Penalized Logistic Regression Analysis for Genetic Association Studies of Binary Phenotypes

ABSTRACT Introduction Increasingly, logistic regression methods for genetic association studies of binary phenotypes must be able to accommodate data sparsity, which arises from unbalanced case-control ratios and/or rare genetic variants. Sparseness leads to maximum likelihood estimators (MLEs) of log-OR parameters that are biased away from their null value of zero and tests with inflated type 1 errors. Different penalized-likelihood methods have been developed to mitigate sparse-data bias. We study penalized logistic regression using a class of log- F priors indexed by a shrinkage parameter m to shrink the biased MLE towards zero. Methods We propose a two-step approach to the analysis of a genetic association study: first, a set of variants that show evidence of association with the trait is used to estimate m ; and second, the estimated m is used for log- F -penalized logistic regression analyses of all variants using data augmentation with standard software. Our estimate of m is the maximizer of a marginal likelihood obtained by integrating the latent log-ORs out of the joint distribution of the parameters and observed data. We consider two approximate approaches to maximizing the marginal likelihood: (i) a Monte Carlo EM algorithm (MCEM) and (ii) a Laplace approximation (LA) to each integral, followed by derivative-free optimization of the approximation. Results We evaluate the statistical properties of our proposed two-step method and compared its performance to other shrinkage methods by a simulation study. Our simulation studies suggest that the proposed log- F -penalized approach has lower bias and mean squared error than other methods considered. We also illustrate the approach on data from a study of genetic associations with “super senior” cases and middle aged controls. Discussion/Conclusion We have proposed a method for single rare variant analysis with binary phenotypes by logistic regression penalized by log- F priors. Our method has the advantage of being easily extended to correct for confounding due to population structure and genetic relatedness through a data augmentation approach.