Random effects meta-analysis for correlated test statistics
Gabriel Hoffman
Icahn School of Medicine at Mount Sinai, New York
Source:
vignettes/remaCor.Rmd
remaCor.RmdStandard 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 Lin
and Sullivan (2009). 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 Han and Eskin (2011), or RE2 for
correlated test statistics from Han et al.
(2016). By default, correlation is set to identity matrix to for
independent test statistics. (In addition, this function computes the
two step RE2C method of Lee, Eskin, and Han
(2017) to further test for heterogenity in effect size after
applying a fixed effect test.)
stat1: statistic testing effect meanstat2: statistic testing effect heterogeneityRE2Cp: 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 iftwoStep==TRUE. (not typically used)QE: test statistic for the test of (residual) heterogeneityQEp: p-value for the test of (residual) heterogeneity-
Isq: I^2 statisticQE,QEpandIsqare 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$se, method="FE")##
## Fixed-Effects Model (k = 2)
##
## I^2 (total heterogeneity / total variability): 0.00%
## H^2 (total variability / sampling variability): 0.02
##
## Test for Heterogeneity:
## Q(df = 1) = 0.0171, p-val = 0.8958
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## -0.0114 0.3080 -0.0370 0.9705 -0.6151 0.5923
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Standard random effects meta-analysis
# of independent effects with metafor pacakge
rma( df$beta, sei=df$se, method="REML")##
## Random-Effects Model (k = 2; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0 (SE = 0.2714)
## tau (square root of estimated tau^2 value): 0
## I^2 (total heterogeneity / total variability): 0.00%
## H^2 (total variability / sampling variability): 1.00
##
## Test for Heterogeneity:
## Q(df = 1) = 0.0171, p-val = 0.8958
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## -0.0114 0.3080 -0.0370 0.9705 -0.6151 0.5923
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Run fixed effects meta-analysis, assume identity correlation
# Use Lin-Sullivan method
LS( df$beta, df$se)## beta se p
## 1 -0.01140088 0.3080247 0.9704747
# Run fixed effects meta-analysis, accounting for correlation
# Use Lin-Sullivan method
LS( df$beta, df$se, C)## beta se p
## 1 -0.01169888 0.3178948 0.9706436
# Run random effects meta-analysis, assume identity correlation
RE2C( df$beta, df$se)## stat1 stat2 RE2Cp RE2Cp.twoStep QE QEp Isq
## 1 0.001369952 0 0.9750027 NA 0.01714325 0.8958289 0
# Run random effects meta-analysis, accounting for correlation
RE2C( df$beta, df$se, C)## stat1 stat2 RE2Cp RE2Cp.twoStep QE QEp Isq
## 1 0.001354322 0 0.9755282 NA NA NA NA## stat1 stat2 RE2Cp RE2Cp.twoStep QE QEp Isq
## 1 0.001354322 0 0.9755282 NA NA NA NASession info
## R version 4.3.0 (2023-04-21)
## Platform: x86_64-apple-darwin22.4.0 (64-bit)
## Running under: macOS Ventura 13.4
##
## Matrix products: default
## BLAS: /Users/gabrielhoffman/prog/R-4.3.0/lib/libRblas.dylib
## LAPACK: /usr/local/Cellar/r/4.3.0_1/lib/R/lib/libRlapack.dylib; LAPACK version 3.11.0
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## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
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## time zone: America/New_York
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] clusterGeneration_1.3.7 MASS_7.3-60 mvtnorm_1.2-1
## [4] metafor_4.2-0 numDeriv_2016.8-1.1 metadat_1.2-0
## [7] Matrix_1.5-4.1 remaCor_0.0.15 ggplot2_3.4.2
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