ratum to whch a ut belogs may be ow after the feld survey oly For example, the age of persos, ther educatoal qualfcatos etc ca ot be ow advace I such cases, we adopt the po ratfcato procedure to crease the precso of the emates ote: Ths topc s to be read after the ext module o rato method of emato Sce t s related to the artfcato, so t s gve here I po ratfcato, draw a sample by smple radom samplg from the populato ad carry out the survey After the completo of survey, ratfy the samplg uts to crease the precso of the emates Assume that the ratum sze m s farly accurately ow Let : umber of samplg uts from th ratum,,,, m ote that m s a radom varable (ad that s why we are ot usg the symbol as earler) Assume s large eough or the ratfcato s such that the probablty that some m 0 s eglgbly small I case, m 0 for some rata, two or more rata ca be combed to mae the sample sze o-zero before evaluatg the fal emates A po ratfed emator of populato mea Y s ow y po y Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 4

25 E( ypo ) E E( y m, m,, m ) E Y Y Var( y po ) E Var( y po m, m,, m ) + Var E( y po m, m,, m ) E w S + Var( Y ) m w E S (Sce Var( Y ) 0) m To fd E, proceed as follows : m Cosder the emate of rato based o rato method of emato as y ˆ y Y, R R x x X We ow that ˆ RS X S XY ER ( ) R X Y X Let x th th f ut belogs to ratum 0 otherwse ad y for all,,, The RR, ˆ ad x S reduces to Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 5

26 Rˆ R y x Y X x S X X Sxy X Y XY 0 Usg these values ER ( ˆ) R, we have Thus ˆ ( )( ) ER ( ) R E ( ) ( )( ) E + ( ) ( ) + ( ) Replacg m place of, we obta ( ) E + m ( ) ow subtute ths the expresso of Var( y po ) as Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 6

27 Var( y ) w E S po m + ( ) + ( ) w w + ( ) w ( w ) + w S ( ) + ( ) w S ( ) ( ) Assumg V( y ) ( w) S po + V y + w S prop ( ) ( ) The secod term s the cotrbuto the varace of y due to m ' s ot beg proportoately po drbuted If S Sw, say for all, the the la term the expresso s ( ) ( ) (Sce ) w Sw S w w S w Var( y ) The crease the varace over Var ( y ) s small f the average sample sze reasoably large prop per ratum s Thus a po ratfcato wth a large sample produces a emator whch s almo as precse as a emator the ratfed samplg wth proportoal allocato Samplg Theory Chapter 4 Stratfed Samplg Shalabh, IIT Kapur Page 7