Suppose that we have a differential system of equations containing variables \((S_{p})_{p\leq P}\) and \(R\), that depends on some parameters. We define a non-linear mixed effects model from this system depending on individual parameters \((\psi_i)_{i\leq N}\) and \((\phi_i)_{i\leq N}\) that defines the parameters from the structural model as individuals. The first part \((\psi_i)_{i\leq N}\) is supposed to derived from a generalized linear model for each parameter \(l\leq m\) : $$h_l(\psi_{li}) = h_l(\psi_{l pop})+X_i\beta_l + \eta_{li}$$ with the covariates of individual \(i\leq N\), \((X_i)_{i\leq N}\), random effects \(\eta_i=(\eta_{li})_{l\leq m}\overset{iid}{\sim}\mathcal{N}(0,\Omega)\), the population parameters \(\psi_{pop}=(\psi_{lpop})_{l\leq m}\) and \(\beta=(\beta_l)_{l\leq m_{re}}\) is the vector of covariates effects on parameters. The rest of the population parameters of the structural model, that hasn't random effetcs, are denoted by \((\phi_i)_{i\leq N}\), and are defined, for each parameters \(l\leq m_{no re}\) as $$f_l(\phi_{li})=\phi_{l pop} + X_i \gamma_l $$
To simplify formula, we write the individual process over the time, resulting from the differential system for a set of parameters \(\phi_i = (\phi_{li})_{l\leq m_{no re}}, \psi_i = (\psi_{li})_{l\leq m_{re}}\) for individual \(i\leq N\), as \(S_{p}(\cdot,\phi_i,\psi_i)=S_{pi}(\cdot)\), \(p\leq P\) and \(R(\cdot,\phi_i,\psi_i)=R_i(\cdot)\). We assume that individual trajectories \((S_{pi})_{p\leq P,i\leq N}\) are observed through a direct observation model, up to a transformation \(g_p\), \(p\leq P\), at differents times \((t_{pij})_{i\leq N,p\leq P,j\leq n_{ip}}\) : $$ Y_{pij}=g_p(S_{pi}(t_{pij}))+\epsilon_{pij} $$ with error \(\epsilon_p=(\epsilon_{pij})\overset{iid}{\sim}\mathcal N(0,\varsigma_p^2)\) for \(p\leq P\). The individual trajectories \((R_{i})_{i\leq N}\) are observed through $K$ latent processes, up to a transformation \(s_k\), \(k\leq K\), observed in \((t_{kij})_{k\leq K,i\leq N,j\leq n_{kij}}\) : $$Z_{kij}=\alpha_{k0}+\alpha_{k1} s_k(R_i(t_{kij}))+\varepsilon_{kij}$$ where \(\varepsilon_k\overset{iid}{\sim} \mathcal N(0,\sigma_k^2)\). The parameters of the model are then \(\theta=(\phi_{pop},\psi_{pop},B,\beta,\Omega,(\sigma^2_k)_{k\leq K},(\varsigma_p^2)_{p\leq P},(\alpha_{0k})_{k\leq K})\) and \(\alpha=(\alpha_{1k})_{k\leq K}\).
Maintainer: Auriane Gabaut aurianegabaut@gmail.com
Authors:
Ariane Bercu
Mélanie Prague
Cécile Proust-Lima