A Bayesian Latent Variable Selection Model for Nonignorable Missingness

Abstract

Missing data are exceedingly common across a variety of disciplines, such as educational, social, and behavioral science areas. Missing not at random (MNAR) mechanism where missingness is related to unobserved data is widespread in real data and has detrimental consequence. However, the existing MNAR-based methods have potential problems such as leaving the data incomplete and failing to accommodate incomplete covariates with interactions, non-linear terms, and random slopes. We propose a Bayesian latent variable imputation approach to impute missing data due to MNAR (and other missingness mechanisms) and estimate the model of substantive interest simultaneously. In addition, even when the incomplete covariates involves interactions, non-linear terms, and random slopes, the proposed method can handle missingness appropriately. Computer simulation results suggested that the proposed Bayesian latent variable selection model (BLVSM) was quite effective when the outcome and/or covariates were MNAR. Except when the sample size was small, estimates from the proposed BLVSM tracked closely with those from the complete data analysis. With a small sample size, when the outcome was less predictable from the covariates, the missingness proportions of the covariates and the outcome were larger, and the missingness selection processes of the covariates and the outcome were more MNAR and MAR, the performance of BLVSM was less satisfactory. When the sample size was large, BLVSM always performed well. In contrast, the method with an MAR assumption provided biased estimates and undercoverage confidence intervals when the missingness was MNAR. The robustness and the implementation of BLVSM in real data were also illustrated. The proposed method is available in the Blimp software application, and the paper includes a data analysis example illustrating its use.

Keywords:

• Missing not at random
• multiple imputation
• Bayesian statistics

Article information

Conflict of interest disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant R305D190002 from the Institute of Education Sciences and Grant R01HD091832 from the National Institute of Child Health and Human Development.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions is not intended and should not be inferred.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 The procedure is that multiplying the two components in Equation (6) and finding a normal distribution for y which has the same kernel as the product.

2 Based on our pilot simulations, if we used noninformative prior for γ (i.e., $p(\gamma )\propto 1$), sometimes we could get converged results but sometimes not, which depended on the data. The default prior for coefficients in probit regression with missing data is $N(0,5)$ in Mplus. We found that prior variance of 5, 10 or 15 did not yield observably different results, and it could ensure convergence results in almost all cases. In addition, $r^{*}$ is scaled as a z-score, and we checked various probit regressions to capture the relation of y and $r^{*}$ under different scenarios. We found that γ was not large across conditions. Therefore, we use the prior variance of 10 in the normal prior of γ, which is still quite large but small enough to induce additional information that facilitates convergence. In real data analysis, researchers can modify this weakly informative prior based on each specific data.