Methods for Combining P-values in Multiple Tests

Abstract

Multiple testing is a standard procedure that arises in applications where many, or even thousands, of hypotheses are tested simultaneously to detect an overall effect. P-value combination methods are flexible and powerful techniques to aggregate evidence from multiple tests. They have recently been applied to the analysis of high-dimensional data, such as in genomics. Several methods are available for combining independent and dependent p-values. However, there is no uniformly most powerful test for combining p-values, which creates an opportunity to develop new methodologies and to study their properties and optimality. This thesis contributes to the growing body of literature on p-value combination methods by developing and evaluating novel methodologies. First, we evaluate the most recent p-value combination test using correlated count data. The findings support its suitability for count data and provide new insights that build upon the existing literature. It remains robust for independent and correlated count data; however, its performance is influenced by the higher-order correlation structure modelled by the multivariate copula and the distribution parameters used to model the count data. Moreover, this thesis proposes two new p-value combination methods designed to combine independent p-values. The first is a novel weighted p-value combination test, which is evaluated in comprehensive simulation studies and comparative analyses. The findings demonstrate that the proposed test achieves higher statistical power while maintaining the type 1 error rate under non-Gaussian alternatives for combining weak signals. In addition, the findings highlight the practical benefit of weighting in increasing sensitivity. The second new combination method is based on a developed distribution. It is a generalised Fisher's test, and its optimality properties are presented. Preliminary assessments focused on understanding the connection with existing combination methods. Initial theoretical insights suggest that it may further enhance power when few true signals exist in sparse settings.