Predictors Using Partially Conditional 2 Stage Response Error Ed Stanek

Transcription

1 Predctor ng Partally Condtonal Stage Repone Error Ed Stane TRODCTO We explore the predctor that wll relt n a mple random ample wth repone error when a dfferent model potlated The model we decrbe here cloely related to the model decrbed n X (and alo to the partally condtonal model dced n c06ed06doc We e the dea for the addtonal random varable gven n c07ed5doc (page 7 propoal The prncpal dfference that we magne that when addng repone error we conder the model Y y W ( where W W W t wt and t t ntead of the model t Y * y W ( where W W t wt t n th model we envon repone error a arng from mple random amplng wth replacement from a fnte et of repone error Samplng of bject va mple random amplng wthot replacement THE SAL FTE POPLATO WTH RESPOSE ERROR ODEL We conder the ettng where a non-tochatc repone repreented by t We defne the average repone a y t t y t for y for the mean repone a μ y Alo we defne the repone devaton a wt yt y and the bject devaton a β y μ o that yt y wt μ β wt Parameter are alo defned ch that σ ( y μ and σ wt for t We ame that the repone that potentally oberved for bject gven by C07ed5doc /9/007 :5 P

2 where Y y W r ndexe poble mltple repone and where ( W w t t t We ame that t repreent an ndcator random varable that tae the vale of one when electon of a repone devaton t and zero otherwe We ame that electon of repone devaton are wth replacement where repreent expectaton wth repect to ( ( amplng of repone devaton A a relt E ( t whle var ( t ( ( Snce amplng wth replacement cov ( t ** t 0 whenever * and/or t t* Then E ( W 0 for all r Alo notce that var ( W σ for all r and var ( W W* * 0 for all ; * ; r ; * ( r We ame r for all o that we repreent a ngle meare of repone for all bject We defne addtonal notaton to repreent mple random wthot replacement amplng of bject ch that ( ( ( y y y (( Y ( Y Y Y (( ( (( ( (( Y y and y YR W W W W W and Y * Y Y R ng th notaton and Y y W R Y * y W Y W Th the al mple random amplng model wth repone error We can conder an alternatve way of expreng th model that cont of prodct of vector Th veron gven by C07ed5doc /9/007 :5 P

3 * Y y W ( y W vec ( y vec ( ( W vec ( We e th repreentaton to motvate expreng the random varable n a more expanded fahon otce that element n y gven by y Let defne a random varable Y y We e the per-crpt arrow to ndcate that the et of random varable are expanded We defne a vector of expanded random varable by Y y ( y y y Y Y Y and note that y Y ( n a mlar manner we defne an expanded et of random varable correpondng to W where we expand element n W gven by W Let defne a random varable W W ng the per-crpt arrow to ndcate an expanded random varable We defne W W ( W W W W W W o that ( Y Y W y W y W y W * We defne the expanded random varable a y y y W W W * y y y W W W vec vec Y or y y y W W W Y * Y W W W W W W W where vec W W W W C07ed5doc /9/007 :5 P

4 Y * vec Y * Y * Y * ( A EW FTE POPLTO ODEL WTH RESPOSE ERROR We defne a lghtly dfferent fnte poplaton model wth repone error next The dfference n the new model the defnton of the random varable repreentng repone error Parameter are defned n an dentcal fahon a n Secton We ame that the repone that potentally oberved for bject gven by Y y W where r ndexe poble mltple repone and where W t wt and we defne ( t t t The term repreent the realzed vale of the random varable Defnton of the random varable dentcal to the defnton n Secton ( t t ( t Snce t t when 0 E 0 0 whle when E ( t When 0 var 0 0 whle when var ( t We ame r for all o that we repreent a ngle meare of repone for all bject We defne addtonal notaton to repreent mple random wthot replacement amplng of bject ch that and ng th notaton and (( Y ( Y Y Y Y R (( W ( W W W W Y y W W W y Y W Y W The vector Y of dmenon where we repreent an element of the vector by Y We next expand the random varable defned by th model To do o we defne an expanded et of random varable correpondng to W where we expand element n W C07ed5doc /9/007 :5 P 4

5 whch gven by W Let defne a random varable W W ng the percrpt arrow to ndcate an expanded random varable We defne W W ( W W W W W W o that Y Y W ( ( y W y W y W We defne the expanded random varable a y y y W W W y y y W W W vec vec Y or y y y W W W Y Y W y y y y y y where vec Y and y y y W W W W W W vec W An alternatve expreon gven by W W W Y vec Y Y Y ( 4 EXPECTED VALE AD VARACE OF Y Let denote the expected vale wth repect to amplng of bject va a bcrpt and the expected vale wth repect to amplng of repone devaton by We develop expreon for the expected vale and varance of the expanded random varable repreented by Y next Expected Vale: Frt note that E Y y ( C07ed5doc /9/007 :5 P 5

6 We evalate the expected vale of term gven by W and W by evalatng the expected vale of each term where a ( t t t t t W t t t t by t We expre th a Let repreent a realzed vale of t helpfl n evalatng the expected vale and varance to conder the jont dtrbton of and a gven n Table Table Example of Jont Dtrbton of t t and for poble repone devaton for each of bject P t ( t t (tme P P ( t t ow E ( W E t t t and ng Table t clear that E ( t A a relt E ( W t t whch eqal to zero nce t 0 for all A a relt t E Y y ( Varance: t ext we develop an expreon for the varance To do o we mae e of an expreon for the condtonal expanon of the varance gven by var ( Y E var Y var E Y E var W var Y C07ed5doc /9/007 :5 P 6

7 A ey component of the varance the expreon for E where vec ( ( W var ( W P ( var for th varance next W W W W W W The element n W vec W W W t t t W We can expre We develop an expreon are gven by W W W W W W W W W W W W vec vec W o W W W W W W that E var W W W W W W W var vec P W W W We frt develop an expreon for W W W W W W var vec W W W W W W W Then Let defne ( ( C07ed5doc /9/007 :5 P 7

8 W W W var W W W vec W W W var ( W cov ( cov ( W W W W cov ( W var co W W v ( W W cov ( cov ( var W W W W W ( otce that W t t Let defne W o that W W ng t thee expreon W W var var W var W W W W or W var var W W W * W W * * W W n a mlar fahon when * cov ( cov W W and * W W replacng W W * W W * * cov ( cov W W W W or * W W C07ed5doc /9/007 :5 P 8

9 * W W * * cov ( cov W W W W * W W W We frt foc or attenton on evalatng var W Th expreon ed to W contrct W var var W W W W W W W W W whch a term on the bloc dagonal of var vec We alo W W W * W W * evalate cov W W for * whch ed to contrct * W W * W W * * cov ( cov W W W W an off-dagonal bloc * W W W W W W W W of var vec W W W C07ed5doc /9/007 :5 P 9

10 ( Frt let evalate var ( W var gven by var cov cov cov var cov var ( W cov cov ( var Each term n th expreon condtonal on whch refer to the random ( varable that can tae on two poble vale zero or one ow t t A a relt when 0 var t t 0 0 whle when ( ( ( ( ( ( var t t t ext we evalate cov * t t t* t* when t t To do o we expre ( ( * * * * ( ( t t t t E t t tt E t t E t* t* cov When 0 whle when ( t t* t t* 0 ( t t t* t* cov 0 E nce E E * when t t ( ( ( t t* t t* ( * * * 0 and E t t t and E t t t o that cov ( t t t t t t ng thee relt when 0 whle when * * * var W 0 0 ( C07ed5doc /9/007 :5 P 0

11 var ( W whch can be expreed a var ( W t t To mmarze we fnd that when 0 var ( 0 W 0 whle when var ( W t t W Thee expreon wll be ed n obtanng an expreon for var W W ext we evalate expreon for ( * * ( * * cov ( W * cov W where * To do o we need to ( * * evalate expreon for the ndvdal term cov W W ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * ( * cov cov cov cov cov cov ( * ( * ( * cov ( * cov ( * cov * ( * Let conder evalatng a dagonal term gven by cov t t t * t where * Th gven by ( * * * cov E E E ( t t t * t ( t t t * t ( t t ( t * t C07ed5doc /9/007 :5 P

12 ( * ( ( * ow E * * * * * t tt t E t t t ( t t t nce t and ( * t are ndependent for * Alo notce that nce the ample of bject elected wthot replacement the prodct * alway eqal to zero A a relt ( * E ( * 0 t t t t ext we evalate E ( ( t E t t t t ( * Combnng term cov ( * 0 * * t t t t t t whch mplfe to ( * cov ( * 0 t t t t nce the prodct * alway eqal to zero ( * ext we conder off-dagonal term gven by cov ( t t t * * t where * * and t t* Th gven by ( * * * cov E E E ( t t t* * t* ( t t t* * t* ( t t ( t* * t* ( * ( ( * ( t t t* * * t ( * t t* t * t* * t * t* nce ow E E ( * and t* are ndependent for * Alo notce that nce the ample of bject elected wthot replacement the prodct * alway eqal to zero A a relt ( * cov ( 0 t t t * * t * We e thee relt to expre cov ( W * W 0 We combne th expreon wth the other expreon e when 0 var W 0 0 whle when var ( W t t W to determne an expreon for var W Th expreon gven by a et of W expreon that depend on the vale of ( where only one of thee term ha a vale of one and all other have a vale of zero We ndcate the vale of the vector by pecfyng whch term ha a vale of one When ( t C07ed5doc /9/007 :5 P

13 t t 0 0 W var W W when W t var W 0 t 0 W etc Recall that W ( W W W and W var var W W ng the prevo expreon W t var t 0 W σ nce t * * * t * * σ t The term σ n th expreon t correpond to the repone error varance for the bject elected n poton We expre th et of condtonal varance n term of ndcator vector We defne e e e where thee vector o a to match the realzed vale of gven by e ( e e e where e 0 when 0 and e when We relate the term σ to thee vector ch that C07ed5doc /9/007 :5 P

14 var ( W e σ * W W * ext we evalate cov W W for * whch ed to contrct * W W * W W * * cov ( cov W W W W an off-dagonal bloc * W W W W W W W W of var vec To evalate th expreon we need to evalate W W W * term le cov ( W W * when * Th gven by * cov W W ( * ( ** ( ** ( ** cov ( * ( * cov cov ( ** ** ( ** cov * cov * cov * ( ** ( ** ( ** cov ( * cov ( * cov ( * Each term n th expreon condtonal on whch refer to the random ( varable that can tae on two poble vale zero or one ow t t A a relt when 0 ( ** ( ** ( ** cov ( ( * * ( ( * E E E ow ( ** ( ( * E ( * ( * * * E ** * ( ( whle E ( ( E ( ** ** * and E ( * A a relt C07ed5doc /9/007 :5 P 4

15 ( ** ** * ** * cov * 0 * cov W W * 0 All of the other term n the expreon are alo zero reltng n * W W * A a relt cov W W * 0 0 and hence cov ( W W 0 We * W W e th expreon pl var ( W e σ to expre W W W W W W var vec W W W ( W W W W W ( W W ( W ov ( W W var cov cov cov var c cov cov W var W W W W ( ( ( e σ 0 0 eσ e eσ We e th expreon to evalate ( σ C07ed5doc /9/007 :5 P 5

16 E var ( W W W W W W W var vec P W W W eσ P eσ P( nce for all and ( We are ntereted n evalatng E var e P σ W Let conder a mple example Frt ppoe that We repreent the permtaton by realzaton of where!! 6 realzaton are poble correpondng to 0 0 the order correpondng to vec( vec the order correpondng to vec( vec the order correpondng to vec( vec the order correpondng to vec( vec the order correpondng to vec( vec the order correpondng to vec( vec 0 0 Each realzaton ha probablty 0 0! The matrx eσ correpond to C07ed5doc /9/007 :5 P 6

17 eσ eσ eσ eσ eσ e 0 0 σ eσ eσ eσ eσ Then for the order σ eσ 0 σ σ for the order σ eσ σ σ for the order C07ed5doc /9/007 :5 P 7

18 σ σ 0 0 eσ σ for the order σ eσ σ σ for the order σ σ 0 0 eσ σ and for the order C07ed5doc /9/007 :5 P 8

19 σ σ ! tme Dvdng th by the eσ 0 0 σ 0 0 Wthn each bloc each dagonal term wll be non zero nmber of poble otcome! E var W eσ P σ Fnally we note that (ee p5 n Argentna007-lecdoc (# n Ro-Gallego ote that var ( Y P y P y A a relt var ( Y E var W var Y σ P y P y We defne Δ y P y Then var ( Y P Δ σ We can expre th n a mlar fahon a we expreed the varance n c06ed0doc (page n that docment the varance of the partally condtonal model wa gven by * var ( Y σ P Δ otce that the dfference n thee two expreon relate to amplng wthot replacement n the docment c06ed0doc and amplng repone error wth replacement n th docment Smmary of odel Relt We mmarze the model relt for the expected vale and varance of expected vale gven by Y The C07ed5doc /9/007 :5 P 9

20 E ( Y y The varance gven by var ( Y P Δ σ where Δ y P y 5 LOSS OF FORATO COLLAPSG TO THE SAL RADO VARABLES n the development of an expanded two tage model we evalated ng the Rao- Bellhoe theorem whether or not we cold collape the expanded random varable to the al random varable wthot long nformaton n contrat n the development of the partally condtonal random varable n c06ed06doc we dd not examne whether nformaton wold be lot bt rather jt proceeded wth the collapng f we collape the random varable we wold mplfy the mean and varance n the followng manner To collape the expanded random varable to a vector of the al random varable we pre-mltply Y by Then Y ( Y vec y y y vec W W W Y W where an element of W gven by W and W t wt Alo we fnd that t E ( Y ( E Y ( ( y μ and var ( Y ( var Y P Δ σ ( e e ( P J σσ ow σ Δ σ σ e where σ e σ and σσ e e σ Let defne σ e σ Then σσ e e σ e ng thee expreon C07ed5doc /9/007 :5 P 0

21 var ( Y σ P Pσe J σe ( ( ( σ P σep J σ e 6 DEVELOPET OF THE BLP WTHOT COLLAPSG RADO VARABLES n c06ed0doc we develop the BLP for an expanded et of random varable wthot collapng the random varable We follow that development to develop the BLP ng the relt on the expreon for the expected vale and varance from ecton 4 Target We ame that there nteret n a target that correpond to the expected vale of repone for a elected bject We defne th a P g Y where g e and where e an vector wth element eqal to one and all other element eqal to zero A a relt P Y Samplng Parttonng We dentfy the ample a the et of the frt n random varable n a permtaton and the remander a the other random varable otaton ntrodced to partton Y nto a L ample and a remander by pre-mltplyng by L L Recall that Y Y W Alo E ( Y E Y y whle var ( Y var Y W var Y var W [ P Δ] σ ( e e ( P J σσ LY Y n general we repreent the relt a LY ng properte of Y L Y Y Y X Y V V we evalate E γ (where γ y and var Y X Alo defne Y V V LY Y Y V V LY where var L Y Y Y V V C07ed5doc /9/007 :5 P

22 n 0 L n ( n Frt let defne L Then nce L 0 n ( n n E ( Y y we fnd that n 0 n ( n E ( LY n y 0 ( n n ( n y ( n n X o that X n We partton the varance by pre and pot mltplyng var ( Y P Δ σ e e P J σσ n 0 L n ( n by L ch that L 0 n ( n n L var ( LY var Y L L L n 0 n ( n n 0 n ( n var ( Y 0 n 0 ( n n ( n n n 0 0 n n ( n n n ( n ( P Δ 0 n ( n n 0 ( n n n 0 0 ( n n ( n n n ( n σ P 0 n 0 ( n n ( n n n ( ( n 0 n n n 0 n n ( J σσ e e 0 n n n 0 ( n n n C07ed5doc /9/007 :5 P

23 ow n 0 n Jn Δ J Δ n ( n n 0 n ( n n ( n ( P Δ 0 n 0 ( n n ( n n n J Δ n n ( n J Δ n whle n 0 n ( n n 0 n ( n P σ 0 n ( n 0 n ( n n n n Jn σ J σ n ( n J σ n n ( n J σ n and n 0 n ( n n 0 n ( n ( J σσ e e 0 n 0 ( n n ( n n n ( Jn n σσ e e J σσ e e n ( n e e ( n n J σσ J σσ e e ( n n Combnng thee term Y V V var Y V V n Jn Δ σ J Δ σ ( n ( n ( ( Jn n σσ e e J σσ e e ( ( n ( n J Δ σ n n ( n J Δ σ n ( ( J σσ e e ( J n n σσ e e ( ( n n ( where C07ed5doc /9/007 :5 P

24 V n Jn Δ σ ( Jn n σσ e e ( ( n Jn Δ n n σ ( n n e e ( J J σσ V J Δ σ J σσ e e n ( n ( ( n ( n and V n J n Δ σ ( J n n σσ e e Alo ( ( n Jn Δ J Δ n ( n Y V V var Y V V J Δ n n ( n J Δ n We are ntereted n predctng a lnear combnaton of the random varable gven by Y P g or P g Y g Y where g ( g g and ( g g ( e e Y where e an vector wth element eqal to one and all other element eqal to zero parttoned to conform We wh to do o baed on the ample data whch repreented by Y Y Y V V V For th reaon we are ntereted n the vector Y where var RY V V V Y Y V V V Y n term of the target we defne P g g g g Y o that P g Y g Y We defne the target a ( g g ( e e where e an vector wth element eqal to one and all other element eqal to zero parttoned to conform A before P Y Contrcton of Predctor and partton g repreentng We wold le the predctor to have coeffcent that depend on the bject n the realzed ample Th wll enable the bject pecfc repone varance to be ed a oppoed to expreon for an average repone varance To allow for th poblty we reqre the predctor to atfy the followng crtera: Lnear n the ample: ˆ P ( a Y C07ed5doc /9/007 :5 P 4

25 nbaed: E ( ˆ P P 0 nmm SE: var ( Pˆ P mnmzed We repreent a (( a ( a a an where (( a ( a a a We lmt or conderaton to the ettng where nbaed Contrant We can expand the nbaed contrant: E ( Pˆ P ( ( ˆ ( E P P a n o that ( g g e 0 ( n X a g γ ch that X a X g X γ n order for th expreon to eqal zero for any vale of γ the nbaed contrant wll be alway be atfed when a X g X 0 Th ntrodced a a contrant ng Lagrangan mltpler when mnmzng the SE We proceed to develop the predctor that atfe thee crtera where the devaton of the predctor gven by ˆ P P a ( Y g Y g Y Y ( a g g Y Y Fndng the nmm SE (followng relt n c06ed06doc: otce that V V V g a var ( ˆ P P (( g a g g V V V g V V V g where V n Jn Δ V J Δ and n ( n V V n n σ ( n n e e ( J J σσ Let defne VR n n σ ( n n e e ( J J σσ Then V V V R ng th expreon C07ed5doc /9/007 :5 P 5

26 var V VR V V g a g a g g V V V g V V V g ( ˆ P P (( V V V g a VR 0 0 g a (( g a g g V V V g (( g a g g g V V V g g g a g ( a V g V a V g V a V g V g ( g a VR ( g a ( a V g V a ( a V g V g ( g a VR ( g a or var ( ˆ P P ( a V g V a ( a V gv g ( g a VR( g a a V a g V a a V g g V g g V g a V g g V a a V a R R R R ( ( a V V a g V gv a g Vg gv R R R a V a g V g V a g V g g V g R R ncldng the contrant a X gx 0 va a Lagrangan mltpler we ee to fnd the vale of a that wll mnmze f ( a λ a ( V a ( g VR g V a ( a X g X λ g V g g V g R Dfferentatng wth repect to a and λ To fnd the vale of f ( a λ ( a f ( a λ ( Xa Xg λ a that mnmze f ( a Va V g V g Xλ and R λ we et thee dervatve to zero mltaneoly and olve for a The etmatng eqaton are gven by f ( aˆ λˆ a V ˆ X a VRg V g 0n f ˆ ( ˆ ˆ a λ X 0 λ Xg 0 λ g C07ed5doc /9/007 :5 P 6

27 V X aˆ VR g V g or eqvalently by ˆ The olton to th eqaton X 0 λ X g gven by aˆ ( ( V V X X V X X ( V ( VR g V g ( V X ( X ( V X X g We can now expre the bet lnear nbaed predctor Recall that P g Y g Y and we P ˆ g a Y where the bet predctor replace a by a ˆ Let defne predct P by ( ˆ α ( X ( V X X ( V Y Then ˆ ( g X ( X ( V X X ( V Y g ( ˆ ˆ ( ˆ VR V Y Xα g Xα V V Y Xα ( a Y g V g V V Y V X X V X X V Y R To mplfy th expreon we need to evalate explctly an expreon for V where V n Jn Δ n n ( n n e e ( J σ J σσ ( n Δ σ σeσ e Jn Δ σ σeσe ( ( ( ( ow when V n A Jn B V n A Jn A B nb A A a relt to evalate and V Recall that we need to evalate ( ( σ e e Δ σ σ ( n Δ σ Δ e e ( ( ( ( σ ( ( σ σ σ σ e e C07ed5doc /9/007 :5 P 7

28 Δ y P y A a relt ( Δ σ σeσ e y ( yy ( σσ ( y σ yy e e y σ e e y σ y σ ( We can e patterned matrx formla to nvert th expreon where ( ( R R R R R * We can mplfy thee expreon by defnng σ σ Then ( * * * Δ σ e e ( y σ σ σ ( e ( e ( ( y σ y σ and ( ( Δ σ σeσ e ( * * ( * * ( y σe ( y σe * * y σ ( y σ y σ y σ * y σ C07ed5doc /9/007 :5 P 8