Chapter 10 Two Stage Sampling (Subsampling)

Transcription

1 Chapter 0 To tage amplg (usamplg) I cluster samplg, all the elemets the selected clusters are surveed oreover, the effcec cluster samplg depeds o sze of the cluster As the sze creases, the effcec decreases It suggests that hgher precso ca e attaed dstrutg a gve umer of elemets over a large umer of clusters ad the takg a small umer of clusters ad eumeratg all elemets th them Ths s acheved susamplg I susamplg - dvde the populato to clusters - elect a sample of clusters [frst stage} - From each of the selected cluster, select a sample of specfed umer of elemets [secod stage] The clusters hch form the uts of samplg at the frst stage are called the frst stage uts ad the uts or group of uts th clusters hch form the ut of clusters are called the secod stage uts or suuts The procedure s geeralzed to three or more stages ad s the termed as multstage samplg For example, a crop surve - vllages are the frst stage uts, - felds th the vllages are the secod stage uts ad - plots th the felds are the thrd stage uts I aother example, to ota a sample of fshes from a commercal fsher - frst take a sample of oats ad - the take a sample of fshes from each selected oat To stage samplg th equal frst stage uts: Assume that - populato cossts of elemets - elemets are grouped to frst stage uts of secod stage uts each, (e, clusters, each cluster s of sze ) - ample of frst stage uts s selected (e, choose clusters)

2 - ample of m secod stage uts s selected from each selected frst stage ut (e, choose m uts from each cluster) - Uts at each stage are selected th RWOR samplg s a specal case of to stage samplg the sese that from a populato of clusters of equal sze m, a sample of clusters are chose If further m, e get RWOR If, e have the case of stratfed samplg : Value of the characterstc uder stud for the j ut;,,, ; j,,, m th j secod stage uts of the th frst stage Y j : mea per d stage ut of th st j stage uts the populato Y Y : mea per secod stage ut the populato j j m : mea per secod stage ut the th frst stage ut the sample m j j m : mea per secod stage the sample m j m j Advatages: The prcple advatage of to stage samplg s that t s more flexle tha the oe stage samplg It reduces to oe stage samplg he m ut uless ths s the est choce of m, e have the opportut of takg some smaller value that appears more effcet As usual, ths choce reduces to a alace etee statstcal precso ad cost Whe uts of the frst stage agree ver closel, the cosderato of precso suggests a small value of m O the other had, t s sometmes as cheap to measure the hole of a ut as to a sample For example, he the ut s a household ad a sgle respodet ca gve as accurate data as all the memers of the household A pctoral scheme of to stage samplg scheme s as follos:

3 Populato ( uts) uts uts clusters uts Populato clusters (large umer) uts uts clusters uts Frst stage sample clusters (small umer) m uts m uts m uts m uts ecod stage sample m uts clusters (large umer of elemets from each cluster) ote: The expectatos uder to stage samplg scheme deped o the stages For example, the expectato at secod stage ut ll e depedet o frst stage ut the sese that secod stage ut ll e the sample provded t as selected the frst stage To calculate the average - Frst average the estmator over all the secod stage selectos that ca e dra from a fxed set of uts that the pla selects - The average over all the possle selectos of uts the pla 3

4 I case of to stage samplg, E( ˆ ) E[ E ( ˆ )] average average average over over over all d all possle stage all st stage selectos from samples samples a fxed set of uts I case of three stage samplg, E( ˆ ) E ˆ E E3( ) To calculate the varace, e proceed as follos: I case of to stage samplg, ˆ ˆ Var( ) E( ) EE ( ˆ ) Cosder E( ˆ ) E ˆ ˆ ( ) E( ) E ˆ ( V ˆ ˆ ( ) E( ) o average over frst stage selecto as ˆ ˆ ˆ ˆ EE ( ) E E ( ) E V( ) EE ( ) E( ) ( ˆ ) ˆ ( ) E E E E V Var( ˆ ) V ˆ ˆ E( ) E V( ) I case of three stage samplg, Var( ˆ ) V E E ˆ ˆ ˆ 3( ) E V E3( ) E E V3( ) 4

5 Estmato of populato mea: Cosder as a estmator of the populato mea Y Bas: Cosder m E( ) E E ( ) d st EE( m ) (as stage s depedet o stage) EE( m ) (as s uased for Y due to RWOR) = E Y Y m Y Thus m s a uased estmator of the populato mea Varace Var( ) EV ( ) V E( / ) EV VE / E V( ) V E( / ) E V Y m E( ) V( c) m (here s ased o cluster meas as cluster samplg) m m here Y Y c j ( ) j Y Y ( ) 5

6 Estmate of varace A uased estmator of varace of ca e otaed replacg ad ther uased estmators the expresso of varace of Cosder a estmator of here as Y s j j s m here s ( j ) m j o Es ( ) EE s EE s ( ) ( ) E E s E E (as RWOR s used) so s s a uased estmator of Cosder s ( ) as a estmator of Y Y ( ) 6

7 o Es ( ) E ( ) ( ) E( s ) E E E( ) EE Var( ) E( ) E E( ) ) Y m E Var( ) E( Y m E Y Y m m E Y m m Y Y Y m m Y Y m m ( ) Y Y m ( ) Y Y m ( ) ( ) m ( ) ( ) m Es ( ) m or Es s m 7

8 Thus ˆ m ˆ Var( ) s s s m m s s m Allocato of sample to the to stages: Equal frst stage uts: The varace of sample mea the case of to stage samplg s ( ) m Var It depeds o ad m,, ad m o the cost of surve of uts the to stage sample depeds o Case Whe cost s fxed We fd the values of ad m so that the varace s mmum for gve cost (I) Whe cost fucto s C = km Let the cost of surve e proportoal to sample sze as C km here C s the total cost ad k s costat C Whe cost s fxed as C C0 usttutg m 0 Var( ), e get k k Var( ) C 0 k C 0 Ths varace s mootoc decreasg fucto of f 0 The varace s mmum he assumes maxmum value, e, C k 0 ˆ correspodg to m 8

9 If 0 (e, traclass correlato s egatve for large ), the the varace s a mootoc creasg fucto of, It reaches mmum he assumes the mmum value, e, C0 ˆ k (e, o susamplg) (II) Whe cost fucto s C k km Let cost C e fxed as C0 k km here k ad k are postve costats The terms k ad k deote the costs of per ut oservatos frst ad secod stages respectvel mze the varace of sample mea uder the to stage th respect to m suject to the restrcto C0 k km We have k C0 Var( ) k k mk m Whe 0, the k 0 ( ) C Var k k mk m hch s mmum he the secod term of rght had sde s zero o e ota mˆ k k The optmum follos from C0 k km as ˆ k C0 k mˆ Whe 0the k C0 Var( ) k k mk m s mmum f m s the greatest attaale teger Hece ths case, he C k k mˆ ˆ 0 C 0 ; ad k k C0 k If C0 kk ; the mˆ ad ˆ k 9

10 If s large, the ( ) k k ˆ m Case : Whe varace s fxed o e fd the sample szes he varace s fxed, sa as V 0 o V0 m m V0 C kmkm V k 0 V0 If 0, C attas mmum he m assumes the smallest tegral value, e, If 0, C attas mmum he mˆ Comparso of to stage samplg th oe stage samplg Oe stage samplg procedures are comparale th to stage samplg procedures he ether () samplg m elemets oe sgle stage or () samplg m frst stage uts as cluster thout su-samplg We cosder oth the cases 0

11 Case : amplg m elemets oe sgle stage The varace of sample mea ased o - m elemets selected RWOR (oe stage) s gve V( R ) m - to stage samplg s gve V( ) m T The traclass correlato coeffcet s ( ) ; () ( ) ad usg the dett ( j Y) ( j Y ) ( Y Y) j j j ( ) ( ) ( ) () here Y, Y j j j j o e eed to fd ad from () ad () terms of From (), e have (3) usttutg t () gves ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) [ ( )] ( )( ) ( ) ( ) ( ) ( ) ( )( ) usttutg t (3) gves

12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) () usttutg ad Var( ) T m( ) m m V( T ) ( ) m ( ) Whe susamplg rate m s small, ad, the V( R ) m V( T ) m m The relatve effcec of the to stage relato to oe stage samplg of RWOR s Var( T ) RE m Var( ) R If ad fte populato correcto s gorale, the, the RE ( m ) Case : Comparso th cluster samplg uppose a radom sample of m clusters, thout further susamplg s selected The varace of the sample mea of equvalet m / clusters s Var( cl ) m The varace of sample mea uder the to stage samplg s Var( T ) m o Var( ) exceedes Var( ) cl T m

13 hch s approxmatel m for large ad 0 ( ) ( ) here ( ) o smaller the m/, larger the reducto the varace of to stage sample over a cluster sample Whe 0 the the susamplg ll lead to loss precso To stage samplg th uequal frst stage uts: Cosder to stage samplg he the frst stage uts are of uequal sze ad RWOR s emploed at each stage Let : value of j m m 0 0 th j secod stage ut of the th frst stage ut : umer of secod stage uts th frst stage uts (,,, ) : : total umer of secod stage uts the populato umer of secod stage uts to e selected from th frst stage ut, f t s the sample m : total umer of secod stage uts the sample ( m) m Y j Y Y u j j m j j Y Y j uy 3

14 The pctoral scheme of to stage samplg th uequal frst stage uts case s as follos: Populato ( uts) uts uts uts Populato clusters clusters uts uts clusters uts Frst stage sample clusters (small) m uts m uts m uts ecod stage sample clusters (small) 4

15 o e cosder dfferet estmators for the estmato of populato mea Estmator ased o the frst stage ut meas the sample: Bas: ˆ Y ( m) E( ) E ( m) E E ( ) ( m) E Y m Y Y Y o s a ased estmator of Y ad ts as s gve [ce a sample of sze s selected out of uts RWOR] Bas ( ) E( ) Y Y Y Y Y ( )( Y Y) Ths as ca e estmated Bas( ) ( m)( ( m) ) ( ) hch ca e see as follos: EBas( ) E E ( m)( ( m) )/ E ( m)( Y ) ( )( Y Y) Y Y here Y 5

16 A uased estmator of the populato mea Y s thus otaed as ( m)( ) ( m) ote that the as arses due to the equalt of szes of the frst stage uts ad proalt of selecto of secod stage uts vares from oe frst stage to aother Varace: Var( ) Var E ( ) E Var( ) Var E Var( ( m) ) here Y Y E m m Y j j The E ca e otaed as E( ) Var( ) Bas( ) Estmato of varace: Cosder mea square etee cluster meas the sample s ( ) m It ca e sho that Es ( ) m Also s ( ) m j ( m) m j E( s ) ( Y) j j o E s m m 6

17 Thus Es ( ) E s m ad a uased estmator of s s s ˆ m o a estmator of the varace ca e otaed replacg ad ther uased estmators as Var( ˆ ˆ ) m Estmato ased o frst stage ut totals: here Y ˆ u u ( m) ( m) Bas E( ) E u( m) E ue ( ( m) ) E uy uy Y Thus s a uased estmator of Y 7

18 Varace: Var( ) Var E( ) E Var( ) Var uy E u ( Var ( m) ) here ( Y) u m j j ( uy Y) j 3 Estmator ased o rato estmator: Y ˆ u u ( m) ( m) u here u, u u Ths estmator ca e see as f arsg the rato method of estmato as follos: Let u ( m) x,,,, e the values of stud varale ad auxlar varale referece to the rato method of estmato The x x u X X 8

19 The correspodg rato estmator of Y s ˆ Y R X x u o the as ad mea squared error of ca e otaed drectl from the results of rato estmator Recall that rato method of estmato, the as ad E of the rato estmator upto secod order of approxmato s ˆ Bas ( R) Y ( Cx CxC) Var( x) Cov( x, ) Y X XY ˆ E ( YR ) Var( ) R Var( x) RCov( x, ) Y here R X Bas: The as of up to secod order of approxmato s Var( x) Cov( x, ) Bas( ) Y X XY here x s the mea of auxlar varale smlar to as x x( m) o e fd Cov( x, ) Cov( x, ) Cov E ux ( m), u ( m) E Cov ux ( m), u ( m) Cov ue( x ( m) ), ue( ( m) ) E u ( Cov x ( m), ( m) ) Cov ux, uy E u m x u m x x here ( u X X)( uy Y) x ( x X )( Y) x j j j 9

20 mlarl, Var( x ) ca e otaed replacg x place of Cov( x, ) as Var( x ) u x x m here ( u X X) x ( x X ) x j usttutg Cov( x, ) ad Var( x ) Bas( ), e ota the approxmate as as x x x x Bas( ) Y u X XY m X XY ea squared error Thus Also E( ) Var( ) R Cov( x, ) R Var( x ) Var( ) u m Var( x ) u x x m Cov( x, ) x u here uy Y ( ) Y R ( j ) j Y X Y m E( ) R x R x u R x R x m x E( ) u Y R X u R x R x m 0

21 Estmate of varace Cosder s u u x x ( ) ( ) x m m s x x ( ) ( ) x j m j m m j It ca e sho that Thus Also Es u Es ( ) x x x m x x o E u sx u x m m ˆ s u s x x x m ˆ s u s x x x m s u s ˆ m E u sx u x m m E u s u m m A cosstet estmator of E of ca e otaed susttutg the uased estmators of respectve statstcs E( ) as E( ) s r sx r sx m ( m) r x( m) u s r sx r sx u s r sx r sx here r m x