Meta-Analytic Mixture Model with a Step-Function Selection Model

Fits the random-effects meta-analytic mixture model of re_mix() combined with a step-function (weight-function) selection model to adjust for publication bias (Maier, 2026). One common selection model is estimated for all studies. Use sel_flexpb() to estimate separate selection models for groups of studies.

Usage

sel_mix(
  y,
  sd,
  M,
  steps = c(0.9, 0.95),
  one_sided = TRUE,
  mu_sd = 1,
  tau_sd = 0.2,
  mean_prior = c("normal", "gap"),
  tau_prior = c("half_normal", "inv_gamma"),
  gap_meanlog = log(0.2),
  gap_sdlog = 0.3,
  gap_min = 0,
  tau_alpha = 2.5,
  tau_beta = 0.15,
  prior_only = FALSE,
  ...
)

Arguments

y: numeric vector of observed effect sizes.
sd: numeric vector of the standard errors of y (same length as y).
M: number of mixture components (a positive integer).
steps: cutoffs of the p-value intervals of the selection model, given as cumulative probabilities of the standard normal distribution (i.e. qnorm(steps) are the cutoffs on the z-statistic; steps = c(0.95, 0.975) corresponds to one-sided p-values of .05 and .025, or two-sided p-values of .10 and .05 in the expected direction). One or two strictly increasing values in (0, 1) are supported. Defaults to c(0.9, 0.95).
one_sided: if TRUE (default) selection only operates on p-values in the expected direction (z-statistic y/sd); to model selection favouring negative effects, negate y before fitting. If FALSE selection is two-sided (abs(y/sd)), in which case all steps must exceed 0.5.
mu_sd: standard deviation of the normal prior on the component means (used by both mean-prior settings: as the prior on every ordered mean when mean_prior = "normal", and as the prior on the first mean when mean_prior = "gap"). Defaults to 1.
tau_sd: standard deviation of the half-normal prior on the heterogeneity SD tau (used when tau_prior = "half_normal"). Defaults to 0.2.
mean_prior: prior family for the component means. "normal" (default) places an independent normal(0, mu_sd) prior on each ordered mean. "gap" uses a lognormal prior on the positive gaps between adjacent means (controlled by gap_meanlog, gap_sdlog, gap_min), which discourages two components collapsing onto the same location.
tau_prior: prior family for the heterogeneity SD tau. "half_normal" (default) uses a half-normal(0, tau_sd) prior. "inv_gamma" places an inverse-gamma(tau_alpha, tau_beta) prior directly on tau.
gap_meanlog: log-scale location of the repulsive lognormal gap prior (only used when mean_prior = "gap"). Defaults to log(0.2).
gap_sdlog: log-scale standard deviation of the repulsive lognormal gap prior (only used when mean_prior = "gap"). Defaults to 0.3.
gap_min: hard lower floor on the gaps between adjacent means: each gap is constrained to be at least gap_min and the lognormal gap prior is truncated to [gap_min, Inf) (only used when mean_prior = "gap"). Defaults to 0.
tau_alpha: shape of the inverse-gamma prior on tau (only used when tau_prior = "inv_gamma"). Defaults to 2.5.
tau_beta: scale of the inverse-gamma prior on tau (only used when tau_prior = "inv_gamma"). Defaults to 0.15.
prior_only: if TRUE, the data are ignored and the prior distribution is sampled instead (useful for prior predictive checks).
...: further arguments passed to rstan::sampling(), e.g. chains, iter, cores, seed, refresh or control.

Value

An object of class stanfit as returned by rstan::sampling(). It contains posterior draws of

mu: vector of length M: the ordered component means.
tau: vector of length M: the between-study standard deviation of each component.
theta: vector of length M: the mixing weights among the published studies.
theta_preselection: vector of length M: the mixing weights corrected for selection, i.e. the estimated proportions of the components among all conducted studies rather than among the published ones.
omega: vector of length length(steps) + 1: the relative publication probabilities of the p-value intervals, non-decreasing with the last element fixed at 1.
omega_raw: the underlying simplex, omega = cumsum(omega_raw).
avg_omega: vector of length M: the average selection weight of the studies attributed to each component, i.e. how strongly each component is affected by selection.
mu1, mu_gap: the underlying parameterisation of the means (the first mean and the M - 1 gaps between adjacent means).
log_tau: log(tau), convenient for funnel diagnostics.
posterior_probs: K by M matrix: the posterior probability that study i belongs to component m.
y_rep, sd_rep: posterior predictive draws of K effect sizes and the standard errors they were generated with.

Use e.g. print(fit, pars = c("mu", "tau", "omega")), summary(fit) or rstan::extract() to access them.

Details

See Maier (2026) for details of the model and the priors.

References

Maier, M. (2026). Addressing heterogeneity with Bayesian meta-analytic mixture modelling. PsyArXiv. doi:10.31234/osf.io/nkyqm_v2

Vevea, J. L., & Hedges, L. V. (1995). A general linear model for estimating effect size in the presence of publication bias. Psychometrika, 60(3), 419–435. doi:10.1007/BF02294384

Maier, M., Bartoš, F., & Wagenmakers, E.-J. (2023). Robust Bayesian meta-analysis: Addressing publication bias with model-averaging. Psychological Methods, 28(1), 107–122. doi:10.1037/met0000405

Examples

# \donttest{
# 30 studies with a true mean of 0.3 under one-sided selection: studies that
# are not significant in the expected direction (two-sided p > .10) are
# published with probability .2, marginally significant ones with .5
set.seed(1)
dat <- sim_mix(K = 30, M = 1, mu = 0.3, tau = 0.1, steps = c(0.95, 0.975),
               weights = c(0.2, 0.5, 1), one_sided = TRUE)

# A single short chain keeps the example fast; use e.g. chains = 4 and
# iter = 2000 (the rstan defaults) for a real analysis.
fit <- sel_mix(dat$y, dat$sds, M = 1, steps = c(0.95, 0.975),
               chains = 1, iter = 400, refresh = 0)
#> Warning: The largest R-hat is NA, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
print(fit, pars = c("mu", "tau", "omega"))
#> Inference for Stan model: selection_mix.
#> 1 chains, each with iter=400; warmup=200; thin=1; 
#> post-warmup draws per chain=200, total post-warmup draws=200.
#> 
#>          mean se_mean   sd 2.5%  25%  50%  75% 97.5% n_eff Rhat
#> mu[1]    0.29    0.01 0.05 0.17 0.26 0.29 0.32  0.38    81 1.02
#> tau[1]   0.06    0.00 0.05 0.01 0.03 0.06 0.09  0.17   112 1.00
#> omega[1] 0.34    0.02 0.17 0.11 0.22 0.31 0.45  0.73   100 1.00
#> omega[2] 0.62    0.02 0.20 0.25 0.48 0.61 0.78  0.97   175 1.00
#> omega[3] 1.00    0.00 0.00 1.00 1.00 1.00 1.00  1.00   169 0.99
#> 
#> Samples were drawn using NUTS(diag_e) at Fri Oct  9 17:03:51 2026.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).
# }