Comparisons of the Variance of Predictors with PPS sampling (update of c04ed26.doc) Ed Stanek

Size: px

Start display at page:

Download "Comparisons of the Variance of Predictors with PPS sampling (update of c04ed26.doc) Ed Stanek"

Alan Howard
5 years ago
Views:

1 Coparo o th Varac o Prdctor wth PPS aplg (updat o c04d6doc Ed Sta troducto W copar prdctor o a PSU a or total bad o PPS aplg Th tratgy to ollow that o Sta ad Sgr (JASA, 004 whr w xpr th prdctor a a lar cobato o th a, plu a dvato ro th a W rt dvlop xpro or th prdctor that ar th coo or Earlr Rult W uarz arlr rult ro c04d8doc that wr dvlopd ug clutr total or th apl ad radr W bg by dg a xpadd rpo vctor wth lt R that ar ordrd by PSU [S c04d4doc or dtal o dvlopt] R R R R R R R R o do, ( wth ( ( ( ( ( R U U y U y U y o do ( ( ( ( U U U ( t t t t ad U a colu vctor W alo d ( ( ( ( R R R R whr R t UU t yt UU t yt UU t yt For th vctor, var ( yy R y t P ( t J P yy y J y whr Pa a J a a To or a vctor o PSU total or th apl ad radr, w d 0 ( ( C R whr ( C For PSU, th 0 ( ar gv by corrpodg lt o ( t t t U ( ( U U t yt t ( ( U U U y whr C05d03doc 7/6/005

2 ( ( ( U U t yt, U U t yt, U t yt ad t t t ( U t yt otc that w alo d ( t c04d8doc, a xpro dvlopd or 0 var ( σ 0 P µ µ µ th xpro, µ yt ad σ ( yt µ t t or,, (whr w u th urvy aplg dto o th paratr σ Ug th xpro, tr o th parttod rado varabl, th total or PSU gv by P g g whr g ad g (, ad ( whr a vctor, ad ha do ( Th prdctor o th total or PSU bad o th rado prutato odl gv by J J ( ( σ P σ P whr ( σ σ σ Wh addto all clutr z ar qual, th prdctor pl to P ( P whr, ad, ( σ,, ad σ σ σ σ σ ot that w d, ot a W u th dto c w wll latr ( pag 6 d grcally a a tr o th or or Dcuo o a prdctor o th PSU paratr th or T c( Coparo o th SE o prdctor o a PSU a wr ad or populato wth T c qual z clutr by xprg prdctor a coo ar, uch that ( C05d03doc 7/6/005

3 ( y t ( Sta ad Sgr (004, whr w d U U t, ad t or grally, uppo w ar trtd prdctg a PSU paratr gv by a lar ( ( ucto o th rado varabl UU t y t gv by P U U t qtyt t z t t t Lt u d q y Th th paratr or PSU gv by P U U z A a xapl, ( t t t wh qt, P corrpod to th a o PSU ; wh q t, P corrpod to th total o PSU otc th drc btw th PSU paratr, ad a quatty gv by Z whr ( Z UU z, ad zt qtyt Th drc du to th act that t t t U a rado varabl, ad hc th xpro rprt a u o a rado ubr o rado varabl W hav avodd th u o uch quatt by dg rado varabl o th or ( UU z, otg that uch a t a xpadd t o rado varabl t t TO HERE /0/005 tr o th xpadd t o rado varabl PSU ad SSU ordr uch xpro W xpr th prdctor a T c ( c c c Th coct, c, drd or drt prdctor A a rult, a coparo o th SE wa valuatd tr o a ucto o c W xpr th prdctor o th total or PSU wh th tadard or wh clutr z ar ot qual, ad cod tag aplg wth PPS Exprg th prdctor o th PSU total th or T c( Th RP odl prdctor o th total or PSU, wh, gv by J J ( P P ( C05d03doc 7/6/005 3

4 ( whr U U t yt, t xpro qual to P ( ( σ σ ( σ σ σ, ad Th ( ( ( ( ( A a rult, ttg c ( ow ( ( ( P, c c c Lt u d cotat q uch that zt qyt Ug th cotat, w d ( Z U U t qyt or,, whr Z Z, uch that t c c c c Z c ( Z Z c Z Z c Z Z q ad c c (, w ca xpr P Z c ( Z Z otc how th xpro paralll to th xpro ltd Tabl (p5 o Sta ad Sgr (004 Th ttg Exprg th prdctor o th PSU a th or T c( wth PPS aplg W xpr th prdctor o th PSU a a lar or Th prdctor gv ( c04d5doc a P ( P, whr w hav aud pp aplg, U U U ( C05d03doc 7/6/005 4

5 ( y t whr U U t t, ad ( σ σ ( σ σ σ ( ( yt, whr U U t h t Alo, w d, or Alo, ad ( U ( yt U U t t ( y t U U t t W d o that Ug th xpro, th a or a PSU wth pp aplg prdctd by P ( P or quvaltly by J J ( P P ( y t whr U U t t (,, ad ( σ σ ( σ σ σ Lt u d cotat q uch that zt qyt Ug th cotat, w d ( Z U U t qyt or,, whr Z Z, uch that t c c c c Z c ( Z Z c Z Z c Z Z Th ttg q ad ( c c, w ca xpr P Z c ( Z Z otc how C05d03doc 7/6/005 5

6 th xpro paralll to th xpro ltd Tabl (p5 o Sta ad Sgr (004 TO HERE /9/005 Exprg th Prdctor or a xd odl ad Scott ad Sth odl a lar Cobato o th Sapl Th xd odl prdctor o a PSU a gv by P ( µ µ µ w, w, σ ad ot that σ σ P ( µ µ w w w w w w ( ( w w whr Fro Sta ad Sgr (004, Scott ad Sth prdctor o th a o PSU wth PPS aplg gv by P ( ( SS µ µ W r-xpr th prdctor a tadard or, T c (, bgg wth th ( y t xd odl prdctor th xpro, w d U U t, ad ot th t qualty o th tr ad uch that quatty by uch that gv by T c( Alo, w d ad rprt th Wth th otato, th prdctor tadard or W wrt th tadard or quvaltly a T c( c ( c c otc that th coct or drt ro th coct or, W xpr a prdctor th or by dg cotat q uch that zt qyt Ug th cotat, w d Z q or,, whr ( P Z c Z Z Th ca b xprd a Z Z, uch that C05d03doc 7/6/005 6

7 ( W d to dtr xpro or q ad P Z c Z Z c ( c c Z Z c uch that P P Oc th tr ar dtrd, w xpr th xd odl prdctor th tadard or va a traorato bad o th valu o q W dvd th xpro or th xd odl prdctor to tr a lar ar a P th tr or uch that P ( µ µ w w w w w w ( ( w w xt, w ubttut th xpro or Z to xpr th xd odl prdctor tr o q Z, rultg w P w( Z ( Z q q P P, th th coct o Z ut b qual th two xpro W t th coct qual to ach othr, ad olv or q, rultg ( w( q c c Th xpro a ucto o c W xpr othr coct, q, (, by quatg th coct o xpro or P ad P Th rult xpro o th or To valuat a xpro or c, w au alo that xpro to th xpro or q abov, w olv or ad hc Z th q w or c q w Equatg th c c Thu, ( w( w c c ( c C05d03doc 7/6/005 7

8 ( w ( w( c ( w( c( w( or c ( w Sc w, c ( w ( ( ( or c ( W u th xpro to olv or th coct, q,,, Wth our ( aupto, q w or all,, ow c c ( ( Th ( c ( ow w Th ( q w c ( w or,, ( ( W ca xpr q or quvaltly, a q Th xd odl prdctor, P valuatd tr o,,, qual to th prdctor gv by ( P Z c ( Z Z whr Z q or,,, q ad c ( C05d03doc 7/6/005 8

9 Suary o Expro or Prdctor o a PSU wh wth PPS Saplg Th prdctor o a PSU total ad a udr PPS aplg ca b xprd a Total: P c RP ( a: P ( c whr whr y t y yt y t t ( σ σ ( ( c ad σ σ σ, ( c ad ( σ σ ( σ σ σ Th xt tp to dvlop xpro to copar th SE o th prdctor W do o a lar ar a wa do c03ddoc Dvlopt o th SE o th Prdctor W dvlop th SE o th prdctor by xprg th prdctor wh grcally a P c( c P ow, P g g whr g ad g (, ad (, or P g A a rult, c P P P g c P g C05d03doc 7/6/005 9

10 var var W xt dvlop th SE o th xpro a a ucto o c To do o, ot that V V, o that V, V c, ( c V V P P P P g V, V g c c c P V P P V, g g V g V V, g g V g c c c P V P P V P V g W coplt th quar to xpr th a a drt ucto ot that b b ac bc a c c b a a a b b ac a a A a rult, ( P P, c c var PVP P V V, g V V, g g V g P V V, g c P V P P V P P V V g V V, g g V g P V P, Exprg th Varac o th Prdctor wth a xd odl or Scott ad Sth odl W dvlop a xpro or th varac or a prdctor gv by C05d03doc 7/6/005 0

11 Pa P Z ca ( Z Z c ap Z whr w t ca c th xpro, ( q q 0 ( Q whr q 0 0 q ow, P g g whr P g A a rult, Th var ( Q Q Q ad Z Q whr ( g ad g (, ad ( Pa P ( cap g c a P g V V c P P P c, a a a P g V, V g, or c a ca c a P V P P V, g g V g, V V g g V g ca PVP c apv c a, P V g whch w xpr a C05d03doc 7/6/005

12 ( P P var a, V V g g V g ca PVP ca PV c a, PV g P V, V g P V P c a P V P P V, V g, V V g g V g P V P Th tr wo t radly cacl wth th tr th xpro or th varac o th rado prutato odl prdctor W xpr th drtly o a to or radly xpr th tr th a ar TO HERE /6/005 C05d03doc 7/6/005

13 For a gv varac, th cod tr th xpro cotat Th rt tr dpd o th dto o c A a rult, wh coparg SE or drt prdctor, th coparo pl to a coparo o th rt tr A a rult, to copar th ESE o two prdctor, var P P var P P ( a ( b P V V, g P V V, g P V P c a c b PVP PVP W actor th tr to ply th xpro ot that a b ( a b( a b Thu, var P P var P P ( a ( b P V V, g P V V, g P V P c a c b PVP PVP P V, V g P VP ( ca cb ca c b P V P Splcato o Drc SE o th Prdctor W ply th xpro or th drc btw th SE o prdctor To do o, w a u o xpro or th varac or th PSU total, or th cald PSU valu Fro c04ddoc, w d that σ V var ( v J, v ( σ σ, ad σ V, var (, ( ( σ σ 0 J J ( ( ( c0ddoc ad c0d4doc or dtal, ad c04d9doc (pag 56 ot drt ordr or J ad c04d0doc addto, V var ( ( 0 0 σ ( ( ( σ ( σ 0 0 ( 0 0 ( c04d0doc Fro c04d5doc, w d that C05d03doc 7/6/005 3

14 σ V v J, v ( σ σ, ad σ V, ( ( σ σ 0 J ( J ad ( ( 0 0 σ ( ( V ( σ ( σ 0 0 ( 0 0 W dvlop a xpro or P V, V g Frt, ot that P V,, V g PV PV PV g σ ow, v PV P PJ v P, whch pl that P V 0, whl P V v P Alo, σ P V g ( P ( σ σ 0 J J g ( (, σ ( ( σ σ P 0 P J J g ( ( ( ( σ σ P 0 g ( g,, ( ( σ σ Sc ( P V g P P V V g PV PV PV g,, ( ( σ σ ( ( σ σ P v P P v xt, w ply th xpro or P V P Sc σ v v Cobg tr, PV P PJ P, P V P v P v Th, ug th xpro ad th prvou xpro, P V, V g v ( ( σ σ v ( ( σ σ P P V P v P v W ply th xpro ug v ( σ σ Subttutg tr or v, C05d03doc 7/6/005 4

15 P V, V g v ( ( σ σ P V P v ( ( σ σ v ( σ σ ( ( σ σ ow ( σ σ ( σ σ σ, o that P V, V g W ca ply th xpro urthr P V P rcallg that c (, o that P V, V g ( P V P or P V, V g P V, c V g c A a rult, P V P P V P W ca u th xpro th drc SE gv by var P P var P P ( a ( b P V V, g P V V, g P V P c a c b PVP PVP Codrg th cod SE a bg or th RP odl, (uch that c c, var c c P P P P P V P c c c c v ca c W ca xpr th a ( ( var a a b C05d03doc 7/6/005 5

16 c c c c c c ca c ca c ca c c ( ca c ca c ( ca c ca c A a rult, ( ( var P var ( a P P P v ca c ca c Wh coparg th drc SE o a clutr a ug PPS aplg btw o othr prdctor ad th RP odl prdctor, w d that ( ( var P var ( a P P P v ca c ca c Splcato o Drc SE wh all clutr ar o qual z Wh all clutr ar o qual z, Sta ad Sgr (004 dvlopd a xpro or th drc SE gv by ( σ var P var ( ( a P P P ca crp σ σ whr crp (,, ad σ σ ot that th xpro, σ σ σ σ ( σ σ σ σ σ Th xpro hould b th a a th xpro wh w had PPS aplg, w t all clutr z qual Th xpro or th drc SE th gv by ( ( var P var ( a P P P v ca c ca c (, ( c v ad σ σ W r-xpr th SE drc aug σ aupto, σ σ σ σ, ad σ σ σ σ ( σ σ ( σ σ σ, ad hc ( whr σ or all clutr Wth th σ σ σ A a rult, σ c c RP Ug th xpro, w d alo that C05d03doc 7/6/005 6

17 ( v σ σ ( σ σ σ σ σ σ σ Subttutg th tr, w d that ( ( σ var P var ( a P P P σ ca crp ca crp Th xpro do ot atch th xpro drvd arlr, but clo to that xpro C05d03doc 7/6/005 7

Estimating the Variance in a Simulation Study of Balanced Two Stage Predictors of Realized Random Cluster Means Ed Stanek

Estimating the Variance in a Simulation Study of Balanced Two Stage Predictors of Realized Random Cluster Means Ed Stanek Etatg th Varac a Sulato Study of Balacd Two Stag Prdctor of Ralzd Rado Clutr Ma Ed Stak Itroducto W dcrb a pla to tat th varac copot a ulato tudy N ( µ µ W df th varac of th clutr paratr a ug th N ulatd