Derivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error.
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1 Drivatio of a Prdictor of Cobiatio # ad th SE for a Prdictor of a Positio i Two Stag Saplig with Rspos Error troductio Ed Stak W driv th prdictor ad its SE of a prdictor for a rado fuctio corrspodig to th sipl avrag rspos for a sigl asur of all SSUs i th i th PSU i th prutatio Backgroud ad otatio Usig asurt rror odl otatio, w assu that rathr tha obsrvig th fixd costat yst for subjct t i clustr s, th pottially obsrvabl quatity at th th k asurt is giv by Ystk yst Wstk This odl is a asurt rror odl W assu that th rado variabl Wstk has xpctd valu qual to zro ad variac giv by Eξ ( W 3 stk st W also assu that ay two asurs, W stk ad W stk for k k ar idpdt Th subscript k idxs th ordrd list of asurts (siilar to th subscripts i ad j but ot th particular valu ralizd upo asurt this ss, w ca thik of th fiit populatio as havig thr lvls of uits, clustrs, subjcts, ad occasios, whr th ubr of occasios for a subjct is vry larg, ad th occasios ar ot listd W assu a sigl rspos is pottially obsrvd o ach subjct, ad us th paratrizatio for y st giv i a odl without asurt rror Rado variabls that aris fro two stag clustr saplig ar itroducd via th two stag rado prutatio odl as bfor Th rsultig vctor of rado variabls is whr Y Y W ad W W W W Takig th xpctd valu ovr rspos rror ad ( s W U U W, ad whr W ( W W W s i i i i rado prutatios, whr E ξξξ Y µ, ad 3 V ( Y ( ( is giv by cd5doc varξξ ξ3 ad (drivd i Sctio of st s t Cd9doc 7// 5:5 A
2 W dvlop a stiat of a rado fuctio corrspodig to th sipl avrag rspos for a sigl asur of all SSUs i th i th PSU i th prutatio, TA gy Uis( µ s Ws, with g dfid by g i s Wstk, ad Ws, t basd o th valus obsrvd o ach rado variabl corrspodig to j,, SSUs i ach of i,, PSUs This liar cobiatio corrspods dirctly to a rado variabl dfid i trs of a udrlyig paratr for a clustr plus avrag rspos rror a supr-populatio cotxt, whr th populatio is a ralizatio of th suprpopulatio, ad th populatio is th targt of itrst, this liar cobiatio ay b th liar cobiatio of itrst ot that w ca xprss this liar cobiatio as th K K K whr su of a sapld part ad raiig part usig K, ad K such that whr ( ( (, KY ( T g Y g Y A g i ad i i i dfid as a colu vctor, ad i Y Y, g, ad whr ( with i i i as a vctor of disio ( TA is a rado variabl du to its dpdc o U ad th avrag rspos rror Dvlopt of th Prdictor W dvlop a prdictor of T A g Y gy usig prvious xprssios for th odl W rquir th stiat of T A to b a liar fuctio of th sapl data, ( g a Y T A, ad dtri a such that th xpctd bias is zro, ad is a iiu 3 ( A A E ξξ ξ T T ( T T var A A ξξ ξ 3 Cd9doc 7// 5:5 A
3 ot that th xpctd valu ad variac of ths rado variabls ar giv by Y X E ξξ ξ 3 µ with X ad X, ad Y X Y V V, varξξ ξ 3 whr ( V ( W Y V, V driv th othr two variac trs xt Dvlopt of xprssios for V ad, V W valuat a xprssio for V V K VK whr,, ( K, ad K ad ( ( ( V ( ( Th ( ( K ( K ( ( ( ( ( ( ( ( ( ( ( ( KK ( ( Also, ( ( ( ad Cd9doc 7// 5:5 A 3
4 ( K K ( ( ( ( ( ( ( ( ( ( ( ( As a rsult, V, V, K VK Fially, w valuat V K VK whr ( K ad ( V ( ( Th ( ( ( K ( K ( ( ( ( ( ( ( ( ( ( ( ( Also, Cd9doc 7// 5:5 A 4
5 K K ( ( ( ( ( ( ( ( ( ( ad ( ( K K ( ( ( ( ( ( ( ( ( ( ( As a rsult, V ( ( ( ( ( ( ( Cotiuatio of Dvlopt of th Prdictor suary, w fid that ( V, V, V, V, Also, ot that ( ( V ( ( ( ( ( Cd9doc 7// 5:5 A 5
6 Collctio of th study data will rsult i ralizig th valus of TA T A ( g a Y g Y g Y gy gy ay gy ay g Y Y ot that Th xprssio for th xpctd bias siplifis to ax g X Th xprssio for th variac ca b xpadd such that var ( ξξ ξ T 3 A TA ava av, g g Vg As a rsult, th Lagrag fuctio to b iiizd is giv by Φ ava av g g V g ax g X Diffrtiatig with rspct to, λ a ad λ, ad sttig th drivativ to zro rsults i th stiatig quatios giv by V X a V, g λ Usig th xprssio for th partitiod ivrs, ad rarragig trs (s Appdix B of cd5v3doc, X X g T, ( A g Y g Xα V V Y Xα whr α X V X X V Y This solutio was prstd i th auscript cd5vdoc cd5vdoc, a additioal xprssio is giv that is supposd to corrspod to a siplificatio of th xprssio for T A o drivatio is giv Th xprssio giv is T A ( f Yi f Y k ( Yi Y, Y ij Yij j i j whr k, Y i, ad Y This allgd siplificatio is rportd i cd5vdoc This stiator is idtical to th stiator giv by Bolfari ad Zacks (99, p6, quatio 79 if variac copots ar s dfid as siilar to ad s Siplificatio of th Prdictor W dvlop a siplificatio of th prdictor giv by T, ( A g Y g Xα V V Y Xα To do so, w first valuat a xprssio for V First, ot that w ca rprst ( V whr ( Usig a la V as Cd9doc 7// 5:5 A 6
7 dvlopd by Fullr ad Batts (973 Th, dfiig D a b c, usig quatio 38 (p63 of Fullr, / D / / a ( a b ( a b c / / ultiplyig this xprssio tis itslf, V D D this xprssio, ( a, b, ad c W xprss ad / / to dtriig V D D As a rsult, prior / / / D a ( a b ( a b c / / ow ultiplyig this xprssio tis itslf, V D D, or V a a / / ( a b ( a b / / ( a b c ( a b c a a b a b c Thus, V a a b a b c Usig th xprssios for a ( whil ( (, b, ad c, a b (, f Cd9doc 7// 5:5 A 7
8 a b c f f f f Usig ths xprssios, V ( f f f Asid to chck th xprssio for V fro Cd6doc cd6doc, w xprss th ivrs as k V k ( whr ( k W s hr if this atchs th ivrs giv abov First, ot that f Th usig th xprssio drivd i th prvious sctio, V or ( ( f f Cd9doc 7// 5:5 A 8
9 V ( f f ( f f ( f f Thus, V k ( f f t rais to show that th ultiplir of siplifis to k ( To show this, w first xprss f f Th f f Cd9doc 7// 5:5 A 9
10 ultiplyig ad dividig by, this xprssio is giv by k which chcks with th prvious xprssio for th ivrs Cotiuatio of Siplificatio of th Prdictor W us this xprssio to valuat f f X V X ad V xt, w valuat ( (, XV, so that f f α X V X X V Y Y V V Usig th xprssio for, (, V V, ( ( ( f ( f f ( ( ( f ( f f ( f f which siplifis to Cd9doc 7// 5:5 A
11 V ( (, V f ( f f f f ow ( f ( ( ( f f f ( As a rsult, Cd9doc 7// 5:5 A
12 ( ( f f ( f f ( ( ( f f f f ( ( ( f f f f ( ( Thus, ( ( V, V f f f ( ( ( f f ( Thus, ( ( ( f ( ( f f f ( ( ( f f ( Cd9doc 7// 5:5 A
13 (, V f ( V ( f f f ( ( f f ( W siplify th xprssio for f f f Exprssig trs ovr a coo doiator, f f f f f f f ( f f ( f ( f ( f ( f f ( f f f f f ( ( ( f ( f f ( f W siplify this furthr by xprssig ( f As a rsult, ( Cd9doc 7// 5:5 A 3
14 f f f f f f ad hc ( V, V f ( ( ( ( f f ( f ( ( f f ( ( ow sic f, f f f Th ( V, V (, ad ( ( ( ( ( ( ( ( k k ( ( ( ( Cd9doc 7// 5:5 A 4
15 or V ( ( k ( V k ( ( ( (, Asid o Chck with a xprssio drivd i cd6doc W rpat th drivatio of this xprssio usig rsults fro cd6doc, which cofirs that th abov drivatio is corrct W chck th solutio to th stiatig quatios First, rcall that k V k ( ad ( ( ( V, Th ( ( ( k, V V k ( ( ( ( ( ( k k ( ( ( k ( ( ( ( ( which siplifis to Cd9doc 7// 5:5 A 5
16 V ( k ( ( k (, V ( ( ( ( k ( ( ( ( ( ( k ( ( k ( ( k k ( k V, V ( ( Th ( k ( ( ( ( Also, k k ( k k ad ( ( ( so ow Cd9doc 7// 5:5 A 6
17 k k ( k k As a rsult, ( ( ( k k ( ( ( ( V, V k ( ( ( This xprssio is qual to th xprssio giv i th prvious sctio Copltio of Siplificatio of th Prdictor Sic α Y, ad α Y X Y, ad as a rsult, k, α V V Y X Y ( ( k Y ( Fially, sic T ( A g Y g Xα V, V Y Xα, ad Cd9doc 7// 5:5 A 7
18 T ( ( A g Y g k Y ( ( k g Y g Y ( ow, sic i g ad g i i, TA i Y i i ( k i Y W attpt to xprss th prdictor i a ar that is siilar to th prdictor drivd o pag 5 of cd4doc for cobiatio # Dfiig f, i ( TA i ( f f k f Y Th, addig ad subtractig th tr ( f, i ( TA i ( f ( f ( f f k f Y or i ( TA i ( f k f Y which siplifis to i ( TA i ( f k f Y Lt us dfi K ( f k f Th A Cd9doc 7// 5:5 A 8
19 i ( TA i K A Y This xprssio is dirctly coparabl to th xprssio for th prdictor of th liar cobiatio dfiig cobiatio # giv i cd4doc (pag 5 Lt us dfi ( Y Y Y Y ijk Y Y ad Yi, ad j f Th Y Y TA ( f i Y f ( i k f i ( i Y Y Y Y ( f iy f( i k f iy ( i ( i Y ow lt us rprst Y, ad ot that Y i Yi Th TA ( f i Y f i Y k ( i i Y i Y Y Th SE of T A W dvlop th SE of T A i a siilar ar to th dvlopt of th SE of T wh thr is ot rspos rror Such a dvlopt was giv i cd9doc To suariz, th stiator was giv by T g g V V V X X V X X V g X X V X X V Y so that ( T T g ay Y whr a V, V ( ( X X V X X V X X V X X V ad varξξ ( Y ( ow i cd9doc (s pag 4, a siplr xprssio is giv for Cd9doc 7// 5:5 A 9
20 A ( ( a whr ( k A ad k Thus, valuatig th SE corrspods to valuatig th variac of th xprssio T T g ( ay Y whr a is giv as abov ad varξξ ( Y ( Th SE is giv by (s pag of cd9doc as var T T ξξ ( ( g g ( ( ( ( This ca b furthr siplifid to ( i i varξξ T T f k i i i i f k ( ( k ( f f i i i i ( i ( i ( ( i i i i i i Wh thr is rspos rror, w show prviously that ( ( k TA g Y g Y ad that ( T g Y g Y As a rsult, A Cd9doc 7// 5:5 A
21 k A TA g Y g Y gy gy T ( ( ( k Y g ( Y ot that w ca xprss ( ( k k ( (, whr ( ( k As a rsult, T k ( k ( ( ow lt A Th ( ( Y TA TA g ( A var ξξ ( A g Sic Y ( ( Y A TA g Y var ξξ ξ3 Y V V, varξξ ξ 3 whr ( V (, Y V, V ( V, V, V, V,, ad ( ( V ( ( ( ( ( Th Cd9doc 7// 5:5 A
22 var V V g A A g V V (, T T ( ( ξξξ 3 A A, g A V A A V V A V g,, g A V V A A V V g W valuat ach tr i this xprssio First, k ( ( AV V (,, k ( ( V V ( ow k k ( ( V ( ( ( ( k ( ( ( ( whil ( ( V ( ( ( As a rsult, Cd9doc 7// 5:5 A
23 ( xt, cobiig AV ad V, ( AV, ( ( AV V, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Thus, AV V, Usig this xprssio, sic ( ( k ( A, ( Cd9doc 7// 5:5 A 3
24 ( k AV V A ( ( ( ( (, ( ( ( ( ( ( ( xt, AV is giv by, k ( (, ( ( ( ( AV k ( ( ( ( ( ( ( ( ( k ( ( ( ( ( ( ( Cobiig trs, w fid that Cd9doc 7// 5:5 A 4
25 AV V A AV V,, ( ( ( ( ( ( ( ( k ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( k ( ( ( ( ( Cd9doc 7// 5:5 A 5
26 or AV V A AV V,, ( k ( ( ( ( ( Sic var ( ξξξ T 3 A TA g,, A V V A A V V g, var ξξξ 3 ( T A TA ( k ( ( g g ( ( ( This xprssio parallls th xprssio for th variac without rspos rror giv i cddoc Cd9doc 7// 5:5 A 6
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