PET-PEESE • bayesma

Introduction

PET-PEESE (Stanley & Doucouliagos, 2014) is a regression-based method for detecting and correcting small-study effects — the tendency for smaller, less precise studies to report larger effects, a pattern consistent with publication bias or inflated estimates in underpowered research.

The key insight is that under publication bias, small-sample studies survive the publication filter only if they produce large effects. This creates a correlation between standard error and effect size. PET and PEESE use regression to estimate the effect that would be reported if a study were infinitely large (i.e., $SE \to 0$ ).

bayesma implements a fully Bayesian version of PET-PEESE via bayesma() with model = "pet_peese".

PET — Precision Effect Test

PET regresses the effect estimate on its standard error:

$y_i = \alpha + \beta \cdot SE_i + \varepsilon_i$

The intercept $\alpha$ estimates the true effect at infinite precision ( $SE_i = 0$ ). The slope $\beta$ captures the association between standard error and effect size: a positive $\beta$ indicates that less precise studies report larger effects.

The error term allows for residual heterogeneity:

$\varepsilon_i \sim \mathcal{N}(0,\; s_i^2 + \tau^2)$

When to use PET: PET is the correct specification when the true effect is zero. Its intercept estimate is unbiased in that case. If the true effect is non-zero, the $SE_i$ term in the regression absorbs some of the real effect, biasing $\hat{\alpha}$ downward.

PEESE — Precision Effect Estimate with Standard Error

PEESE replaces the linear $SE_i$ term with $SE_i^2$ :

$y_i = \alpha + \beta \cdot SE_i^2 + \varepsilon_i$

The quadratic specification is better calibrated when the true effect is non-zero: the $SE_i^2$ term changes more slowly for imprecise studies, leaving more of the true effect in the intercept. PEESE is therefore preferred as the primary bias-correction model when prior evidence suggests a non-zero effect.

Priors

$\alpha \sim \mathcal{N}(0,\; 1), \qquad \beta \sim \mathcal{N}(0,\; 1), \qquad \tau \sim \text{Half-Cauchy}(0,\; 0.5)$

These are the defaults. The prior on $\beta$ is weakly informative and symmetric — it does not enforce the direction of small-study effects.

Fitting PET-PEESE in bayesma

#| eval: false
fit_pet   <- bayesma(data, model = "pet")
fit_peese <- bayesma(data, model = "peese")

summary(fit_peese)

summary() reports the posterior for $\alpha$ (bias-corrected effect), $\beta$ (small-study effect slope), and $\tau$ (residual heterogeneity).

The conditional PET-PEESE rule

Stanley & Doucouliagos (2014) proposed a two-step decision rule:

Fit PET. If the PET intercept $\alpha$ is clearly different from zero (95% CI excludes 0), proceed to PEESE.
Report the PEESE intercept as the bias-corrected effect estimate.

The Bayesian version of this rule uses the posterior probability that $\alpha_\text{PET} \neq 0$ :

#| eval: false
# Fit both
fit_pet   <- bayesma(data, model = "pet")
fit_peese <- bayesma(data, model = "peese")

# Posterior probability of non-zero effect (PET intercept)
interpret(fit_pet, parameter = "alpha")

# Report PEESE if PET provides clear evidence of non-zero effect
summary(fit_peese)

In bayesma, Bayes factors for $\alpha$ are computed using the Savage-Dickey density ratio, providing a continuous measure of evidence for vs. against a non-zero effect.

Interpreting the output

Parameter	Interpretation
`alpha`	Bias-corrected pooled effect ( $SE \to 0$ )
`beta`	Small-study effect coefficient
`tau`	Residual between-study heterogeneity
`BF_alpha`	Bayes factor for $H_1: \alpha \neq 0$ vs $H_0: \alpha = 0$

Limitations

PET-PEESE assumes that small-study effects are the only source of funnel asymmetry. Genuine small-study effects (e.g., smaller studies targeting different populations) can mimic publication bias.
The method has low power when $k < 20$ and performs poorly when heterogeneity is large relative to the precision range of the included studies.
If true heterogeneity and publication bias both exist, PET-PEESE may overcorrect.

For a more comprehensive approach that models multiple sources of bias simultaneously, see Robust Bayesian Meta-Analysis (RoBMA), which averages over PET-PEESE and selection model assumptions.

References

Stanley TD, Doucouliagos H (2014). Meta-regression approximations to reduce publication selection bias. Research Synthesis Methods, 5(1), 60–78.