Estimating Linear Mixed Effects Models with Truncated Normally Distributed Random Effects

TL;DR

This paper extends Linear Mixed Effects models by incorporating sign constraints on coefficients through a symmetric doubly truncated Normal distribution, enabling more realistic modeling in economics, business, and medicine.

Contribution

It introduces a novel constrained LME model using SDTN distribution for random effects, with likelihood-based estimation methods for intractable distributions.

Findings

Constrained models improve interpretability of coefficients.

Simulation shows satisfactory model fit performance.

Enhanced applicability in real-world economic and medical data.

Abstract

Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the classical (unconstrained) LME models to allow for sign constraints on its overall coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN) distribution on the random effects instead of the unconstrained Normal distribution which is often found in classical literature. With the aforementioned change, difficulty has dramatically increased as the exact distribution of the dependent variable becomes analytically intractable. We then develop likelihood-based approaches to estimate the unknown model parameters utilizing the approximation of its exact distribution. Simulation studies have shown that the proposed constrained model not only improves real-world interpretations of results, but also achieves satisfactory performance on model fits as compared to the existing model.