Chapter 10 Two Stage Sampling (Subsampling)

Transcription

1 Chapter 0 To tage amplg (usamplg) I cluster samplg, all the elemets the selected clusters are surveyed oreover, the effcecy cluster samplg depeds o sze of the cluster As the sze creases, the effcecy decreases It suggests that hgher precso ca e attaed y dstrutg a gve umer of elemets over a large umer of clusters ad the y 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 survey - 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 fshery - 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) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page

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 y : Value of the characterstc uder study for the j ut;,,, ; j,,, m th j secod stage uts of the th frst stage Y y : mea per d stage ut of th st j j stage uts the populato Y y y Y : mea per secod stage ut the populato j j y m y : mea per secod stage ut the th frst stage ut the sample m j j m y y y y : 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 opportuty 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 very closely, 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 amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page

3 A pctoral scheme of to stage samplg scheme s as follos: 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 y the pla amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 3

4 I case of to stage samplg, E( ˆ θ) E[ E ( ˆ θ)] 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, { ˆ θ } E( ˆ θ) E E E3( ) To calculate the varace, e proceed as follos: I case of to stage samplg, ˆ ˆ Var( θ) E( θ θ) EE ( ˆ θ θ) Cosder E( ˆ θ θ) E ˆ ˆ ( θ ) θe( θ) + θ { E ˆ ( θ} V ˆ ˆ ( θ) + θe( θ) + θ o average over frst stage selecto as ˆ θ θ ˆ ˆ ˆ θ + θ θ θ + θ EE ( ) E E ( ) E V( ) EE ( ) E( ) { ˆ } ( θ) θ ˆ ( θ) E E + E V Var( ˆ θ) V ˆ ˆ ( ) + ( ) E θ E V θ I case of three stage samplg, { } { } { } Var( ˆ θ) V E E ˆ ˆ ˆ 3( θ) + E V E3( θ) + E E V3( θ) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 4

5 Estmato of populato mea: Cosder y y as a estmator of the populato mea Y Bas: Cosder m [ ] E( y) E E ( y ) d st E E( ym ) (as stage s depedet o stage) E E( ym ) (as y s uased for Y due to RWOR) E Y Y m Y Thus y m s a uased estmator of the populato mea Varace [ ] Var( y) E V ( y ) + V E( y /) E V y + V E y / E V( y ) V E( y /) + E V Y m + E( ) + V( yc) m (here y s ased o cluster meas as cluster samplg) + + m m here Y Y c ( ) j ( ) j Y Y ( ) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 5

6 Estmate of varace A uased estmator of varace of y ca e otaed y replacg ad y ther uased estmators the expresso of varace of y Cosder a estmator of here as y Y s ( ) j j s m here s ( yj y ) m j o ( ) Es ( ) EE s so EE s E E s E E ( ) ( ) s s a uased estmator of (as RWOR s used) Cosder s y y ( ) as a estmator of Y Y ( ) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 6

7 o Thus Es ( ) E ( y y) ( ) E( s ) E y y E y E( y ) E E y Var( y) + { E( y) } E E( y ) ) + + Y m E { Var( y ( ) ) + E( y } + + Y m E + Y + + Y m m E + Y + m + Y m + Y Y m + + m + Y Y m + + m ( ) + Y Y m ( ) + Y Y m ( ) + ( ) m ( ) + ( ) m Es ( ) + m or E s s m ˆ m ˆ Var( y) ω + s + s s m m s + s m amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 7

8 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 y + It depeds o ad m,, ad m o the cost of survey 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 survey 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( y), e get k k Var( y) + C 0 k C 0 Ths varace s mootoc decreasg fucto of f he assumes maxmum value, e, C k 0 ˆ correspodg to m > 0 The varace s mmum If < 0 (e, traclass correlato s egatve for large ), the the varace s a mootoc C0 creasg fucto of, It reaches mmum he assumes the mmum value, e, ˆ k (e, o susamplg) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 8

9 (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 respectvely 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( y) + k + k + mk + m Whe > 0, the k 0 ( ) C Var y 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 + kmˆ Whe 0 the k C0 Var( y) + k + k + mk + m s mmum f m s the greatest attaale teger Hece ths case, he C0 C ˆ ˆ 0 k+ k ; m ad k+ k C0 k If C0 k+ k ; the mˆ ad ˆ k If s large, the ( ρ) ρ k k ρ ˆ m amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 9

10 Case : Whe varace s fxed o e fd the sample szes he varace s fxed, say as V 0 o V0 + m + m V0 + C km km + 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 m () samplg frst stage uts as cluster thout su-samplg We cosder oth the cases Case : amplg m elemets oe sgle stage The varace of sample mea ased o - m elemets selected y RWOR (oe stage) s gve y V( yr ) m - to stage samplg s gve y V( yt ) + m amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 0

11 The traclass correlato coeffcet s ( ) ρ ; ρ () ( ) ad usg the detty ( yj Y) ( yj Y ) + ( Y Y) j j j ( ) ( ) + ( ) () here Y y, Y y j j j j o e eed to fd ad from () ad () terms of From (), e have ρ + (3) usttutg t () gves ( ) ( ) + ( ) ρ ( ) + ( )( ) ρ( )( ) ( ) [ + ( )] ρ ( )( ) ( ) ( ) + ( ) ρ ( )( ) [ ( ) ρ] usttutg t (3) gves ( ) ( ) ( ) ( ) + ( ) ( ) ρ ( ) ( ) ( ) ( ) ρ ( ) ( )( ) (ρ) [ ρ] amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page

12 usttutg ad Var( y ) T m( ) m m V( yt ) ρ ( ) m + ( ) Whe susamplg rate m s small, ad, the V( yr ) m V( yt ) ρ m m + The relatve effcecy of the to stage relato to oe stage samplg of RWOR s Var( yt ) RE + ρ m Var( y ) R If ad fte populato correcto s gorale, the, the RE + ρ( 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( ycl ) m The varace of sample mea uder the to stage samplg s Var( yt ) + m o Var( y ) exceedes Var( y ) y cl m hch s approxmately T m ρ for large ad > 0 ( ) here + ( ) ( ρ) [ ρ ] amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page

13 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 employed at each stage Let y : value of j th j secod stage ut of the th frst stage ut m m 0 0 : 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 y( m) yj m j Y yj j Y y Y u y Y Y j j uy amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 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) amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 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 y y ( m) E( y ) E y( m) E E ( y( m) ) E Y m Y Y Y o y s a ased estmator of Y ad ts as s gve y Bas ( y ) E( y ) Y Y [ce a sample of sze s selected out of uts y RWOR] Y Y ( )( Y Y) Ths as ca e estmated y Y Bas( y ) ( m)( y( m) y ) ( ) hch ca e see as follos: E Bas( y ) E E {( m)( y( m) y ) / } E ( m)( Y y) ( )( Y Y) Y Y here y Y amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 5

16 A uased estmator of the populato mea Y s thus otaed as y + ( m)( y y ) ( m) ote that the as arses due to the equalty of szes of the frst stage uts ad proalty of selecto of secod stage uts vares from oe frst stage to aother Varace: Var( y) Var E( y ) + E Var ( y ) Var y E Var( y ( m) ) + + E m + m ( ) here Y Y y Y j j ( ) The E ca e otaed as [ ] E( y ) Var( y ) + Bas( y ) Estmato of varace: Cosder mea square etee cluster meas the sample s y y ( ) ( ) m It ca e sho that Es ( ) + m Also s ( y y ) m j ( m) m j E( s ) ( y Y) j j o E s m m amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 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 y replacg ad y ther uased estmators as Var( y ˆ ˆ ) + m Estmato ased o frst stage ut totals: here Bas Y ˆ u y y uy ( m) ( m) E( y ) E uy ( m) E ue ( y( m) ) E uy uy Y Thus y s a uased estmator of Y Varace: Var( y) Var E( y ) + E Var( y ) Var uy E u ( Var y( m) ) + + u m amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 7

18 here ( y Y) j j ( uy Y) j 3 Estmator ased o rato estmator: Y ˆ y y u y uy ( m) ( m) u here u, u u Ths estmator ca e see as f arsg y the rato method of estmato as follos: Let y uy ( m) x,,,, e the values of study varale ad auxlary varale referece to the rato method of estmato The y y y x x u X X The correspodg rato estmator of Y s ˆ y y Y R X y x u o the as ad mea squared error of y ca e otaed drectly 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 amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 8

19 ˆ Bas ( yr) Y ( Cx ρcxcy) Var( x) Cov( x, y) Y X XY ˆ E ( YR ) Var( y) + R Var( x) RCov( x, y) Y here R X Bas: The as of y up to secod order of approxmato s Var( x) Cov( x, y) Bas( y ) Y X XY here x s the mea of auxlary varale smlar to y as x x ( m) o e fd Cov( x, y ) Cov( x, y ) Cov E ux ( m), u y( m) E Cov ux ( m), u y( m) + Cov ue( x ( m) ), ue( y( m) ) + E u ( Cov x ( m), y( m) ) Cov u X, uy E u + m xy + u m xy xy here ( u X X)( uy Y) xy ( x X )( y Y) xy j j j mlarly, Var( x ) ca e otaed y replacg x place of y Cov( x, y ) as Var( x ) u x + x m here ( ux X) x ( x X ) x j amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 9

20 usttutg Cov( x, y ) ad Var( x ) Bas( y ), e ota the approxmate as as x xy x xy Bas( y) Y + u X XY m X XY ea squared error Thus Also E( y ) Var( y ) R Cov( x, y ) + R Var( x ) Var( y ) u y + y m Var( x ) u x + x m Cov( x, y ) xy + u here uy Y y ( ) y Y R y ( j ) j Y Y X m E( y ) ( y Rxy + R x ) + u ( y Rxy+ R x ) m xy E( y ) u ( Y R X ) + u ( y R xy + R x ) m Estmate of varace Cosder s uy y ux x ( ( ) )( ( ) ) xy m m s x x y y ( ( ))( ( )) xy j m j m m j amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page 0

21 It ca e sho that Thus Also E( s ) u Es ( ) xy xy + xy m xy xy o E u sxy u xy m m ˆ s u s xy xy xy m ˆ s u s x x x m s u s ˆ y y y m E u sx u x m m E u sy u y m m A cosstet estmator of E of y ca e otaed y susttutg the uased estmators of respectve statstcs E( y ) as E( y ) ( sy r sxy + r sx ) + + m ( y( m) r x ( m) ) ( ) u sy rsxy r sx y + u ( sy rsxy+ r sx ) here r m x amplg Theory Chapter 0 To tage amplg halah, IIT Kapur Page