Random effects meta-analysis for correlated test statistics
In remaCor: Random Effects Meta-Analysis for Correlated Test Statistics

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Standard approaches to meta-analysis assumes that effect sizes are statistically independent. Here we provide methods for fixed and random effects meta-analysis when the correlation between effect sizes are known.

Fixed effects meta-analysis

LS() implements fixed effect meta-analysis for correlated test statistics using method of @lin2009meta. By default, correlation is set to identity matrix to for independent test statistics.

Random effects meta-analysis

RE2C() implements random effect meta-analysis for correlated test statistics that jointly tests deviation of the mean from zero as well as effect size heterogenity. This method uses the RE2 method of @han2011random, or RE2 for correlated test statistics from @han2016general. By default, correlation is set to identity matrix to for independent test statistics. (In addition, this function computes the two step RE2C method of @lee2017increasing to further test for heterogenity in effect size after applying a fixed effect test.)

stat1: statistic testing effect mean
stat2: statistic testing effect heterogeneity
RE2Cp: RE2 p-value accounting for correlelation between tests. (This is the p-value appropriate for most questions)
RE2Cp.twoStep: two step RE2C test after fixed effect test. Only evaluated if twoStep==TRUE. (not typically used)
QE: test statistic for the test of (residual) heterogeneity
QEp: p-value for the test of (residual) heterogeneity
Isq: I^2 statistic

QE, QEp and Isq are only evaluted if correlation is diagonal

Examples

library(remaCor)
library(metafor)
library(mvtnorm)
library(clusterGeneration )

# sample size
n = 30

# number of response variables
m = 2

# Error covariance
Sigma = genPositiveDefMat(m)$Sigma

# regression parameters
beta = matrix(0, 1, m)

# covariates
X = matrix(rnorm(n), ncol=1)

# Simulate response variables
Y = X %*% beta + rmvnorm(n, sigma = Sigma)

# Multivariate regression
fit = lm(Y ~ X)

# Correlation between residuals
C = cor(residuals(fit))

# Extract effect sizes and standard errors from model fit
df = lapply(coef(summary(fit)), function(a) 
  data.frame(beta = a["X", 1], se = a["X", 2]))
df = do.call(rbind, df)

# Standard fixed effects meta-analysis
# of independent effects with metafor pacakge
rma( df$beta, sei=df$se, method="FE")

# Standard random effects meta-analysis
# of independent effects with metafor pacakge
rma( df$beta, sei=df$se, method="REML")

# Run fixed effects meta-analysis, assume identity correlation  
# Use Lin-Sullivan method
LS( df$beta, df$se)

# Run fixed effects meta-analysis, accounting for correlation  
# Use Lin-Sullivan method
LS( df$beta, df$se, C)

# Run random effects meta-analysis, assume identity correlation  
RE2C( df$beta, df$se)

# Run random effects meta-analysis, accounting for correlation 
RE2C( df$beta, df$se, C)