Heckman imputation models for binary or continuous MNAR outcomes and MAR predictors

Abstract

Background: Multiple imputation by chained equations (MICE) requires specifying a suitable conditional imputation model for each incomplete variable and then iteratively imputes the missing values. In the presence of missing not at random (MNAR) outcomes, valid statistical inference often requires joint models for missing observations and their indicators of missingness. In this study, we derived an imputation model for missing binary data with MNAR mechanism from Heckman's model using a one-step maximum likelihood estimator. We applied this approach to improve a previously developed approach for MNAR continuous outcomes using Heckman's model and a two-step estimator. These models allow us to use a MICE process and can thus also handle missing at random (MAR) predictors in the same MICE process.

Methods: We simulated 1000 datasets of 500 cases. We generated the following missing data mechanisms on 30% of the outcomes: MAR mechanism, weak MNAR mechanism, and strong MNAR mechanism. We then resimulated the first three cases and added an additional 30% of MAR data on a predictor, resulting in 50% of complete cases. We evaluated and compared the performance of the developed approach to that of a complete case approach and classical Heckman's model estimates.

Results: With MNAR outcomes, only methods using Heckman's model were unbiased, and with a MAR predictor, the developed imputation approach outperformed all the other approaches.

Conclusions: In the presence of MAR predictors, we proposed a simple approach to address MNAR binary or continuous outcomes under a Heckman assumption in a MICE procedure.

Trial registration: ClinicalTrials.gov NCT00799760.

Keywords: Heckman’s model; Missing data; Missing not at random (MNAR); Multiple imputation by chained equation (MICE); Sample selection method.

Conflict of interest statement

Ethics approval and consent to participate

All the data have already been published in: “Efficacy of oseltamivir-zanamivir combination compared to each monotherapy for seasonal influenza: a randomized placebo-controlled trial.” (http://dx.doi.org/10.1371/journal.pmed.1000362). This study was approved on July 18, 2008 by the Ethics Committee of Ile de France 1 (“CPP Ile de France 1”) and the French drug administration (AFSSAPS). We used already analysed data and a pre-specified secondary outcome on compliance to antiviral treatment (Trial registration: http://www.ClinicalTrials.govNCT00799760).

Consent for publication

All the data have already been published in: “Efficacy of oseltamivir-zanamivir combination compared to each monotherapy for seasonal influenza: a randomized placebo-controlled trial.” (http://dx.doi.org/10.1371/journal.pmed.1000362). This study was approved on July 18, 2008 by the Ethics Committee of Ile de France 1 (“CPP Ile de France 1”) and the French drug administration (AFSSAPS). We used already analysed data and a pre-specified secondary outcome on compliance to antiviral treatment (Trial registration: http://www.ClinicalTrials.govNCT00799760).

Competing interests

The authors declare that they have no competing interests.

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Figures

Fig. 1

Binary outcome Boxplot of β1 estimates on the 1000 simulations associated to Table 1 (plot a), Table 3 (plot b), Table 5 left (plot c) and Table 5 right (plot d)

Fig. 2

Continuous outcome Boxplot of β1 estimates on the 1000 simulations associated to Table 2 (plot a), Table 4 (plot b), Table 6 left (plot c) and Table 6 right (plot d)