INFERENCE PAR APPROXIMATION NORMALE DE L A POSTERIORI DANS LES MODELES DYNAMIQUES A EFFETS MIXTES

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Huguette Eveline Goudreau
il y a 7 ans
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1 INFERENCE PAR APPROXIMATION NORMALE DE L A POSTERIORI DANS LES MODELES DYNAMIQUES A EFFETS MIXTES 1,2,3, J. Guedj 4, J. Drylewicz 5, D. Commenges 1,2,3 and R. Thiébaut 1,2,3 1 Univ. Bordeaux, ISPED, Bordeaux, FR 2 INSERM U897 Épidémiologie et Biostatistique, Bordeaux, FR 3 INRIA, SISTM, Bordeaux, FR 4 INSERM UMR 738, Paris Diderot, FR 5 Department of Immunology, Univ. Med. Center Utrecht, NL SFDS Mai 2013

State of the art How can we estimate parameters in mixed effects models based on differential equations? -> Semi-parametric functional analysis (Ramsay et al.

2 State of the art How can we estimate parameters in mixed effects models based on differential equations? -> Semi-parametric functional analysis (Ramsay et al. 2007) -> Laplace methods FOCE (Pinheiro et Bates 1995) -> SAEM (Kuhn et al ; Lavielle et al 2007) -> Bayesian MCMC (Huang et al ; Lunn et al. 2000) An alternative algorithm and program : -> Penalized maximum likelihood (Guedj et al ; Drylewicz et al 2010) -> NIMROD : Normal approximation Inference in Models with Random effects based on Ordinary Differential equations (Prague et al 2013) 1/18

3 Mathematical model -> One Compartment with absorption pharmacokinetics da GI dt = k a A GI, dc P dt = ka V 0 A GI k e C P, A GI (0) = dose 0, C P (0) = 0. 2/18

4 Statistical model -> Mixed effect model : populational approach -> Reparametrization k i a = k a0 + u i k a with, u i k a N (0, σ ka ) k i e = k e0 + u i k e with, u i k e N (0, σ ke ) Ṽ i 0 = Ṽ00 + βweight + u i V 0 with, u i V 0 N (0, σ V0 ) 3/18

5 Observational model -> Longitudinal Data -> Discrete time observation -> Normality and Homoscedasticity of residuals Y i 1 (t ij ) = [C i P(t ij )] ɛ ij1, ɛ ij1 N ( 0, σ 2 C P ). 4/18

6 Change the model 5/18

7 Optimization Algorithm In the illustration, we have to estimate 8 parameters θ = [ k a, k e ; V 0 ; β; σ ka ; σ ke ; σ V0 ; σ 2 C P ] R m. 6/18

8 Penalized likelihood computation (Guedj et al. 2007) -> Individual likelihood : ODE Solver L Fi u i = ( 1 1 Y ijm g m (X(t ij, ξ ) 2 i )) σ j,m m (2π) 2 σ m -> Observed Individual likelihood : Adaptive Gaussian Quadrature L Oi = L i(u)φ(u)du Fi u avec φ N (0, I q) R q -> Penalized log-likelihood : L P O = i n L Oi + log(prior) 7/18

9 Estimation algorithm (Guedj et al. 2007) -> Scores computation : ODE Solver / Adaptive Gaussian Quadrature U O (θ k ) = n ( LOi i=1 θ θ k ) + log(prior) θ -> Robust Variance Scoring (Commenges et al. 2006) : Hessian approximation H(θ k ) G(θ k ) = i n ( UOi (θ k )U O i (θ k ) ) + ν n U(θ k)u (θ k ) + 2 log(prior) θ 2 8/18

10 ILLUSTRATION : Evaluation du NAP (Bernstein von Mises theorem - Van der Vaart 2000) ω V0 ω ke ω ka 9/18

11 Convergence criteria Stabilization of parameters estimates : Stabilization of log-likelihood : Relative Distance to Maximum : θ (k+1) θ k < η 1 L P O(θ (k+1) ) L P O(θ k ) < η 2 RDM(θ k ) = U(θ k)g 1 (θ k )U (θ k ) m < η 3 10/18

12 ILLUSTRATION : Convergence criteria Assessment RDM final value is correlated with precision 11/18

13 Choice of model criterion Approximation of the likelihood cross-validation criterion : -> Estimator of the expected Kullback-Leibler criterion, -> The lower the better, -> Difference > 0.1 is significative. LCV a = n 1 [L(ˆθ) Tr(H 1 L P (ˆθ)H L (ˆθ))], 12/18

14 ILLUSTRATION : APV Pharmacokinetics Model choice in HIV infected patients (PUZZLE - Raguin et al. 2004) 13/18

15 ILLUSTRATION : Methods comparison on simulations scenarios (Prague13) 14/18

16 Dans la pratique...

17 Compilation/Utilisation 16/18

18 Outputs : Estimations, PEB and graphics 17/18

19 For further informations... -> -> et al (Computer Methods and Programs in Biomedicine.) : NIMROD : A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations. 18/18

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