Collapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder
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1 Collapg to Saple ad Reader Mea Ed Staek Collapg to Saple ad Reader Average order to collape the expaded rado varable to weghted aple ad reader average, we pre-ultpled by ( M C C ( ( M C ( M M M ( M M M, where C ad ( M ( ( M C ( Let u defe M M M ( M M M C M M M ad C The M M M C ( M C C C ( ( M where C C C ( ( M C ( ad C6ed9doc 5//6 3:7 PM
2 C ( ( M Y Y C Y C ( The vector C C ( Y Y C Y Y C Y th expreo, (, ad ( C Y Y C ( ( M Y Y C ( Y C ( ( M Y C ( A a reult, C C C C otce that ( ( ( ad that C C C C C C ( ( ( ( where J J J J J J CC J J J C6ed9doc 5//6 3:7 PM
3 addto, ce C C C C C C, otce that J J J J J J C C C C J J J Coder the upper left had ter the ecod part of th expreo gve by J f for all,,, the J, ad C C C C P Let u expre C C C C P Otherwe, th expreo doe ot plfy J J J J J J + J J J J o that C6ed9doc 5//6 3:7 PM 3
4 ( C C C C C C C C + C ( C + P + J J J + J J J J J J J Predctg the PSU Mea We are tereted predctg the PSU ea Recall that th wa poble wth PPS aplg by ug aple ad reader total baed o data that were trafored by dvdg by the cluter ze Suppoe that we dvde by the cluter aple ze a oppoed to the cluter ze We ca decrbe the reult a lar aer a whe we dvded by the cluter ze Traforg by dvdg by the PSU aple Sze order to collape the expaded rado varable to weghted aple ad reader total, we trafor y t to y wt y t Apart fro th chage, everythg ele the ae otce that M M yt y wt t M t M ad that the total M M ; larly, M y y t wt t t we defe M ywt w M t C6ed9doc 5//6 3:7 PM 4
5 Let u coder the expreo for the expected value of the trafored parttoed data Th gve by Y where w ad w X w E ξξ Yw X w M w A a reult, M w All of M w ( M the odelg detcal, wth the excepto that paraeter are cotructed ug y wt tead of y t Wll the ubaed cotrat be atfed whe ecod tage aplg PPS? We exae th by aug PPS aplg, reultg f w f w ( M f f ( M Wth PPS aplg, E f Y X X Whe f f ( M w w Y w X w X ξξ predctg a total, we requre the predctor of P gy + gy to be a lear fucto of the aple rado varable, P ˆ LY, ad to be ubaed ow g e whle g ( e e ( M ( (( L g g Th ple that ˆ Y E ξξ P P E ξξ Sce Y where Y X E ξξ Y X C6ed9doc 5//6 3:7 PM 5
6 X ( ad X Th ple that ( M ( ˆ X Eξξ P (( P L g g or X LX g X + g X ow g X ( ( M ad g X ( e ( ( M A a reult, g X + gx M ( ( ( Th ple that L ( ( M Thu, wthout further retrcto, t ot poble to develop a ubaed lear predctor ug aple ad reader total that doe ot deped o the paraeter We defe the total ter of weghted repoe uch that P g Yw + gy w ad a predctor gve by P ˆw LY w w Wth PPS aplg, the ubaed cotrat gve by f ( ˆ E P (( w P X ξξ L w g g f f ( M whch wll be atfed whe X f X Thu, we ca cotruct ubaed predctor whe f X f ( M (( L w g g there PPS aplg However, wthout PPS aplg, uch cotructo ot poble TO HERE 4/8/6 C6ed9doc 5//6 3:7 PM 6
7 Suppoe ow that we for the aple ea, ad the reader ea for each cluter, but cotruct thee ea baed o trafored data A a frt tep, recall that the target paraeter defed by the lear cobato of ea or total that we wat to predct a P g Y where g b ( c M c M c M where c for all,, for total, ad c for all,, for ea ad where M (( b b a vector of cotat We reove the depedece of c o the cluter by aocatg wth Y a weght w lar to c We requre the value of w to be defed M uch that w To do o, let u defe c c c, ad et w c Suppoe that we are tereted a paraeter correpodg to the ea of a PSU We defe th by ettg b e where e a vector wth all eleet equal to zero, except for eleet whch ha the value of oe, ad ettg c We repreet th rado M varable va the otato P g Y where g e ( c M c M c M ter of the weght, w c c, g ce ( w M w M w M We expre th a lghtly dfferet aer to reove the depedece of g o ( g ce w w w M M M ( ce w M M M M w To do o, otce that C6ed9doc 5//6 3:7 PM 7
8 Let u defe g w ce ( M M M The P g w wm Y We coder the expreo for wm Y ore detal Cobg the Weght wth the Populato Paraeter otce that a ple populato, we defe yw w y a clutered populato, for cluter we defe M yt The paraeter M t y a ple populato coparable to the paraeter a clutered populato Let u defe ywt w yt The M M w w y y Suppoe that we develop predctor baed o repoe w t wt M t M t that correpod to the trafored populato wth value gve by ywt w yt We troduce repreetato of a expaded et of rado varable for the trafored populato ext We defe y, (( y ( y y y w wt w w wm (( ( y y y y y, w w w w w M M ( ( wj jt wt ad ( jt t Y Y Y U y w U y t t ( Y U y, ( ( Y w w wj (( ( U Y U Y U Y U Y U Y w wj w w wm ( ( w Y Y Y Y or w wj w w wm C6ed9doc 5//6 3:7 PM 8
9 Y Y Y Y Y, (( U ( U U U w w w w w Y Y Y Y Y (( ( w w w w w Ug thee defto, ce ( (( U ( U U U Y Y Y Y Y, ( ( Y Y Y Y Y, ad Y w wy wm w M w M w M Y Y Y Y UwY UwY UwY UY w UY w U Y w Y w A a reult, P g w w M w w Y g Y Dvdg the RP to a Saple ad Reader We partto the rado varable to the aple ad reader uch that Y w C C ( M + Yw where C C, C, Yw C ( M C ( C ( M ad C M otce that wth th defto, M C ( C C C TO HERE 4/7/6 Sce Y X + Z( M+ B + E, β Y C C C X + Z( M+ B + E Y C β C C C6ed9doc 5//6 3:7 PM 9
10 or C X where X, C X Y X Z E + ( M+ B + Y X β Z E X ad ( C Z X ad Z where ( ( M ( M C Z ( + Z ad Z Alo, ce E ( ξξ ( Y X, β M E Y X ξξ Y X β Collapg Weghted Expaded Rado Varable We coder collapg weghted expaded rado varable to a total for the aple ad for the reader otce that the ae weght are ued for each copoet, where the weght wll be et equal to w ad c Let u defe ( M C C ( ( M, uch that C ( ( ( M C6ed9doc 5//6 3:7 PM
11 ( M ( ( ( M ( ( ( M ( ( ( ( C ( M ( ( ( ( ( M M M P ( M + P ( ( M ( P ( ( M Y P g wyw gw gw, Yw ertg th expreo to ( w Rather tha defg Of prcpal teret the lear cobato that repreet the total (or the ea of PSU, defed by ettg b e where e a vector wth all eleet equal to zero, except for eleet whch ha the value of oe We repreet ether of thee rado varable va the otato P g Y where g e ( c M c M c M C6ed9doc 5//6 3:7 PM
12 ow ( C Y Y C ( ( M Y Y C ( Y P g Y g g g Y + g Y ad ( Y where we aue the value of c ad hece g e whle g ( e e ( ow M g Y g Y g C ( C Y + g ( P Y + J J J + J J J g J J J J Y ow g C e C e C Y Y,, ( g ( P, g J e J e ad C6ed9doc 5//6 3:7 PM
13 J J J J J J g J J J J J J J J J e J J J e e A a reult, gy ey + e e Y Hece, C6ed9doc 5//6 3:7 PM 3
14 gy ey + e Y + P ad Slarly, ( ( ( ( ( ( ( ( M ( M + P Ug thee expreo, M M ( M C6ed9doc 5//6 3:7 PM 4
15 ( M ( ( ( M ( ( ( M ( ( ( M ( ( ( ( M + ( ( ( ( ( M ( ( M M P ( M P ( ( M ( P ( ( M ( ( ( ( M ( ( ( ( M ( ( M ( ( ( ( ( ( M M ( ( M M P ( M P + ( ( M ( P ( ( M ( M C Sce C ( ( M, C ( ( ( M ( C6ed9doc 5//6 3:7 PM 5
16 ( M ( ( ( M ( ( ( M ( ( ( ( C ( M ( ( ( ( ( M M M P ( M + P ( ( M ( P ( ( M Y P gy g g, Y ertg th expreo to ( ( ( ( ( P Y g g ( C ( M Y ( ( ( ( ( M M M P ( M ( P Y + g g ( ( M ( P Y ( ( M C6ed9doc 5//6 3:7 PM 6
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