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Modèle nonlinéaire à processus latent 69Nonlinear Model with Latent Process for Cognitive Evolution 1017where θ is the complete vector of parameters containing thetransformation parameters η ′ k =(η 1k,η 2k ),k =1,...,K, thefixed parameters μ, β, γ 1 ,...,γ K , and the variance–covarianceparameters vec(D),σ w ,σ α1 ,...,σ αK ,σ e1 ,...,σ eK . J(y; θ) isthe Jacobian of the transformation given the data and thevector of parameters θ. For the beta transformation, theJacobian is defined byJ(y i ; θ) =n K∏ ∏ ikk=1 j=1y η 1k−1ijk(1 − y ijk ) η 2k−1. (5)B(η 1k ,η 2k )Formulae of the Jacobian for the other potential transformationsare given in the Appendix.L(ỹ i ; θ) is the log likelihood of the transformed data for subjecti. Let Zi k =(Z(t i1k),...,Z(t inik k)) T be the n ik × (p +1)matrix of time polynomials for subject i and test k;X1i k =(X 1i(t i1k ),...,X 1i (t inik k)) T and X2i k =(X 2i(t i1k ),...,X 2i (t inik k)) T are, respectively, the n ik × q 1 matrix of timedependentcovariates for the latent process and n ik × q 2 matrixof time-dependent covariates for the psychometric tests.Let I n be the identity matrix of size n, and J n , the matrix ofsize n where all the elements equal 1. Then, the density of ỹ iis a multivariate Gaussian density of size n i = ∑ Kn k=1 ik withmean E i =(E T i1 , ..., E iK T )T and covariance matrix V i given byE ik = Zi k μ + X1iβ k + X2iγ k k (6)V i =⎛⎜⎝Z 1 i. .Z K i⎞⎟⎠ D ( Z 1Ti⎛⎞Σ 1 0 0)··· ZiKT ⎜ . + Vw + ⎝ 0 .. ⎟ 0 ⎠ ,0 0 Σ Kwith Σ k = σ 2 α kJ nik + σ 2 ɛ kI nik (7)and V w the covariance matrix for the Brownian process withargument σ 2 w(min(t l , t m )) for (l, m) ∈ [1, n i ] 2 . The contributionof subject i to the log likelihood of the transformed dataL(ỹ i ; θ) is the logarithm of this multivariate density taken atthe observation values. The log likelihood (4) has a closedform (except for the computation of the beta CDFs for whichstandard routines are available) and is maximized using amodified Marquardt algorithm (Marquardt, 1963), which is aNewton–Raphson-like algorithm. The vector of parameters θis updated until convergence usingθ (l+1) = θ (l) − δ ( ˜H(l) ) −1∇( L( y; θ(l) )) . (8)The step δ equals 1 by default but can be modified to ensurethat the likelihood is improved at each iteration. The matrix˜H is a diagonal-inflated Hessian to ensure positive definiteness.∇(L(y; θ (l) )) is the gradient of the log likelihood (4) atiteration l. First and second derivatives are computed by finitedifferences. The program is written in Fortran90 and is availableon the web site http://www.isped.u-bordeaux2.fr.This algorithm is less computationally demanding than alternativeMonte Carlo approaches such as in Arminger andMuthén (1998), who proposed a Bayesian approach for latentvariable models with nonlinear relationships between the latentvariables. Nevertheless, it is computationally intensiveand, for example, with a sample of 563 subjects (8227 observations)and a model with 36 parameters (the final model inthe application), the CPU time is around 15 minutes using aBi-Xeon 3.06 GHz 1024 MB RAM.Moreover, after convergence, standard error estimates ofthe parameter estimates are directly obtained using the inverseof the Hessian. A bootstrap method using 200 resamplesof the N subjects is also performed for obtaining standard errorsof g k (y, ˆη k ), where y is in the range of the psychometrictest k.4. Assessment of the FitAn unsolved question in mixed modeling is the assessment ofthe goodness of fit. In this work, we propose two approachesto evaluate the adequacy of the model, a residual-based approachand a prediction-based approach. The residual-basedapproach consists of evaluating the Gaussian distribution ofthe standardized marginal residuals ˆɛ i given byˆɛ i = U i (ỹ i − Êi), (9)where U i is the upper triangular matrix of the Cholesky transformationof V −1iand Êi = E ˆθ(ỹ i ) is obtained by replacing theparameters by their MLE in (6). A normal quantile plot withthe 95% confidence bands computed using the Kendall andStuart formula (Kendall and Stuart, 1977, p. 251) is then displayedto evaluate whether the empirical distribution of thestandardized residuals ˆɛ ijk is close to the theoretical N(0, 1)distribution.To evaluate the fit of the data on the natural scale ofthe tests, we plot the observed mean evolution of each testversus the estimated marginal mean evolution or the conditionalmean evolution, which includes random effects es-−1timates. The marginal estimated means E ˆθ(g k(ỹ ijk )) and the−1conditional estimated means E ˆθ(g k(ỹ ijk ) | û i , ˆα ik , ŵ i ) are computedby numerical integration of g −1k(ỹ ik ) over the marginaldistribution of ỹ ik ,N(E ik (ˆθ); V i (ˆθ)), or over the conditionaldistribution N(E ik (ˆθ)+Ŵik ;ˆσ k I nik ). Here the marginal expectationand variance of ỹ ik is given by (6) and (7) andŴ ijk = Z i (t ijk ) T û i +ŵ i (t ijk )+ˆα ik is the empirical Bayes estimateof the subject-specific deviation from the model.5. Application: Cognitive Evolution in the Elderly5.1 The DataThe aim of this analysis is to describe the decline with age ofthe global cognitive ability measured by several psychometrictests and to evaluate the association of covariates, especiallyApolipoprotein E (apoE) genotype, with the latent cognitiveprocess. Indeed, the presence of one or two ɛ4 alleles of apoEis associated with a higher risk of Alzheimer’s disease (Farreret al., 1997) but it is not well established whether the ɛ4 alleleis more generally associated with cognitive ageing (Winnocket al., 2002).The data came from the French prospective cohort studyPAQUID, initiated in 1988 to study normal and pathologicalageing (Letenneur et al., 1994). Subjects included in thecohort were 65 years and older at the initial visit and werefollowed six times with intervals of 2 or 3 years. At each visit,a battery of psychometric tests was completed and an evaluationof whether the person satisfied the criteria for a diagnosisof dementia was carried out. Measurements at the initialvisit were excluded because of a first passing effect (Jacqmin-Gadda et al., 1997). In the analysis, we included subjects who