Deprmen of ppled Sscs Johnnes Kepler Unversy Lnz IFS Reserch Pper Seres 04-67 Smplfed rnce Esmon for Three-Sge Rndom Smplng ndres Quember Ocober 04
Smplfed rnce Esmon for Three-Sge Rndom Smplng ndres Quember IFS - Deprmen of ppled Sscs Johnnes Kepler Unversy (JKU) Lnz, usr I Inroducon o he Problem When populon U under sudy s proned no C clusers (=,,C) s prmry smplng uns (P), ech cluser self s proned no M subclusers j (j=,,m ) s secondry smplng uns (S), nd ech subcluser j conns N j elemens (=,,N j ) s erry smplng uns (T), he ol of sudy vrble y (for nsnce, cern pr of coss per un) cn be wren s j = C M N y = j= = j. () In (), y j denoes he vlue of y for he -h elemen n he j-h subcluser of he -h cluser. Here nd n he followng, he bsc lerure used s Särndl e l. (99), rdlly nd Tllé (006), Lohr (00), nd Quember (04). If proporon P of nd he ol of noher vrble (for nsnce, he overll coss of un) j = C M N = j= = j, s of neres, hen P s gven by P =. For he prl coss y j per un, y j = q j j pples. To esme ol of vrble y unbsedly, he generl Horvz-Thompson (HT) esmor s gven by n y HT = = wh, he frs-order smple ncluson probbles of elemens (=,,n). Is vrnce s gven by N N y yl ( HT ) = = l= l l Ths pper s produc of he conrc projec wh he sme nme (vlble on [0/6/04]: hp://www.ju./conen/e63/e6099/e6086/e7379/?vew=prod&fed=98&fp_d=355)
wh he covrnce of he smple ncluson ndcors I nd I l : l = l l. Theren, l denoes he second-order smple ncluson probbles of wo elemens nd l of he populon. Ths heorecl vrnce s esmed unbsedly by n n l y yl ( HT ) = = l= l l (see, for nsnce, Quember 04, p. 4ff, or Särndl e l., p. 4ff). Hence, he proporon P s esmed by HT P =. Is vrnce s gven by ( P) = ( ) HT, whch s esmed unbsedly by ( P) = ( ) HT. II Three-sge elemen smplng In he cse of hree-sge smplng scheme n populon U proned no clusers nd subclusers s descrbed he begnnng of Secon I, probbly smple of sze c s seleced he frs sge wh cluser ncluson probbles I (=,,C). he nex sge, probbly smple of m subclusers j s drwn whn ech of he c smple clusers wh condonl subcluser ncluson probbles j (j=,,m ). he hrd sge, whn ech of he smpled subclusers j of cluser of he second sge, probbly smple of sze n j s drwn wh condonl ncluson probbles j for ll elemens (=,,N j ). Wh hese erms, he HT esmor of he hree-sge process (3s) s gven by c m n y j j 3s == j=. = j For proporonl o sze whou replcemen smplng (PS), he ncluson probbles he dfferen sges e on he vlues x c =, C x = he frs smplng sge wh uxlry sze vrble x, u j m =, j M u j= j he second smplng sge wh uxlry sze vrble u, nd z nj =, j N j z = he hrd smplng sge wh uxlry sze vrble z. 3
In such hree-sge probbly smple wh frs-order elemen ncluson probbles =, j j j he Horvz-Thompson (HT) esmor s defned n he prevous secon yelds c m n y j j c m n y j j c m HT, j 3s = = j= = = = j= = = =. j= j j j j Hence, hs HT esmor cn be wren s HT, j HT, c HT, 3s = () = wh HT,, he HT esmor of, he ol of y n cluser. The HT esmor HT,j esmes unbsedly he ol of y n he j-h subcluser of he -h cluser. For x M Nj = j= = j, he sum of he coss of vrble n he -h cluser, u N j j = = j, he sum of he coss whn he j-h subcluser of he -h cluser, z =, j he coss of he -h elemen of he j-h subcluser whn he -h cluser, j = n j j, N j nd = j Nj = j m j M Nj j= = j = M Nj M Nj j= = j j= = = c = c C M Nj = j= = j pples. Hence, for 3s, c m nj 3s = y = j= = j c m nj j q pples. j For he esmon of P, 3s P = s used. furher mprovemen wh respec o he esmon of P my be cheved by ro esmor 4 j
wh P r = 3s = c m n j = j= = c m n. j The vrnce of he HT esmor 3s ccordng o () n hree-sge probbly smple s wren by 3s = P + S + T. (3) Obvously, he vrnce s proned no hree componens reflecng he hree sges of smplng s hree dfferen sources of vron of 3s. In (3), C C ' P = = ' = ' ' wh, he covrnce of he smple ncluson probbly of clusers nd. denoes he ol of y n cluser. Ths s he vron wh respec o smplng he frs sge of he process. Furhermore, S, he vron of he esmor due o he second smplng sge, s gven by C S =. = The vrnce denoes he vrnce of m j wh j= j, he ol of y n subcluser j of j cluser (see Formul (4.4.9) n Särndl e l. 99, p.48). Evenully, he hrd-sge conrbuon o he overll vron of 3s s gven by T = M j= C j. = j Theren, j denoes he vrnce of he HT esmor HT,j ccordng o (4.4.8) n bd., p.48. Ths vrnce componen complees he clculon of 3s. I s hs vrnce h hs o be esmed, when he resuls of smple survey re o be presened n form of n pproxme confdence. I s good prcce n he reporng of survey resuls o supply he pon esmes wh her esmed sndrd errors, h s he squre roo of he esmed vrnces (bd., p. 50). rnce (3) s unbsedly esmed by 3s c c ' HT, HT, ' c = = + ' = = ' ' P (4) wh 5
nd m m jj ' HT, j HT, j ' m j = j= + j ' = j= jj ' j j ' j y y. n ' ' = n j j j j = ' = ' j j ' j From hese formule, he sscl properes of wo-sge smplng process cn mmedely be derved (see, for nsnce, Quember 04, ch. 6). The clculon of vrnce esme ccordng o (4) my be hrd. In prculr, he clculon of he second-order ncluson probbles of selecon uns he dfferen sges of he smplng process cn be cumbersome or even mpossble for cern smplng procedures ppled whn he hree sges of smplng. In prculr, hs pples for PS smplng. One possbly o cope wh hs problem s he esmon of hese probbles (cf. Berger 004). Bu, ng no ccoun he heorecl nd prccl effor of hs pproch n hree-sge smplng, smpler vrnce expresson hn (4) hs o be consdered. III Four Opons for Smplfed rnce Esmon III. Opon I The smplfed opon I, pplcble s n esmor of he vrnce of 3s, uses only he frs erm P of he vrnce esmor (4): c c ' HT, HT, ' I = = ' = ' ' (5) Ths mens h only he covrnce of he smple ncluson ndcor on cluser level nd he cluser second-order ncluson probbles re needed. In fc, P overesmes P, bu does no cover ll oher componens of 3s. Ths mens h P provdes negvely bsed esmor of 3s. Bu, experence shows h n mny cses he moun of underesmon s smll, especlly, when he s re smll. compenson of he negve bs usng subsmplng from he smples fer he frs sge ws dscussed by Srnh nd Hdroglou (980). Bu, lso hs smplfed bsed vrnce esmor needs he second-order ncluson probbles he cluser level o be clculed, whch s cumbersome, for exmple, for PS smplng. For fxed sze frs-sge probbly smple, P cn be wren s I = c c ' HT, HT, ' = ' = ' ' (6) 6
(cf. Särndl e l. 99, p.53). III. Opon II noher opon (II) of smplfed vrnce esmor for whou replcemen smplng schemes s delvered by dpng he vrnce esmor h would hve been obned when he clusers would hve been seleced by wh-replcemen smplng desgn. Usully, hs wll resul n n overesmon of he rue vrnce when he smplng s cully done whou replcemen. In mul-sge smplng desgn, he specfc vrnce esmor s gven by II c HT, = = 3s c ( c ) p (7) wh p =, he probbly for cluser o be seleced n he nex sep of he whreplcemen selecon process (cf. Särndl e l. 99, p.54). Ths expresson ncorpores c lso he vrnce due o he nd nd 3 rd sge of smplng by he vrnce of he weghed HT esmors HT, of he cluser ols wh weghs p. Theren, HT, s gven by m n y j j HT, = j= = s presened n Secon. j j Sysemc probbly proporonl o sze smplng whou replcemen followng from rndomly ordered populon s n exmple of smple selecon mehod, for whch II overesmes 3s The resul s conservve confdence nervl h cn esly be clculed becuse no second-order ncluson probbles from ny of he sges re needed. For smll smplng frcon of clusers, c/c, he dfference beween 3s, nd II wll be neglgble. Särndl e l. (99) delver n exmple for he cul clculon of II (bd., p.5f). Becuse of he wh-replcemen smplng, s possble o ge dfferen subsmples from he sme cluser, nd II cpures boh prs of he vrnce 3s : he one due o he selecon of he clusers nd he pr of 3s rsng from he esmon of he cluser ols he followng sges. III.3 Opon III hrd opon III for he smplfed esmon of he vrnce of 3s uses n esmon of he desgn effec of he smplng desgn defned s 3s deff =, SI where SI s he vrnce of he HT esmor of n smple rndom smplng whou replcemen (SI). bsed esmor of deff s gven by 7
deff =, 3s SI he ro of wo vrnce esmes. Hence, bsed esmor III of 3s cn be defned by III = deff SI. (8) For equl numbers n of elemens observed whn he smple clusers, SI smplng he dfferen sges, nd lrge C, he desgn effec s esmed by deff + ρ ( n ) wh ρ, he esmed nr-clss correlon coeffcen mesurng he homogeney of uns whn he sme clusers (cf., for nsnce, rdlly nd Tllé 006, p.6). For lrge populon sze N compred o he number of clusers C n he populon, hs mesure hs rnge from zero o one. I reches he vlue one for complee homogeney whn he clusers, whch s he wors cse of smplng wh clusers wh respec o he vrnce of he HT esmor (cf. Särndl e l. 99, p.3). For ρ 0, menng h ech cluser hs he sme heerogeney wh respec o he sudy vrble, he desgn effec pproxmely equls one nd he vrnce of he hree-sge process cn be esmed by he SI vrnce formul. Neverheless, one needs no only n esme of deff bu lso n esme of he vrnce of he HT esmor wh SI smplng. Usng d of he hree-sge smple o esme S, he vrnce of y n he populon, lhough he cul smplng ws no SI elemen smplng, delvers bsed esme of he rue S. Hence, SI wll hve bs of unnown exen. To e lso ccoun of possble unequl ncluson probbles (le n PS smplng), n esmor of he overll desgn effec s clculed by he desgn effec deffc due o cluserng wh clusers of unequl smple szes n nd desgn effec due o he unequl ncluson probbles deffp. Ksh (987) descrbed n esmor of deffc wh he men vlue n of he whn-cluser smple szes, whch subsues he equl smple szes n whn clusers n he formul bove by deffc + ρ ( n ) (cf., for nsnce, Gbler e l. 999, or Gnnnger e l. 007). The pr of he desgn effec wh respec o unequl ncluson probbles s esmed by = deffp = n ( L w n = ) L w n wh w, he unque desgn weghs of he weghng clss of L weghng clsses (see lso Gbler e l. 999). Then, he vrnce of he hree-sge HT esmor of s esmed by III = deffp deffc SI. (9) For SI smplng whn ech sge nd equl numbers n of elemens observed whn he smple clusers, deffp = pples. 8
III.4 Opon I noher vrnce esmon opon I mes use of resmplng mehods. These compuernensve mehods use compuer power nsed of hevy clculons. One of hese mehods s he boosrp. Ths resmplng procedure ws orgnlly developed for..d. suons (Efron 979). For s pplcon n sscl surveys, dfferen pproches re proposed (see, for nsnce, Sho nd Tu 995, p.46ff). The pproch h drecly mmcs he orgnl de mes use of he generon of boosrp populon, from whch he resmples re drwn by he orgnl smplng scheme (see, for nsnce, Quember 04b, p.89ff). For hs purpose, n hree-sge desgn, he generon of he boosrp populon from he orgnl smple d hs o consder ll hree sges. Therefore, whn he smpled second sge clusers, he smple uns re replced ccordng o her hrd sge ncluson probbles j. Ths resuls n se-vlued esmors of he second-sge smple subclusers j wh respec o he neresng vrbles. Then, hese second sge uns j hve o be replced ccordng o her second sge ncluson probbles j. Ths resuls n se-vlued esmors of he frs-sge smple clusers. Evenully, by replcng ech genered cluser ccordng o s ncluson probbly, he generon of boosrp populon s se-vlued esmor of he enre populon s fnshed. From hs populon, whch cn be clled pseudo-populon (Quember 04b), number of B resmples re drwn wh he sme smplng scheme s he one orgnlly used n he survey nd n ech of hese B resmples he esmor c HT, 3 s, b = = s clculed ccordng o () esmed by (b=,,b). Then, he heorecl vrnce (3) of () s (0) B I = ( 3 s, b 3s ) B b= wh B 3s 3 s, b B b = =, he men vlue of he esmors 3s,b from he B boosrp smples. For (0) o be n ccure esmor of he rue vrnce (3), he smple szes hve o be lrge enough ll hree sges becuse f hs s no he cse, only smll number of uns re replced ll sges nd resmples re drwn from only smll number of dfferen vlues. I Inervl Esmon Wh one of he opons for he esmon of he vrnce of 3s (or P ) presened n Secon III, one cn clcule n pproxme ( α)-confdence nervl for he rue by 9
± u () 3s α / Theren, denoes he used vrnce esme nd u α/ he ( α/)-qunle of he sndrd norml dsrbuon. For hs nervl o be vld, he cenrl lm heorem mus hold nd should be conssen esmor of he rue vrnce ( 3s ) ccordng o (3). Consderng he compuonl nd echncl effors of he dfferen opons dscussed n Secon III, Opon II seems o be of neres for s use s n (), f he smplng frcon c/c he frs sge s smll. In hs cse, he overesmon of he rue vrnce (3) by he wh-replcemen vrnce (7) wll be unmporn (Särndl e l. 99, p. 54) nd one could s well drw wh-replcemen smples, so h he vrnce formul relly fs o he ppled smplng scheme. The vrnce esmor self s desgn-bsed. The quly of Opon III (Formul (9)) for s use s n (), depends mnly on he quly of he esmon of he SI-vrnce wh he d from he observed smple. Therefore, hs opon s model-bsed. If P s esmed by P, he pproxme confdence nervl s gven by P ± u α /. When he ro esmor P r from Secon II s used, he nervl P r ± u α / wll be conservve, f he model holds h y nd re srongly posvely correled. Wys o Improve he Precson of he Esmon of Tol There re dfferen wys o mprove he precson of smple survey resuls regrdng ols or funcons of ols such s P (Secon I). The frs fcor s he smplng scheme h s used o selec he smple elemens from he gven populon. n mporn componen n hs drecon s he choce of he frs-order smple ncluson probbles for ll elemens n he populon ( U). The bes of choces for hese probbles s o deermne hem proporonl o he sze of he sudy vrble y. In hs cse, when y = n, N y = every smple even of sze n = would provde he perfec Horvz-Thompson esmor of becuse y y N HT = = = y = = y N y = 0
pples. Of course, s he y s re unnown n he populon, he probbles cnno be deermned n hs wy. Bu, he esmor HT would lso hve smll vrnce, when he s cn be deermned ccordng o nown uxlry vrble, whch s pproxmely proporonl o y. Furher mprovemen of he precson of he esmon of cn be cheved by more effcen esmor compred o P (Secon I). One possbly s he ro esmor P r (Secon II). These fcors for beer performnce of n esmor re lredy mplemened n he process descrbed n he secons bove. hrd fcor s he srucurng of he populon. On he one hnd, for gven n, cluserng s ofen source for decrese of precson. Neverheless, dfferen specs such s rvel coss my cernly ndce s use. One he oher hnd, cern vrns of srfcon dfferen sges of hree-sge process my ncrese he precson of he esmor HT. Srfed smple rndom smplng wh proporonl llocon of he smple sze n on he sr, for nsnce, s more effcen hn n unresrced smple rndom smple s long s he whn-srum men vlues of he sudy vrble y dffer. The opmum llocon s cheved when he smple could be lloced on he sr proporonl o he sndrd devons of y n he sr. Hence, more uxlry nformon would be necessry (cf. Särndl e l. 99, Secon 3.7.3). noher possbly h hs he poenl o mprove he ccurcy of gven esmor s possrfcon (cf. bd., Secon 7.6). s n exmple, my hppen h, fer smple rndom smple s drwn from he populon whou srfcon nd he vrble y of neres s observed heren, urns ou h he men vlues of y dffer beween cern groups. For nsnce, n survey on ncome, he smple mens would dffer beween men nd women ncluded n he smple rndom smple. Ths mens h regrdng o he effcency of he esmon of he overll men would hve been beer o srfy he smple proporonl o he szes of hese wo groups n he populon lredy n he desgn-sge of he survey. Possrfcon mens he mplemenon of hs de n he esmon-sge of he survey process. If oo mny men re rndomly seleced for he smple rndom smple, lower weghs should be ssgned o hem. If here re oo few women, hgher weghs should be ssgned o hem o ncrese he mpornce of hs oo smll smple group. Obvously, he effcency of pos-srfcon of n unresrced smple rndom smple les somewhere beween srfed nd n unresrced smple rndom smple. For hree-sge process s descrbed n Secons I nd II, wh proporonl o sze whou replcemen smplng nd cluserng ll hree sges, proporonl nd opmum llocon nd pos-srfcon re dffcul o mplemen. The mos probble wy o nclude her effecs n he esmon of he vrnce of he esmor would be bsed on smulon (Secon III.4). Hence, he mos effecve nsrumen of mprovng he precson of he esmon of prmeer or P n he gven hree-sge process would be o ncrese he smple sze.
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