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Annexes 185Estimation of linear mixed models with a mixture of distribution for the random effects 1672.2. LikelihoodFollowing the previous works [3,4], we definew ig the unobserved variable indicating if thesubject i belongs to the component g. We haveP(w ig = 1) = g . The density for the vector y i canthen be written as:f i (y i ) =G∑ g f(y i |w ig = 1) (6)g=1Givenw ig , y i follows a linear mixed model, andthe densityf(y i |w ig = 1) denoted by ig is the multivariateGaussian density with meanE ig and covariancematrix V i given by:E ig =E(Y i |w ig = 1) = X 1iˇ + X 2i ı g + Z 2i g andV i = var(Y i |w ig = 1) = Z i DZ ′ i +2 I ni (7)Let now be the vector of the m parametersof the model. contains with ′ =(ˇ′,(ı g ) ′ g=1,G , ( g) ′ g=1,G , Vec(D)′ , 2 ) and the vectorof the G − 1 first component probabilities( g ) g=1,G−1 . Note that G is entirely determined by as 1 − ∑ G−1g=1 g. Vec(D) represents the vector ofthe upper triangular elements of D. The estimatesof are obtained as the vector ˆ that maximizesthe observed log-likelihood:⎛⎞N∑ N∑ G∑L(Y; ) = ln(f i (y i )) = ln ⎝ g ig (y i ) ⎠=i=1N∑i=1i=1g=1− n i2 ln(2) − 1 2 ln(|V i|)⎛G∑+ ln ⎝g=1 g e − 1 2 (Y i−E ig ) ′ V −1i2.3. Estimation procedure(Y i −E ig )⎞⎠ (8)We propose to maximize directly the observedlog-likelihood (8) using a modified Marquardt optimizationalgorithm [9], a Newton—Raphson-likealgorithm [10]. The diagonal of the Hessian atiteration k, H (k) , is inflated to obtain a positivedefinite matrix as: H ∗(k) = (H ∗(k)ij) with H ∗(k)ii=H (k)ii+[(1 −)|H (k)ii|+tr(H (k) )] and H ∗(k)ij=H (k)ijif i ≠j. Initial values for and are = 0.01 and = 0.01. They are reduced when H ∗ is positivedefinite and increased if not. The estimates (k) arethen updated to (k+1) using the current modifiedHessian H ∗(k) and the current gradient of theparametersg( (k) ) according to the formula: (k+1) = (k) −˛H ∗(k)−1 g( (k) ) (9)where if necessary, ˛ which equals 1 by defaultis modified to ensure that the log-likelihood isimproved at each iteration.To ensure that the covariance matrix D is positive,we maximize the log-likelihood on the nonzeroelements of U, the Cholesky factor of D (i.e.U ′ U = D) [7]. Furthermore, to deal with the constraintson (4), we use the transformed parameters( g ) g=1,G−1 with :( )g g = ln(10) GStandard errors of the elements of D and( g ) g=1,G−1 are computed by the -method [11],while standard errors of the other parameters aredirectly computed using the inverse of the observedHessian matrix.The convergence is reached when the threefollowing convergence criteria are satisfied:∑ mj=1( (k)j− (k−1)j) 2 ≤ a , |L (k) −L (k−1) |≤ b andg( (k) ) ′ H (k)−1 g( (k) ) ≤ d . The default values are a = 10 −5 , b = 10 −5 and d = 10 −8 .As the log-likelihood of a mixture model mayhave several maxima [8], we use a grid of initial valuesto find the global maximum. The multimodalityof the log-likelihood in mixture models has been oftendiscussed and some authors proposed differentstrategies to choose the set of initial values [12].However, none of them seems to be optimal in ageneral way. We have observed, in our experience,that the results were mainly sensitive to initial valuesof ( g ) g=1,G−1 and ( g ) g=1,G and less sensitiveto the other parameters (Vec(U), ˇ and) for whichestimates of the homogeneous mixed models weregood initial values.A mixture model is estimated with a fixed numberof components G, otherwise the number of parametersin the model is unknown. To choose theright number of components, one has to estimatemodels with different values for G and select thebest model according to a test or a criterion. Someworks favor a bootstrap approach to approximatethe asymptotic distribution of the likelihood ratiotest between models with different number ofcomponents [13] but this approach is very heavyin particular for mixture models with random effects.Criteria such as Akaike’s Information Criterion(AIC) [14] or Bayesian Information Criterion(BIC) [15] are often preferred. We use these selectioncriteria to select the optimal number ofcomponents.