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

Consider the conventional multilevel model Y = C γ + Zu + e where γ represents fixed effects and ( u , e ) are multivariate normal random effects. The continuous outcomes Y and covariates C are fully observed with a subset Z of C . The parameters are θ = ( γ , var ( u ) , var ( e ) ) . Dempster, Rubin and Tsutakawa framed the estimation as a missing data problem, where ( Y , 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 C is partially observed or subject to measurement error, it is natural to formulate a multilevel model for C that includes random effects, ν . In this article, we extend this thinking to allow estimation of the joint distribution of data Y = ( Y , C ) = ( Y o , Y m ) and random effects b = ( u , v ) from observed data