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This article focuses on Bayesian estimation of a hierarchical linear model (HLM) from incomplete data assumed missing at random where continuous covariates C and discrete categorical covariates $D$ have interaction effects on a continuous response $R$. Given small sample sizes, maximum likelihood estimation is suboptimal, and existing Gibbs samplers are based on a Bayesian joint distribution compatible with the HLM, but impute missing values of $C$ and the underlying latent continuous variables $D^*$ of $D$ by a Metropolis algorithm via proposal normal densities having constant variances while the target conditional distributions of $C$ and $D$ have nonconstant variances. Therefore, the samplers are neither guaranteed to be compatible with the joint distribution nor ensured to always produce unbiased estimation of the HLM. We assume a Bayesian joint distribution of parameters and partially observed variables, including correlated categorical $D$, and introduce a compatible Gibbs sampler that draws parameters and missing values directly from the exact posterior distributions. We apply our sampler to incompletely observed longitudinal data from the small number of patient-physician encounters during office visits, and compare our estimators with those of existing methods by simulation., Comment: arXiv admin note: text overlap with arXiv:2405.21020

We consider Bayesian estimation of a hierarchical linear model (HLM) from partially observed data, assumed to be missing at random, and small sample sizes. A vector of continuous covariates $C$ includes cluster-level partially observed covariates with interaction effects. Due to small sample sizes from 37 patient-physician encounters repeatedly measured at four time points, maximum likelihood estimation is suboptimal. Existing Gibbs samplers impute missing values of $C$ by a Metropolis algorithm using proposal densities that have constant variances while the target posterior distributions have nonconstant variances. Therefore, these samplers may not ensure compatibility with the HLM and, as a result, may not guarantee unbiased estimation of the HLM. We introduce a compatible Gibbs sampler that imputes parameters and missing values directly from the exact posterior distributions. We apply our Gibbs sampler to the longitudinal patient-physician encounter data and compare our estimators with those from existing methods by simulation.

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[subscript obs] and C[subscript 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[subscript obs], C[subscript obs] |u) is normally distributed and u is a low-dimensional normal random vector; we approximate h(R[subscript obs], C[subscript obs]) =[integral]h(R[subscript obs], C[subscript obs]¦u)g(u)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. [This paper will be published in "Journal of Computational and Graphical Statistics."]

We analyze adolescent BMI and middle-age systolic blood pressure (SBP) repeatedly measured on women enrolled in the Fels longitudinal study (FLS) between 1929 and 2010 to address three questions: Do adolescent-specific growth rates in BMI and menarche affect middle-age SBP? Do they moderate the aging effect on middle-age SBP? Have the effects changed over historical time? To address the questions, we propose analyzing a growth curve model (GCM) that controls for age, birth-year cohort and historical time. However, several complications in the data make the GCM analysis non-standard. First, the person-specific adolescent BMI and middle-age SBP trajectories are unobservable. Second, missing data are substantial on BMI, SBP and menarche. Finally, modeling the latent trajectories for BMI and SBP, repeatedly measured on two distinct sets of unbalanced time points, are computationally intensive. We adopt a bivariate GCM for BMI and SBP with correlated random coefficients. To efficiently handle missing values of BMI, SBP and menarche assumed missing at random, we estimate their joint distribution by maximum likelihood via the EM algorithm where the correlated random coefficients and menarche are multivariate normal. The estimated distribution will be transformed to the desired GCM for SBP that includes the random coefficients of BMI and menarche as covariates. We demonstrate unbiased estimation by simulation. We find that adolescent growth rates in BMI and menarche are positively associated with and moderate the aging effect on SBP in middle age, controlling for age, cohort and historical time, but the effect sizes are at most modest. The aging effect is significant on SBP, controlling for cohort and historical time, but not vice versa. [This paper was published in "Biometrical Journal" v62 n7 p1702-1716 2020.]

In this paper, we analyze a two-level latent variable model for longitudinal data from the National Growth of Health Study where surrogate outcomes or biomarkers and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to reexpress the desired model as a joint distribution of variables, including the biomarkers, that are subject to missingness conditional on all of the covariates that are completely observed, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. We show that the over-identified joint model produces biased estimation of the latent variable model, and describe how to impose constraints on the joint model so that it has a one-to-one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation-maximization (EM) algorithm. [This paper was published in "Statistics in Medicine" v35 n26 p4729-4745 2016.]

Hierarchical organization of schooling in all nations insures that international large-scale assessment data are multilevel where students are nested within schools and schools are nested within nations. Longitudinal follow-up of these students adds an additional level. Hierarchical or multilevel models are appropriate to analyze such data. A ubiquitous problem, however, is that explanatory as well as outcome variables may be subject to missingness at any of the levels, posing the data analyst with a challenge. This chapter explains how to efficiently analyze a two-level hierarchical linear model given incompletely observed data where students at level 1 are nested within schools at level 2. This social setting may also apply to occasions nested within individuals, students nested within nations, and schools nested within nations. The efficient missing data method we use in this chapter aims to analyze all available data. The "all available data" include children with item as well as unit nonresponse as they belong to a school and a nation having observed data and thus add information to strengthen inferences at higher levels. [This chapter was published in: Rutkowski, L., von Davier, M., & Rutkowski, D. (Eds.), "Handbook of International Large-Scale Assessment: Background, Technical Issues, and Methods of Data Analysis" (pp. 451-479). New York, NY: CRC Press.]

This paper 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 "q"th 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. [This paper was published in "International Journal of Biostatistics" v9 n1 p109-133 2013.]