Accounting for bias due to outcome data missing not at random: comparison and illustration of two approaches to probabilistic bias analysis: a simulation study

Abstract

Background: Bias from data missing not at random (MNAR) is a persistent concern in health-related research. A bias analysis quantitatively assesses how conclusions change under different assumptions about missingness using bias parameters which govern the magnitude and direction of the bias. Probabilistic bias analysis specifies a prior distribution for these parameters, explicitly incorporating available information and uncertainty about their true values. A Bayesian approach combines the prior distribution with the data's likelihood function whilst a Monte Carlo approach samples the bias parameters directly from the prior distribution. No study has compared a Monte Carlo approach to a fully Bayesian approach in the context of a bias analysis to MNAR missingness. Methods: We propose an accessible Monte Carlo probabilistic bias analysis which uses a well-known imputation method. We designed a simulation study based on a motivating example from the UK Biobank study, where a large proportion of the outcome was missing and missingness was suspected to be MNAR. We compared the performance of our Monte Carlo probabilistic bias analysis to a principled Bayesian probabilistic bias analysis, complete case analysis (CCA) and missing at random implementations of inverse probability weighting (IPW) and multiple imputation (MI). Results: Estimates of CCA, IPW and MI were substantially biased, with 95% confidence interval coverages of 7-64%. Including auxiliary variables (i.e., variables not included in the substantive analysis which are predictive of missingness and the missing data) in MI's imputation model amplified the bias due to assuming missing at random. With reasonably accurate and precise information about the bias parameter, the Monte Carlo probabilistic bias analysis performed as well as the fully Bayesian approach. However, when very limited information was provided about the bias parameter, only the Bayesian approach was able to eliminate most of the bias due to MNAR whilst the Monte Carlo approach performed no better than the CCA, IPW and MI. Conclusion: Our proposed Monte Carlo probabilistic bias analysis approach is easy to implement in standard software and is a viable alternative to a Bayesian approach. We caution careful consideration of choice of auxiliary variables when applying imputation where data may be MNAR.