Comparisons among GLM, LME and BML. (a) The seed-based correlation results at 21 ROIs from 124 subjects were fitted with LME (using the R package lme4) and BML (using Stan with 4 chains and 1,000 iterations) separately, in which overall ToMI was an explanatory variable. Random effects under LME correspond to group/entity-level effects plus family specific parameters (standard deviation σ for the residuals ϵij) under BML, while fixed effects under LME correspond to population-level effects under BML. The column headers SD, QI, and ESS are short for standard deviation, quantile interval, effect sample size, respectively. The parameter estimates from the LME and BML outputs (columns in gray) are very similar, even though priors were injected into BML. All values under BML were less than 1.1, indicating that all the 4 chains converged well. The effective sizes for the population- and group/entity-level effect of ToMI were 468 and 947, respectively, enough to warrant quantile accuracy in summarizing the posterior distributions. Comparisons between GLM and BML. (b) To directly compare with BML, the Bayesianized version of GLM (15) was fitted with the data, and the higher predictive accuracy of BML is seen here with its substantial lower out-of-sample deviance measured by the leave-one-out information criterion (LOOIC), the widely applicable (or Watanabe-Akaike) information criterion (WAIC) through leave-one-out cross-validation, and the corresponding standard error (SE).
