Extending multivariate-t linear mixed models for multiple longitudinal data with censored responses and heavy tails

Abstract

The analysis of complex longitudinal data is challenging due to several inherent features: (i) more than one series of responses are repeatedly collected on each subject at irregularly occasions over a period of time; (ii) censorship due to limits of quantification of responses arises left- and/or right- censoring effects; (iii) outliers or heavy-tailed noises are possibly embodied within multiple response variables. This article formulates the multivariate- t linear mixed model with censored responses (MtLMMC), which allows the analysts to model such data in the presence of the above described features simultaneously. An efficient expectation conditional maximization either (ECME) algorithm is developed to carry out maximum likelihood estimation of model parameters. The implementation of the E-step relies on the mean and covariance matrix of truncated multivariate- t distributions. To enhance the computational efficiency, two auxiliary permutation matrices are incorporated into the procedure to determine the observed and censored parts of each subject. The proposed methodology is demonstrated via a simulation study and a real application on HIV/AIDS data.

Influence diagnostics for censored regression models with autoregressive errors

“…The proposed methods can also be easily applied to other substantial areas in which the data being analyzed have censored observations. In this context, intended future work includes extending the proposed methods to accommodate missing values and censored observations using hybrid Bayesian sampling procedures (Wang & Fan 2012) and multivariate outcomes (Wang, Lin & Lachos 2015). i = 1 2 ŷ − Xβ @M −1 @ i (φ), i = 1,…, p.…”

The t linear mixed model: model formulation, identifiability and estimation

“…(23) to find outlying subjects. Wang and Lin (2014) and Wang, Lin, and Lachos (2018) complement the approaches above with a quantitative threshold-based method to distinguish outliers, where the threshold is…”

“…In these publications the authors discussed also the prediction of future values, which is a natural consequence in the Bayesian formulation. Multivariate extensions of the tLMM have been studied in the frequentist and Bayesian frameworks Fan 2011, 2012) and extended by other authors for handling missing values, censored data and heavy tails (Ho and Lin 2010;Matos et al 2013;Wang 2013a;Lin 2014, 2015;Wang, Lin, and Lachos 2018;Lin and Wang 2019). Besides allowing robust inference, the tLMM enables the detection of outliers, intended as observations that do not comply with normality (and therefore with the LMM) (Pinheiro, Liu, and Wu 2001;Lin and Lee 2007;Ho and Lin 2010;Fan 2011, 2012;Matos et al 2013;Wang and Lin 2014;Wang 2017).…”

Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

“…To introduce robustness against outliers, Frühwirth-Schnatter and Kaufmann (2008) consider a multivariate t distribution and Juárez and Steel (2010) introduce a skew t distribution for the error term to capture conditional skewness. Wang (2013) and Wang et al (2015) extend the aforementioned approaches in a mixed-effects regression framework. Between-and within-subject variations through subject-specific random effects and intra-subject errors can be modeled, as well as missingness and censoring can be easily dealt with in such a framework.…”