Model selection is an essential task for many applications in scientific discovery. The most common approaches rely on univariate linear measures of association between each feature and the outcome. Such classical selection procedures fail to take into account nonlinear effects and interactions between features. Kernel-based selection procedures have been proposed as a solution. However, current strategies for kernel selection fail to measure the significance of a joint model constructed through the combination of the basis kernels. In the present work, we exploit recent advances in post-selection inference to propose a valid statistical test for the association of a joint model of the selected kernels with the outcome. The kernels are selected via a step-wise procedure which we model as a succession of quadratic constraints in the outcome variable.

BACKGROUND: Part of the missing heritability in Genome Wide Association Studies (GWAS) is expected to be explained by interactions between genetic variants, also called epistasis. Various statistical methods have been developed to detect epistasis in case-control GWAS. These methods face major statistical challenges due to the number of tests required, the complexity of the Linkage Disequilibrium (LD) structure, and the lack of consensus regarding the definition of epistasis. Their limited impact in terms of uncovering new biological knowledge might be explained in part by the limited amount of experimental data available to validate their statistical performances in a realistic GWAS context. In this paper, we introduce a simulation pipeline for generating real scale GWAS data, including epistasis and realistic LD structure. We evaluate five exhaustive bivariate interaction methods, fastepi, GBOOST, SHEsisEpi, DSS, and IndOR. Two hundred thirty four different disease scenarios are considered in extensive simulations. We report the performances of each method in terms of false positive rate control, power, area under the ROC curve (AUC), and computation time using a GPU. Finally we compare the result of each methods on a real GWAS of type 2 diabetes from the Welcome Trust Case Control Consortium. RESULTS: SNPs and 15,000 samples in a couple of hours using a GPU. CONCLUSION: This study confirms that computation time is no longer a limiting factor for performing an exhaustive search of epistasis in large GWAS. For this task, using DSS on SNP pairs with limited LD seems to be a good strategy to achieve the best statistical performance. A combination approach using both DSS and GBOOST is supported by the simulation results and the analysis of the WTCCC dataset demonstrated that this approach can detect distinct genes in epistasis. Finally, weak epistasis between common variants will be detectable with existing methods when GWAS of a few tens of thousands cases and controls are available.