Predicting Random Effects in Group Randomized Trials. Edward J. Stanek III. PredictingRandomEffects1-stanek.doc 10/29/2003 8:55 AM

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1 Predicting Random Effect in Group Randomized Trial Edward J. Stanek III

2 Abtract The main objective in a group randomized trial i a comparion between treatment. Often, there i interet in the expected repone for a particular group receiving a treatment. Since a random ample of group i included in the trial, the parameter for uch a group i repreented a a random effect. In thi context, predicting the expected repone for a group require predicting a random effect. We review propoed olution to thi problem focuing on difference in the model aumption in a imple etting. We conclude that while popular computing olution exit, fundamental open quetion remain that impact interpretation. Keyword: Mixed model, random effect, variance component, uper-population model, model baed, deign baed, Bayeian etimation, Empirical Baye.

3 Introduction The main objective in a group randomized trial i a comparion of the expected repone between treatment, where the expected repone for a given treatment i defined a the average expected repone over all group in the population. Often, there may be interet in the expected repone for a particular group included in the trial. Since a random ample of group i included in the trial, thi i uually repreented a a random effect. In thi context, predicting the expected repone for an individual group require predicting a random effect. For example, in a tudy of the impact of teaching paradigm on ubtance ue of high chool tudent in New Haven, Connecticut, high chool were randomly aigned to an intervention or control condition. The main evaluation of the intervention wa a comparion of tudent repone between intervention and control averaged over all high chool. There wa alo interet in tudent repone at particular high chool. Since high chool were randomly aigned to condition, the difference in repone for a particular high chool from the population average repone i a random effect. Background Since the early work on analyi of group randomized trial [-3] model for repone have included both fixed and random effect. Such model are called mixed model. Fixed effect appear directly a parameter in the model, while group effect are 3

4 included a random variable. The random effect have mean zero and a non-zero variance. An advantage of the mixed model i the imultaneou incluion of population parameter for treatment while accounting for the random aignment of group in the deign. Hitorically, ince the main focu of a trial i comparion of treatment over all group, and group are aigned at random to a treatment, the effect of an aigned group wa not conidered to be of interet. Thi perpective ha reulted in many author [4-8] limiting dicuion to etimation of fixed effect. Support for thi poition tem from the fact that a group cannot be guaranteed to be included in the tudy. Thi fact, plu the reult that the average of the group effect i zero, ha been ufficient for many to limit dicuion of group effect to etimating the group variance, not particular group effect. Neverthele, when conducting a group randomized trial, it i natural to want to etimate repone for a particular group. Due to the limitation of random effect in a mixed model, ome author [9, 0] ugget that uch etimate can be made, but they hould be baed on a different model. The model i conditional on the group aignment, and repreent group a fixed effect. With uch a repreentation, repone for individual group can be directly etimated. However, the fixed effect model would not be uitable for etimating treatment effect, ince the evaluation would not be baed on the random aignment of group to treatment. The apparent neceity of uing different model to anwer two quetion in a group randomized trial ha prompted much tudy []. A baic quetion i whether a mixed model can be ued to predict the mean of a group that ha been randomly aigned to a treatment. If uch prediction i poible, the mixed model would provide a unified 4

5 framework for addreing a variety of quetion in a group randomized trial, including prediction of combination of fixed and random effect. A number of worker [-5] argue that uch prediction i poible. The predictor i called bet linear unbiaed predictor (BLUP) [6]. Moreover, ince the BLUP minimize the expected mean quared error (EMSE), it i optimal. In light of thee reult, it may appear that method for prediction of random effect are on firm ground. There are, however, ome problem. The BLUP of a realized group i cloer to the overall ample treatment mean than the ample group mean, a feature called hrinkage. Thi hrinkage reult in a maller average mean quared error. The reduction in EMSE i often attributed to borrowing trength from the other ample obervation. However, the bet linear unbiaed predictor of a realized group i biaed, while the ample group mean i unbiaed. The apparent contradiction in term i due to two different definition of bia. The BLUP are unbiaed in the ene that there i zero average bia over all poible ample (i.e. unconditionally). The ample group mean i unbiaed conditional on the realized group. A imilar paradox may occur when conidering the EMSE, where the mean quared error (MSE) for the bet linear unbiaed predictor may be larger than the MSE of the group mean for a realized group. Thi can occur ince for the BLUP, the EMSE i bet in the ene that the average MSE i minimum over all poible ample (i.e. unconditionally), while for the ample group mean, the MSE i evaluated for a realized group [7]. Thee difference lead to etting, uch a when the ditribution of group mean i bi-modal, where the BLUP eem inappropriate for certain group [8]. 5

6 For uch reaon, predictor of random effect are omewhat controverial. We do not reolve uch controverie here. Intead, we attempt to clearly define the problem, and then decribe ome competing model and olution. Statitical reearch in thi area i active, and new inight and controverie may till emerge. Method We provide a imple framework that may be helpful for undertanding thee iue in the context of a group randomized trial. We firt define a tudy population, along with population/group parameter. Thi provide a finite population context for the group randomized trial. Next, we explicitly repreent random variable that arie from random aignment of group, and ampling ubject within a group. Thi context provide a framework for dicuion of variou mixed model and aumption. We then proceed to outline development of predictor of random effect. We limit thi development to the implet etting. Finally, we conclude with a broader dicuion of other iue. Modeling Repone for a Subject Suppoe a group randomized trial i to be conducted to evaluate the impact of a ubtance abue prevention program in high chool. We aume that for each tudent, a meaure of the perception of peer ubtance abue repone can be obtained from a et of item of a quetionnaire adminitered to the tudent a a continuou repone. We denote the th k meaure of repone (poibly repeated) of tudent t in high chool by 6

7 Ytk = yt + Wtk () indexing tudent by t =,..., M in each of N high chool, indexed by =,..., N. Meaurement error (correponding to tet, re-tet variability) i repreented by the random variable W tk repone tudent t ) from Y tk, and ditinguihe y t (a fixed contant repreenting the expected. The ubcript k indicate an adminitration of the quetionnaire, where potentially k >. The average expected repone over tudent in chool i defined a M µ = yt, while the average over all chool i defined a M t= N µ = µ. We will refer to µ a the latent value of chool. We limit dicuion to N = the implet etting where each chool ha the ame number of tudent, an equal number of chool are aigned to each intervention, an equal number of tudent are elected from each aigned chool and a ingle meaure of repone i made on each elected tudent. Thee definition provide the context for defining the impact of a ubtance abue prevention program. To do o, we imagine that each tudent could be meaured both with and without the intervention. If no ubtance abue program i implemented, let the expected repone of a tudent be y t ; if an intervention program i in place, let the expected repone be * y t. The difference, y y = δ repreent the effect of the * t t t intervention on tudent t in high chool. The average of thee effect over tudent in M chool i defined a δ = δ t, while the average over all chool i defined a M t= 7

8 N δ N = δ =. The parameter δ i the main parameter of interet in a group randomized trial. To emphaize the effect of the chool and of the tudent in the chool, we define ( ) β = µ µ a the deviation of the latent value of chool from the population mean and ε ( y µ ) = a the deviation of the expected repone of tudent t (in chool ) t t from the chool latent value. Uing thee definition, we repreent the expected repone of tudent t in chool a y t = µ + β + εt (). Thi model i called a derived model [9]. The effect of the intervention on a tudent can * * * be expreed in a imilar manner a δ = δ + δ + δ where δ = δ δ and t t δ = δ δ. Combining thee term, the expected repone of tudent t in chool * t t receiving the intervention i y = µ + δ + β + ε where * * * t t * β = β + δ and ε = ε + δ. * t t t When the effect of the intervention i equal for all tudent (i.e., δ t the expected repone for tudent t (in chool ) by y = µ + x δ + β + ε t t = δ ), we repreent where x i an indicator of the intervention, taking a value of one if the tudent receive the intervention, and zero otherwie. We aume the effect of the intervention i equal for all tudent to implify ubequent dicuion. The latent value for chool under a given condition i given by µ + xδ + β. Random Aignment of Treatment and Sampling 8

9 The firt tep in a group randomized trial i random aignment of group to treatment. We aume that there are two treatment (intervention and control), with n group (i.e., chool) aigned to each treatment. A imple way to conduct the aignment i to randomly permute the lit of chool, aigning the firt n chool to the control, and the next n chool to intervention. We index the chool poition in the permutation by i =,..., N, referring to control chool by i =,..., n, and to the intervention chool by i = n+,...,n. A ample of tudent in a chool can be repreented in a imilar manner by randomly permuting the tudent in a chool, and then including the firt j =,..., m tudent in the ample. We repreent the expected repone (over meaurement error) of a tudent in a chool a Y ij, uing the indice for the poition in the permutation of tudent and chool. Note that Y ij i a random variable ince the actual chool and tudent correponding to the poition will differ for different permutation. Once a permutation (ay of chool) i elected, the chool that occupie each poition in the permutation i known. Given a elected permutation, the chool i a fixed effect; model that condition on the chool in the ample portion of a permutation of chool will repreent chool a fixed effect. School are repreented a random effect in a model that doe not condition on the permutation of chool. The reulting model i an unconditional model. We repreent the expected repone for a chool aigned to a poition explicitly a a random variable, and account for the uncertainty in the aignment by a et of indicator random variable, U i, =,..., N where U i take the value of one when chool i aigned to 9

10 poition i, and the value zero otherwie. Uing thee random variable, N = U µ i repreent a random variable correponding to the latent value of the chool aigned to poition i. Uing β ( µ µ ) = and noting that for any permutation, N Ui =, = N Uiµ = µ + Bi where = B N = U β repreent the random effect of the chool i i = aigned to poition i in the permutation of chool. We ue the random variable U i to repreent permutation of chool, and a imilar et of indicator random variable, U that take on a value of one when the ( ) jt th j poition in a permutation of tudent in chool i tudent t, and zero otherwie to relate y t toy ij. For eae of expoition, we refer to the chool that will occupy poition i in the permutation of chool a primary ampling unit (PSU) i, and to the tudent that will occupy poition j in the permutation of tudent in a chool a econdary ampling unit (SSU) j. PSU and SSU are indexed by poition (i and j ), wherea chool and tudent are indexed by label ( and t ) in the finite population. A a conequence, the random variable correponding to PSU i and SSU j i given by N M ( ) =. Y U U y ij i jt t = t= Uing the repreentation of expected repone for a SSU in a PSU, Y = µ + xδ + B + E ij i i ij 0

11 noting that M ( ) U jt =, t= δ N Ui x = xiδ ince the treatment aigned to a poition = depend only on the poition, and N M ( ) E = U U ε. Adding meaurement error, the ij i jt t = t= model i given by Y = µ + xδ + B + E + W * ijk i i ij ijk N M ( ) * where W = U U W. Thi model include fixed effect (i.e., µ and δ ) and ijk i jt tk = t= random effect (i.e., B i and mixed model. E ij ) in addition to meaurement error, and hence i called a Intuitive Predictor of the Latent Value of a Realized Group Suppoe that there i interet in predicting the latent value of a realized group (i.e. chool), ay the firt elected group, µ B U µ N + =, where i = i i =. Before randomly aigning chool, ince we do not know which chool will be firt, the expected value of the firt PSU i a random variable repreented by the um of a fixed (i.e. µ ) and a random effect (i.e. B i where i = ). Once the chool correponding to the firt PSU ha been randomly aigned, the random variable U i for i = and =,..., N will be realized. If chool * i aigned to the firt poition, then the realized value, i.e., U = u, =,..., N, are u = 0 when * and u = when = * ; the parameter

12 for the realized random effect will correpond to β *, the deviation of the latent value of chool * from the population mean. We dicu method for predicting the latent value of a realized PSU aigned to the control condition (i.e. i n) in the implet etting when there i no meaurement error. The model for SSU j in PSU i i the imple random effect model, Y = µ + B + E (3). ij i ij The latent value of PSU i i repreented by the random variable µ + Bi. The parameter for the latent value of chool i M µ = yt. We wih to predict the latent value for M t= the chool correponding to PSU i which we refer to a the latent value of the realized PSU. It i valuable to develop ome intuitive idea about the propertie of predictor. For chool, we can repreent the latent value a the um of two random variable, i.e., µ m M = Yj + Yj M j= j= m+ where M ( ) Y = U y repreent repone for SSU j in j jt t t= chool. Let Y I m m j = = Y and j Y II M Yj repreent random variable j m = M m = + correponding to the average repone of SSU in the ample and remainder, repectively. Then fy ( ) µ = + f Y, where the fraction of tudent elected in a I II chool i given by f m =. If chool i a control chool (i.e. one of the firt n PSU), M then the average repone for tudent elected from the chool, Y I, will be realized after ampling and the only unknown quantity in the expreion for µ will be Y II, the average

13 repone of tudent not included in the ample. Framed in thi manner, the eential problem in predicting the latent value of a realized PSU i predicting the average repone of the SSU not included in the ample. The predictor of the latent value of a realized PSU will be cloe to the ample average for the PSU when the econd tage ampling fraction i large. For example, repreenting a predictor of Y II by Y ˆII, and auming that f = 0.95, the predictor of the latent value of chool i ˆ µ ˆ = YI + YII. Even poor predictor of Y II will only modetly affect the predictor of the latent value. Thi obervation provide ome guidance for aeing different predictor of the latent value of a realized random effect. A the econd tage ampling fraction i allowed to increae, a predictor hould be cloer and cloer to the average repone of the ample SSU for the realized PSU. Model and Approache We provide a brief dicuion of four approache that have been ued to predict the latent value of a realized group, limiting ourelve to the implet etting given by (3). The four approache correpond to Henderon approach [, 0], a Bayeian approach [6], a uper-population model approach [, ], and a random permutation model approach []. An influential paper by Robinon [3] and it dicuion brought prediction of realized random effect into the tatitical limelight. Thi paper identified different application (uch a electing ire for dairy cow breeding, Kalman filtering in time erie analyi, Kriging in geographic tatitic) where the ame baic problem of prediction of realized random effect aroe, and directed attention to the bet linear 3

14 unbiaed predictor (BLUP) a a common olution. Other author have dicued iue in predicting random effect under a Bayeian etting [3], random effect in logitic model [4-6], in application to blood preure [7], choleterol, diet and phyical activity [7]. Textbook preentation of predictor are given by [6, 8-30]. A general review in the context of longitudinal data i given by [3]. Henderon Mixed Model Equation Henderon developed a et of mixed model equation to predict fixed and random effect []. The ample repone for model (3) i organized by SSU in PSU, reulting in the model where Y i an nm repone vector, Y = Xα + ZB+ E (4) X= i the deign matrix for fixed effect, and nm = Z In m i the deign matrix for random effect, where nm denote an nm column vector of one, I n i a n n identity matrix, and A A repreent the Kronecker product formed by multiplying each element in the matrix A by A [3]. The parameter α = E( Y ij ) i a fixed effect correponding to the expected repone, while the random effect are contained in B = ( ). We aume that ( ) 0 var ( ) = B G, ( ) 0 ij B B B n E E =, and var ( E) = R o that ( ) E B =, Σ = var ZB + E = ZGZ + R. Henderon propoed etimating α (or a linear function of the fixed effect) by a linear combination of the data, ay. Requiring the etimator to be unbiaed and have minimum i 4

15 variance lead to the generalized leat quare etimator ( ) ˆα = X Σ X XΣ Y. Difficultie in inverting Σ (in more complex etting) lead Henderon to expre the etimating equation a the olution to two imultaneou equation known a Henderon mixed model equation [0] αˆ ˆ XR X + XR ZB= XR Y ( ) ˆα ˆ ZR X + ZR Z+ G B= ZR Y. Thee equation are eaier to olve than the generalized leat quare equation ince the invere of R and G are often eaier to compute than the invere of Σ. The mixed model equation were motivated by computational need and aroe from a matrix identity [33]. The vector ˆB wa a byproduct of thi identity, and had no clear interpretation. If ˆB could be interpreted a the predictor of a realized random effect, then the mixed model equation would be more than a computational device. Henderon [34] provided the motivation for uch an interpretation by developing a linear unbiaed predictor of α +Β i that had minimum prediction quared error in the context of the joint ditribution of Y and B. Uing Y Xα E = B 0 n and var Y = Σ ZG B GZ G, (5) Henderon howed that the bet linear unbiaed predictor of α + Β i i ˆ α + Bˆ i where B ˆi i the th i element of ˆ = ( ˆα ) B GZΣ Y X. Uing a variation of Henderon matrix identity, one may how that thi i identical to the predictor obtained by olving the econd of Henderon mixed model equation for ˆB. With the additional aumption that G = σ I and n = σ e nm R I where var ( B ) σ i var Eij σ e = and that ( ) =, the expreion 5

16 for ˆi B implifie to B ( ) ˆi k Y i Y =, with k = σ σ + σ e /, m Y i m m j = = Y, and ij Y n Yi n i = =. The coefficient k i alway le than one, and hrink the ize of the deviation of ample mean for the i th PSU from the ample PSU average. The predictor of α +Β i i given by ( ) α + Bˆ = Y + k Y Y. ˆ i i Henderon mixed model equation arie from pecifying the joint ditribution of Y and B. Only firt and econd moment aumption are needed to develop the predictor. A dicued by [6], Henderon tarting point wa not the ample likelihood. However, if normality aumption are added to (5), then the conditional mean, E( α Y y ) α k( y α) predictor of +Β = = +. Replacing α by the ample average, y, the i i i i α +Β i conditional on Y i y k( yi y) +. Bayeian Etimation The ame predictor can be obtained uing a hierarchical model with Bayeian etimation [6]. Beginning with (4), we claify the term in the model a obervable (including Y, X, and Z ), and unobervable (including α, B, and E ). We conider the unobervable term a random variable, and hence ue different notation to ditinguih α from the correponding random variable A. The model i given by Y = XA + ZB+ E 6

17 and a hierarchical interpretation arie from conidering the model in tage [8, 9]. At the firt tage, aume that the cluter correponding to the elected PSU are known, o that A = α0 and B = β 0. At thi tage, the random variable E arie from election of the SSU. At the econd tage, aume A and B are random variable, and have ome joint ditribution. The implet cae (which we conider here) aume that the unobervable term are independent and that A N( α, τ ), B N ( 0, G) n, and ( ) E N 0, R nm, where G = σ I and n R = σ e I nm. Finally, we pecify the prior ditribution for α, τ, σ, and σ e. We condition on Y in the joint ditribution of the random variable to obtain the poterior ditribution. The expected value of the poterior ditribution i commonly ued to etimate parameter. We implify the problem coniderably by auming that σ, and σ e are contant. We repreent the lack of prior knowledge of the ditribution of A by etting τ =. With thee aumption, the Bayeian etimate of B i the expected value of the poterior ditribution, i.e., ˆ = ( ˆα ) B GZΣ Y X. Thi predictor i identical to the predictor defined in Henderon mixed model equation. Super-population Model Predictor Predictor of the latent value of a realized group have alo been developed in a finite population urvey ampling context []. To begin, uppoe we conider a finite population of M tudent in each of N chool a the realization of a potentially very large group of tudent, called a uper-population. We do not identify tudent or chool 7

18 explicitly in the uper-population, and refer to them intead a PSU and SSU. Neverthele, there i a correpondence between a random variable, Y ij, and the correponding realized value for PSU i and SSU j, y ij, which would identify a tudent in a chool. Predictor are developed in the context of a probability model pecified for the uper-population. Thi general trategy i referred to in urvey ampling a model baed inference. The random variable correponding to the latent value for PSU i (where i n) given by Y i M = Yij, can be divided into two part, ii, M j = Y and Y iii, uch that ( ) Yi = fyi, I + f Yi, II, where Y, ii m m j = = Y correpond to the average repone of ij SSU that will potentially be oberved in the ample, Y, iii = M Yij i the average M m j = m + repone of the remaining random variable, and f m =. Predicting the latent value for M a realized chool implifie to predicting an average of random variable not realized in the ample. For chool that are elected in the ample, only repone for the tudent not ampled need be predicted. For chool not included in the ample, a predictor i needed for all tudent repone. Scott and Smith aume a neted probability model for the uper-population, repreenting the variance between PSU a σ, and the variance between SSU a σ e. They derive a predictor for the average repone of a PSU that i a linear function of the ample, unbiaed, and ha minimum mean quared error. For elected PSU, the predictor implifie to a weighted um of the ample average repone, and the average 8

19 m M m predicted repone for SSU not included in the ample, y ˆ i + Y, M M i II, where Y ˆ = y+ k y y and iii, ( i ) k = σ σ + σ e /. Thi ame reult wa derived by Scott and m Smith under a Bayeian framework. For a ample chool, the predictor of the average repone for tudent not included in the ample i identical to Henderon predictor and the predictor reulting from Bayeian etimation. For PSU not included in the ample, Scott and Smith predictor reduce to the imple ample mean, y. There i a ubtantial conceptual difference between Scott and Smith predictor and the mixed model predictor. The difference i due to the direct weighting of the predictor by the proportion of SSU that need to be predicted for a realized PSU. If thi proportion i mall, the reulting predictor will be cloe to the PSU ample mean. On the other hand, if only a mall portion of the SSU are oberved in a PSU, Scott and Smith predictor will be very cloe to the mixed model predictor. Random Permutation Model Predictor An approach cloely related to Scott and Smith uper-population model approach can be developed from a probability model that arie from ampling a finite population. Such an approach i baed on the two-tage ampling deign and i deign baed [35]. Since election of a two tage ample can be repreented by randomly electing a two-tage permutation of a population, we refer to model under uch a approach a random permutation model. Thi approach ha the advantage of defining random variable directly from ampling a finite population. Predictor are developed 9

20 that have minimum expected mean quare error under repeated ampling in a imilar manner a thoe developed by Scott and Smith. In a ituation comparable to that previouly decribed for Scott and Smith, predictor of the realized latent value are nearly identical to thoe derived by Scott and Smith. The only difference i the ue of a lightly different hrinkage contant. The predictor i given by m M m y ˆ i + Y M M * i, II, where Y ˆ = y+ k y y, * * iii, ( i ) k * = σ σ * * + σ e /, and σ m σ = σ. M * e Practice and Extenion All of the predictor of the latent value of a realized group include hrinkage contant in the expreion for the predictor. An immediate practical problem in evaluating the predictor i etimating thi contant. In a balanced etting, imple method of moment etimate of variance parameter can be ubtituted for variance parameter in the hrinkage contant. Maximum likelihood, or retricted maximum likelihood etimate for the variance are alo commonly ubtituted for variance parameter in the prediction equation. In the context of a Bayeian approach, the reulting etimate are called empirical Baye etimate. Replacing the variance parameter by etimate of the variance will inflate the variance of the predictor uing any of the approache. Several method have been developed that account for the larger variance [36-40]. In practice, group in the population are rarely of the ame ize, or have identical within group variance. The firt three approache can be readily adapted to account for uch unbalance when predicting random effect. The principal difference in the predictor 0

21 i replacement of σ e by σ ie, the SSU variance within SSU i, when evaluating the hrinkage contant. The implicity in which the method can account for uch complication i an appeal of the approache. Predictor of realized random effect can alo be developed uing a random permutation model that repreent the two tage ampling. When SSU ampling fraction are equal, the predictor have a imilar form with the hrinkage contant contructed from variance component etimate imilar to component ued in two-tage cluter ampling variance [4]. Different trategie are required when econd tage ampling fraction are unequal [4]. In many etting, a imple repone error model will apply for a tudent. Such repone error can be included in a mixed model uch a (4). When multiple meaure are made on ome tudent in the ample, the repone error can be eparated from the SSU variance component. Bayeian method can be generalized to account for uch additional variability by adding another level to the model hierarchy. Super-population model [43] and random permutation model [35] can alo be extended to account for repone error. Practical application often involve much more complicated population and reearch quetion. Additional hierarchy may be preent in the population (i.e., chool ditrict). Variable may be available for control correponding to ditrict, chool, and tudent. Meaure may be made over time on the ame ample of tudent, or on different ample. Repone variable of primary interet may be continuou, categorical, or ordinal. Some general prediction trategie have been propoed and implemented [9, 44, 45] uing mixed model and generalized mixed model [30], often following a Bayeian paradigm. Naturally, the hypothee and approache in uch etting are more

22 complex. There i active reearch in thee area which hould lead to clearer guidance in the future. Dicuion and Concluion The latent value of a group i a natural parameter of interet in a group randomized trial. While uch a parameter may be readily undertood, development of an inferential framework for predicting uch a parameter i not eay. Many worker truggled with idea underlying interpretation of random effect in the mid 0 th century. Predictor have emerged largely baed on computing trategie and Bayeian model in the pat twenty year. Such trategie have the appeal of providing anwer to quetion that have long puzzled tatitician. Computing oftware (baed on mixed model and Bayeian approache) i widely available and flexible, allowing multi-level model to be fit with covariate at different level for mixed model. Although flexible oftware i not yet available for uper-population model or random permutation model approache, there i ome evidence that when ampling fraction are mall (< 0.5), predictor and their MSE are very imilar [35]. Whether the different approache predict the latent value of a realized group i a baic quetion that can till be aked. All of the predictor have the property of hrinking the realized group mean toward the overall ample mean. While the predictor are unbiaed and have minimum expected MSE, thee propertie hold over all poible ample, not conditionally on a realized ample. Thi ue of the term unbiaed differ from the popular undertanding. For example, for a realized group, an unbiaed etimate

23 of the group mean i the ample group mean, while the BLUP i a biaed etimate of the realized group mean. The rationale for preferring the biaed etimate i that in an average ene (over all poible random effect), the mean quared error i maller. Since thi property refer to an average over all poible random effect, it doe not imply maller MSE for the realized group [8]. In an effort to mitigate thi effect, Raudenbuh and Bryk [8] ugget including covariate that model realized group parameter to reduce the potential biaing effect. Alternative trategie, uch a conditional modeling framework have been propoed [46-48], but increae in complexity with the complexity of the problem. While there i increaing popularity of model that reult in BLUP for realized latent group, the baic quetion that plagued reearcher about interpretation of the predictor in the late 0 th century remain for the future. 3

24 Reference. Scheffe, H. (956). A 'mixed model' for the analyi of variance. Annal of Mathematical Statitic, 7, Kempthorne, O. (975). Fixed and mixed model in the analyi of variance. Biometric, 3, Wilk, M.B. and O. Kempthorne. (955). Fixed, mixed, and random model. American Statitical Aociation Journal, Scheffé, H. (959). Analyi of Variance. Wiley, New York. 5. Murray, D.M. (998). Deign and Analyi of Group-randomized Trial. Oxford Univerity Pre, New York. 6. Searle, S.R., G. Caella, and C.E. McCulloch. (99). Chapter 7. Prediction of Random Variable, in Variance Component, John Wiley: New York. 7. Kirk, R.E. (995). Experimental Deign. Procedure for the Behavioral Science. 3rd Edition ed. Brook/Cole Publihing Company, New York. 8. Donner, A. and N. Klar. (000). Deign and Analyi of Cluter Randomized Trial in Health Reearch. Arnold. 9. Eienhart, C. (947). The aumption underlying the analyi of variance. Biometric, 3, Searle, S. (97). Linear Model. John Wiley and Son, New York.. Samuel, M.L., G. Caella, and G.P. McCabe. (99). Interpreting block and random factor. Journal of the American Statitical Aociation, 86, Henderon, C.R. (984). Application of linear model in animal breeding. Univerity of Guelph, Guelph, Canada. 3. Robinon, G.K. (99). That BLUP i a good thing: the etimation of random effect. Statitical Science, 6, Harville, D.A. (976). Extenion of the Gau-Markov theorem to include the etimation of random effect. Annal of Statitic, 4, Littell, R.C., G.A. Milliken, W.W. Stroup, and R.D. Wolfinger. (996). SAS Sytem for Mixed Model. SAS Intitute, Cary, North Carolina. 6. Goldberger, A.S. (96). Bet linear unbiaed prediction in the generalized linear regreion model. American Statitical Aociation Journal, 57, Stanek, E.J.I., A. Well, and I. Ockene. (999). Why not routinely ue bet linear unbiaed predictor (BLUP) a etimate of choleterol, per cent fat from Kcal and phyical activity? Statitic in Medicine, 8, Verbeke, G. and G. Molenbergh. (000). Linear mixed model for longitudinal data. Springer, New York. 9. Hinkelmann, K. and O. Kempthorne. (994). Deign and analyi of experiment. Volume. Introduction to experimental deign. John Wiley and Son, New York. 0. Henderon, C.R., O. Kempthorne, S.R. Searle, and C.M. von Kroigk. (959). The etimation of environmental and genetic trend from record ubject to culling. Biometric, Scott, A. and T.M.F. Smith. (969). Etimation in multi-tage urvey. Journal of the American Statitical Aociation, 64,

25 . Stanek, E.J.I. and J.M. Singer. (003). Etimating cluter mean in finite population two tage clutered deign. International Biometric Society Eatern North American Region. 3. Cox, D.R. (000). The five face of Bayeian tatitic. Calcutta Statitical aociation bulletin, 50, Ten Have, T.R. and A.R. Localio. (999). Empirical baye etimation of random effect parameter in mixed effect logitic regreion model. Biometric, 55, Ten Have, T.R., J.R. Landi, and S.L. Weaver. (995). Aociation model for periodontal dieae progreion: a comparion of method for clutered binary data. Statitic in Medicine, 4, Heagerty, P.J. and S.L. Zeger. (000). Marginalized multilevel model and likelihood inference. Statitical Science, 5, Rabinowitz, D. and S. Shea. (997). Random effect analyi of children' blood preure data. Statitical Science,, Raudenbuh, S.R. and A.S. Bryk. (00). Hierarchical linear model: application and data analyi method. nd ed. Sage Publication, London. 9. Goldtein, H. (003). Multilevel tatitical modeling, 3rd Edition. Kendall' Library of Statitic 3. Arnold, London. 30. McCulloch, C.E. and S.R. Searle. (00). Generalized, Linear, and Mixed Model. John Wiley and Son, New York. 3. Singer, J.M. and D.F. Andrade. (000). Analyi of longitudinal data, in Handbook of Statitic, E. P.K. Sen and C.R. Rao, Elevier Science B.V.: New York, New York. 3. Graybill, F.A. (983). Matrice with application in tatitic. Wadworth International, Belmont, California. 33. Henderon, H.V. and S.R. Searle. (98). On deriving the invere of a um of matrice. SIAM Review, 3, Henderon, C.R. (963). Selection index and expected genetic advance, in Statitical Genetic and Plant Breeding, National Academy of Science - National Reearch Council. 35. Stanek, E.J.I. and J.M. Singer. (003). Predicting random effect from finite population clutered ample with repone error, Dept. of Biotat/Epid., U of Ma., Amhert, Ma. 36. Kackar, R.N. and D.A. Harville. (984). Approximation for tandard error of etimator of fixed and random effect in mixed linear model. Journal of the American Statitical Aociation, 79, Ka, R.E. and D. Steffey. (989). Approximate Bayeian inference in conditionally independent hierarchical model (Parametric empirical Baye model). Journal of the American Statitical Aociation, 84, Harville, D.A. and D.R. Jeke. (99). Mean quared error of etimation or prediction under general linear model. Journal of the American Statitical Aociation, 87, Da, K., J. Jiang, and J.N.K. Rao. (00). Mean quared error of empirical predictor (Technical Report), School of Mathematic and Statitic, Carleton Univerity, Ottawa, Canada. 5

26 40. Wang, J. and W.A. Fuller. (003). The mean quared error of mall area predictor contructed with etimated area variance. Journal of the American Statitical Aociation, 98, Cochran, W. (977). Survey Sampling. John Wiley. 4. Stanek, E.J.I. and J.M. Singer. (003). Predicting realized cluter parameter from two tage ample of an unequal ize clutered population, Dept. of Biotat/Epid., U of Ma., Amhert, Ma. 43. Bolfarine, H. and S. Zack. (99). Prediction Theory for Finite Population. Springer-Verlag, New York. 44. Zhou, X.-H., A.J. Perkin, and S.L. Hui. (999). Comparion of oftware package for generalized linear multilevel model. The American Statitician, 53, de Leeuw, J. and I. Kreft. (00). Software for multilevel analyi, in Multilevel modelling of health tatitic, A. Leyland and H. Goldtein, Wiley: Chicheter. 46. Dawid, A.P. (979). Conditional independence in tatitical theory. Journal of the Royal Statitical Society, 4, Stanek, E.J.I. and J.R. O'Hearn. (998). Etimating realized random effect. Communication in Statitic- Theory and Method, 7, Verbeke, G., B. Spieen, and E. Leaffre. (00). Conditional linear mixed model. The American Statitician, 55,