508 The Professional Counselor | Volume 10, Issue 4 Data Analysis Following the data collection, the observed data was entered into and analyzed using SAS statistical software system to run the Markov chain Monte Carlo procedure with the Metropolis–Hastings algorithm to generate the estimated models. A Bayesian theoretical approach was taken to use prior information elicited from Newhart et al.’s (2020) publication, “Factors Influencing Publication Rates Among Counselor Educators.” It was determined that a Poisson regression analysis was appropriate for determining the relationship between a predictor variable and an outcome variable characterized in the form of a frequency count. One assumption of Poisson models is that the mean and the variance are equal (homogeneity of conditional means). If this assumption is violated, a negative binomial model can account for a large difference between the variance and mean by estimating a dispersion parameter (Agresti, 2007). The test for the assumption of equal conditional mean and variance was violated, indicating overdispersion. Overdispersion occurs when the data has greater variability. The following negative binomial model was used to run a negative binomial regression (where D is the dispersion parameter): E(Y) = μ, Var(Y) = μ+Dμ2. Next, the self-reported data collected from Newhart et al. (2020) was used to determine prior information to distinguish the differences between Carnegie classification and publication rates using R1 institutions as a baseline (see Table 2). The logarithm of the ratio was used for the prior mean of the distribution. Table 2 Newhart et al. (2020) Self-Reported Total Publications by Carnegie Classification Total Publications Carnegie Classification M SD Ratio R1: Doctoral Universities – Very High 25.78 26.15 - R2: Doctoral Universities – High 19.74 18.53 0.766 D/PU: Doctoral/Professional Universities – Moderate 13.31 14.78 0.516 Master’s Universities 7.98 7.46 0.309 Note. Ratio reflects multiplicative factor in relation to baseline R1. Results A negative binomial regression was used to (a) determine what the posterior probability publishing rates of non-R1 programs were compared to programs at R1 institutions and (b) examine if Newhart et al.’s (2020) self-reported data was plausible given the observed data. An initial model contained 10,000 burn-ins with a total of 100,000 iterations; however, this model lacked efficient information as determined by the effective sample size and efficiency. Next, Gibbs sampling with the Jeffreys prior was used and produced similar posterior parameter estimates and increased efficiency, indicating robustness; however, the effective sample size did not increase. Therefore, a sum-to-zero constraint was used to re-parametrize the model by centering the parameters. This resulted in coefficients representing group deviations from the grand mean, where in prior models the coefficients represented group deviations from the reference group. The following results are reported using the WAMBS checklist procedure for reporting Bayesian
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