Multivariate meta-analysis

Introduction

Many systematic reviews report multiple outcomes per study — for example, both a primary efficacy endpoint and a safety endpoint, or the same outcome measured at multiple time-points. Analysing each outcome independently discards the within-study correlation between outcomes and can yield inconsistent conclusions (e.g., significant benefit on the primary endpoint but not the secondary, when the two are strongly correlated within studies).

Multivariate meta-analysis (MVMA) models all outcomes jointly, exploiting the within-study correlation to improve estimation efficiency and providing coherent inference across outcomes.

Model specification

Let $\mathbf{y}_i = (y_{i1}, \ldots, y_{iP})^\top$ be the vector of $P$ effect estimates from study $i$, with known within-study covariance matrix $\mathbf{S}_i$. The MVMA model is

$$\mathbf{y}_i \mid \boldsymbol{\theta}_i \sim \mathcal{N}_P(\boldsymbol{\theta}_i,\, \mathbf{S}_i)$$

$$\boldsymbol{\theta}_i \sim \mathcal{N}_P(\boldsymbol{\mu},\, \boldsymbol{\Sigma})$$

where:

• $\boldsymbol{\mu} = (\mu_1, \ldots, \mu_P)^\top$ is the vector of pooled effects
• $\boldsymbol{\Sigma} = \text{diag}(\boldsymbol{\tau}) \cdot \Omega \cdot \text{diag}(\boldsymbol{\tau})$ is the between-study covariance matrix
• $\tau_p$ is the between-study standard deviation for outcome $p$
• $\Omega$ is the between-study correlation matrix

Priors:

$$\mu_p \sim \mathcal{N}(0, \sigma_p^2), \quad \tau_p \sim \text{Half-Cauchy}(0, 0.5), \quad \Omega \sim \text{LKJ}(\eta)$$

Within-study covariance

The within-study covariance $\mathbf{S}_i$ is typically unknown and must be approximated. bayesma supports:

1. Known correlations. If the within-study correlation $\rho_\text{within}$ is known (e.g., from individual participant data or published correlation matrices), $\mathbf{S}_i$ is constructed from the marginal variances $s_{ip}^2$ and $\rho_\text{within}$.

2. Imputed correlations. When $\rho_\text{within}$ is not reported, a plausible value (typically 0.5) is imputed. Sensitivity to this choice should be reported.

3. Riley approximation. Riley et al. (2008) proposed marginalising over $\rho_\text{within}$ with a uniform prior, avoiding the need for a point estimate.

Fitting multivariate models

fit_mv <- bayesma_mv(
  data,
  outcomes  = c("lnOR_primary", "lnOR_safety"),
  se_cols   = c("se_primary", "se_safety"),
  rho_within = 0.5
)

The outcomes argument names the columns containing the per-outcome effect estimates; se_cols names the standard error columns. rho_within accepts a scalar (applied uniformly) or a list of study-specific values.

Borrowing strength

The key benefit of MVMA is borrowing strength across outcomes. When outcome $p$ is observed in only a subset of studies, the model uses the between-study correlation to impute $\theta_{ip}$ for studies that did not report it. This is only valid when the missing outcomes are missing at random — i.e., the probability of reporting does not depend on the unreported outcome value.

When univariate meta-analysis is adequate

Multivariate meta-analysis adds complexity. It is most useful when:

• outcomes are strongly correlated within studies ($|\rho_\text{within}| > 0.3$)
• some outcomes are partially missing across studies
• joint inference across outcomes is required (e.g., benefit–risk trade-off)

When outcomes are weakly correlated and fully observed, separate univariate analyses produce nearly identical estimates.

Limitations

• The between-study correlation $\Omega$ is estimated from between-study variation. With few studies ($k < 10$) or many outcomes ($P > 3$), $\Omega$ is poorly identified and posterior estimates depend heavily on the LKJ prior.
• Robust heterogeneity models (Student-$t$ RE, skew-normal RE) for the multivariate case are computationally demanding and not currently supported in all model-type combinations.