Informative Prior Distributions in Multilevel/Hierarchical Linear Growth Models: Demonstrating the Use of Bayesian Updating for Fixed Effects
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This study demonstrates a fully Bayesian approach to multilevel/hierarchical linear growth modeling using freely available software. Further, the study incorporates informative prior distributions for fixed effect estimates using an objective approach. The objective approach uses previous sample results to form prior distributions included in subsequent samples analyses, a process referred to as Bayesian updating. Further, a method for model checking is outlined based on fit indices including information criteria (i.e., Akaike information criterion, Bayesian information criterion, and deviance information criterion) and approximate Bayes factor calculations. For this demonstration, five distinct samples of schools in the process of implementing School-Wide Positive Behavior Interventions and Supports (SWPBIS) collected from 2008 to 2013 were used with the unit of analysis being the school. First, the within-year SWPBIS fidelity growth was modeled as a function of time measured in months from initial measurement occasion. Uninformative priors were used to estimate growth parameters for the 2008-09 sample, and both uninformative and informative priors based on previous years' samples were used to model data from the 2009-10, 2010-11, 2011-12, 2012-13 samples. Bayesian estimates were also compared to maximum likelihood (ML) estimates, and reliability information is provided. Second, an additional three examples demonstrated how to include predictors into the growth model with demonstrations for: (a) the inclusion of one school-level predictor (years implementing) of SWPBIS fidelity growth, (b) several school-level predictors (relative socio-economic status, size, and geographic location), and (c) school and district predictors (sustainability factors hypothesized to be related to implementation processes) in a three-level growth model. Interestingly, Bayesian models estimated with informative prior distributions in all cases resulted in more optimal fit indices than models estimated with uninformative prior distributions.