Regression Models Involving Nonlinear Effects With Missing Data: A Sequential Modeling Approach Using Bayesian Estimation

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a joint normal distribution, the default in many statistical software packages. This distribution will in general be misspecified if the predictors with missing data have nonlinear effects (e.g., x(2)) or are included in interaction terms (e.g., x . z). In the present article, we discuss a sequential modeling approach that can be applied to decompose the joint distribution of the variables into 2 parts: (a) a part that is due to the model of interest and (b) a part that is due to the model for the incomplete predictors. We demonstrate how the sequential modeling approach can be used to implement a multiple imputation strategy based on Bayesian estimation techniques that can accommodate rather complex substantive regression models with nonlinear effects and also allows a flexible treatment of auxiliary variables. In 4 simulation studies, we showed that the sequential modeling approach can be applied to estimate nonlinear effects in regression models with missing values on continuous, categorical, or skewed predictor variables under a broad range of conditions and investigated the robustness of the proposed approach against distributional misspecifications. We developed the R package mdmb, which facilitates a user-friendly application of the sequential modeling approach, and we present a real-data example that illustrates the flexibility of the software. Translational Abstract Regression models testing whether two predictor variables interact to produce an effect on the outcome variable are commonly used in psychology. Often a portion of the participants do not fully complete their responses so their data are missing on one or both of the predictor variables. Although more modern methods of addressing missing data typically lead to more accurate results, the performance of these methods may be greatly diminished when regression models contain interactions or other nonlinear effects. We describe a new sequential modeling approach using multiple imputation that separates the problem into two parts: (a) the substantive regression model of interest and (b) the imputation model; this approach theoretically identifies when the two parts are compatible. When the two parts are compatible, Bayesian estimation can be used to produce accurate results. We show the improved performance of this sequential modeling approach relative to other forms of multiple imputation in four simulation studies under a broad range of conditions. The simulation studies considered most of the types of predictor variables commonly considered in research: normally distributed continuous, skewed continuous, binary, and latent. We developed the R package mdmb, which facilitates a user-friendly application of the sequential modeling approach, and we present a real-data example that illustrates the flexibility of the software. Annotated R computer script for the main analyses is presented in the online supplemental material.