Stan code — RE mixture model (heterogeneity)

Model description

The random-effects mixture model for heterogeneity (distinct from the publication bias mixture model) represents the distribution of true study effects as a mixture of two Gaussian components. This captures bimodal heterogeneity — for example, when two distinct populations of studies exist that produce qualitatively different effect sizes.

This is the same model as the mixture RE distribution but framed explicitly as a heterogeneity assessment tool rather than a distributional sensitivity analysis.

Mathematical specification

Likelihood:

$$y_i \mid \theta_i \sim \mathcal{N}(\theta_i,\, s_i^2)$$

Mixture RE prior:

$$p(\theta_i) = \pi \cdot \mathcal{N}(\theta_i \mid \mu_1,\, \tau_1^2) + (1 - \pi) \cdot \mathcal{N}(\theta_i \mid \mu_2,\, \tau_2^2)$$

Marginal likelihood (integrated over $\theta_i$):

$$p(y_i) = \pi \cdot \mathcal{N}(y_i \mid \mu_1,\, \tau_1^2 + s_i^2) + (1-\pi) \cdot \mathcal{N}(y_i \mid \mu_2,\, \tau_2^2 + s_i^2)$$

Priors:

$$\pi \sim \text{Beta}(2,\, 2), \qquad \mu_j \sim \mathcal{N}(0,\, 1), \qquad \tau_j \sim \text{Half-Cauchy}(0,\, 0.5)$$

with the ordering constraint $\mu_1 \leq \mu_2$ to resolve label switching.

Stan code

data {
  int<lower=1> N;
  vector[N] y;
  vector<lower=0>[N] se;
}

parameters {
  ordered[2] mu;
  vector<lower=0>[2] tau;
  real<lower=0, upper=1> pi_mix;
}

model {
  target += beta_lpdf(pi_mix | 2, 2);
  target += normal_lpdf(mu   | 0, 1);
  target += cauchy_lpdf(tau  | 0, 0.5);

  for (i in 1:N) {
    target += log_mix(
      pi_mix,
      normal_lpdf(y[i] | mu[1], sqrt(square(tau[1]) + square(se[i]))),
      normal_lpdf(y[i] | mu[2], sqrt(square(tau[2]) + square(se[i])))
    );
  }
}

generated quantities {
  real b_Intercept = pi_mix * mu[1] + (1 - pi_mix) * mu[2];
  real b_pi        = pi_mix;
}

How bayesma calls this model

bayesma(
  data,
  model_type = "random_effect",
  re_dist    = "mixture"
)

Parameterisation notes

The ordered[2] mu declaration enforces $\mu_1 \leq \mu_2$, which fully resolves label switching for the component means. The mixing weight $\pi$ is still identified up to the relabelling $(\pi, \mu_1, \mu_2) \leftrightarrow (1-\pi, \mu_2, \mu_1)$, but the ordering constraint breaks this symmetry.

b_Intercept is the mixture-weighted mean effect — the expected true effect for a randomly drawn study.

Distinguishing from the Maier publication bias mixture

Both the RE mixture and the Maier model use two Gaussian components. The key difference is their purpose and constraints:

Feature	RE mixture	Maier mixture
Purpose	Characterise heterogeneity	Correct publication bias
Constraint	$\mu_1 < \mu_2$ (ordering)	$\mu_b > \theta$ (direction)
Interpretation	Two populations of studies	Biased vs unbiased studies

Known sampling difficulties

Same as the mixture RE model: use adapt_delta = 0.99 and inspect per-chain traces.