Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models

Abstract

Multivariate nonlinear mixed-effects models (MNLMM) have received increasing use due to their flexibility for analyzing multi-outcome longitudinal data following possibly nonlinear profiles. This paper presents and compares five different iterative algorithms for maximum likelihood estimation of the MNLMM. These algorithmic schemes include the penalized nonlinear least squares coupled to the multivariate linear mixed-effects (PNLS-MLME) procedure, Laplacian approximation, the pseudo-data expectation conditional maximization (ECM) algorithm, the Monte Carlo EM algorithm and the importance sampling EM algorithm. When fitting the MNLMM, it is rather difficult to exactly evaluate the observed log-likelihood function in a closed-form expression, because it involves complicated multiple integrals. To address this issue, the corresponding approximations of the observed log-likelihood function under the five algorithms are presented. An expected information matrix of parameters is also provided to calculate the standard errors of model parameters. A comparison of computational performances is investigated through simulation and a real data example from an AIDS clinical study.

Bayesian semiparametric modeling for HIV longitudinal data with censoring and skewness

“…21 in the context of a nonlinear mixed effects-model from a Bayesian perspective. From a maximum likelihood viewpoint, Lin and Wang 22 analyzed the viral load and CD4+ cells simultaneously by using a multivariate SN linear mixed model, whereas Wang 23 applied a multivariate nonlinear mixed model to analyze this same dataset. Further, and still using this dataset, Matos et al.…”

“…This dataset was previously analyzed by Wu 20 in the context of a LMEC from a classic point of view and recently by Lachos et al 21 in the context of a nonlinear mixed effects-model from a Bayesian perspective. From a maximum likelihood viewpoint, Lin and Wang 22 analyzed the viral load and CD4 þ cells simultaneously by using a multivariate SN linear mixed model, whereas Wang 23 applied a multivariate nonlinear mixed model to analyze this same dataset. Further, and still using this dataset, Matos et al 24 proposed a censored nonlinear mixed-effects model using a damped exponential correlation structure for the error term.…”

Score Tests for Intercept and Slope Parameters of Doubly Multivariate Linear Models with Skew-Normal Errors

“…No attempt was made therein to develop the Fisher information matrix for a wide array of cases where was structured since the monograph had other aims in mind. Some structured covariance matrices are considered in the multivariate linear mixed models developed in Kheradmandi et al [14], Lachos et al [15], Lin and Wang [18], and Wang [35]; however, the Fisher information matrices were unreported or undeveloped since these papers aimed at fitting their models and recommending variants of the EM algorithm. In fact, it appears that no systematic development of the Fisher information matrix of parameters of a multivariate linear model with a skew-normal errors and structured covariance matrices has been performed, and only mentions of approximating the observed information matrix numerically are made.…”

Multivariate longitudinal data analysis with censored and intermittent missing responses

“…A simulation study was conducted by Wang and Fan to investigating how the separate analysis of correlated responses affect the estimation performance. There are other proposals relevant to the extensions and applications of MLMMs, including Sammel et al, Roy and Lin, Song et al, Marshall et al, and Wang, among others. However, there has been relatively few studies dealing with multivariate longitudinal data in the presence of censored and missing responses simultaneously.…”