Analyzing genome-wide association studies with an FDR controlling modification of the Bayesian information criterion

The prevailing method of analyzing GWAS data is still to test each marker individually, although from a statistical point of view it is quite obvious that in case of complex traits such single marker tests are not ideal. Recently several model selection approaches for GWAS have been suggested, most of them based on LASSO-type procedures. Here we will discuss an alternative model selection approach which is based on a modification of the Bayesian Information Criterion (mBIC2) which was previously shown to have certain asymptotic optimality properties in terms of minimizing the misclassification error. Heuristic search strategies are introduced which attempt to find the model which minimizes mBIC2, and which are efficient enough to allow the analysis of GWAS data. Our approach is implemented in a software package called MOSGWA. Its performance in case control GWAS is compared with the two algorithms HLASSO and GWASelect, as well as with single marker tests, where we performed a simulation study based on real SNP data from the POPRES sample. Our results show that MOSGWA performs slightly better than HLASSO, whereas according to our simulations GWASelect does not control the type I error when used to automatically determine the number of important SNPs. We also reanalyze the GWAS data from the Wellcome Trust Case-Control Consortium (WTCCC) and compare the findings of the different procedures

Dissipative eigenvalue problems for a Sturm-Liouville operator with a singular potential

In this paper we consider the Sturm-Liouville operator d2/dx2 - 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L2(a, 0) and L2(0, b) of the space L2(a, b) = L2(a, 0) ⊕ L2(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L2(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition limx→0+ (f'(x)-f'(-x)) = γf(0) with γ ∈ C+ ∪ R or by the Dirichlet condition f(0+) = f(0-) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L2(a, b).

False positives under the global null hypothesis.

<p> refers to the total number of SNPs. The methods analyzed are MOSGWA (MOS), HLASSO with three different choices of the parameter , GWASelect with three different choices of the stable-selection-threshold , and single marker tests (SM) with Benjamini Hochberg procedure at level .</p

Illustration of the simulation results under the total null hypothesis.

<p>The average number of false positives for MOSGWA is compared with HLASSO (Fig 1a) and with GWASelect (Fig 1b), for which false positives were clustered. Simulations were performed for four different numbers of chromosomes, resulting in different numbers of SNPs plotted on the x-axis.</p

Simulation results under an alternative with causal SNPs.

<p>The four panels (Fig 4a, Fig 4b, Fig 4c, Fig 4d) show the average power, number of false positives, misclassification rate and false discovery rate as a function of the threshold value which determines if a detection is a true or a false positive. The performance of MOSGWA is compared with single marker tests, HLASSO, and with GWASelect using three different parameters for stability selection.</p

Summary of simulation results for complex traits.

<p>The average over 200 simulation runs is reported for the number of detected associations (Size), the estimated power, the number of false positive detections (FP), the estimated false discovery rate (FDR) and the average number of misclassifications (Mis). GWASelect performed with parameters is abbreviated as GS , MOSGWA as MOS, HLASSO as HL, and single marker tests as SM.</p