Bayesian random-effects meta-analysis of study effect sizes

Pools study-level effect sizes, such as the \(\delta_{AB}\) posteriors of bc_smd_bayes(), in the brms model yi | se(sei) ~ 1 + (1 | study), adding (1 | study:case) when several estimates come from one study and moderators when given.

Usage

meta_bayes(
  es_data,
  yi = "yi",
  sei = "sei",
  study = "study",
  case = NULL,
  moderators = NULL,
  priors = NULL,
  metric = c("smd", "lrr"),
  iter = 4000,
  chains = 4,
  mcid = NULL,
  yi_stat = c("median", "mean"),
  control = list(),
  ...
)

Arguments

es_data: Data frame with one row per effect size, or a named list of bc_smd_bayes() results (names are the studies).
yi, sei: Effect size and standard error columns.
study: Study column.
case: Optional column identifying several effects within a study.
moderators: Optional character vector of moderator columns.
priors: brms priors; NULL uses the defaults in Details.
metric: "smd" (standardized mean difference, including BC-SMD) or "lrr" (log response ratio); sets the default prior scale on the mean effect and moderator slopes.
iter, chains: Sampler settings.
mcid: Optional minimal clinically important difference on the effect size scale; reports \(\Pr(\mu > \mathrm{MCID})\).
yi_stat: "median" or "mean"; the per-study posterior summary used as yi when es_data is a list of bc_smd_bayes() results.
control: Sampler control list passed to brms::brm(), merged over the defaults adapt_delta = 0.99 and max_treedepth.
...: Passed to brms::brm().

Value

A list with summary (data frame with columns quantity ("mu", "tau", "tau_within" when case is given, "prediction"), estimate (posterior median), sd, lower, upper), study_effects (data frame: study, estimate, sd, lower, upper), pd (probability of direction of \(\mu\)), prob_mcid (\(\Pr(\mu > \mathrm{MCID})\), NA without mcid), priors (the priors used), convergence (list: converged, max_rhat, min_ess_bulk, min_ess_tail, n_divergent), diagnostics (the compute_convergence_diagnostics() result), and model (the brmsfit).

Details

Orientation. Every yi must point the same way, with positive values meaning improvement. Effects on outcomes the intervention aims to decrease need their sign reversed first (bc_smd_bayes() does this with intervention_goal = "decrease").

Inputs. Each study enters through a normal likelihood with known standard error, the normal-theory approximation of Bayesian meta-analysis. es_data is a data frame, or a list of bc_smd_bayes() results named by study; for a list, yi is the posterior median (yi_stat = "median") or mean ("mean") of the finite \(\delta_{AB}\) draws and sei their SD. This two-stage approximation has three consequences. A skewed per-study posterior is summarized by two numbers only. The per-study summaries already carry the first-stage priors of bc_smd_bayes(), so the pooled posterior is not the posterior of one joint model. And the precision weights \(1/(\tau^2 + \mathrm{sei}^2)\) can correlate with the estimates, because sei grows with \(|\delta_{AB}|\); such an association biases average effects and confidence interval coverage in the meta-analysis of within-case effect sizes. Pooling the full per-study draws (a mixture approach) is not implemented.

Priors. The intercept and moderator slopes get a fixed, weakly informative normal(0, s) on the effect size metric, set before seeing the data: \(s = 2\) for standardized mean differences (metric = "smd", including BC-SMD), which keeps 95\ mass within \(\pm 4\) SD units, and \(s = 1\) for log response ratios (metric = "lrr"), which keeps 95\ 0.14 to 7. Moderator slopes are per unit of the moderator, so center and scale continuous moderators. Effects on any other scale need explicit priors. Each between-study (and within-study) SD gets a half-\(t_3\)(0, 0.5) prior; the half-\(t\) family lets the data dominate when the likelihood is informative. Supply priors to override, and refit with a wider and a narrower \(s\) as a prior sensitivity check.

Convergence. The fit is checked with compute_convergence_diagnostics(): R-hat < 1.01, bulk and tail ESS above 100 per chain, and no divergent transitions. A warning names the failed checks.

Sampler. The sampler runs with adapt_delta = 0.99 and max_treedepth from the scdbayes.max_treedepth option (default 12): with few studies the posterior of \(\tau\) has a funnel near 0 that causes divergent transitions at the brms default. Entries of control replace these defaults one at a time, so list(max_treedepth = 15) keeps adapt_delta = 0.99.

Outputs. The mean effect is b_Intercept (the effect at moderator values of 0). The prediction interval is the posterior predictive distribution of the true effect in a new study, \(\mu + N(0, \tau^2)\) per draw. When case is given the within-study SD is added, so it is the prediction for a new case (or measure) in a new study. Study effects are the shrunken study means \(\mu + u_{study}\) (plus moderator terms), averaged over a study's rows.

Examples

if (interactive()) {
  es <- data.frame(study = paste0("S", 1:6), yi = c(0.8, 1.2, 0.4, 1.5, 0.9, 0.6),
                   sei = c(0.3, 0.4, 0.35, 0.5, 0.3, 0.45))
  meta_bayes(es)$summary
}