Comparison of Two Meta-Analysis Methods: Inverse-Variance-Weighted Average and Weighted Sum of Z-Scores

Abstract

The meta-analysis has become a widely used tool for many applications in bioinformatics, including genome-wide association studies. A commonly used approach for meta-analysis is the fixed effects model approach, for which there are two popular methods: the inverse variance-weighted average method and weighted sum of z-scores method. Although previous studies have shown that the two methods perform similarly, their characteristics and their relationship have not been thoroughly investigated. In this paper, we investigate the optimal characteristics of the two methods and show the connection between the two methods. We demonstrate that the each method is optimized for a unique goal, which gives us insight into the optimal weights for the weighted sum of z-scores method. We examine the connection between the two methods both analytically and empirically and show that their resulting statistics become equivalent under certain assumptions. Finally, we apply both methods to the Wellcome Trust Case Control Consortium data and demonstrate that the two methods can give distinct results in certain study designs.

Keywords: fixed effects model; genome-wide association study; inverse variance-weighted average; meta-analysis; optimality; weighted sum of z-scores.

Fig. 1. Power test of IVW, SZ_SE, and SZ_N. Total 100,000 simulated meta-analysis data sets of five studies were generated, each with 500 case and 500 control. (A) We assumed the same MAFs of 0.3 for five studies. (B) We assumed varying MAFs of 0.1, 0.2, 0.3, 0.4, and 0.5 for five studies. IVW, inverse variance-weighted average method; SZ_SE, weighted SZ whose weights are given as inverse standard error; SZ_N, SZ whose weights are given as the square root of sample size; MAF, minor allele frequency.

Fig. 2. Ratio of standard errors of RA and T1D association analyses. Left panel shows log10 values of the ratio of the two standard errors from the two studies participating in a meta-analysis (RA and T1D). Right panel shows the −log10 values of p-values of two different meta-analysis methods (IVW and SZ_N) for combining the two studies (RA and T1D). (A) Both RA and T1D analyses used logistic regression. (B) Both RA and T1D analyses used linear mixed model using GEMMA. (C) RA analysis used linear mixed model and T1D analysis used logistic regression. RA, rheumatoid arthritis; T1D, type 1 diabetes; IVW, inverse variance-weighted average method; SZ_N, SZ whose weights are given as the square root of sample size.

Fig. 3. Ratio of standard errors of RA and T1D association analyses, when RA analysis used the standard linear regression and T1D analysis used linear mixed model. RA, rheumatoid arthritis; T1D, type 1 diabetes; SE, standard error.