One-stage and two-stage models for binary outcomes

Introduction

Meta-analyses of binary outcomes (events vs. non-events) can be conducted using either a two-stage or a one-stage approach. The distinction matters when studies are small, baseline event rates vary widely, or the outcome is rare.

Two-stage: each study is first summarised to a log-scale effect estimate $y_i$ and standard error $s_i$, then the meta-analysis uses a Gaussian likelihood on those summaries. This is the default in bayesma and adequate for most applications.

One-stage: the original binomial counts are modelled directly, avoiding the normal approximation and handling studies with zero events without continuity corrections. bayesma implements the one-stage model following Jackson et al. (2018).

Two-stage binary meta-analysis

Effect size computation

For study $i$ with treatment arm $(r_{1i}, n_{1i})$ and control arm $(r_{0i}, n_{0i})$, the log odds ratio is

$$y_i = \log\!\left(\frac{r_{1i}/(n_{1i}-r_{1i})}{r_{0i}/(n_{0i}-r_{0i})}\right)$$

with approximate variance

$$s_i^2 = \frac{1}{r_{1i}} + \frac{1}{n_{1i}-r_{1i}} + \frac{1}{r_{0i}} + \frac{1}{n_{0i}-r_{0i}}$$

Log-risk ratios and risk differences are supported via the effect_measure argument.

Meta-analysis model

The two-stage random-effects model applies the standard Gaussian hierarchy to the log-scale summaries:

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

See Gaussian Random-Effects Model for full details.

Limitations

• The normal approximation to the log-OR is poor when cell counts are small ($r_i < 5$).
• Studies with zero events require a continuity correction, which shifts the estimate in ways that depend on the correction chosen.
• Within-study information about baseline event rates is discarded.

One-stage model (Jackson et al.)

The one-stage model replaces the Gaussian approximation with an exact binomial likelihood. The formulation follows Jackson et al. (2018).

Model specification

For study $i \in \{1, \ldots, k\}$ and arm $j \in \{0\;\text{(ctrl)},\; 1\;\text{(trt)}\}$:

$$r_{ij} \sim \text{Binomial}(n_{ij},\; \pi_{ij})$$

$$\text{logit}(\pi_{ij}) = \gamma_i + \theta_i \cdot j$$

where $\gamma_i$ is the study-specific log-odds on control (baseline log-odds) and $\theta_i$ is the study-specific log-odds ratio.

Treatment effects are exchangeable across studies:

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

Baseline log-odds receive weakly informative priors:

$$\gamma_i \sim \mathcal{N}(0,\; \sigma_\gamma^2), \quad \sigma_\gamma = 2$$

covering a wide range of baseline event rates on the logit scale.

Priors

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

These match the defaults for the two-stage model. See Prior Predictive Checks for outcome-specific guidance.

Why the one-stage model handles zero events

A study with $r_{ij} = 0$ contributes a valid binomial log-likelihood term — the probability $\text{Binomial}(0 \mid n, \pi)$ is defined and informative. The nuisance parameters $\gamma_i$ are integrated out by MCMC, so uncertainty about baseline risk propagates into the posteriors for $\mu$ and $\tau$.

Fitting in bayesma

#| eval: false

# Two-stage (default for binary outcomes with pre-computed summaries)
fit_2s <- bayesma(
  data,
  outcome        = "binary",
  effect_measure = "log_or",
  model_stage    = "two_stage"
)

# One-stage (Jackson et al.) — expects raw count columns r1, n1, r0, n0
fit_1s <- bayesma(
  data,
  outcome        = "binary",
  effect_measure = "log_or",
  model_stage    = "one_stage"
)

summary(fit_1s)

When to prefer the one-stage model

Situation	Recommendation
Large studies, moderate event rates	Two-stage is adequate
One or more studies have zero events	One-stage preferred
Small $k$ or sparse data	One-stage preferred
Baseline event rates vary substantially	One-stage preferred
Computational speed matters	Two-stage is faster

For most systematic reviews with reasonably sized studies and no zero-event cells, the two approaches yield very similar estimates.

Comparing estimates

#| eval: false
compare_models(fit_2s, fit_1s, labels = c("Two-stage", "One-stage"))

Non-trivial differences between the two approaches indicate that the normal approximation is influencing the two-stage result. In that case, the one-stage estimate should be reported as the primary analysis.

References

Jackson D, Law M, Stijnen T, Viechtbauer W, White IR (2018). A comparison of 7 random-effects models for meta-analyses that estimate the summary odds ratio. Statistics in Medicine, 37(7), 1059–1085.