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Annexes 184166 C. Proust, H. Jacqmin-GaddaSpiessens and Verbeke [3] recently proposed a freeSAS-macro (HETNLMIXED) using the EM algorithmand the NLMIXED procedure for the optimization inthe M-step. This SAS-macro is an extension of theSAS-macro HETMIXED which was developed earlierfor estimating heterogeneous linear mixed modelsusing the MIXED procedure [6]. To our knowledge,HETNLMIXED and its first version HETMIXED are theonly free available programs developed for estimatingheterogeneous mixed models. The first versionHETMIXED was proved to be very slow and limitedto small samples due to very large matrices handlingand prohibitive computation; it will not beexpanded in this work. HETNLMIXED was developedto reduce these computational problems and to allowestimation of both linear and generalized linearmodels. However, in the linear case, this SAS-macrohas the drawback of computing numerically an integralacross the random effects while it has a closedform, and thus the macro is limited to a small numberof random effects. We have also observed convergenceproblems when using the macro with largesamples except for very simple models.Moreover, the EM algorithm, which is used inthese macros, has some general drawbacks. In particular,it does not have any good convergence criteria;the convergence is only built on a lack ofprogression of the likelihood or the parameter estimates[7]. Furthermore, the convergence is slow[8] and the EM algorithm does not provide directestimates of the variance of the parameters. In theparticular case of an heterogeneous mixed model,the M-step also requires the estimation of an homogeneousmixed model which is computationallyexpensive.Therefore, the first aim of this paper is to proposea program for estimating more general heterogeneouslinear mixed models suitable for largesamples. The proposed program HETMIXLIN is writtenin Fortran90 and uses a direct maximization ofthe likelihood via a Marquardt optimization algorithm.The second objective of this paper is to illustratethe use of heterogeneous linear mixed modelthrough a study of the different patterns of evolutionin cognitive ageing.2. Computational methods and theory2.1. The heterogeneous linear mixed modelLet Y i = (Y i1 ,...,Y ini ) be the response vector forthe n i measurements of the subject i with i =1,...,N. The linear mixed model [1] for the responsevector Y i is defined as:Y i = X iˇ + Z i u i + i (1)X i is a n i ×p design matrix for the p-vector offixed effects ˇ, and Z i is an i ×q design matrix associatedto the q-vector of random effects u i whichrepresents the subject specific regression coefficients.The errors i are assumed to be normallydistributed with mean zero and covariance matrix 2 I ni , and are assumed to be independent from thevector of random effects u i .In an homogeneous mixed model [1], u i is normallydistributed with meanand covariance matrixD, i.e.u i ∼N(, D) (2)In the heterogeneous mixed model [2–4], u i is assumedto follow a mixture of G multivariate Gaussianswith different means ( g ) g=1,G and a commoncovariance matrix D, i.e.u i ∼G∑ g N( g , D) (3)g=1Each component g of the mixture has a probabilityg and the ( g ) g=1,G verify the following conditions:0 ≤ g ≤ 1∀g = 1,G andG∑ g = 1 (4)g=1In this work, we propose a slightly more generalformulation of the model described in (1) in whichthe effect of some covariates may depend on thecomponents of mixture and some of the randomeffects may have a common mean whatever thecomponent of mixture. Thus, the X i design matrix issplit in X 1i associated with the vector ˇ of fixed effectswhich are common to all the components andX 2i associated with the vectors ı g of fixed effectswhich are specific to the components. The Z i designmatrix is also split in Z 1i associated with the vectorv i of random effects following a single Gaussiandistribution and Z 2i associated with the vector u iof random effects following a mixture of Gaussiandistributions. The model is then written as:Y i = X 1iˇ +G∑ g X 2i ı g + Z 1i v i + Z 2i u i + i (5)g=1where v i ∼N(0,D v ) and u i ∼ ∑ Gg=1 g N( g ,D u );given the component ( g, ) the conditional (( ) distributionof the vector is N , D with)vi0u i ( ) gDv DD = vu.D uv D u