Predicting Random Effects from a Finite Population of Unequal Size Clusters based. on Two-Stage Sampling. Edward J. Stanek III

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1 Predctng Random Effect from a Fnte Populaton of Unequal Sze Cluter baed on To-Stage Samplng Edard J. Stanek III Department of Publc Health 40 Arnold Houe Unverty of Maachuett 75 orth Pleaant Street Amhert, MA USA tanek@choolph.uma.edu Julo M. Snger Departamento de Etatítca Unverdade de São Paulo São Paulo, Brazl jmnger@me.up.br C07ed3.doc /9/007 :37 PM

2 ABSTRACT Predcton of random effect an mportant problem th expandng applcaton. In the mplet context, the problem correpond to predcton of the latent value (the mean) of a realzed cluter elected va to-tage amplng. Bet lnear unbaed predctor developed from mxed model are dely ued, but ther development requre dtrbutonal aumpton or an nfnte populaton frameork. When the number and ze of cluter fnte, uper populaton model have been ued to predct the contrbuton of the unoberved unt to a realzed cluter mean. Recently, predctor developed from a to-tage amplng model have been hon to out-perform the modelbaed predctor. Hoever, the random permutaton model underlyng thee predctor lmted to ettng here cluter ze and ample ze per cluter are equal. We preent a ne to-tage amplng model that can be ued hen cluter ze and ample ze per cluter dffer. The model expand the et of random varable from the to-tage amplng model to accommodate the dfferent ze cluter. The reultng bet lnear unbaed predctor out perform any of the competng predctor, even hen cluter and ample ze per cluter are equal. The reducton n expected mean quared error relatve to predctor developed under mxed model or uper populaton model aumpton lkely due to the pecfcty of the expanded random permutaton model to the problem. Th ugget fathfully capturng the tochatc apect of a problem more mportant than mplfyng aumpton n developng optmal oluton. Many other problem may be amenable to mproved oluton baed on extenon of th approach. C07ed3.doc /9/007 :37 PM

3 Contact Addre: Edard J. Stanek III Department of Publc Health 40 Arnold Houe Unverty of Maachuett at Amhert Amhert, Ma. 000 Phone: Fax: Emal: KEYWORDS: uperpopulaton, bet lnear unbaed predctor, random permutaton, optmal etmaton, degn-baed nference, mxed model. ACKOWLEDGEMET. Th ork a developed th the upport of the Conelho aconal de Deenvolvmento Centífco e Tecnológco (CPq), Fundação de Amparo à Pequa do Etado de São Paulo (FAPESP), Brazl and the atonal Inttute of Health (IH-PHS-R0-HD36848, R0-HL0788-0), USA. C07ed3.doc /9/007 :37 PM

4 . ITRODUCTIO There often nteret n the mean unt repone of a cluter here repone oberved for only a ubet of the cluter unt. In t mplet form, th problem often addreed n the context of a mxed model, and correpond to predctng the mean of a realzed cluter n a to tage mple random ample. There are many applcaton here uch predcton of nteret. Henderon (984) a ntereted n predctng a co mlk producton, and tuded lactaton record of dary co. In the current U.S. educatonal envronment, hgh chool tandardzed tet core are of nteret, th chool havng hgh take n the reult due to ther ue n reource allocaton. The trength of a famly emotonal upport of nteret n tude of copng th ubequent llne n agng (Yorgaon, et al. 006). A patent bomarker level may be mportant n quantfyng rk of deae or ubequent nterventon. Although applcaton vary dely, each problem can be defned ung a common clutered populaton frameork, here the target parameter can be defned a the average expected repone of unt n a realzed cluter. In the example, a cluter may correpond to a co, a chool, a famly, or a patent, hle the unt may correpond to lactaton perod, tudent, famly member, or tme perod, repectvely. Ideally, the number and ze of cluter knon, nce the practcal relevance and nterpretaton of the tattcal nference enhanced by a clear problem defnton. It rarely poble to oberve the expected repone on all unt n each cluter. Intead, repone oberved on a ubet of unt for a ubet of cluter. An mportant C07ed3.doc /9/007 :37 PM

5 queton ho to ue uch nformaton to gue the average repone of unt n a pecfed cluter. To aner th queton, the oberved data are lnked to the populaton va an aumed probablty model, objectve crtera are defned that can be ued to evaluate competng guee, and the bet gue then determned. Ideally, the problem defned n the populaton hould be dentcal to the problem anered va the tochatc model. The typcal ue of mxed model to gue the mean repone for a cluter an example here the problem anered va the probablty model a lghtly dfferent problem. In a mxed model, ho bet to etmate a parameter correpondng to a cluter mean tranlated nto the queton of ho bet to predct a realzed random effect. The cluter dentfed n the populaton repreented a a random varable n the tochatc model. Th tranlaton change the problem, nce the dentty of the cluter lot n the uual repreentaton of the tochatc model. The dfference ubtle, nce the ubcrpt that dentfe a cluter n the populaton uually replaced by a ubcrpt repreentng the poton of a elected cluter n the ample (Cael, Sarndal, and Wretman 979). For example, the frt cluter n a populaton ltng not the ame a the frt ample cluter, hch e refer to a the frt prmary amplng unt, or PSU (a n Stanek and Snger, 004). A mlar ubttuton occur for the ubcrpt dentfyng unt n a cluter, and ample unt for a ample cluter, hch e refer to a econdary amplng unt, or SSU. The dfference beteen a cluter and a PSU are often a ource of confuon n nterpretng predctor of realzed random effect. C07ed3.doc /9/007 :37 PM

6 Model baed approache to predct a realzed random effect are characterzed by a gap beteen the problem tated n term of dentfable cluter and unt n the fnte populaton and the random effect n a mxed model that do not dentfy cluter or unt. Snce multple probablty model can be poted, dfferent optmal oluton can be clamed for the ame bac problem. Each oluton uffer from the lmtaton that the underlyng model can not be connected back to the dentfable fnte populaton. Thu, hle each oluton optmal for t theoretcal frameork, none of the frameork match realty, and t unclear hch optmal oluton bet. There are other dffculte th model-baed approache that occur hen cluter ze dffer n the populaton. A problem occur hen tryng to ue a model to repreent ample SSU n a PSU. Such a repreentaton mportant for the model to nclude charactertc of unt (thn cluter varable) that may be related to repone. The problem occur nce vector repreentng ample SSU for a ample PSU are of dfferent ze (correpondng to dfferent number of ampled unt). Interpretaton of repone for an element n a ample vector then depend on the orderng of the PSU, nce vector repreentng dfferent ample PSU have dfferent length. The mxed model aumpton that PSU are elected at random conflct th the condtonng needed to nterpret repone for a SSU. In part to detep th dffculty, model baed approache may aume the number of SSU nfntely large (a n the mxed model approache of Goldberger 96; Henderon 984, McLean, Sander and Stroup 99; Robnon 99; and McCulloch and Searle 00), or condton on cluter ze (a n the uperpopulaton model baed approache of Scott and Smth 969; Ghoh and Lahr 987; Bolfarne and C07ed3.doc /9/007 :37 PM 3

7 Zach 99; Vallant, Dorfman and Royall, 000). Such aumpton den the gap beteen the problem defned n the fnte populaton, and the oluton to the problem defned n the model. Mxed model method are dely accepted and ued n pte of thee ue, but the foundaton for uch method not completely ecure. We preent a ne expanded to tage random permutaton (RP) model that overcome ome of the lmtaton of prevou probablty model. A key feature of the expanded model multaneouly retanng the cluter dentfy and the PSU poton n the model, hle dtnguhng for a PSU the relevant contrbuton of ample SSU, and non-ampled SSU to a target random varable uch a a cluter mean. The model extend the expanded model for mple random amplng of Stanek, Snger, and Lencna (004) to the to-tage RP model ued by Stanek and Snger (004), hle allong for poble unequal ze cluter. We develop the bet lnear unbaed predctor (BLUP) of a PSU mean ung th model, and ho that the predctor ha maller mean quared error than other predctor, ncludng the uual mxed model predctor. Begnnng th an explct repreentaton of random varable underlyng a to tage RP of a fnte populaton, e ho that hen predctng a PSU mean va a lnear unbaed predctor, the expanded model can not be further reduced thout lo of nformaton. Snce the expanded model retan th nformaton, the bet lnear unbaed predctor (developed n an dentcal manner a the development n Stanek and Snger (004)), ha maller expected MSE than any prevouly developed predctor (ncludng commonly ued mxed model predctor, uper populaton model predctor). We characterze the extent of the reducton n the theoretcal expected mean quared error, C07ed3.doc /9/007 :37 PM 4

8 and llutrate reult for emprcal predctor va mulaton relatve to mxed model and uper populaton model predctor. We conclude by hghlghtng model feature that have conequence n extendng th ork and related ork that offer promng poblte for future mprovement.. A EXPADED RP MODEL FOR A FIITE CLUSTERED POPULATIO We frt defne notaton and termnology for a clutered fnte populaton. Let a fnte populaton be defned by a ltng of M unt labeled t =,..., M n each of cluter, labeled =,...,, here the non-tochatc repone for unt t n cluter gven by y t. We aume that the repone for a unt can be oberved thout error, and correpond to the expected value for the unt. The fnte populaton parameter correpondng to the mean and varance of cluter, =,...,, are defned by M M M μ = yt and σ ( ) = yt μ M t= M M t= (here e ue the urvey amplng defnton of the parameter σ ). Smlarly, the populaton mean, and the varance beteen cluter mean are defned a μ μ = =. = = and σ ( μ μ) Ung thee parameter, e repreent the potentally obervable repone for unt t n cluter a y t = μ + β + ε t C07ed3.doc /9/007 :37 PM 5

9 here β = ( μ μ the devaton of the mean for cluter from the overall mean, and ) ε t ( y μ ) = the devaton of unt t repone (n cluter ) from the mean for t y = ( y y y ) cluter. Defnng model can be ummarzed a here y = y y y, the y = Xμ + Zβ + ε ( ) M here X =, Z = M = a column vector of one,,, and = ( ) = = M = β β β β. Here, an A denote a block dagonal matrx th block gven by a A (Graybll 983), and ε defned mlarly to y. one of the term n the model are random varable... Random Varable and The To Stage Random Permutaton Model We explctly defne a vector of random varable that repreent a to tage RP of the populaton. Aumng that each realzaton of the permutaton equally lkely, th probablty! M! =, the random varable formally repreent to-tage amplng (Cochran 977). We aume that the ample cluter are n the frt n poton n a permutaton of cluter. Smlarly, e aume that the ample unt n cluter correpond to the unt n the frt m poton n a permutaton of the cluter unt. C07ed3.doc /9/007 :37 PM 6

10 ote that thee defnton repreent random varable a a equence a oppoed to the more uual et notaton. When all cluter are of equal ze uch that M = M for all =,...,, Stanek and Snger (004) defned ndcator random varable to explctly repreent a random vector of dmenon M correpondng to a to-tage permutaton of the populaton. We follo a mlar trategy hen cluter ze dffer, but note to dfference. Frt, to retan the dentty of cluter and PSU n the vector repreentng a RP, e expand the number of random varable from to. Second, e nclude a et of fxed knon eght aocated th the SSU n a cluter. Dfferent target parameter (uch a the cluter mean, total, or a cluter regreon parameter) can be formed ealy by changng the eght. The expanded model ue a larger et of random varable than the random varable ued n Stanek and Snger model, or n uper-populaton model approache. Thee random varable are feer than the random varable that ould reult from a further expanon that ould retan the dentty of unt and SSU, and even feer than the very general repreentaton of the model ued by Godambe (955). We attrbute the better effcency of predctor (to be hon) under the expanded model n part to the larger number of random varable ued to repreent the bac problem. Alo notce that the to-tage RP model fathfully repreent a to-tage cluter amplng degn, o that the method are degned baed. We defne the eghted expanded random varable, and llutrate the need for the expanon next. C07ed3.doc /9/007 :37 PM 7

11 Sample ndcator random varable relate the repone for unt t n cluter, to the repone for a unt n poton j of a cluter n poton n a to tage permutaton of cluter and unt. We defne U a an ndcator random varable that ( ) jt y t, take on a value of one hen SSU j n cluter unt t, and zero othere, and ue t to repreent the random varable correpondng to SSU j n cluter gven by Y j M = U t= ( ) jt y. Let t denote a fxed non-tochatc eght for =,...,, j =,..., M, j and defne the eghted repone a Y = Y j j j. For example, M Y j correpond to a j= cluter total hen = for all j =,..., M, or a cluter mean hen j j = for M all j =,..., M. otce that e can defne ( ) = y U here Y j j j ( ) ( ) ( ) ( ) ( ) U j = U j U j U jm, and Y = ( Y Y Y M ). The vector Y repreent a permutaton of eghted repone for SSU n cluter. A permutaton of cluter defned ung the ndcator random varable =,..., and =,...,, that take on a value of one hen PSU cluter, and a value of zero othere. If all cluter ere equal n ze, e could repreent a U for permutaton of SSU for PSU by expreon are defned only hen = U Y. When cluter ze dffer, element n th j mn( M, =,..., ). Mxed model and C07ed3.doc /9/007 :37 PM 8

12 uperpopulaton model repreentaton of to tage cluter amplng do not formally account for th range retrcton n lnkng the random varable to the fnte populaton.. We drectly account for dfferent ze cluter, and avod the requrement of range contrant for ubcrpt by expandng the number of random varable ued to repreent a permutaton of cluter and unt. For PSU, the expanded random varable are defned by the vector Y = ( U Y ) = U Y U Y U Y ( ) ( ), th a to tage random permutaton of the populaton repreented by the vector, Y = Y = Y Y Y. An element of Y gven by UY j. (( )) ( ) A mple example llutrate the notaton. Suppoe the populaton cont of three cluter ( = 3), here the frt to cluter have to unt ( M = M = ), the thrd cluter ha three unt ( M 3 = 3 ), = 7, and = for all =,...,, j =,..., M. We repreent a permutaton of unt for cluter by Y, =,...,3. Suppoe the frt permutaton of cluter reult n cluter =, =, and = 3 n poton repectvely, hle a econd permutaton reult n cluter j =,...,3 = 3, =, and = n poton =,...,3 repectvely. The repreentaton of the random varable realzed by the frt permutaton of PSU the random vector ( Y Y Y 3), and by the econd permutaton of PSU the random vector ( Y 3 Y Y ). Although both vector are of dmenon, the thrd SSU n the frt permutaton n PSU =, hle the thrd SSU n the econd permutaton n PSU =. The poton of a SSU n the permuted C07ed3.doc /9/007 :37 PM 9

13 populaton not uffcent to retan the dentty of the PSU for the SSU. In contrat, ung the expanded random varable repreentaton, the random varable realzed by the frt permutaton of PSU are repreented by ( Y Y Y3), hle thoe realzed by the econd, here permutaton are repreented by ( 0 0 Y 3 0 Y 0 3 Y 0 03) 0 a an a vector th all element equal to zero. Th notaton preerve the dentty of the PSU for each SSU, avodng the ambguty that are n mxed model and uperpopulaton model.. A Mxed Effect Model for the Expanded Random Varable We repreent a mxed model for the expanded RP model ndexng repreentng expectaton th repect to permutaton of cluter th the ubcrpt ξ and expectaton th repect to permutaton of unt n a cluter th the ubcrpt ξ. For PSU, e expre here Eξξ ( ) = Y = E Y + E Y E Y ( ) ( ) ( ) +E ξξ ξξ ξξ Y = μ (( )) ( ) j M, E ξ ξ ( ) Y = μ U =, E = Y E ξ ξ ( Y ) = =, μ = ( μ ) = μ μ μ and (( )) ( ( ) ( ) U = U = U U U ). The fxed effect are gven by μ, the vector of, C07ed3.doc /9/007 :37 PM 0

14 cluter mean. The random effect, Eξ ξ( Y ) Eξ ξ( Y ), are defned a the devaton from the fxed effect of the expected repone condtonal on a realzed PSU. In the RP model of Stanek and Snger (004), the random effect for the mean of PSU a defned explctly a Uβ, th the random varable U explctly lnkng the cluter to = PSU. In the expanded RP model, random effect are defned for SSU j n PSU a ( j μ U Eξ ( U )). For both model, the expected value of the random effect (th repect to ξ ) zero, but th reult are from qute dfferent crcumtance. The devaton of repone from the expected repone thn a PSU repreented by E. We combne thee expreon arrvng at the expanded RP mxed model gven by Y = + μ vec( Eξ ( )) + = μ I = U U E () here var ξξ U = ( U U U ). The varance of random effect gven by I μ vec( U Eξ ( U) ) = P μ P μ = = = hle varξξ ( ) M M σ E = I j M j = P, here j= j= Pa = Ia J a and a J a denote an a a matrx th all element equal to one..3. Defnng Random Varable of Interet C07ed3.doc /9/007 :37 PM

15 Model () an expanded veron of a mxed model that retan the dentty of cluter, hle accountng for a to tage RP. Our nteret n predctng a lnear combnaton of thee random varable defned by T = gy, here g non-tochatc. Although many lnear combnaton are poble, e lmt the dcuon to lnear combnaton gven by T = gy here g = c and c a vector of contant. In partcular, e focu on the ettng here c = e an vector th all element equal to zero, except for element hch ha the value of one. Of prncpal nteret the ettng here n, uch that the cluter of nteret realzed n the ample. When j = for all =,...,, j =,..., M, the target random varable, M M T = U jy j the mean of PSU ; hen j = for all =,...,, = j= M j =,..., M, the target random varable, T = U jy j the total of PSU. = j= 3. PREDICTIG A PSU MEA USIG A EXPADED RP MODEL The bac trategy for developng a predctor under a model-baed approach gven n many place (Scott and Smth 969; Royall 976; Bolfarne and Zack 99; Vallant et al. 000), and appled n a degn-baed frameork to a balanced to tage cluter amplng by Stanek and Snger (004). Stanek and Snger hoed that the predctor of a realzed PSU mean from the RP model outperform the mxed model and C07ed3.doc /9/007 :37 PM

16 uperpopulaton model predctor. We develop mlar reult ung the expanded RP model, and llutrate that the reultng predctor of a realzed PSU mean ha even loer expected MSE relatve to the predctor gven by Stanek and Snger (004). We aume that the element n the ample porton of Y ll be oberved, and expre T a the um of to part, one hch a functon of the ample, and the other hch a functon of the remanng random varable. Then, requrng the predctor to be a lnear functon of the ample random varable and to be unbaed, coeffcent are evaluated that mnmze, the expected value of the MSE gven by var ( ˆ ξξ T T). Whle n theory, an optmal predctor can be expreed follong th predcton recpe, n practce, the hgh dmenonalty of the vector from the expanon of random varable may reult n ngularte that prevent unque oluton (a n Stanek, Snger, and Lencna (004)). In part for th reaon, e project the random varable nto a loer dmenonal pace that retan the neceary nformaton for an optmal oluton, mplfyng the problem. The projecton dffer from the projecton ued hen all cluter have equal ze. 3.. Partal Collapng of the Expanded RP Random Varable Rao and Bellhoue (978) (ee Theorem.) provde a ay of determnng hether the optmal lnear unbaed predctor of a target random varable, T = gy be obtaned a the optmal lnear unbaed predctor of T = g Y baed on Y = CY, a vector of random varable that pan a loer dmenonal pace. We apply th p p p can C07ed3.doc /9/007 :37 PM 3

17 ( ) m 0 M m = = theorem hen C = ( 0 m ) M m = = g p = g C CC and ( ), here m repreent the number of ample unt n cluter. The addtonal ubcrpt p denote the partal collapng of the expanded random varable. The collapng um the SSU for the ample and remander n each cluter for each PSU, reducng the number of random varable from to. Snce P = I C CC C C M that = [ ] p Y = C CC Y + P Y, here ( ) p C ( ), e can expre ( ) g c, gp Y = 0 unbaed predctor of T baed on g = ( P Y ) 0, T ˆ ll be optmal for C C Y p g Y = g CCC Y + g PY. otce and T p p p C = g Y. Defnng Tˆ a the optmal lnear, and ˆB a a lnear unbaed predctor of T determne condton under hch ( ) E Tˆ T Bˆ ξξ = 0. We = gy f and only f ( ) E Tˆ T Bˆ ξξ = 0 by ung the unbaed contrant of the predctor, and expreng E ( ˆ ξξ T T) Bˆ a a functon of E ξξ ( YY ) Smplfyng term, e fnd that hen j = for all,..., j = M, E ( ˆ ) ˆ ξξ T T B = 0 (ee c06ed54.doc for detal). Th mean that e can obtan the optmal predctor ung the partally collaped random varable a long a thn each cluter, the eght are equal for SSU.. C07ed3.doc /9/007 :37 PM 4

18 We aume that j = for all j =,..., M n ubequent development, and develop the bet lnear unbaed predctor of T = g py p baed on Y p. The vector Y p contan random varable. The frt random varable are of the form UmY I, hle the remanng random varable are of the form ( ) U M m Y II, here Y I = m Y j and m j = Y II = M Y j. j m M m = + p It natural to conder hether or not t uffcent to predct T = g py p here [ ] g = c ung ( ) * Yp = C Y p here * C = I. ote that * T = g Y * here * * * * * g = gp C C C. ote that ( *) = * C C C I and g = c. We refer to the random varable n Y a the collaped random varable. Th et p of random varable mlar to thoe ued by Stanek and Snger (004) for a populaton th equal ze cluter and equal ze ample per cluter. When there no repone error, the collaped random varable can be ued to develop the ame predctor of a lnear combnaton of PSU mean a that obtaned by Stanek and Snger (004). Our goal to ee hether or not e loe nformaton n dong th collapng by applyng the Rao-Bellhoue theorem. Frt, e fnd that an unbaed predctor of T ung Y p can be obtaned only f amplng of cluter th probablty proportonal to ze. baed on. We proceed n a mlar manner further collapng of the random varable to a et of random varable, th one varable for the ample SSU and one for the remanng SSU n each PSU. Th can be accomplhed by er(would t be poble to collape thee random varable further?) p 3.. Predctng a Parameter for a PSU Ung Collapng Expanded RP Random Varable C07ed3.doc /9/007 :37 PM 5

19 Y I We partton Y p nto the frt n random varable correpondng to the ample,, and the remanng random varable, Y II to predct T = gy I I + giiyii, here g I = ci and II = ( II ) c = c c, here ci an n vector of contant. Explctly, the parttoned RP model defned by Y I XI Y I Y I E I = μ + Eξ E ξξ + YII XII YII YII EII g c c, and ( I II ) n m = here XI = n m = and XII =, random effect ( M m) = In fd vec( UI Eξ ( U )) I = Y I Y I are gven by E E n fd vec( II Eξ ( )) ξ II ξξ = I U U =, YII YII I ( f) d vec Eξ ( ) = U U m E I Y I Y I here f =, d = Mμ, = E ξ M EII YII Y, and U = ( UI U II ), II UI = (( U) ) = ( U U U n ) and UII = (( U )) = ( Un+ Un+ U ). We Y VI VI, II I partton the varance n a mlar manner, repreentng t by varξξ = Y II VIII, VII * here V I = n n fd fd + n f e = = = I J P I v,, V III, J n ( ) n ( n) fd f d I 0 J n ( n) n P = = = * fd fd P n f ve = = I 0 n ( n) = C07ed3.doc /9/007 :37 PM 6

20 I n J n n 0 I J ( n) n ( n) fd P fd fd ( f) d = = P = = I n 0 I n ( n) n * VII = 0 n ( n) + 0 f ( ) v n n e = J P I ( n) n I I n ( f) d ( f) d P = = ( f) dp fd = = * f Mσ and ve =. f We develop an expreon for the bet lnear unbaed predctor of T next. The predctor a lnear functon of the ample, T ˆ = LY. Snce ˆ Y I T T = ( L g I g II ), the unbaed contrant gven by YII ( ) 0 L g I XI g IIX II =. Mnmzng var ( ˆ ξξ T T) accountng for the unbaed contrant ung Lagrange multpler, reult n the famlar oluton, I ( ), ( ) ˆ L = gi + VI VI XI XI VI X I XI VI VI IIgII + VI XI XI VI XI XIIgI I. Th reult mplfe to (ee c06ed56.doc, p4 and c07ed0.doc p) ˆ k n L = P c + + * n I c n cii n = f n = f n ( ) here k k k k d, * * = * d * = k k d = d v * + ( ) e, k k = =, c II = c and n = + n c = = c. The predctor T ˆ = LY ˆ I can be expreed a (ee page 6 of c07ed0.doc) C07ed3.doc /9/007 :37 PM 7

21 n ( ) ( ) ( ) n Tˆ = c Yˆ Yˆ + c I M Y + c I M f Y I II I = n = n = here * = MkYI = Yˆ U, Yˆ n = Yˆ, n = I n = U an ndcator ncluon random = varable for cluter n the ample, and Y I m = Y j m j =. An expreon for the expected mean quared error (EMSE) of the predctor can be developed drectly ung expreon for the parttoned varance, and mplfy to (ee c07ed7.doc, p5) n ( ˆ ) ( ) ( c ) * * * I kd kd, d e e = = n = c c ( c nci ) σ d = n varξξ T T = c c σ σ + k v + v + + here c I n n = = c, μd = d, = μ * kd kd =, * =, σ = ( kd μ ) kd kd = σd = ( d ) * μ and σ ( )( ) d kd, d = kd μkd d μd =. = When predctng a PSU mean, =, and the predctor mplfe to M ˆ ( ) ˆ T = IYI + Y Yˆ f n, and to ˆ T = I( YI) n = n hen > n. The EMSE of the = ample PSU mean predctor (hen n) mplfe to (ee c07ed5.doc, page 3) ( ˆ n * * varξξ T T) = ( ) ( ) k k n μ μ = =, * σ + + ( n ) k ( f) n m = C07ed3.doc /9/007 :37 PM 8

22 hle the EMSE of a PSU not n the ample gven by 4. COMPARISO OF EMSE ( ˆ n + T T) = σ + ( f ) σ varξξ. n n = m 5. EMPIRICAL PREDICTOR SIMULTIO 6. APPLICATIO 7. DISCUSSIO OLD STUFF 4. APPLICATIO Table. Predctor of latent value of a realzed PSU Mean baed on Dfferent model Predctor Target Sample SSU Remanng SSU Y Smple Mean μ U = * * cy + ( c ) Y P ˆMM Mxed Model P ˆSS Scott&Smth Model P P ˆ μ + k ( ˆ Y μ ) f Y + ( f ) ˆ * * ( ˆ * μ + k Y μ ) P ˆ* RP (PPS) P * fy + ( f ) Y + k ( Y Y) C07ed3.doc /9/007 :37 PM 9

23 P ˆ RP (General) P cy + ( c) Y + k ( Y Y) C07ed3.doc /9/007 :37 PM 0

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25 Graybll, F. A. (983), Matrce th applcaton n tattc, Belmont, Calforna: Wadorth Internatonal. Henderon, C.R. (984), Applcaton of Lnear Model n Anmal Breedng, Guelph, Canada: Unverty of Guelph. Henderon, C. R., Kempthorne, O., Searle, S. R. and von Krogk, C. M., (959), The Etmaton of Envronmental and Genetc Trend from Record Subject to Cullng, Bometrc, 5, 9-8. Hnkelmann, K., and Kempthorne, O. (994), Degn and Analy of Experment, Vol., Introducton to Expermental Degn, e York: Wley. L, W. (003), Ue of random Permutaton Model n rate Standardzaton and Calbraton, unpublhed doctoral the, Unverty of Maachuett, Maachuett. McCulloch, C. E. and Searle, S. R. (00), Generalzed, Lnear, and Mxed Model, e York: John Wley and Son. McLean, R. A., Sander, W. L. and Stroup, W. W. (99), A Unfed Approach to Mxed Lnear Model, The Amercan Stattcan, 45(), C07ed3.doc /9/007 :37 PM

26 Ockene, I. S., Hebert, J. R., Ockene, J. K., Sapera, G. M., colo, R., Merram, P.A. and Hurley, T. G. (999), Effect of Phycan-delvered utrton Counelng Tranng and an Offce-upport Program on Saturated Fat Intake, Weght, and Serum Lpd Meaurement n a Hyperlpdemc Populaton: Worceter Area Tral for Counelng n Hyperlpdema (WATCH), Archve of Internal Medcne, Apr, 59(7), Rao, J..K. (003), Small Area Etmaton, e York: Wley. Robnon, G. K. (99). That BLUP a Good Thng: the Etmaton of Random Effect, Stattcal Scence, 6(), 5-5. Royall, R. M. (976), The Lnear Leat-quare Predcton Approach to To-tage Samplng, Journal of the Amercan Stattcal Aocaton, 7, Sarndal, C-E, Senon, B., and Wretman, J. (99), Model Ated Survey Samplng, e York: Sprnger-Verlag. Scott, A. and T. M. F. Smth (969), Etmaton n Mult-tage Survey, Journal of the Amercan Stattcal Aocaton, 64(37), C07ed3.doc /9/007 :37 PM 3

27 Searle, S. R., Caella, G. and McCulloch, C. E. (99), Varance Component. e York: Wley. Stanek, E. J. III and Snger, J. M. (004), Predctng Random Effect from Fnte Populaton Clutered Sample th Repone Error, Journal of the Amercan Stattcal Aocaton, 99, Vallant, R., Dorfman, H. A. and Royall, R. M. (000), Fnte Populaton Samplng and Inference, e York: Wley. Verbeke, G., and Molenbergh, G. (000), Lnear Mxed Model for Longtudnal Data, e York: Sprnger-Verlag. C07ed3.doc /9/007 :37 PM 4