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Modèle nonlinéaire à processus latent 681016 Biometrics, December 2006for the test k; ɛ ijk are independent Gaussian errors with mean0 and variance σ 2 ɛ k.As in Dunson (2003), the random effect α ik accounts for thefact that for a same value of the latent process, two subjectscan score differently in the cognitive domain associated withpsychometric test k. The contrasts γ k make the relationshipbetween the outcomes and the latent process more flexible byallowing some covariates to be differently associated with thevarious outcomes. The sum of the contrasts over the K testsfor a given covariate equals 0. Thus, parameters β in (1) capturethe mean association with the covariates contained bothin X 1i (t) and X 2i (t), while parameters γ k in (2) capture thevariability of the association for each test around this meanvalue.2.3 The Choice of the Family of Functions GFor all the outcomes, the transformations g k (y; η k ) come fromthe same family of functions G. The choice of the family is akey aspect of the model; it determines the flexibility of thelink between the joint outcomes with various behaviors andthe underlying latent process. The transformations must bemonotonic and increasing functions of y and depend on fewparameters to make the estimation of the model easier. So,the choice of the family G is a compromise between flexibilityand parsimony.The first transformation considered here is the beta cumulativedistribution function (CDF), which can take very differentshapes, including concave, convex, and sigmoid, accordingto the parameters, as illustrated in Figure 1. It is defined fory ∈ [0, 1], η 1k > 0, and η 2k > 0byg k (y; η 1k ,η 2k )=∫ y0x η 1k−1 (1 − x) η 2k−1B(η 1k ,η 2k )dx. (3)As the beta CDF is defined in [0, 1], for each psychometrictest, a preliminary step consists of rescaling the tests to theunit interval.The main drawback of this transformation is its computationalcomplexity. As a consequence, simpler transformationshave also been considered to compare the fits of the models:the linear transformation, the logit transformation combinedwith a linear transformation, and the Weibull cumulative distributionfunction (details in the Appendix). When using alinear transformation, the model is a multivariate linear mixedmodel similar to Roy and Lin (2000) or Rabe-Hesketh et al.(2004), with an additional Brownian motion term. In thatcase, constraints have to be added to make the model identifiable:we assume the intercept μ 0 equals 0 and the variance ofthe random intercept u 0i equals 1. In contrast, when using aCDF, the requirement that g k (y) is in [0, 1] avoids additionalconstraints on the latent process.3. EstimationParameter estimation is achieved using maximum likelihoodtechniques assuming that missing data are missing atrandom. A nonstandard aspect of the model is the presenceof parameters both in the nonlinear transformationg k of the outcome and in the model for the transformedresponse ỹ i =(ỹ i11 ,...,ỹ ini1 1,...,ỹ ijk ,...,ỹ i1K ,...,ỹ iniK K) T ,where ỹ ijk = g k (y ijk ). The log likelihood of interest is the loglikelihood of the outcomes in their natural scale, and thusincludes the Jacobian of the transformations g k . It is given byL(y; θ) =L(ỹ; θ) + ln(J(y; θ))N∑N∑= L(ỹ i ; θ)+ ln(J(y i ; θ)), (4)i=1i=110.8Beta(0.5;2)Beta(3;3)0.6beta(x)0.4Beta(2;0.7)Beta(0.6;0.5)0.200 0.2 0.4 0.6 0.8 1xFigure 1.Examples of beta transformations for various pairs of parameter values.