Fast selection of nonlinear mixed effect models using penalized likelihood

Nonlinear Mixed effects models are hidden variables models that are widely used in many field such as pharmacometrics. In such models, the distribution characteristics of hidden variables can be specified by including several parameters such as covariates or correlations which must be selected. Recent development of pharmacogenomics has brought averaged/high dimensional problems to the field of nonlinear mixed effects modeling for which standard covariates selection techniques like stepwise methods are not well suited. This work proposes to select covariates and correlation parameters using a penalized likelihood approach. The penalized likelihood problem is solved using a stochastic proximal gradient algorithm to avoid inner-outer iterations. Speed of convergence of the proximal gradient algorithm is improved by the use of component-wise adaptive gradient step sizes. The practical implementation and tuning of the proximal gradient algorithm is explored using simulations. Calibration of regularization parameters is performed by minimizing the Bayesian Information Criterion using particle swarm optimization, a zero order optimization procedure. The use of warm restart and parallelization allows to reduce significantly computing time. The performance of the proposed method compared to the traditional grid search strategy is explored using simulated data. Finally, an application to real data from two pharmacokinetics studies is provided, one studying an antifibrinolitic and the other studying an antibiotic.

Penalized Poisson model for network meta-analysis of individual patient time-to-event data

Network meta-analysis (NMA) allows the combination of direct and indirect evidence from a set of randomized clinical trials. Performing NMA using individual patient data (IPD) is considered as a gold standard approach as it provides several advantages over NMA based on aggregate data. For example, it allows to perform advanced modelling of covariates or covariate-treatment interactions. An important issue in IPD NMA is the selection of influential parameters among terms that account for inconsistency, covariates, covariate-by-treatment interactions or non-proportionality of treatments effect for time to event data. This issue has not been deeply studied in the literature yet and in particular not for time-to-event data. A major difficulty is to jointly account for between-trial heterogeneity which could have a major influence on the selection process. The use of penalized generalized mixed effect model is a solution, but existing implementations have several shortcomings and an important computational cost that precludes their use for complex IPD NMA. In this article, we propose a penalized Poisson regression model to perform IPD NMA of time-to-event data. It is based only on fixed effect parameters which improve its computational cost over the use of random effects. It could be easily implemented using existing penalized regression package. Computer code is shared for implementation. The methods were applied on simulated data to illustrate the importance to take into account between trial heterogeneity during the selection procedure. Finally, it was applied to an IPD NMA of overall survival of chemotherapy and radiotherapy in nasopharyngeal carcinoma.

Stochastic Proximal Gradient Algorithms for Penalized Mixed Models

Motivated by penalized likelihood maximization in complex models, we study optimization problems where neither the function to optimize nor its gradient have an explicit expression, but its gradient can be approximated by a Monte Carlo technique. We propose a new algorithm based on a stochastic approximation of the Proximal-Gradient (PG) algorithm. This new algorithm, named Stochastic Approximation PG (SAPG) is the combination of a stochastic gradient descent step which - roughly speaking - computes a smoothed approximation of the past gradient along the iterations, and a proximal step. The choice of the step size and the Monte Carlo batch size for the stochastic gradient descent step in SAPG are discussed. Our convergence results cover the cases of biased and unbiased Monte Carlo approximations. While the convergence analysis of the Monte Carlo-PG is already addressed in the literature (see Atchade et al. [2016]), the convergence analysis of SAPG is new. The two algorithms are compared on a linear mixed effect model as a toy example. A more challenging application is proposed on non-linear mixed effect models in high dimension with a pharmacokinetic data set including genomic covariates. To our best knowledge, our work provides the first convergence result of a numerical method designed to solve penalized Maximum Likelihood in a non-linear mixed effect model.

A SAEM Algorithm for Fused Lasso Penalized Non Linear Mixed Effect Models: Application to Group Comparison in Pharmacokinetic

Non linear mixed effect models are classical tools to analyze non linear longitudinal data in many fields such as population Pharmacokinetic. Groups of observations are usually compared by introducing the group affiliations as binary covariates with a reference group that is stated among the groups. This approach is relatively limited as it allows only the comparison of the reference group to the others. In this work, we propose to compare the groups using a penalized likelihood approach. Groups are described by the same structural model but with parameters that are group specific. The likelihood is penalized with a fused lasso penalty that induces sparsity on the differences between groups for both fixed effects and variances of random effects. A penalized Stochastic Approximation EM algorithm is proposed that is coupled to Alternating Direction Method Multipliers to solve the maximization step. An extensive simulation study illustrates the performance of this algorithm when comparing more than two groups. Then the approach is applied to real data from two pharmacokinetic drug-drug interaction trials.