| Summary: | In bioinformatics applications samples of biological variables of interest can take a variety of structures. For instance, in this thesis we consider vector-valued observations of multiple gene expression and genetic markers, curve-valued gene expression time courses, and graph-valued functional connectivity networks within the brain. This thesis considers three problems routinely encountered when dealing with such variables: detecting differences between populations, detecting predictive relationships between variables, and detecting association between variables. Distance-based approaches to these problems are considered, offering great flexibility over alternative approaches, such as traditional multivariate approaches which may be inappropriate. The notion of distance has been widely adopted in recent years to quantify the dissimilarity between samples, and suitable distance measures can be applied depending on the nature of the data and on the specific objectives of the study. For instance, for gene expression time courses modeled as time-dependent curves, distance measures can be specified to capture biologically meaningful aspects of these curves which may differ. On obtaining a distance matrix containing all pairwise distances between the samples of a given variable, many distance-based testing procedures can then be applied. The main inhibitor of their effective use in bioinformatics is that p-values are typically estimated by using Monte Carlo permutations. Thousands or even millions of tests need to be performed simultaneously, and time/computational constraints lead to a low number of permutations being enumerated for each test. The contributions of this thesis include the proposal of two new distance-based statistics, the DBF statistic for the problem of detecting differences between populations, and the GRV coefficient for the problem of detecting association between variables. In each case approximate null distributions are derived, allowing estimation of p-values with reduced computational cost, and through simulation these are shown to work well for a range of distances and data types. The tests are also demonstrated to be competitive with existing approaches. For the problem of detecting predictive relationships between variables, the approximate null distribution is derived for the routinely used distance-based pseudo F test, and through simulation this is shown to work well for a range of distances and data types. All tests are applied to real datasets, including a longitudinal human immune cell M. tuberculosis dataset, an Alzheimer's disease dataset, and an ovarian cancer dataset.
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