Introduction. Generation of the Population Data

Transcription

1 Ovrvw of th Smulaton Study for Prformanc of Unqual Sz Clutrd Populaton Prdctor of a Ralzd Random Clutr an (updat of c05d30.doc wth mplr notaton) Ed Stank ntroducton W dcrb a mulaton tudy mlar to th tudy rportd by Pfffrmann and athan (98, JASA ). Th mulaton nabl valuaton of prdctor n two tag clutr amplng contxt whr th clutr z ar not qual, and thr may (or may not) b cond tag ampl lctd wth probablty proportonal to z. Th mulaton tudy dvdd nto part.. Th populaton gnratd n th SAS program cd05p.a wth th macro POPUE (whch llutratd n th program cd05p0.a). ncludd n th populaton th ampl z for ach clutr.. Paramtr ar valuatd for th populaton. Th ar llutratd n cd05p.a, and gnratd n a macro namd POPP n cd05p3.a. 3. Sampl ar lctd, and prdctor ar valuatd. Th mulaton lct ampl from th populaton, and valuat prdctor. Th mulaton contand n th program cd05p5.a. W dcrb th mulaton n th documnt. Th dcrpton of gnratng th populaton, and valuatng th paramtr contand n th documnt c05d7.doc. Gnraton of th Populaton Data Data from th populaton th ba of th mulaton. W frt gnrat th populaton. Th populaton pcfd accordng to clutr z and ampl z for clutr. Th clutr man can b t to corrpond to unform prcntl of a gvn dtrbuton. Smlar crtra can b t for th man wthn a clutr. A lt of paramtr n for th ovrall populaton ar gvn n Tabl a. Paramtr pcfc to a clutr ar gvn n Tabl b. Th paramtr ar contand n th data t pop.a7bdat that cratd a output va th program cd05p0.a. C05d44.doc /4/006 8:55 A

2 Tabl a. Paramtr n th Populaton Data Populaton Contant Varabl SAS am Dcrpton bn, &bn umbr of Clutr n th Populaton bm_m Avrag # unt n Clutr m m m_m Avrag # of unt n ampl of clutr m f f_bar Ovrall proporton n ampl max m m_max, &m ax # Unt n ampl for clutr max ( ) ( ) bm_max, &bm ax # Unt n clutr ( ) v_c, &popv Varanc of Clutr an v_ Avrag of Unt Varanc for Clutr ( y ) v_ Unt Varanc for clutr t t Tabl b. Varabl n th Orgnal Populaton Data pcfc to Clutr. Clutr Spcfc Contant Varabl SAS am Dcrpton Clutr labl m m Sampl z for unt n clutr bm Clutr z y t,t, tc Unt valu t yt mu_ Clutr man t Evaluaton of Populaton Paramtr Th cond tp n th mulaton valuaton of populaton paramtr. W lt th populaton paramtr n Tabl a and b. Th paramtr ar dfnd for th prdctor n c05d37.doc. An llutraton program gvn by cd05p.a, wth a macro vron gvn by cd05p3.a. C05d44.doc /4/006 8:55 A

3 Tabl a. Paramtr n th Populaton Data Populaton Contant Varabl SAS am Dcrpton R R β β β β R βr mun Avrag ampl wghtd naught man mun Avrag rmandr wghtd naught man mu Avrag ampl wghtd tar man mu Avrag rmandr wghtd tar man mur Avrag wghtd man β β bnmu Avrag ampl wghtd naught dvaton. bnmu Avrag rmandr wghtd naught dvaton. β bmu Avrag ampl wghtd tar dvaton β bmu Avrag rmandr wghtd tar dv. β brmu Avrag wghtd dvaton β β, β β vbn Var of ampl wghtd naught dvaton. vbn Var of rmandr wghtd naught dvaton. vbn Covar of rmandr wghtd naught dvaton. β vb Var of ampl wghtd tar dvaton β vb Var of rmandr wghtd tar dv., ββ vb Covar of rmandr wghtd tar dvaton ( ) R R R vbr Var of wghtd dvaton C05d44.doc /4/006 8:55 A 3

4 f ( f ) r f f ( ) ( f ) ( f ) vn_ Unt varanc for naught (total) f v_ Unt varanc for tar f r r r r R R vd_ Unt varanc for ovrall dot vd_ Unt varanc for ampl dot vd_ Unt varanc of rmandr dot vnr Adjutd unt var for naught vr Adjutd unt var for tar vdr Adjutd unt var for dot m R k m R + kn Shrnkag contant for naught m k + m k Shrnkag contant for tar m k + m kd Shrnkag contant for dot C05d44.doc /4/006 8:55 A 4

5 Tabl b. Tmporary trm ud n Calculaton of Paramtr for Populaton Data Populaton Contant Varabl SAS am Dcrpton β β bn_ Wghtd naught dv product β β b_ Wghtd tar dv product ββ ββ bnmu Av Wghtd naught dv product bmu Av Wghtd tar dv product β vbntot Total for var of ampl wghtd naught dvaton. β vbntot Total for var of rmandr wtd naught dvaton. β vbtot Total var of ampl wghtd tar dvaton β vbtot Total var of rmandr wghtd tar dv. f f f ( f ) ( f ) ( f ) ( f ) r f r r r r vn_ Unt varanc for naught (total) v_ Unt varanc for tar vd_ Unt varanc for ovrall dot vd_ Unt varanc for ampl dot vd_ Unt varanc of rmandr dot C05d44.doc /4/006 8:55 A 5

6 Tabl c. Varabl n th Orgnal Populaton Data pcfc to Clutr. Clutr Spcfc Contant Varabl SAS am Dcrpton m f f_ Samplng fracton for Clutr v v + v_mm Varanc wght for and SS m k k v k_v Shrnkag for and SS m r m r_ Sampl z rato for clutr m r m r_ Rmandr z rato for clutr r r_ Clutr z rato for clutr r mun_ Sampl wghtd naught man for clutr r mun_ Rmandr wghtd naught man for clutr r r mu_ Sampl wghtd tar man for clutr r r mu_ Rmandr wghtd tar man for clutr R r mur_ Wghtd man for clutr β bn_ Sampl wghtd naught dvaton for clutr β bn_ Rmandr wghtd naught dv. for clutr β b_ Sampl wghtd tar dvaton for clutr β b_ Rmandr wghtd tar dv. for clutr β br_ Wghtd dvaton for clutr R R R Slcton of Two Stag Clutr Sampl Slcton of a Sampl of Clutr Th thrd modul n th mulaton lct a two tag mpl random ampl from th clutrd populaton (llutratd n cd05p4.a). Frt, ung a lt of clutr labl, a mpl random ampl of clutr labl dntfd. Th proc rpatd many tm (wth ach rptton calld a tral ). Onc th ampld clutr ar dntfd, thy ar combnd wth th populaton data, and from th data, a ampl of unt n ach clutr lctd. nput to th modul a fnt lt of fxd valu (varabl) corrpondng to th valu for unt n th populaton. C05d44.doc /4/006 8:55 A 6

7 W dcrb contructon of th bac prmutaton n mor dtal. A an xampl, uppo that thr ar clutr n th lt to b prmutd. Lt ach clutr b aocatd wth a labl. Alo, lt u t th valu of ach clutr to b qual to th clutr labl. To form a prmutaton, w frt lct on of th clutr. For th frt poton n th prmutaton, any on of th clutr wll hav th am chanc of bng lctd,... W mak th lcton by dvdng th 0- ntrval nto qual z ntrval. Th tartng and ndng pont for p p ntrval p gvn by and. A unform random numbr gnrator ud to lct a numbr btwn 0 and, and th rult ud to dntfy th clutr that pckd n th lcton. W plac th paramtr for th clutr n poton n a lt of prmutd valu. Aftr lcton of th frt clutr, thr wll b rmanng clutr n th populaton. Pror to lctng th nxt clutr n th prmutaton, w r-agn th labl to th clutr rmanng n th populaton. For ach clutr n th lt wth a labl gratr than th lctd clutr, w rduc th labl by on. For xampl, uppo that thr ar 5 clutr n th populaton, and that th clutr wth labl 3 wa lctd a th frt clutr n a prmutaton. Thn th rmanng 4 clutr would b r-agnd uch that clutr valu appard a th valu,, 4, 5. W lct th cond clutr n a prmutaton by dvdng th 0- ntrval nto qual z ntrval. For th cond lcton,. Th tartng and ndng pont for ntrval p p p gvn by and. A unform random numbr gnrator ud to lct a numbr btwn 0 and, and th rult ud to dntfy th clutr that pckd n th lcton. W plac th paramtr for th clutr n poton n a lt of prmutd valu. Contnung n th proc, w may form a prmutaton of all th valu n th populaton. Altrnatvly, w may top aftr gnratng th prmutd valu for,..., npoton. n th mulaton program, w only gnrat th frt n poton for ach prmutaton, nc th poton dfn th ampl. Onc a t of clutr lctd for a ampl, w aocat wth ach lctd clutr th unt valu for th clutr. A mlar proc thn followd for lctng a mpl random ampl of unt valu from th lctd clutr. Th rult of th lcton ar wrttn to a data t for a two-tag clutr ampl (rfrrd to a a tral). Th proc rpatd for many tral. Th frt part of th lcton macro lct a ampl of clutr. A data t wth on rcord pr lctd clutr pr tral (C) cratd. Th data t partcularly mpl, wth thr varabl corrpondng to th tral numbr, th poton n th ampl, and th labl of th lctd clutr. Th data t ubquntly combnd wth th populaton data (P3) to form th data t SAP pror to lctng unt n a clutr. Varabl Dfnd n Slctng Sampl for a tral Th macro varabl that ar ud n rpatdly lctng ampl ar ltd n Tabl 3. Tabl 3. acro varabl Ud n Smulatng a Sampl of Clutr and a ampl of unt pr lctd clutr. C05d44.doc /4/006 8:55 A 7

8 Varabl SAS am Dcrpton nd &nd am of populaton data t. (. pop) Tral &traln umbr of ndpndnt ampl lctd from th populaton. &bn # of clutr n th populaton &bm # of unt for a clutr n &n umbr of clutr lctd n a ampl. mm &mm ax (n populaton) of # of unt lctd n a lctd clutr. c &c Contant n lnar combnaton that fxd chk &chk # of rplcaton of prmutaton of unt (for chckng) out &out Varabl to control dcrptv output about mulatd ampl clutr (0non, mnmal, dtald) n th proc of prmutng th clutr, w crat om varabl tmporarly for th procng. Th varabl ar ltd n Tabl 4. Th varabl contand n th data t wth th lctd clutr ar ltd n Tabl 5. Tabl 4. Tmporary Varabl ud n Prmutng th Clutr n th Populaton. Varabl SAS am Dcrpton u Tmporary labl for clutr n th populaton that agnd a th valu for a clutr for prmutng clutr. rm_p Th numbr of rmanng clutr to prmut n th populaton. p p Th rlatv poton of th clutr rmanng to b prmutd n th populaton. f thr clutr, thn p ntally p,,. Aftr lctng clutr, p,,-. rm_ rm_ Th numbr of clutr rmanng to lct n a ampl. rn rn Th random numbr lctd from th 0- gnrator lct lct An ndcator havng a valu of 0 f clutr n all prvou poton hav not bn chon for a partcular lcton, or a valu of f a clutr n th currnt or prvou poton ha bn lctd. x-x x-x An array of clutr valu t qual to th labl to. Tabl 5. Varabl n Sampl Tral Data t C Varabl SAS am Dcrpton tral tral Tral numbr Poton of lctd clutr n a ampl Labl of lctd clutr C05d44.doc /4/006 8:55 A 8

9 Slcton of a Sampl of Unt n a Clutr Onc th clutr ar lctd for a tral, w combn th data wth th populaton data. Th rultng data t namd SAP. Th cond tag ampl lctd from th lctd clutr n th data t SAP. Th proc of lcton mlar to th lcton of clutr. Th rultng ampl tord n th data t OUTD. Th data t th mulatd two tag ampl data. Addtonal ummary varabl ar contand n a parat data t OUTD. Tabl 6. Tmporary Varabl ud n Prmutng th Unt n Clutr to lct th Scond Stag Sampl Varabl SAS am Dcrpton ck ndx n loop for unt ud to t chkt( ) to unt valu chk &chk umbr of rplcaton of lcton of SSU j n PSU to chck mulaton. Th allow for multpl valu rpon for a lctd unt. chkt-chkt( ) Valu of th Unt n a clutr. Th array ud to r-t th unt valu aftr lcton of a unt. p ndx of 0/ ntrval gmnt ud to dntfy SSU j rm_pc umbr of rmanng unt not yt lctd n a PSU rm_c umbr of unt rmanng to b lctd n a PSU rn Random numbr ud to lct unt labl for a SSU tu ndx of numbr of tm a unt lctd (uually ) lctc ndcator of unt lctd n poton j for an SSU t-t( ) Orgnal unt valu, not rtand unt ndx of unt n th lcton of a unt Addtonal varabl ar valuatd n th proc of lctng a ampl of unt for ach PSU that ar ndd to valuat prdctor. Th varabl corrpond to ampl tattc for PSU. Th varabl (ltd n Tabl 7) ar tord n a data t namd OUTD.a7bdat, and procd furthr. C05d44.doc /4/006 8:55 A 9

10 Tabl 7. Sampl tattc for ach PSU that ar Cratd n Slctng SSU from a PSU. Sampl Stattc Varabl SAS am Dcrpton / v v and SS /v() ( / v ) v and SS (/v())(/v()) m Y U Yj m j y_bar Sampl PSU man m YR Ur Yj m j yr_bar Av for r tm ampl man for PSU v Y yv and SS /v() tm av ( / ) Th ampl PSU tattc n Tabl 7 ar ud to form avrag, and total for a ampl. Th avrag and total ar ltd n Tabl 8. Tabl 8. Avrag and Total of Sampl tattc for ach PSU that ar Cratd n Slctng SSU from a PSU n data t OUTDA.a7bdat. Sampl Stattc Varabl SAS am Dcrpton n / v v_um Sum of and SS /v() n ( / v ) v_um Sum of and SS (/v())(/v()) Y Y R n n Y n n y_barb Av of Sampl PSU man Y yr_barb Av of Total for ampl SSU for PSU n R ( / v) Y yv_um Sum of and SS /v() tm av Ung th rult n Tabl 8 for a ampl, w valuat th addtonal tattc ud n mxd modl and Scott and Smth modl prdctor. Th tattc gvn n Tabl 9. All of th ampl tattc n Tabl 8 and 9 ar contand n th data t OUTDB.a7bdat, wth a ngl rcord pr tral. Tabl 9. Wghtd an Evaluatd from Sampl Stattc for ach Sampl (n OUTDB.a7bdat). C05d44.doc /4/006 8:55 A 0

11 Sampl Stattc Varabl SAS am Dcrpton n ( / v ) ˆ Y n / v yv_b Wghtd Av for and SS Prd. Fnally, w combn th tattc n OUTDB.a7bdat wth th data t for th ampl (OUTD.a7bdat). n th proc, on mor tattc valuatd, corrpondng to th wght that ud n valuatng prdctor n xd odl and Scott and Smth modl. Th llutratd n Tabl 0. Th rultng data tord n OUTE.a7bdat. Tabl 0. Wght valuatd for and SS modl n OUTE.a7bdat. Sampl Stattc Varabl SAS am Dcrpton ( / v ) w n w Wght and SS Prd. / v Prdctor n th Smulaton Th nxt tp to form prdctor for ach tral. To dvlop a prdctor, w u quantt calculatd for ach lcton of a clutr (PSU), and avrag of th quantt ovr lctd clutr (PSU) n th ampl (tral). Th prdctor obtand by combnng th rult. Th prdctor ar dcrbd n Tabl. C05d44.doc /4/006 8:55 A

12 Tabl a. Prdctor Dvlopd for ach PSU and Tral Sampl Stattc Varabl SAS am Dcrpton Pˆbar Y p_bar Sampl an Prdctor Pˆ ˆ + k Y ˆ p_mm Prdctor ( ) ( ) ( ) + ( ) + ( ) + ( ) + ( ) ( ) ( ) Pˆ fy + f ˆ + k Y ˆ SS ( ) ˆ P fy f Y k Y Y R R R R P fy f Y k Y Y R R R R p_ p_n p_nb SS Prdctor RP odl prdctor of total Av SSU total prdctor ˆ P fy + f Y + k Y Y p_ RP Prdctor wghtd by c f c ultplr for RP Av Prd. bad on avrag amplng fracton. ˆ P cy + ( c) Y + k ( Y Y) p_df RP Prdctor bad on c f Tabl b. Prdctor Dvlopd for ach PSU whr Contant Dpnd on th Ralzd PSU. Sampl Stattc Varabl SAS am Dcrpton c f c ultplr for RP Av Prd. bad on clutr pcfc amplng fracton. ˆ P cy + ( c) Y + k ( Y Y) & c f ˆ ( ˆ P P ) p_nb RP odl prdctor of an Expron for th thortcal SE ar prntd n c05d37.doc. W valuat th xpron for ach PSU along wth th prdctor. Th xpron along wth th varabl that dfn thm ar gvn n Tabl. Tabl. Expron for th Thortcal SE for Prdctor of Random Varabl for PSU n Two Stag Samplng. Sampl Stattc Varabl SAS am Dcrpton C05d44.doc /4/006 8:55 A

13 ( Y P ) ( Pˆ P ) ( Pˆ P ) ( ˆ P P ) ( ˆ P P ) var tm_bar PSU SE mpl PSU man. ξξ var tm_n PSU SE RP Total Prd ξξ varξξ tm_nb PSU SE RP Av Prd varξξ tm_ PSU SE RP Wt Prd varξξ tm_df PSU SE RP dot wth av f Tabl. Expron for th SE ummarzd n Tabl 0. ( Y P ) ( c) varξξ + whr c arbtrarly t btwn 0 and. m m ( ˆ n varξξ P ) ( ) ( ) P f + k R nm n ( ˆ n varξξ P ) ( ) ( ) P f + k R nm n ( ˆ n varξξ P ) ( ) ( ) P f + k mn n ( ˆ n f + f var ξξ P ) ( ) ( ) ( ) P c f f + k + f mn n f + f f f + f n f + f n n + f ( f ) + ( k ) + ( f ) mn n n n + f ot on th Evaluaton of th SE of th Smpl an whn Samplng Fracton Vary Th thortcal SE of th mpl clutr ampl man dpnd upon how th wght ar agnd to th ampl man and th rmandr man n dfnng th valu to b prdctd. Th th dfnton of c. For any dfnton of c btwn zro and on, th xpctd valu (ovr SSU) of th prdctd valu wll b qual to th PSU man. Whn targt random varabl wll qual th PSU man. n mulaton, w t. c f U f, th c f U f otc that w xpand th dcuon n c05d46.doc. otc that whn th valu of c chang wth th clutr that wll b ralzd, th thortcal SE wll alo chang wth th clutr that ralzd. Th man that th thortcal C05d44.doc /4/006 8:55 A 3

14 SE wll not b contant. n ordr to compar th mulatd SE wth th thortcal SE, w avrag th thortcal SE ovr th ralzd clutr. A a rult, th thortcal SE for th mpl ralzd clutr man gvn by th avrag of th SE for all,...,, var ξξ ( Y ) ( ) P U c +. m m ot on Evaluaton of th SE for and SS Prdctor. Whn th multplr or hrnkag contant chang wth th ralzd PSU, t not pobl to dvlop th xpctd SE drctly ung th xpron for th uncondtonal varanc. W valuat th xpctd SE n th ca drctly va th mulaton. Summarzng Rult n th Smulaton Th lat tp n th mulaton to ummarz th rult and dplay th ummary. Th ummary valuat th avrag SE for ach prdctor, and th varanc n th SE valu. W alo valuat th avrag of ach prdctor, and th ba. To valuat th quantt, w form array of valu ( Tabl ). Each array contan nn valu (ltd n Tabl ). Th array contan thr varabl that ar cumulatd ovr th mulaton tral bad on th prdctor and th tru valu of th random varabl. W valuat th SE for ach lctd PSU by ubtractng th prdctor from th tru PSU man, and quarng th valu. Th an tmat of th SE, ay m _ t ( ˆ ) t P t P t, whr ttral numbr, and PSU n th tral. W contruct th avrag of th tmat, and th # tral n varanc of th tmat. W contruct th avrag ba by m _ b t (# ) ba, th n tral t # tral n avrag SE by m _ v _ t (# ) m t and th varanc of th SE by n tral t # tral n m _ ( ) ( ) m _ tt m _ vt n # tral t # #. tral n tral n m _ tt m _ tt n( # tral) t n( # tral ) t C05d44.doc /4/006 8:55 A 4

15 Tabl. Elmnt n Array ud to Summarz Smulaton Rult n Array for Prd and Tru Sampl Stattc Prdctd Tru ndx Varabl Random Var SAS am Dcrpton Pˆbar Y p_bar Sampl PSU man P ˆ p_mm Prdctor 3 P ˆSS p_ SS Prdctor 4 Pˆ 5 P 6 P ˆ τ p_n Av SSU total prdctor (nought) p_nb RP Prdctor of an (nought/bm) p_ RP Prdctor of an (tar) 7 P ˆ p_df RP Prdctor bad on av PSU f 8 P ˆ p_df RP odl prdctor of total ung f 9 ˆ P ( P ) p_n RP Total (nought) ovr Ralzd Tabl 3. Array of Valu ud to Cumulat Sum and Summarz Smulaton Rult SAS Array Dcrpton Bac Varabl prd( ) Prdctor tru( ) Tru valu of PSU random varabl Cumulatng Varabl ba ( ) Ba of prdctor prd ( ) tru ( ) m_t ( ) Squard valu of th ba SE for ampl PSU m_t ( ) Squard valu of th quar of th ba Summarzng Varabl m_b ( ) Avrag Ba of prdctor m_v ( ) Avrag Varanc of Prdctor m_( ) Standard Error of Smulaton SE Th Smulaton Th populaton cratd n th program c05d3.a. Th dvlopmntal mulaton run n c05d4.doc. C05d44.doc /4/006 8:55 A 5