Model-robust regression and a Bayesian ``sandwich'' estimator

TL;DR

This paper introduces a Bayesian method for robust linear regression that offers uncertainty estimates with properties similar to the classical sandwich estimator, providing a Bayesian perspective on a widely used frequentist tool.

Contribution

It offers a Bayesian derivation and justification for the sandwich estimator, enhancing understanding of its robustness and interpretation in non-linear data contexts.

Findings

The Bayesian approach replicates the robustness of the sandwich estimator.

Simulation studies confirm the method's effectiveness.

Application to healthcare data demonstrates practical utility.

Abstract

We present a new Bayesian approach to model-robust linear regression that leads to uncertainty estimates with the same robustness properties as the Huber--White sandwich estimator. The sandwich estimator is known to provide asymptotically correct frequentist inference, even when standard modeling assumptions such as linearity and homoscedasticity in the data-generating mechanism are violated. Our derivation provides a compelling Bayesian justification for using this simple and popular tool, and it also clarifies what is being estimated when the data-generating mechanism is not linear. We demonstrate the applicability of our approach using a simulation study and health care cost data from an evaluation of the Washington State Basic Health Plan.