Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

Abstract

We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data $R_{\text{obs}}$ and $C_{\text{obs}}$. However, if the model for R given C includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” u such that $h(R_{\text{obs}},C_{\text{obs}}|u)$ is normally distributed and u is a low-dimensional normal random vector; we approximate $h(R_{\text{obs}},C_{\text{obs}})=\int h(R_{\text{obs}},C_{\text{obs}}|u)g(u)\mathit{\text{du}}$ via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting u, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler. Supplementary materials for this article are available online.

KEYWORDS:

• Adaptive Gauss Hermite quadrature
• Compatibility
• Missing at random
• Provisionally known random effects
• The EM algorithm

Supplementary Materials

The supplementary files that contain R and C executable codes, data sets and initial values to reproduce Tables 2 and 3 may be downloaded. See README_supplementarymaterials.txt for instruction.

Acknowledgments

We thank two anonymous reviewers and an associate editor for their helpful comments, Craig Enders and Brian Keller for providing Blimp simulation codes, and Dongho Shin for helping Blimp simulation in R environment.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305D210022. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education.