Meta-analytic Models

Overview

The meta-analytic family implements Meta-Analytic Gaussian Network Aggregation (MAGNA): a one-stage random-effects framework for pooling correlation (or covariance) matrices from multiple studies into a single model, while accounting for between-study heterogeneity. The pooled structure can be modeled directly as a correlation matrix (meta_varcov()) or as a Gaussian Graphical Model (meta_ggm()), yielding a meta-analytic partial-correlation network.

Typical use cases include aggregating published correlation matrices of symptom measures into a pooled symptom network, or combining the correlation matrices of several (possibly heterogeneous) samples collected by different labs.

The Random-Effects Model

For each study $s$, the vector of sample correlations $\boldsymbol{r}_s$ is modeled as:

$$\boldsymbol{r}_s = \boldsymbol{\rho} + \boldsymbol{u}_s + \boldsymbol{e}_s$$

• $\boldsymbol{\rho}$: the pooled (population-average) correlation structure — the quantity of interest, optionally parameterized as a GGM ($\boldsymbol{\Omega}_y$, matrix omega_y).
• $\boldsymbol{u}_s$: random study effects with covariance matrix $\boldsymbol{\Sigma}_{\text{RE}}$ (matrix sigma_randomEffects), capturing between-study heterogeneity.
• $\boldsymbol{e}_s$: sampling error with (approximately) known covariance $\boldsymbol{V}_s$, computed from the study sample sizes (see Vmethod).

Both the pooled structure and the random-effects covariance can be parameterized as cov, chol (default for random effects), prec, ggm, or cor. The square roots of the diagonal of $\boldsymbol{\Sigma}_{\text{RE}}$ are the heterogeneity standard deviations of each correlation, analogous to $\tau$ in univariate meta-analysis.

Data Input

meta_varcov() and meta_ggm() accept three kinds of input:

Sampling-Variance Methods (Vmethod)

The sampling covariance matrices $\boldsymbol{V}_s$ of the study correlations must be estimated. The Vmethod argument controls how:

The related argument Vestimation ("averaged", the default, or "per_study") controls whether the sampling-error part of the likelihood uses an averaged or per-study weight matrix.

Example: Meta-analytic GGM on Simulated Studies

We simulate 12 studies whose data come from a common chain network ($V_1 - V_2 - V_3 - V_4$, all partial correlations 0.3) with mild between-study heterogeneity, and pool them with meta_ggm(). This example is fully reproducible and runs in seconds:

library("psychonetrics")
library("dplyr")

# --- Simulate per-study correlation matrices ---
set.seed(123)
nStudy <- 12
nv <- 4

# True partial-correlation network (chain): V1 - V2 - V3 - V4
Omega <- matrix(0, nv, nv)
Omega[1, 2] <- Omega[2, 3] <- Omega[3, 4] <- 0.3
Omega <- Omega + t(Omega)

# Implied correlation matrix:
Delta <- diag(1 / sqrt(diag(solve(diag(nv) - Omega))))
trueSigma <- Delta %*% solve(diag(nv) - Omega) %*% Delta

cors <- list(); nobs <- numeric(nStudy)
for (s in seq_len(nStudy)) {
  nobs[s] <- round(runif(1, 200, 600))
  # Study-specific population (between-study heterogeneity):
  P <- cov2cor(trueSigma + crossprod(matrix(rnorm(nv * nv, 0, 0.08), nv, nv)))
  X <- matrix(rnorm(nobs[s] * nv), nobs[s], nv) %*% chol(P)
  cors[[s]] <- cor(X)
  colnames(cors[[s]]) <- rownames(cors[[s]]) <- paste0("V", 1:nv)
}

# --- Fit the meta-analytic GGM ---
mod_meta <- meta_ggm(cors = cors, nobs = nobs,
                     Vmethod = "OS_pooled") %>% runmodel

# Pooled partial-correlation network (recovers the true chain):
round(getmatrix(mod_meta, "omega_y"), 2)

# Heterogeneity SDs per correlation:
round(sqrt(diag(getmatrix(mod_meta, "sigma_randomEffects"))), 3)

# --- Prune to a sparse pooled network ---
mod_meta_pruned <- mod_meta %>% prune(alpha = 0.05, matrices = "omega_y")
compare(saturated = mod_meta, pruned = mod_meta_pruned)

The pooled network recovers the true chain structure (edges ≈ 0.30), and pruning correctly removes the three null edges without significant loss of fit (p = 0.78).

Related Functions: meta_lvm, meta_gvar

• meta_lvm(): single-stage meta-analytic latent variable models — a CFA/SEM for the pooled correlation/covariance structure combined with a random-effects model for between-study heterogeneity. Supports all standard parameterizations for the latent, residual, and random-effects structures.
• meta_var1() / meta_gvar(): meta-analytic VAR(1) models pooling temporal and contemporaneous network structures across multiple time-series studies. Accepts raw time-series data (data + studyvar) or pre-computed Toeplitz covariance matrices (covs + nobs).
• meta_ggm() is shorthand for meta_varcov(..., type = "ggm").

Summary

• Meta-analytic models pool correlation/covariance structures across studies in a one-stage random-effects framework (MAGNA)
• meta_ggm() yields a pooled partial-correlation network with heterogeneity estimates per correlation
• Input can be correlation matrices, covariance matrices (genuinely supported since 0.16), or raw data with a study indicator
• New Olkin–Siotani Vmethod options are recommended when heterogeneity variances are of interest
• The usual workflow applies: runmodel → prune → compare
• Latent variable (meta_lvm) and time-series (meta_gvar) extensions are available

Further Reading

• Epskamp, S., Isvoranu, A. M., & Cheung, M. W.-L. (2022). Meta-analytic Gaussian network aggregation. Psychometrika, 87(1), 12–46.