Second, the way the data are coded, larger effect sizes are interpreted as more negative outcomes associated with children having been spanked. We prefer conditions associated with lower values for things like Child aggression and Adult mental health problems.
First, with the possible exception of Adult support for physical punishment, all of the outcomes are negative. \[y_i \sim \text (0, 1)\) as the default prior for the group-level standard deviations, it makes sense to use it here for our meta-analytic \(\tau\) parameter. The basic version of a Bayesian meta-analysis follows the form Other possible estimands are the effect size in any of the observed studies and the effect size in another, comparable (exchangeable) unobserved study. The first potential estimand of a meta-analysis, or a hierarchically structured problem in general, is the mean of the distribution of effect sizes, since this represents the overall ‘average’ effect across all studies that could be regarded as exchangeable with the observed studies. A third, more general, possibility is that we regard the studies as exchangeable but not necessarily either identical or completely unrelated in other words we allow differences from study to study, but such that the differences are not expected a priori to have predictable effects favoring one study over another.… This third possibility represents a continuum between the two extremes, and it is this exchangeable model (with unknown hyperparameters characterizing the population distribution) that forms the basis of our Bayesian analysis… A second possibility is that the studies are so different that the results of any one study provide no information about the results of any of the others. The first possibility is that we view the studies as identical replications of each other, in the sense we regard the individuals in all the studies as independent samples from a common population, with the same outcome measures and so on. Our focus is on estimating meaningful parameters, and for this objective there appear to be three possibilities, accepting the overarching assumption that the studies are comparable in some broad sense. We’ll let Gelman and colleagues introduce the topic:ĭiscussions of meta-analysis are sometimes imprecise about the estimands of interest in the analysis, especially when the primary focus is on testing the null hypothesis of no effect in any of the studies to be combined. And since McElreath’s text doesn’t directly address meta-analyses, we’ll take further inspiration from Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin’s Bayesian data analysis, Third edition. Thus, you can use brms::brm() to fit Bayesian meta-analyses, too.īefore we proceed, I should acknowledge that this section is heavily influenced by Matti Vourre’s great blog post, Meta-analysis is a special case of Bayesian multilevel modeling. As it turns out, meta-analyses are often just special kinds of multilevel measurement-error models. If your mind isn’t fully blown by those measurement-error and missing-data models, let’s keep building.