Consider the conventional multilevel model Y=C[gamma]+Zu+e where [gamma] 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 [theta]=([gamma],var(u),var(e)). Dempster, Rubin and Tsutakawa (1981) framed the estimation as a…

In Bayesian statistics, the most widely used criteria of Bayesian model assessment and comparison are Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC). A multilevel mediation model is used as an illustrative example to compare different types of DIC and WAIC. More specifically, the study compares the…

We consider Bayesian estimation of a hierarchical linear model (HLM) from small sample sizes. The continuous response Y and covariates C are partially observed and assumed missing at random. With C having linear effects, the HLM may be efficiently estimated by available methods. When C includes cluster-level covariates having interactive or other…

This article presents a latent class model for multilevel data to identify latent subgroups and estimate heterogeneous treatment effects. Unlike sequential approaches that partition data first and then estimate average treatment effects (ATEs) within classes, we employ a Bayesian procedure to jointly estimate mixing probability, selection, and…

The methodology of single-case experimental designs (SCED) has been expanding its efforts toward rigorous design tactics to address a variety of research questions related to intervention effectiveness. Effect size indicators appropriate to quantify the magnitude and the direction of interventions have been recommended and intensively studied for…

The Bayesian information criterion (BIC) can be useful for model selection within multilevel modeling studies. However, the formula for BIC requires a value for N, which is unclear in multilevel models, since N is observed in at least two levels. The present study uses simulated data to evaluate the rate of false positives and power when using a…

Piecewise mixed-effects models are useful for analyzing longitudinal educational and psychological data sets to model segmented change over time. These models offer an attractive alternative to commonly used quadratic and higher-order polynomial models because the coefficients obtained from fitting the model have meaningful substantive…

Multilevel data are a reality for many disciplines. Currently, although multiple options exist for the treatment of multilevel data, most disciplines strictly adhere to one method for multilevel data regardless of the specific research design circumstances. The purpose of this Monte Carlo simulation study is to compare several methods for the…

Specialized imputation routines for multilevel data are widely available in software packages, but these methods are generally not equipped to handle a wide range of complexities that are typical of behavioral science data. In particular, existing imputation schemes differ in their ability to handle random slopes, categorical variables,…

The ability to generate novel hypotheses is an important problem-solving capacity of humans. This ability is vital for making sense of the complex and unfamiliar world we live in. Often, this capacity is characterized as an inference to the best explanation--selecting the "best" explanation from a given set of candidate hypotheses.…

Hierarchical linear modeling (HLM) is a useful tool when analyzing data collected from groups. There are many decisions to be made when constructing and estimating a model in HLM including which estimation technique to use. Three of the estimation techniques available when analyzing data with HLM are maximum likelihood, restricted maximum…

The traditional approach to estimating the consistency of school effects across subject areas and the stability of school effects across time is to fit separate value-added multilevel models to each subject or cohort and to correlate the resulting empirical Bayes predictions. We show that this gives biased correlations and these biases cannot be…

This article proposes a latent variable regression four-level hierarchical model (LVR-HM4) that uses a fully Bayesian approach. Using multisite multiple-cohort longitudinal data, for example, annual assessment scores over grades for students who are nested within cohorts within schools, the LVR-HM4 attempts to simultaneously model two types of…

When fitting hierarchical regression models, maximum likelihood (ML) estimation has computational (and, for some users, philosophical) advantages compared to full Bayesian inference, but when the number of groups is small, estimates of the covariance matrix (S) of group-level varying coefficients are often degenerate. One can do better, even from…

When fitting hierarchical regression models, maximum likelihood (ML) estimation has computational (and, for some users, philosophical) advantages compared to full Bayesian inference, but when the number of groups is small, estimates of the covariance matrix [sigma] of group-level varying coefficients are often degenerate. One can do better, even…