REMixed-package {REMixed} | R Documentation |
REMixed : Regularisation & Estimation for Mixed effect model
Description
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}
.
Author(s)
Maintainer: Auriane Gabaut auriane.gabaut@inria.fr
Authors:
Ariane Bercu
Mélanie Prague
Cécile Proust-Lima