Multiple Imputation to Estimate Hierarchical Models From Data Missing at Random: Latent Covariates,Random Coefficients,and Statistical Interactions

Abstract

Consider the conventional multilevel model $\mathrm{Y}=\mathrm{C}\mathrm{\gamma }+\mathrm{Zu}+\mathrm{e}$ where $\mathrm{\gamma }$ represents fixed effects and $(\mathrm{u},\mathrm{e})$ are multivariate normal random effects. The continuous outcomes $\mathrm{Y}$ and covariates $\mathrm{C}$ are fully observed with a subset $\mathrm{Z}$ of $\mathrm{C}$ . The parameters are $\mathrm{\theta }=(\mathrm{\gamma },\mathrm{var}(\mathrm{u}),\mathrm{var}(\mathrm{e}))$ . Dempster, Rubin and Tsutakawa framed the estimation as a missing data problem, where $(\mathrm{Y},\mathrm{u})$ are the complete data and the random effects $u$ are conceived as missing data. Viewed in this way, the Expectation-Maximization (EM) algorithm has proven to be a natural and popular approach to estimation. However, when $\mathrm{C}$ is partially observed or subject to measurement error, it is natural to formulate a multilevel model for $\mathrm{C}$ that includes random effects, $\nu$ . In this article, we extend this thinking to allow estimation of the joint distribution of data $\mathrm{Y}=(\mathrm{Y},\mathrm{C})=({\mathrm{Y}}_{\mathrm{o}},{\mathrm{Y}}_{\mathrm{m}})$ and random effects $\mathrm{b}=(\mathrm{u},\mathrm{v})$ from observed data ${\mathrm{Y}}_{\mathrm{o}}=({\mathrm{Y}}_{\mathrm{o}},{\mathrm{C}}_{\mathrm{o}})$ and to generate multiple imputations of missing data $({\mathrm{Y}}_{\mathrm{m}},\mathrm{b})$ based on the estimated distribution under the assumption that the data $\mathrm{Y}$ are missing at random. This approach contributes to the literature on multiple imputation in three ways: (a) it allows random effects $\nu$ to be conceived as latent covariates, thus addressing measurement errors of $\mathrm{C}$ ; (b) it allows non-linearities, including random coefficients, interaction effects, and other polynomial effects involving partially observed covariates; (c) it imputes $({\mathrm{Y}}_{\mathrm{m}},\mathrm{b})$ using two-step importance sampling. In these cases, the joint distribution of $\mathrm{Y}$ is not analytically tractable even if the analytic multilevel model of interest to the analyst follows a multivariate normal distribution. We prove that our method of maximizing the likelihood and imputing missing data ensures compatibility of the nonnormal joint distribution with the analytic normal theory multilevel model via provisionally known random effects. We present and evaluate a sufficient condition under which the produced imputations are compatible with the analytic model.