[Paper Review] Missing data analysis and imputation via latent Gaussian Markov random fields

Virgilio Gómez‐Rubio, Michela Cameletti+1 more|arXiv (Cornell University)|Dec 23, 2019

Statistical Methods and Bayesian Inference6 citations

TL;DR

This paper proposes a fully Bayesian approach to missing data imputation in regression models by modeling covariate imputation as a latent Gaussian Markov random field (GMRF) within the INLA framework. The method enables fast, joint modeling of data, imputed covariates, and missingness mechanisms, supporting sensitivity analysis under ignorable and non-ignorable missingness, with efficient computation and extensibility to spatial and multivariate settings.

ABSTRACT

In this paper we recast the problem of missing values in the covariates of a regression model as a latent Gaussian Markov random field (GMRF) model in a fully Bayesian framework. Our proposed approach is based on the definition of the covariate imputation sub-model as a latent effect with a GMRF structure. We show how this formulation works for continuous covariates and provide some insight on how this could be extended to categorical covariates. The resulting Bayesian hierarchical model naturally fits within the integrated nested Laplace approximation (INLA) framework, which we use for model fitting. Hence, our work fills an important gap in the INLA methodology as it allows to treat models with missing values in the covariates. As in any other fully Bayesian framework, by relying on INLA for model fitting it is possible to formulate a joint model for the data, the imputed covariates and their missingness mechanism. In this way, we are able to tackle the more general problem of assessing the missingness mechanism by conducting a sensitivity analysis on the different alternatives to model the non-observed covariates. Finally, we illustrate the proposed approach with two examples on modeling health risk factors and disease mapping. Here, we rely on two different imputation mechanisms based on a typical multiple linear regression and a spatial model, respectively. Given the speed of model fitting with INLA we are able to fit joint models in a short time, and to easily conduct sensitivity analyses.

Motivation & Objective

• To address the gap in INLA methodology for handling missing values in covariates by integrating imputation into the latent GMRF framework.
• To enable joint modeling of the analysis model, imputed covariates, and missingness mechanism within a fully Bayesian framework.
• To support sensitivity analysis on different missingness mechanisms (e.g., MCAR, MNAR) using the computational speed of INLA.
• To extend the applicability of INLA to models with missing covariates, including spatial and multivariate settings.
• To provide a computationally efficient alternative to MCMC-based imputation methods for missing data in regression models.

Proposed method

• Model the imputation of missing covariates as a latent GMRF effect within a hierarchical Bayesian model.
• Formulate the imputation sub-model as a Gaussian Markov random field with a proper precision matrix, enabling efficient computation via INLA.
• Integrate the imputation model with the main analysis model into a single joint model, allowing uncertainty propagation from imputation to parameter estimation.
• Use the INLA framework for fast posterior computation, avoiding the need for time-consuming MCMC sampling.
• Extend the approach to spatially correlated covariates using the SPDE approach or other GMRF-based correlation structures.
• Implement the method in the R-INLA package via the MIINLA package, using the rgeneric framework for custom latent effects.

Experimental results

Research questions

• RQ1 Can missing covariates in regression models be effectively imputed using a latent GMRF structure within the INLA framework?
• RQ2 How does the joint modeling of imputation and analysis models under INLA improve inference accuracy and uncertainty quantification compared to separate imputation and analysis?
• RQ3 To what extent can the proposed method support sensitivity analysis on different missingness mechanisms (e.g., MCAR vs. MNAR) with minimal computational cost?
• RQ4 Can the method be extended to handle multiple missing covariates or categorical variables using GMRF-based imputation?
• RQ5 How does the performance of the INLA-based imputation compare to traditional multiple imputation or MCMC approaches in terms of speed and accuracy?

Key findings

• The proposed method successfully enables INLA to fit models with missing covariates by embedding imputation within a latent GMRF structure, filling a key methodological gap in the INLA framework.
• Joint modeling of the analysis model, imputed covariates, and missingness mechanism allows for proper uncertainty propagation and sensitivity analysis under different missingness assumptions.
• The computational speed of INLA allows for rapid fitting of multiple models under different missingness mechanisms, facilitating comprehensive sensitivity analysis.
• Posterior marginals of imputed values under MNAR and MCAR mechanisms showed distinct patterns, with MNAR marginals closer to the true values, confirming the method’s ability to detect and reflect the actual missingness mechanism.
• The method is extensible to spatial models and multivariate imputation, with implementation available in the R-INLA and MIINLA packages for practical use.
• The approach supports imputation of continuous covariates via GMRFs and provides a pathway for handling categorical covariates through Bayesian model averaging or extended latent effects.