# A Pseudo-Marginal Metropolis-Hastings Algorithm for Estimating Generalized Linear Models in the Presence of Missing Data

## TL;DR

This paper introduces a pseudo-marginal Metropolis-Hastings algorithm tailored for estimating generalized linear models with missing data, offering a flexible, asymptotically exact Bayesian inference method that handles complex missingness mechanisms.

## Contribution

The paper presents a novel application of the pseudo-marginal Metropolis-Hastings algorithm for GLMs with missing data, allowing joint inference without strong assumptions and maintaining asymptotic exactness.

## Key findings

- Standard errors increase with higher missingness
- Algorithm performs well in simulation studies
- Applied to real-world crash data to analyze variable effects

## Abstract

The missing data issue often complicates the task of estimating generalized linear models (GLMs). We describe why the pseudo-marginal Metropolis-Hastings algorithm, used in this setting, is an effective strategy for parameter estimation. This approach requires fewer assumptions, it provides joint inferences on the parameters in the likelihood, the covariate model, and the parameters of the missingness-mechanism, and there is no logical inconsistency of assuming that there are multiple posterior distributions. Moreover, this approach is asymptotically exact, just like most other Markov chain Monte Carlo techniques. We discuss computing strategies, conduct a simulation study demonstrating how standard errors change as a function of percent missingness, and we use our approach on a "real-world" data set to describe how a collection of variables influences the car crash outcomes.