Bias-adjusted model (Verde 2021) • bayesma

Introduction

Verde (2021) proposed a Bayesian hierarchical model that explicitly represents the risk of bias at the study level. Unlike a simple RoB subgroup analysis (which discards high-risk studies) or a meta-regression on RoB score (which estimates a correlation, not a causal adjustment), Verde’s model treats potential bias as a latent quantity and partially adjusts for it within the model.

The model shares the same hierarchical structure as Verde and Jung (2022; see Bias-adjusted models (Jung & Aloe 2026)), but the source and characterisation of the bias parameter differ.

Model specification

Let $y_i$ be the observed effect from study $i$ and $b_i$ the latent bias in that study. The model decomposes the observed effect into a true effect and a bias component:

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

$\theta_i \sim \mathcal{N}(\mu,\, \tau^2)$

$b_i \mid \rho_i \sim \mathcal{N}(\rho_i \cdot \xi,\, \sigma_b^2)$

where:

$\mu$ is the bias-adjusted pooled true effect
$\tau$ is the between-study heterogeneity in true effects
$b_i$ is the study-specific bias
$\rho_i \in [0, 1]$ is an observed risk-of-bias indicator (or composite score)
$\xi$ is the mean bias per unit of $\rho_i$
$\sigma_b$ is the residual variability in bias across studies with the same $\rho_i$

When $\rho_i = 0$ (low risk of bias), the expected bias is zero and $b_i$ is centred at zero with SD $\sigma_b$ . When $\rho_i = 1$ (high risk), the expected bias is $\xi$ .

Prior distributions

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

$\xi \sim \mathcal{N}(0, 0.5), \qquad \sigma_b \sim \text{Half-Normal}(0, 0.5)$

The prior on $\xi$ is weakly informative and centred at zero, allowing the data to determine whether high-risk studies are inflated or deflated relative to low-risk studies.

Relationship to Verde and Jung (2022)

Verde (2021) and the Jung & Aloe (2026) variant share the same decomposition $y_i = \theta_i + b_i$ . The key difference lies in how $b_i$ is parameterised:

Feature	Verde (2021)	Jung & Aloe (2026)
Bias driver	Observed RoB score $\rho_i$	Domain-specific binary indicators
Bias mean	$\rho_i \cdot \xi$	$\sum_d \xi_d \cdot R_{id}$
Bias variability	$\sigma_b$	Domain-specific $\sigma_{b_d}$

Verde (2021) is more parsimonious; Jung & Aloe (2026) provides domain-specific bias estimates.

Fitting the model

fit_verde <- bayesma(
  data,
  model_type  = "bias_corrected",
  bias_source = "verde_2021",
  rho_col     = "rob_composite_score"
)

summary(fit_verde)

rho_col should name a column in data containing the composite risk-of-bias score for each study, scaled to $[0, 1]$ .

Interpreting results

The posterior for $\mu$ is the bias-adjusted pooled effect. The posterior for $\xi$ quantifies the magnitude of bias per unit of the RoB score. A 95% credible interval for $\xi$ that excludes zero provides evidence that high-risk studies report systematically different effects than low-risk studies.

The posterior for $b_i$ gives the estimated bias for each study, which can be plotted alongside the forest plot to show which studies are most affected.

Sensitivity to RoB scale

The composite RoB score $\rho_i$ is typically constructed from domain ratings. Different aggregation schemes (simple average, weighted, domain-level indicators) yield different values of $\rho_i$ and hence different adjustments. Reporting results under two or more aggregation schemes is recommended.

Assumptions

Bias acts additively on the effect scale.
The direction of bias is the same across studies (positive $\xi$ means high-risk studies overestimate; negative means they underestimate).
The relationship between $\rho_i$ and bias is linear. Non-linear relationships require an extended model.