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Annexes 214Appendix : model specificationWe consider K neuropsychological tests. For each test k, k=1,…,K, each subject i, i=1,…,Nand each occasion j, j=1,…,n ik , the measure of the neuropsychological test y ijk is collected attime t ijk , t ijk being different for each test and each subject. The latent process which representsthe common factor of the K neuropsychological tests is modelled using the following linearmixed model including a quadratic function of time and a Brownian motion (w i (t)) t≥0 withvariance term σ w ²×t :ΛTTT2( t) = ( X β + µ + u ) + ( X β + µ + u ) × t + ( X β + + u ) × t w ( t)i i 0 0 0ii 1 1 1ii 2 µ 2 2i+The vector of random effects u i =(u 0i ,u 1i ,u 2i ) T follows a multivariate normal distribution withmean vector 0 and variance covariance matrix D. The mean evolution of the common factor isrepresented by the fixed effects µ 0 , µ 1 and µ 2 . The vector of covariates X i can either beassociated with the mean latent common factor level through the vector b 0 or the meancommon factor evolution through the vectors b 1 and b 2 .The observed score value y ijk is linked to the value of the common factor at the time ofmeasurement Λ i (t ijk ) through a nonlinear link function h k which is a Beta CumulativeDistribution Function depending on two test-specific parameters h k =(η 1k ,η 2k ). This leads tothe following measurement model :hkT( yijk; ηk) = Λi( tijk) + Xi( tijk) γk+ αik+ εijkwhere X i (t ijk ) is a vector of covariates including the quadratic function of time and associatediwith the neuropsychological tests through the vector of parameters g k . The vector g krepresents the differential association (or contrast) of X i (t ijk ) with the neuropsychological test∑ k = mkm1Kk after adjusting on the common factor value Λ i (t ijk ) ( γ = 0, ∀ ). The test-specificrandom intercept α ik follows a Gaussian distribution with mean 0 and variance σ αk ². It takesinto account the residual individual variability between tests after adjustment on the latent16