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Stan code — PET-PEESE

Source: vignettes/stan-pet-peese.qmd

Model description

PET-PEESE (Stanley & Doucouliagos, 2014) corrects for publication bias by regressing effects on their standard errors (PET) or squared standard errors (PEESE). The intercept estimates the effect at infinite precision — the unbiased estimate.

Mathematical specification

PET likelihood:

yi∣α,β∼𝒩(α+β⋅si,si2+τ2) y_i \mid \alpha, \beta \sim \mathcal{N}(\alpha + \beta \cdot s_i,\, s_i^2 + \tau^2)

PEESE likelihood:

yi∣α,β∼𝒩(α+β⋅si2,si2+τ2) y_i \mid \alpha, \beta \sim \mathcal{N}(\alpha + \beta \cdot s_i^2,\, s_i^2 + \tau^2)

Priors:

α∼𝒩(0,1),β∼𝒩(0,1),τ∼Half-Cauchy(0,0.5) \alpha \sim \mathcal{N}(0,\, 1), \qquad \beta \sim \mathcal{N}(0,\, 1), \qquad \tau \sim \text{Half-Cauchy}(0,\, 0.5)

Stan code (PEESE) 

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

parameters {
  real alpha;
  real beta;
  real<lower=0> tau;
}

model {
  target += normal_lpdf(alpha | 0, 1);
  target += normal_lpdf(beta  | 0, 1);
  target += cauchy_lpdf(tau   | 0, 0.5);

  target += normal_lpdf(y | alpha + beta * square(se),
                           sqrt(square(se) + square(tau)));
}

generated quantities {
  real b_Intercept = alpha;
}

For the PET model, replace square(se) with se in both the mean and (if needed) a different specification.

How bayesma calls this model 

bayesma(data, model_type = "pet_peese", pet_peese_form = "peese")
bayesma(data, model_type = "pet_peese", pet_peese_form = "pet")

The default is PEESE, which is preferred when the meta-analytic effect is expected to be non-zero. PET is preferred under the null or when testing whether there is any effect after adjusting for publication bias.

Parameterisation notes

  • alpha is the bias-corrected pooled effect (effect at si=0s_i = 0).
  • beta is the publication-bias slope: the rate at which effects grow as standard error increases.
  • tau captures residual between-study heterogeneity not explained by the precision-effect relationship.
  • b_Intercept = alpha is the estimand reported by bayesma_output().

Identifiability

PET-PEESE is identified when there is meaningful variation in sis_i across studies. When all studies have similar precision, the slope β\beta is weakly identified and the intercept α\alpha is uncertain. This is a limitation of the method, not a model specification error.

Known sampling difficulties

PET-PEESE is a linear regression model and samples efficiently. No divergences are expected. The only potential issue is when τ\tau is near zero and the posterior becomes very flat; a slightly more informative prior on τ\tau helps.