Inequality Constrained Multilevel Models

TL;DR

This paper introduces a Bayesian approach for inequality constrained multilevel linear models, enabling researchers to incorporate substantive theories with inequality constraints and perform model selection using posterior probabilities.

Contribution

It develops a Bayesian formulation for inequality constrained multilevel models and proposes an augmented Gibbs sampler for parameter estimation and model comparison.

Findings

Bayesian formulation for inequality constrained multilevel models

Augmented Gibbs sampler for efficient estimation

Posterior probabilities for model selection

Abstract

Multilevel or hierarchical data structures can occur in many areas of research, including economics, psychology, sociology, agriculture, medicine, and public health. Over the last 25 years, there has been increasing interest in developing suitable techniques for the statistical analysis of multilevel data, and this has resulted in a broad class of models known under the generic name of multilevel models. Generally, multilevel models are useful for exploring how relationships vary across higher-level units taking into account the within and between cluster variations. Research scientists often have substantive theories in mind when evaluating data with statistical models. Substantive theories often involve inequality constraints among the parameters to translate a theory into a model. This chapter shows how the inequality constrained multilevel linear model can be given a Bayesian formulation, how the model parameters can be estimated using a so-called augmented Gibbs sampler, and how posterior probabilities can be computed to assist the researcher in model selection.