Penalized Logistic Regression Analysis for Genetic Association Studies of Binary Phenotypes

Abstract

<b><i>Introduction:</i></b> 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 I errors. Different penalized likelihood methods have been developed to mitigate sparse data bias. We study penalized logistic regression using a class of log-<i>F</i> priors indexed by a shrinkage parameter <i>m</i> to shrink the biased MLE toward zero. <b><i>Methods:</i></b> We proposed 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 <i>m</i>; second, the estimated <i>m</i> is used for log-<i>F</i>-penalized logistic regression analyses of all variants using data augmentation with standard software. Our estimate of <i>m</i> 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 and (ii) a Laplace approximation to each integral, followed by derivative-free optimization of the approximation. <b><i>Results:</i></b> We evaluated 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-<i>F</i>-penalized approach has lower bias and mean squared error than other methods considered. We also illustrated the approach on data from a study of genetic associations with “Super Senior” cases and middle-aged controls. <b><i>Discussion/Conclusion:</i></b> We have proposed a method for single rare variant analysis with binary phenotypes by logistic regression penalized by log-<i>F</i> 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.