Bias-adjusted models (Jung & Aloe) • bayesma

Introduction

Publication bias and selection models address systematic inflation driven by the publication process. A different but equally important problem is within-study bias: some studies may report inflated effects due to methodological limitations (low risk-of-bias items, selective reporting, inadequate blinding) rather than because they were selectively published.

Jung & Aloe (2026) extend the bias-adjusted framework of Verde (2021) to handle this problem. The model assigns each study a latent bias indicator and estimates how much the bias inflates the reported effect, while still recovering a valid estimate of the true underlying effect from the unbiased component.

bayesma implements this model via bayesma() with model = "jung_aloe".

Model specification

Observation model

Each study $i$ has a latent bias indicator $I_i \in \{0, 1\}$ , where $I_i = 1$ means the study is biased. The observed effect is:

$y_i \mid I_i \sim \mathcal{N}\!\left(\theta_i^B,\; s_i^2\right)$

where the apparent effect under bias is

$\theta_i^B = \theta_i + I_i \cdot \beta_i$

$\theta_i$ is the true study effect (what the study would have reported under ideal conditions).
$\beta_i$ is the bias magnitude for study $i$ (zero for unbiased studies, positive for upwardly biased ones).
$I_i$ is the binary bias indicator.

Random-effects structure

True effects are exchangeable across studies:

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

Bias magnitudes are also treated as random:

$\beta_i \mid \mu_\beta, \tau_\beta \sim \mathcal{N}(\mu_\beta,\; \tau_\beta^2), \quad \mu_\beta \geq 0$

The constraint $\mu_\beta \geq 0$ encodes the assumption that bias inflates rather than deflates effects.

Bias prevalence:

$I_i \mid \phi \sim \text{Bernoulli}(\phi), \qquad \phi \sim \text{Beta}(1,\; 4)$

The Beta(1, 4) prior places most prior mass on low bias prevalence, reflecting the assumption that most studies in a well-conducted systematic review are not severely biased.

Priors

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

$\mu_\beta \sim \text{Half-Normal}(0,\; 0.5), \qquad \tau_\beta \sim \text{Half-Cauchy}(0,\; 0.5)$

Incorporating risk-of-bias information

The key extension in Jung & Aloe (2026) is that the bias indicator probability $\phi_i$ can be informed by risk-of-bias assessments (e.g., from the Cochrane RoB 2 tool or INSPECT-SR). High-risk-of-bias studies receive a higher prior probability of being biased:

$\text{logit}(\phi_i) = \alpha + \gamma \cdot \text{RoB}_i$

where $\text{RoB}_i$ is a numeric risk-of-bias score for study $i$ .

#| eval: false
fit_jung <- bayesma(
  data,
  model      = "jung_aloe",
  rob_scores = data$rob_score   # numeric RoB score per study
)

Without rob_scores, all studies share the same $\phi$ (Verde’s original formulation).

Fitting the model

#| eval: false
# Without RoB scores (Verde 2021 version)
fit_verde <- bayesma(data, model = "jung_aloe")

# With risk-of-bias scores (Jung & Aloe 2026)
fit_jung <- bayesma(
  data,
  model      = "jung_aloe",
  rob_scores = data$rob_score
)

summary(fit_jung)

Key estimands

Parameter	Interpretation
`mu`	True pooled effect (bias-corrected)
`tau`	Between-study heterogeneity in true effects
`phi`	Posterior probability of bias prevalence
`mu_beta`	Mean bias magnitude across biased studies
`tau_beta`	Between-study variation in bias magnitudes
`gamma`	Log-odds increase in bias probability per unit RoB (if RoB scores supplied)

The posterior for mu is the primary quantity of interest: it estimates the pooled true effect after removing the contribution of study-level bias.

Comparing with the uncorrected model

#| eval: false
fit_re   <- bayesma(data)
fit_jung <- bayesma(data, model = "jung_aloe", rob_scores = data$rob_score)

compare_models(fit_re, fit_jung, labels = c("Random-effects", "Jung & Aloe"))

A downward shift in mu from the standard RE model to the bias-adjusted model indicates that within-study bias is inflating the naive pooled estimate.

Integration with INSPECT-SR

bayesma integrates with INSPECT-SR assessments. Risk-of-bias scores from inspect_sr() can be passed directly:

#| eval: false
rob <- inspect_sr(data)
fit_jung <- bayesma(data, model = "jung_aloe", rob_scores = rob$rob_score)

See INSPECT-SR Assessment for details on computing rob_score.

Limitations

The model requires informative priors on $\phi$ (or per-study RoB scores) to identify the bias component. Without any prior information about bias prevalence, the bias parameters are weakly identified.
The assumption $\beta_i \geq 0$ (upward bias) should be checked against subject-matter knowledge. Downward bias (e.g., measurement attenuation) requires setting $\mu_\beta \leq 0$ .
The model assumes that the bias process is independent of the true effect. This may not hold if more biased studies are conducted precisely when true effects are small.

References

Verde PE (2021). A bias-corrected meta-analysis model for combining studies of different types and quality. Biometrical Journal, 63(2), 406–422.

Jung Y, Aloe AM (2026). Bayesian bias-adjusted meta-analysis with risk-of-bias information. Research Synthesis Methods.