Linear mixed models for skew‐normal/independent bivariate responses with an application to periodontal disease

Abstract

Bivariate clustered (correlated) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed model framework with underlying normality assumptions of the random effects and within-subject errors. However, such normality assumptions might be questionable if the data-set particularly exhibit skewness and heavy tails. Using a Bayesian paradigm, we use the skew-normal/independent (SNI) distribution as a tool for modeling clustered data with bivariate non-normal responses in a linear mixed model framework. The SNI distribution is an attractive class of asymmetric thick-tailed parametric structure which includes the skew-normal distribution as a special case. We assume that the random effects follows multivariate skew-normal/independent distributions and the random errors follow symmetric normal/independent distributions which provides substantial robustness over the symmetric normal process in a linear mixed model framework. Specific distributions obtained as special cases, viz. the skew-t, the skew-slash and the skew-contaminated normal distributions are compared, along with the default skew-normal density. The methodology is illustrated through an application to a real data which records the periodontal health status of an interesting population using periodontal pocket depth (PPD) and clinical attachment level (CAL).

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

“…To alleviate such limitations, it is natural to replace the multivariate normally-distributed random effects and within-subject errors of the MNLMM by a broader family, such as the multivariate skew-normal distribution [51], the multivariate skew-t distribution [52], the multivariate skew-elliptical distribution [53], or the multivariate skew-normal independent distribution [54,55]. The proposed methods are readily extendable to carry out ML estimation of the multivariate version of skew-family nonlinear mixed models.…”

Bayesian Bivariate Linear Mixed-Effects Models with Skew-Normal/Independent Distributions, with Application to AIDS Clinical Studies

“…An SNI distribution can be defined by a process of the m-dimensional random vector as follows Bandyopadhyay et al, 2010):…”

“…In this article, we propose a parametric modeling of BLME model for robust estimation using skewnormal/independent (SNI) distributions under a Bayesian paradigm. We assume an SNI distribution for the random-effects and a symmetric normal/independent (NI) distribution for the within-subject errors (Lange and Sinsheimer, 1993;Bandyopadhyay et al, 2010), so that the SNI-BLME model is defined. The multivariate SNI distribution used in this article is developed primarily from the multivariate SN density proposed by Sahu et al (2003) for Bayesian regression problems; it is different from the multivariate SNI distribution developed by Lachos et al (2010) which is motivated from the SN version proposed by Azzalini and Dalla-Valle (1996).…”

Maximum-likelihood estimation and influence analysis in multivariate skew-normal reproductive dispersion mixed models for longitudinal data

“…Many studies focused on relaxing the normality assumption of the random effect over years. Particularly, the skew-normal (SN) linear mixed models have received considerable attention in recent years (e.g., see [10][11][12]). However, to the best of our knowledge, there is little literature on a reproductive dispersion mixed model (RDMM) with the random effect following the SN distribution, which is referred to as a multivariate skew-normal reproductive dispersion mixed model (SNRDMM).…”