A novel variational Bayes multiple locus Z-statistic for genome-wide association studies with Bayesian model averaging

Abstract

Motivation: For many complex traits, including height, the majority of variants identified by genome-wide association studies (GWAS) have small effects, leaving a significant proportion of the heritable variation unexplained. Although many penalized multiple regression methodologies have been proposed to increase the power to detect associations for complex genetic architectures, they generally lack mechanisms for false-positive control and diagnostics for model over-fitting. Our methodology is the first penalized multiple regression approach that explicitly controls Type I error rates and provide model over-fitting diagnostics through a novel normally distributed statistic defined for every marker within the GWAS, based on results from a variational Bayes spike regression algorithm.

Results: We compare the performance of our method to the lasso and single marker analysis on simulated data and demonstrate that our approach has superior performance in terms of power and Type I error control. In addition, using the Women's Health Initiative (WHI) SNP Health Association Resource (SHARe) GWAS of African-Americans, we show that our method has power to detect additional novel associations with body height. These findings replicate by reaching a stringent cutoff of marginal association in a larger cohort.

Availability: An R-package, including an implementation of our variational Bayes spike regression (vBsr) algorithm, is available at http://kooperberg.fhcrc.org/soft.html.

Fig. 1.

The family-wise error rate (FWER) and power for six different strategies for choosing the model size associated with the z_vb statistic, and for single-marker analysis (χ²_sma) for simulations of 10⁴ independent genotypes with differing sample sizes and heritabilities. For the FWER, the red horizontal line shows a FWER of 0.05 and the blue horizontal lines show the 95% CIs for controlling FWER to 0.05. The six different strategies are: choice based on minimum of KL diagnostic statistic (z_vb^a), expectation of the diagnostic statistic (z_vb^b), minimum plus one standard error (z_vb^c), expectation plus one standard error (z_vb^d), minimum plus two standard errors (z_vb^e) and expectation plus two standard errors (z_vb^f)

Fig. 2.

The estimated expected regression coefficients as a function of the penalty parameter ℓ₀ for the analysis of height. As the penalty parameter increases in magnitude, the size of the model increases until it becomes over-fit. The position in the path for the four different strategies is shown with the vertical bars, with the chosen strategy (z_vb^b) in blue. The features that were significant for z_vb^b at are also shown in blue along the entire path