The gold standard in Genome Wide Association (GWA) studies is to bring interesting findings forward to replication in independent cohorts.  With unique phenotypes this approach is not possible. We suggest an approach based on non-parametric bootstrap. It allows adjustment for the effect known as winner’s curse, i.e. a bias towards over-estimation of effect sizes among the most significant Single Nucleotide Polymorphisms (SNPs).

We used a modified version of the double bootstrap re-sampling method (BR-squared) proposed by Faye et.al. in Statistics in Medicine 2011.  In brief this method uses a weighted average over bootstrap replicates of the difference between the within-sample effect size and the out-of-sample effect size to estimate the over-estimation. The difference was adjusted for the negative correlation between the two effect sizes (disjoint events). Since the bias correction in this context may be large, the standard deviation of the original effect size can be a poor estimate of the standard deviation of the bias-corrected effect size. To solve this, we performed a double bootstrap.

The results from the bootstrap are presented as a biased reduced effect size estimate with a corresponding (1-α/2)100% CI based on Efron’s bootstrap percentile interval. A (1-α/2)100% CI not covering zero corresponds to rejecting the null hypothesis at significance level α. In case of multiple testing α was adjusted using Bonferroni correction.