Quantifying posterior effect size distribution of susceptibility loci by common summary statistics

Abstract

Testing millions of single nucleotide polymorphisms (SNPs) in genetic association studies has become a standard routine for disease gene discovery. In light of recent re-evaluation of statistical practice, it has been suggested that p-values are unfit as summaries of statistical evidence. Despite this criticism, p-values contain information that can be utilized to address the concerns about their flaws. We present a new method for utilizing evidence summarized by p-values for estimating odds ratio (OR) based on its approximate posterior distribution. In our method, only p-values, sample size, and standard deviation for ln(OR) are needed as summaries of data, accompanied by a suitable prior distribution for ln(OR) that can assume any shape. The parameter of interest, ln(OR), is the only parameter with a specified prior distribution, hence our model is a mix of classical and Bayesian approaches. We show that our method retains the main advantages of the Bayesian approach: it yields direct probability statements about hypotheses for OR and is resistant to biases caused by selection of top-scoring SNPs. Our method enjoys greater flexibility than similarly inspired methods in the assumed distribution for the summary statistic and in the form of the prior for the parameter of interest. We illustrate our method by presenting interval estimates of effect size for reported genetic associations with lung cancer. Although we focus on OR, the method is not limited to this particular measure of effect size and can be used broadly for assessing reliability of findings in studies testing multiple predictors.

Keywords: approximate Bayes methods; p-values; prior distributions; strength of evidence.

Figure 1:. Conjugate Normal and Non-conjugate Skewed Prior Distributions for log(OR).

Symmetric density is the normal prior with the variance set to approximate the occurrence of OR>2 at about 1 in 20,000 SNPs. The asymmetric density represents non-conjugate skewed prior. The graph plots the two distributions on the same scale with the overlap shown in purple.

Figure 2:. Simulation scatter plots.

Each point in the graphs is an association result selected on the basis of a SNP with the minimum P-value out of 10,000 tests.