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- ItemOn multivariate chi-square distributions and their applications in testing multiple hypotheses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dickhaus, Thorsten; Royen, ThomasWe are considered with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and provide formulas for computing multiplicity-adjusted p-values under the respective global hypothesis.
- ItemOn the Simes inequality in elliptical models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bodnar, Taras; Dickhaus, ThorstenWe provide necessary and sufficient conditions for the validity of the inequality of Simes (1986) in models with elliptical dependencies. Necessary conditions are presented in terms of sufficient conditions for the reverse Simes inequality. One application of our main results concerns the problem of model misspecification, in particular the case that the assumption of Gaussianity of test statistics is violated. Since our sufficient conditions require nonnegativity of correlation coefficients between test statistics, we also develop exact tests for vectors of correlation coefficients.
- ItemSimultaneous Bayesian analysis of contingency tables in genetic association studies(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dickhaus, ThorstenGenetic association studies lead to simultaneous categorical data analysis. The sample for every genetic locus consists of a contingency table containing the numbers of observed genotype-phenotype combinations. Under case-control design, the row counts of every table are identical and fixed, while column counts are random. The aim of the statistical analysis is to test independence of the phenotype and the genotype at every locus. We present an objective Bayesian methodology for these association tests, utilizing the Bayes factor proposed by Good (1976) and Crook and Good (1980). It relies on the conjugacy of Dirichlet and multinomial distributions, where the hyperprior for the Dirichlet parameter is log-Cauchy. Being based on the likelihood principle, the Bayesian tests avoid looping over all tables with given marginals. Hence, their computational burden does not increase with the sample size, in contrast to frequentist exact tests. Making use of data generated by The Wellcome Trust Case Control Consortium (2007), we illustrate that the ordering of the Bayes factors shows a good agreement with that of frequentist p-values. Furthermore, we deal with specifying prior probabilities for the validity of the null hypotheses, by taking linkage disequilibrium structure into account and exploiting the concept of effective numbers of tests. Application of a Bayesian decision theoretic multiple test procedure to The Wellcome Trust Case Control Consortium (2007) data illustrates the proposed methodology. Finally, we discuss two methods for reconciling frequentist and Bayesian approaches to the multiple association test problem for contingency tables in genetic association studies.
- ItemSelf-concordant profile empirical likelihood ratio tests for the population correlation coefficient: A simulation study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dickhaus, ThorstenWe present results of a simulation study regarding the finite-sample type I error behavior of the self-concordant profile empirical likelihood ratio (ELR) test for the population correlation coefficient. Three different families of bivariate elliptical distributions are taken into account. Uniformly over all considered models and parameter configurations, the self-concordant profile ELR test does not keep the significance level for finite sample sizes, albeit the level exceedance monotonously decreases to zero as the sample size increases. We discuss some potential modifications to address this problem.
- ItemComputing and approximating multivariate chi-square probabilities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Stange, Jens; Loginova, Nina; Dickhaus, ThorstenWe consider computational methods for evaluating and approximating multivariate chi-square probabilities in cases where the pertaining correlation matrix or blocks thereof have a low-factorial representation. To this end, techniques from matrix factorization and probability theory are applied. We outline a variety of statistical applications of multivariate chi-square distributions and provide a system of MATLAB programs implementing the proposed algorithms. Computer simulations demonstrate the accuracy and the computational efficiency of our methods in comparison with Monte Carlo approximations, and a real data example from statistical genetics illustrates their usage in practice.
- ItemOn an extended interpretation of linkage disequilibrium in genetic case-control association studies(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dickhaus, Thorsten; Stange, Jens; Demirhan, HaydarWe are concerned with statistical inference for 2 x 2 x K contingency tables in the context of genetic case-control association studies. Multivariate methods based on asymptotic Gaussianity of vectors of test statistics require information about the asymptotic correlation structure among these test statistics under the global null hypothesis. We show that for a wide variety of test statistics this asymptotic correlation structure is given by the linkage disequilibrium matrix of the K loci under investigation. Three popular choices of test statistics are discussed for illustration.