Abstract: Evaluating Methods for Moderation Analysis with Missing Data in the Predictors

Evaluating Methods for Moderation Analysis with Missing Data in the Predictors Qian Zhang and Lijuan Wang Department of Psychology, University of Notre Dame The most widely used statistical model for conducting moderation analysis is the moderated multiple regression (MMR) model. While conducting moderation analysis using MMR models, missing data could pose a challenge, mainly because of the nonlinear interaction term. In the study, we considered a simple MMR model, where the effect of predictor X on the outcome Y was moderated by a moderator U. The primary interest was to find ways of estimating and testing the moderation effect with the existence of missing data in the predictor X. We mainly focused on cases when X was missing completely at random (MCAR), and missing at random (MAR) depending on U. Four methods were compared: (1) listwise deletion (LD); (2) normal-distribution-based maximum likelihood estimation (NML); (3) normal-distributionbased multiple imputation (NMI); and (4) Bayesian estimation (BE). For NML and NMI, the distribution for XU was mis-specified because XU was typically non-normal. In contrast, for BE, data augmentation was used for modeling X, and the normality assumption for XU was not needed. A simulation study was conducted for comparing the performance of the methods. In this study, manipulated factors included: distributions for X (normal and chi-squared), moderation effect sizes (null, small, and medium, measured by Cohen’s f 2), sample sizes (n = 50, 100, 200, 500, and 1,000), missing rates in X (15%, 30%, and 60%), missing data mechanisms (MCAR and MAR). Additionally, for the MAR condition, the extent to which missingness in X was dependent on U was manipulated (lower or higher). The authors are grateful to Dr. Ke-Hai Yuan and SMEP sponsor Dr. Scott Maxwell for their valuable input in this work. Correspondence concerning this abstract should be addressed to Qian Zhang, 118 Haggar Hall, University of Notre Dame, Notre Dame, IN 46556. E-mail: qzhang3@nd.edu Color versions of one or more of the figures in the article can be found online at www.tandfonline. com/hmbr. We had several findings. For MCAR data, consistent with the finding from Enders et al. (2014), NML and NMI had accurate moderation effect estimates with sample sizes greater than 100. However, for MAR data, when X was either normal or chi-squared and the moderation effect sizes were nonzero, NML and NMI yielded biased moderation effect estimates when sample sizes were greater than 100 (Figure 1). Moreover, we also found through analytical derivations that with normally distributed X and U, NML estimates were inconsistent with nonzero moderation effect sizes, and the asymptotic bias increased with increasing moderation effect sizes. Yuan and Savalei (2014) and Yuan (2009) had similar conclusions for estimating means and covariances when the variables with missing data and the observed variables responsible for missingness were nonlinearly related. In addition, bias was larger when the extent to which missingness in X was more dependent on U. For the types of missing data in our study, LD had accurate yet less powerful moderation effect estimates. In contrast, BE had accurate and the most powerful estimates of moderation effects with sample sizes greater than 100 when the marginal distribution for X was correctly specified. We also found, however, that BE was more sensitive to the violation of the distribution assumption of X under some conditions (e.g., medium Cohen’s f 2) than the other methods. In future studies, more robust BE needs to be developed for conducting moderation analysis with missing data.