Common-effect model • bayesma

Introduction

The common-effect model is the simplest Bayesian meta-analysis model. It assumes that every study in the synthesis estimates the same underlying true effect $\theta$ , and that observed differences between studies arise entirely from sampling error.

This assumption is strong. It implies that if every study had infinite sample size, all estimates would converge on the same value. In practice it is most defensible when:

studies share the same population, intervention, comparator, and outcome
$k$ is small and between-study variation cannot be reliably estimated
a sensitivity check against random-effects estimates is wanted

When heterogeneity is plausible, prefer a random-effects model with informative priors on $\tau$ .

Model specification

Let $y_i$ denote the effect estimate from study $i = 1, \ldots, k$ , and $s_i$ its known standard error. The common-effect likelihood is

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

A weakly informative prior is placed on the shared effect:

$\theta \sim \mathcal{N}(0,\, \sigma_\theta^2)$

where $\sigma_\theta$ is set to reflect plausible effect magnitudes on the analysis scale (log-OR, SMD, etc.). The default in bayesma is $\mathcal{N}(0, 1)$ .

The posterior is

$\theta \mid \mathbf{y} \propto \mathcal{N}(0, \sigma_\theta^2) \prod_{i=1}^{k} \mathcal{N}(y_i \mid \theta, s_i^2)$

For a conjugate Gaussian prior the posterior is also Gaussian:

$\theta \mid \mathbf{y} \sim \mathcal{N}\!\left(\hat{\mu},\, \hat{\sigma}^2\right)$

$\hat{\sigma}^{-2} = \sigma_\theta^{-2} + \sum_{i=1}^{k} s_i^{-2}, \qquad \hat{\mu} = \hat{\sigma}^2 \sum_{i=1}^{k} \frac{y_i}{s_i^2}$

This is precision-weighted averaging: studies with smaller standard errors contribute more to the posterior.

The two-stage common-effect model

In two-stage meta-analysis the raw outcome data are first summarised to $(y_i, s_i)$ and then combined using the Gaussian likelihood above. This is the default in bayesma for continuous outcomes and log-scale binary outcomes.

The one-stage common-effect model

For binary outcomes, two-stage analysis discards within-study information by treating $s_i$ as known. The one-stage common-effect model replaces the normal approximation with the exact binomial likelihood:

$r_{ij} \sim \text{Binomial}(n_{ij},\, \pi_{ij}), \quad j \in \{\text{ctrl}, \text{trt}\}$

$\text{logit}(\pi_{ij}) = \gamma_i + \theta \cdot \mathbb{1}[j = \text{trt}]$

where $\gamma_i$ are study-specific baseline log-odds (nuisance parameters) and $\theta$ is the common log-odds ratio. Because the $\gamma_i$ appear as fixed effects rather than random effects, this is the one-stage common-effect model — all heterogeneity in baseline risk is absorbed into the study dummies.

Estimands

The posterior for $\theta$ lives on the analysis scale (log-OR, log-RR, SMD, etc.). bayesma reports the posterior median and equal-tailed 95% credible interval on this scale and, where applicable, back-transforms to the natural scale (OR, RR, etc.).

Prior sensitivity

Because the common-effect model has a single parameter, the posterior is sensitive to $\sigma_\theta$ only when $k$ is very small or standard errors are large. A sensitivity check with wider and narrower priors (e.g., $\mathcal{N}(0, 0.5)$ and $\mathcal{N}(0, 2)$ ) is recommended.

Limitations

The common-effect model does not estimate between-study heterogeneity. If heterogeneity exists, the pooled estimate $\hat{\mu}$ is still consistent but its posterior credible interval is too narrow — it quantifies uncertainty about the study-average effect, not the distribution of effects across a target population of studies.

For most applied meta-analyses, a random-effects model is the preferred starting point. The common-effect model is best viewed as a boundary case and a diagnostic tool.