Multiarm models • bayesma

Introduction

A multiarm (or multi-treatment) trial assigns participants to three or more arms — typically a control and two or more active interventions. Including multiarm studies in a pairwise meta-analysis raises two problems:

Unit of analysis error. Using each active arm as a separate comparison to control double-counts the shared control arm, inflating precision and producing correlated residuals.
Missing comparisons. If only one arm-pair is extracted, information from the other arms is discarded.

The one-stage approach in bayesma handles both problems by modelling the full multiarm data structure jointly.

Model specification

Let study $i$ have $A_i$ arms. Arm $j$ in study $i$ contributes $r_{ij}$ events out of $n_{ij}$ participants (binary outcome). The treatment indicator for arm $j$ is $t_{ij} \in \{0, 1, \ldots, T\}$ where $T$ is the number of active treatments.

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

$\text{logit}(\pi_{ij}) = \gamma_i + \sum_{t=1}^{T} \theta_t \cdot \mathbb{1}[t_{ij} = t] + \epsilon_{ij}$

where:

$\gamma_i$ is the study-specific baseline log-odds (control arm)
$\theta_t$ is the pooled log-OR for treatment $t$ vs control
$\epsilon_{ij}$ is a study-by-treatment random effect capturing heterogeneity

The $\epsilon_{ij}$ for arms within the same study are correlated. bayesma uses an LKJ prior on the correlation matrix $\Omega$ among random effects:

$\boldsymbol{\epsilon}_i \sim \mathcal{N}(\mathbf{0},\, \text{diag}(\boldsymbol{\tau}) \cdot \Omega \cdot \text{diag}(\boldsymbol{\tau}))$

$\Omega \sim \text{LKJ}(\eta)$

The LKJ prior with $\eta = 1$ is uniform over correlation matrices. Larger $\eta$ concentrates mass near the identity (i.e., independent arms).

Continuous outcomes

For continuous outcomes the model is analogous with a normal likelihood:

$y_{ij} \sim \mathcal{N}(\gamma_i + \sum_t \theta_t \cdot \mathbb{1}[t_{ij} = t] + \epsilon_{ij},\, \sigma^2)$

Why the one-stage approach avoids unit-of-analysis error

In the one-stage model the control arm appears once per study ( $\gamma_i$ ), and each active arm contributes an additional $\theta_t + \epsilon_{ij}$ term. The shared control arm enters the likelihood a single time, so its contribution to precision is not double-counted. The within-study correlation between arm-level estimates is handled through the covariance structure of $\boldsymbol{\epsilon}_i$ .

Fitting multiarm models

fit <- bayesma(
  data,
  model_type = "random_effect",
  stage      = "one_stage",
  multiarm   = TRUE,
  rho_prior  = lkj(eta = 1)
)

The multiarm = TRUE flag activates the correlated random-effects structure. The data must include a arm column identifying each arm and a treatment column coding the treatment label.

Estimands from multiarm models

bayesma() returns a posterior for each $\theta_t$ (treatment vs control). Pairwise treatment contrasts $\theta_a - \theta_b$ can be derived from the joint posterior:

bayesma_extract(fit, contrast = c("treatment_a", "treatment_b"))

Limitations

The LKJ prior on $\Omega$ has limited influence when the number of multiarm studies is small ( $< 5$ ). In this setting the within-study correlation is poorly estimated and the model reduces approximately to independent arm-level analyses.
Models with many treatments ( $T > 4$ ) may be better handled by network meta-analysis frameworks (e.g., gemtc, multinma).
Multiarm models are computationally heavier than pairwise models due to the correlation matrix. Increase adapt_delta (default 0.95 for multiarm) if divergent transitions occur.