Multivariate Metaanalysis (I)
Reference: Chen, Y, Cai, Y, Hong, C, and Jackson, D. (2016) Inference for correlated effect sizes using multiple univariate metaanalyses, Statistics in Medicine, 35(9): 14051422.
Introduction
Multivariate metaanalysis jointly analyzes multiple and possibly correlated outcomes in a single analysis. The main challenge that is common to these multivariate methods is that they require the knowledge of the withinstudy correlations, whose calculation may not be easy and may sometimes require more computationally intensive methods. The goal of this tutorial is to explain a simple and noniterative method proposed in the paper by the R packages step by step.
R Packages and Analysis Function
In this paper, we analyze the example data using the proposed MMoM method in R.
The proposed method is implemented in the ’mmeta’ function in our R software package ‘xmeta’, and can be installed from CRAN ( the official R package archive. Make sure that you load them before trying to run the examples on this page. If you do not have the package installed, run:
Structure:
Below is the R code command for the MMoM (Multivariate Method of Moments with working independence assumption) method.
Input:

ys: effect sizes (y1 and y2) from n studies in a matrix format with two columns: y1 and y2

vars: an (nx2) matrix with column 1 being variance of y1 and column 2 being variance of y2

Note: when y1 or y2 is missing, we put y1 or y2 as 0 and corresponding variance as 10ˆ6.
Output:

beta.hat (the pooled effect sizes for two outcomes)

sigma.hat (covariance matrix of the estimates)
Implementation of Working Example
Now, let's apply this method to a working example. Below is the sample data.
To apply the MMoM method to this working example, run:
And you will get the following result:
You can obtain the estimate and standard error of delta = beta1beta2 by running:
The results are:
To obtain the estimate and standard error of beta.average = (beta1+beta2)/2, run:
And you will get: