Multivariate Meta-analysis (I)
Reference: Chen, Y, Cai, Y, Hong, C, and Jackson, D. (2016) Inference for correlated effect sizes using multiple univariate meta-analyses, Statistics in Medicine, 35(9): 1405-1422.
Multivariate meta-analysis 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 within-study 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 non-iterative 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:
Below is the R code command for the MMoM (Multivariate Method of Moments with working independence assumption) method.
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.
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 = beta1-beta2 by running:
The results are:
To obtain the estimate and standard error of beta.average = (beta1+beta2)/2, run:
And you will get: