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Methodology of rare variant analysis underwent a rapid devel-opment since the application of collapsing tests to predefined bins was proposed (Li and Leal, 2008). The basic collapsing test is the 1 d.f. χ² test dichotomizing carriers and non-carriers of rare variants. An individual is considered a carrier if at least one minor allele is present in the analysis unit (bin).Although a multitude of statistical tests exist by now, the major-ity relies on a twofold parameterization of the dataset: the choice of the threshold minor allele frequency (MAFT) to define ‘rare’ and ‘common’ variants, and the definition of analysis bins (‘bin-ning’). The choice of parameters is an educated guess at best and implies a certain Type II error rate due to uncertain signal-no-noise ratio in predefined analysis units.The Variable-Threshold (VT) method (Price et al., 2010) elim-inates the choice of a specific MAFT by analyzing the space of all possible thresholds in each bin, finding the optimal MAFT and using permutations for multiple testing correction. The COLL VT method is computationally feasible even for large genome-wide datasets with ~20,000–100,000 bins (Drichel et al., 2014).We investigated the possibility of elimination of a priori bin-ning in an analogous fashion. In this scenario, all possible contigu-ous bins, i.e. bins with all possible combinations of start and end positions are tested for association. Although the number of com-binations is very large (n(n+1)/2 per chromosome with n rare vari-ants), in the case of COLL, the number of bins with distinct sets of carriers is much smaller due to the characteristic ‘individual-wise’ collapsing condition.We applied the Variable-Binning method (VB) implemented in INTERSNP-RARE to an imputed case-control dataset contain-ing 9832 controls and 3332 late-onset Alzheimer’s disease (AD) cases, with 2,279,783 rare variants (MAF

Meta-analysis refers to the statistical synthesis of the results from a series of related studies. Meta-analysis of summary statistics from Genome-wide association studies (GWAS) may lead to dis-covery of new susceptibility loci without the need to exchange gen-otype data.Well-known p-value combination methods can be applied to an arbitrary association test. However, these methods do not pro-vide summary effects and are known to be underpowered for high-dimensional models. The broadly used methods for meta-analysis of individual GWAS results address mostly one parameter models. Results of multiple regression analysis applied, for instance, to ge-nome-wide interaction analysis have rarely been used due to the complexities underlying the process of their synthesis. For high-dimensional models the gain of power can be achieved when meta-analysis methods incorporate information on study-specific properties of parameter estimates, their effect directions, standard errors and covariance structure. In this context, a method for the synthesis of regression slopes (MSRS) represents a perspec-tive approach of meta-analysis for advanced GWAS. MSRS pro-vides meta-analysis p-values and common parameter estimates of multiple regression models for an arbitrary number of parameters, and can be used to test the homogeneity of studies results. We introduce an efficient, powerful and freely available soft-ware tool METAINTER, which implements MSRS and three fur-ther meta-analysis methods: Fisher’s method, Stouffer’s method with weights and Stouffer’s method with weights and effect direc-tions. METAINTER enables meta-analysis of tests for and under gene-gene interaction, single-SNP association tests with several degrees of freedom, global haplotype tests etc. Simulation study shows that MSRS has correct type I error and its power comes close to that of the joint sample analysis. We have conducted a real data analysis of six GWAS of type 2 Diabetes. For each study, a genome-wide interaction analysis of all SNP pairs has been performed by logistic regression tests. The results have then been meta-analysed with METAINTER.

A possible reason for the phenomenon of missing heritability encountered in the genetics of complex diseases is the omission of interaction effects in genome-wide association studies (GWAS). Even pairwise SNP-SNP-interaction may be a too simple assump-tion in the presence of polygenic influences. In order to address multi marker models, we recently devel-oped the test for supra-multiplicativity (SMT) that is applicable to SNP sets of up to 1,000. The SMT requires only one degree of free-dom and is designed to detect amplifying interaction as implied by liability threshold models. In general, the SMT allows deviation from multiplicativity. A drawback of the original approach is the need to use a Bonferroni-correction to adjustment for multiple testing.Here, we present an extended SMT (ESMT), an improved ver-sion of the original SMT. By implementation of a generalized re-gression equation, the ESMT avoids multiple testing and provides more sophisticated modeling of amplifying interaction. In a simu-lation study, we show that the ESMT uniformly has higher power than its predecessor. In addition, computing time is improved sub-stantially. The ESMT is implemented in the interaction analysis tool INTERSNP.

It has recently been suggested that so-called liability mod-els may define the architecture of the genetics of complexdiseases [Zuk et al., 2012]. For individuals with less thank risk alleles a moderate baseline penetrance f0kcan be as high as 50%-80%.Such models in general do not lead to epistasis that is de-tectable by pair-wise testing but exhibit interaction of allhigher orders. For a phenotype with m a priori confirmedSNPs, it is an important question whether they follow athreshold model. To this purpose, we present a simple 1d.f. regression test. We conducted an extensive simulationfor the threshold models with n SNPs, n∈{5,10,20,50} andliability threshold corresponding to a portion of 70%–85%causal alleles. Our models lead to marginal effects that arecompatible with typical GWAS effect sizes and pair-wiseinteraction that is hard to detect. However, with moderatesample size (3000 individuals) the power to detect the pres-ence of a threshold model is effectively 100%, given that allSNPs of the threshold model are known. For the more realis-tic situation, in which the number of already detected SNPsm is smaller than the number n of SNPs in the unknownmodel, we can show that for m>2/3n there is decent powerto proof the involvement in a threshold model. We presentresults for Alzheimer’s and Type II Diabetes GWAS datasets

Various strategies to identify causal SNPs in GWAS stud-ies under assumption of possible interactions are evalu-ated. Series of “realistic” disease models are defined on a2-SNP-genotype table by specification of allele frequencies,penetrances and a minimal distance between available andcausal SNPs. We compare the standard single-marker anal-ysis with multi-marker analysis; investigate relative per-formance of tests for interaction and tests including bothmarginal and interaction effects, so-called tests allowing forinteraction. We compare case-only with case-control de-sign and contrast allelic and genotypic models. Tests in-cluding marginal effects may become significant becauseof the marginal effect of just one SNP from an interac-tion pair. Since our goal is to detect both SNPs we embedtests allowing for interaction in a two-step hybrid strategy:genome-wide interaction analysis with a combined case-only-interaction/marginal-effects test; follow-up analysisof the significant pairs with a test for interaction, exclud-ing the strongest marginal effect but allowing for marginaleffect of the second SNP. For 5% of settings the most effi-cient strategy is single-marker analysis, typically when al-lele frequencies are high or causal variant tagging is poor.For another 5% of models the most suitable strategy is test-ing for interaction without marginal effects provided that acase-only test is used. In the remaining m ajority of scenariosa hybrid strategy is the most powerful

In this thesis the main boundary value problems, namely, the Schwarz, the Dirichlet, the Neumann and the Robin boundary value problems are studied for the homogeneous and inhomogeneous Cauchy-Riemann equation in a circular ring domain. The main tool to treat boundary value problems for the Poisson equation is related Green functions. Besides the known Green function for a circular ring domain connected with the Dirichlet problem, the Neumann (Green function of second order) and the Robin (Green function of third order) functions are found here for a circular ring. On their basis the integral representation formulas for the solutions of the Neumann and Robin problem for the Poisson equation are derived. The Dirichlet and Schwarz problem for the Bitsadze equation are also solved explicitly. At last, a biharmonic Green function is constructed for a circular ring domain leading to the solution of the respective Dirichlet problem for the biharmonic equation., Die Theorie von komplexen Randwertproblemen ist wegen ihrer Bedeutung für ebene Probleme in Physik und Technik vor allem in einfach zusammenhängenden Gebieten sehr eingehend untersucht. Vor allem sind es das Riemannsche Kopplungsproblem, wo zu vorgegebener Randfunktion Funktionen bestimmter Klassen – im einfachsten Fall analytische Funktionen – zu finden sind, die auf dem Rand den vorgegebenen Sprung aufweisen, und das Riemann-Hilbertsche Randwertproblem, dessen einfachster Fall – das Schwarz Problem – darin besteht, zu am Rand vorgegebenem Realteil eine analytische Funktion zu finden, deren Realteil am Rand diese Werte erreicht. In dieser Arbeit werden Randwertprobleme in einem speziellen zweifach zusammenhängenden Gebiet, einem konzentrischen Kreisring, untersucht. Zunächst werden vier grundlegende Randwertprobleme für analytische Funktionen auf Grundlage einer modifizierten Cauchyschen Formel explizit gelöst, das Schwarzsche, das Dirichletsche, das Neumannsche und ein Robinsches Randwertproblem. Neben den Lösungen werden Lösbarkeitsbedingungen angegeben, die auch sichern, dass die Lösungen einwertig sind, was für mehrfach zusammenhängende Gebiete im Allgemeinen nicht zutrifft. Anschliessend werden inhomogene Cauchy-Riemannsche Gleichungen untersucht und die Ergebnisse auf diese erweitert. Schliesslich werden im Hinblick auf die Poissonsche Gleichung harmonische Green, Neumann und Robin Funktionen explizit für den Ring konstruiert, von denen nur die Greenfunktion in modifizierter Form in der Literatur bekannt ist. Diese Fundamentallösungen dienen dazu, die entsprechenden Randwertprobleme für die Poisson Gleichung im Ring zu lösen und gegebenenfalls Lösbarkeitsbedingungen zu finden. Ein Dirichlet und das Schwarz Problem werden auch für die Bitsadze Gleichung behandelt. Dies ist eine weitere Modellgleichung zweiter Ordnung, im Gegensatz zum Laplace Operator ist der Bitsadze Operator aber nur schwach elliptisch, was Konsequenzen für Randwertprobleme hat. Als Letztes wird eine biharmonische Greensche Funktion explizit konstruiert, die sich als Faltung der harmonischen Greenschen Funktion mit sich selbst ergibt. Diese Funktion ist eine weitere biharmonische Greenfunktion, die sich durch ihr Randverhalten von einer kürzlich für den Ring konstruierten anderen biharmonischen Greenschen Funktion unterscheidet.