Flexible linear mixed models with improper priors for longitudinal and survival data

TL;DR

This paper introduces a Bayesian framework for hierarchical linear mixed models with flexible error distributions, including heavy tails and asymmetry, suitable for robust longitudinal and survival data analysis.

Contribution

It provides a formal justification for Bayesian inference using improper priors in a broad class of mixed models with flexible residual and random effects distributions.

Findings

Models are robust to outliers due to heavy-tailed error distributions.

Posterior propriety is established even with censored data.

Framework covers a wide range of models with minimal prior elicitation.

Abstract

We propose a Bayesian approach using improper priors for hierarchical linear mixed models with flexible random effects and residual error distributions. The error distribution is modelled using scale mixtures of normals, which can capture tails heavier than those of the normal distribution. This generalisation is useful to produce models that are robust to the presence of outliers. The case of asymmetric residual errors is also studied. We present general results for the propriety of the posterior that also cover cases with censored observations, allowing for the use of these models in the contexts of popular longitudinal and survival analyses. We consider the use of copulas with flexible marginals for modelling the dependence between the random effects, but our results cover the use of any random effects distribution. Thus, our paper provides a formal justification for Bayesian inference in a very wide class of models (covering virtually all of the literature) under attractive prior structures that limit the amount of required user elicitation.