Efficient analysis of Q-level nested hierarchical general linear models given ignorable missing data

This article extends single-level missing data methods to efficient estimation of a Q-level nested hierarchical general linear model given ignorable missing data with a general missing pattern at any of the Q levels. The key idea is to reexpress a desired hierarchical model as the joint distribution of all variables including the outcome that are subject to missingness, conditional on all of the covariates that are completely observed and to estimate the joint model under normal theory. The unconstrained joint model, however, identifies extraneous parameters that are not of interest in subsequent analysis of the hierarchical model and that rapidly multiply as the number of levels, the number of variables subject to missingness, and the number of random coefficients grow. Therefore, the joint model may be extremely high dimensional and difficult to estimate well unless constraints are imposed to avoid the proliferation of extraneous covariance components at each level. Furthermore, the over-identified hierarchical model may produce considerably biased inferences. The challenge is to represent the constraints within the framework of the Q-level model in a way that is uniform without regard to Q; in a way that facilitates efficient computation for any number of Q levels; and also in a way that produces unbiased and efficient analysis of the hierarchical model. Our approach yields Q-step recursive estimation and imputation procedures whose qth-step computation involves only level-q data given higher-level computation components. We illustrate the approach with a study of the growth in body mass index analyzing a national sample of elementary school children.