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- ItemMore Specific Signal Detection in Functional Magnetic Resonance Imaging by False Discovery Rate Control for Hierarchically Structured Systems of Hypotheses(San Francisco, California, US : PLOS, 2016) Schildknecht, Konstantin; Tabelow, Karsten; Dickhaus, ThorstenSignal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to increase statistical power. We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family at this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses. In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.
- ItemSimultaneous statistical inference for epigenetic data(San Francisco, California, US : PLOS, 2015) Schildknecht, Konstantin; Olek, Sven; Dickhaus, ThorstenEpigenetic research leads to complex data structures. Since parametric model assumptions for the distribution of epigenetic data are hard to verify we introduce in the present work a nonparametric statistical framework for two-group comparisons. Furthermore, epigenetic analyses are often performed at various genetic loci simultaneously. Hence, in order to be able to draw valid conclusions for specific loci, an appropriate multiple testing correction is necessary. Finally, with technologies available for the simultaneous assessment of many interrelated biological parameters (such as gene arrays), statistical approaches also need to deal with a possibly unknown dependency structure in the data. Our statistical approach to the nonparametric comparison of two samples with independent multivariate observables is based on recently developed multivariate multiple permutation tests. We adapt their theory in order to cope with families of hypotheses regarding relative effects. Our results indicate that the multivariate multiple permutation test keeps the pre-assigned type I error level for the global null hypothesis. In combination with the closure principle, the family-wise error rate for the simultaneous test of the corresponding locus/parameter-specific null hypotheses can be controlled. In applications we demonstrate that group differences in epigenetic data can be detected reliably with our methodology.
- 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.
- ItemSimultaneous statistical inference for epigenetic data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Schildknecht, Konstantin; Olek, Sven; Dickhaus, ThorstenEpigenetic research leads to complex data structures. Since parametric model assumptions for the distribution of epigenetic data are hard to verify we introduce in the present work a nonparametric statistical framework for two-group comparisons. Furthermore, epigenetic analyses are often performed at various genetic loci simultaneously. Hence, in order to be able to draw valid conclusions for specific loci, an appropriate multiple testing correction is necessary. Finally, with technologies available for the simultaneous assessment of many interrelated biological parameters (such as gene arrays), statistical approaches also need to deal with a possibly unknown dependency structure in the data. Our statistical approach to the nonparametric comparison of two samples with independent multivariate observables is based on recently developed multivariate multiple permutation tests. We adapt their theory in order to cope with families of hypotheses regarding relative effects. Our results indicate that the multivariate multiple permutation test keeps the pre-assigned type I error level for the global null hypothesis. In combination with the closure principle, the family-wise error rate for the simultaneous test of the corresponding locus/parameter-specific null hypotheses can be controlled. In applications we demonstrate that group differences in epigenetic data can be detected reliably with our methodology.
- 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.
- ItemMore specific signal detection in functional magnetic resonance imaging by false discovery rate control for hierarchically structured systems of hypotheses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Schildknecht, Konstantin; Tabelow, Karsten; Dickhaus, ThorstenSignal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to increase statistical power. We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family in this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses. In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.
- ItemUncertainty quantification for the family-wise error rate in multivariate copula models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Stange, Jens; Bodnar, Taras; Dickhaus, ThorstenWe derive confidence regions for the realized family-wise error rate (FWER) of certain multiple tests which are empirically calibrated at a given (global) level of significance. To this end, we regard the FWER as a derived parameter of a multivariate parametric copula model. It turns out that the resulting onfidence regions are typically very much concentrated around the target FWER level, while generic multiple tests with fixed thresholds are in general not FWER-exhausting. Since FWER level exhaustion and optimization of power are equivalent for the classes of multiple test problems studied in this paper, the aforementioned findings militate strongly in favour of estimating the dependency structure (i. e., copula) and incorporating it in a multivariate multiple test procedure. We illustrate our theoretical results by considering two particular classes of multiple test problems of practical relevance in detail, namely, multiple tests for components of a mean vector and multiple support tests.
- ItemUtilizing anatomical information for signal detection in functional magnetic resonance imaging(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Neumann, André; Peitek, Norman; Brechmann, André; Tabelow, Karsten; Dickhaus, ThorstenWe are considering the statistical analysis of functional magnetic resonance imaging (fMRI) data. As demonstrated in previous work, grouping voxels into regions (of interest) and carrying out a multiple test for signal detection on the basis of these regions typically leads to a higher sensitivity when compared with voxel-wise multiple testing approaches. In the case of a multi-subject study, we propose to define the regions for each subject separately based on their individual brain anatomy, represented, e.g., by so-called Aparc labels. The aggregation of the subject-specific evidence for the presence of signals in the different regions is then performed by means of a combination function for p-values. We apply the proposed methodology to real fMRI data and demonstrate that our approach can perform comparably to a two-stage approach for which two independent experiments are needed, one for defining the regions and one for actual signal detection.
- 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.