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This thesis has two themes: (1) the predictive potential of principal components in regression, and (2) methodological developments in sufficient dimension reduction. For the first theme, several research papers have established a number of results showing that, under some uniformity assumptions, higher-ranking principal components of a predictor vector tend, across a range of datasets, to have greater squared correlation with a response variable than lower-ranking ones. This is despite the procedure being unsupervised. This thesis reviews these results and greatly extends them by showing that analogues hold in the setting where nonlinear principal component analysis with general predictors is applied. For the second theme, research in the past 10 years has led to a measure-theoretic framework for sufficient dimension reduction, inspired by the measure-theoretic formulation of sufficient statistics, which permits nonlinear reductions. This thesis extends this framework to allow for some of the predictors to be categorical. A new estimator, partial generalised sliced inverse regression, is proposed and its properties and effectiveness are explored.

The human gastrointestinal tract is colonised by trillions of microbes that actively impact the health of the host. This ecosystem, known as the gut microbiota, varies in composition between individuals and is associated with outcomes of many diseases. However, the underlying physiology of these associations remains largely unknown, preventing development of targeted microbiota therapeutics and clinical interventions. Animal models are essential for studying these host-microbiota interactions and for causally linking them to health outcomes; mice are the most widely used models for achieving this. Despite this, the mouse gut microbiota is poorly characterised compared to humans, limiting translation of research between these hosts. This thesis describes the generation of the largest and most comprehensive catalogue of high-quality mouse gut-derived genomes to date: the Mouse Gastrointestinal Bacteria Catalogue (MGBC). By facilitating the comparison of the human and mouse gut microbiotas, the MGBC demonstrates that only 2.58% of species are shared between these biomes, although over 80% of predicted functions are conserved. Application of species-level taxonomic mapping of these functions to predict the closest functionally-related species between the gut microbiotas of humans and mice demonstrates that these taxa are not necessarily the same as the closest genetically-related species. These analyses were implemented as a bioinformatic toolkit to enable other researchers to identify functionally equivalent species according to their functions of interest, with the aim of improving translation of gut microbiome research between the clinical and laboratory contexts. The MGBC improves coverage of the mouse gut microbiota, yielding metagenome classification rates of over 95%, and enables better resolution for mouse gut metagenomic studies. Nearly 2,500 mouse gut-derived shotgun metagenomes were used to characterise the taxonomic structure of the global mouse gut microbiota and identify factors associated with variation between studies. These analyses found that virtually every species of the mouse gut microbiota varied significantly between study institutes, therefore potentially representing an underlying basis for the irreproducibility crisis observed between mouse studies. This thesis further investigates the role of the gut microbiota in determining outcomes of bacterial enterocolitis caused by Salmonella Typhimurium and Clostridioides difficile. Through this work, Enterocloster clostridioformis was identified as a novel resistance-inducing microbe against Salmonella infection. This phenotype is potentially mediated through induction of protective epithelial responses and expansion of regulatory T cells in the caecal mucosa. Furthermore, by applying the MGBC to explore the association of intra-institutional microbiota variation with outcomes of C. difficile infection, this work identifies taxonomic and functional correlates of resistance and susceptibility. These findings validate the importance of the MGBC for facilitating shotgun metagenomic analyses in mice and highlight the impact of the gut microbiota on outcomes of health and disease. Mouse models will continue to be essential for developing microbiota-directed diagnostic and therapeutic interventions as well as enabling their introduction to clinical practice. This thesis therefore represents multiple advances towards understanding and tackling the obstacles posed by host-specific microbiotas. To this end, I have provided a starting point for efficient and informed translation of gut microbiota research between humans and mice, as well as the means for experimental validation of these analyses.

Cluster Randomised Controlled Trials involve randomising groups of participants, rather than the individual participants themselves, whilst the outcomes are measured on the participants. Whilst there are a number of practical and methodological advantages to such a design, there are also statistical implications, both in terms of study design and sample size calculation, and in analysis. The methodology underpinning the cluster randomised design is now well-established in the statistical literature. However, the overwhelming majority of methodological developments to date have been within the frequentist paradigm, and as such, there is an opportunity to explore methodological developments in the context of Bayesian approaches to the design and analysis of Cluster Randomised Controlled Trials, which is the focus of this thesis. This thesis begins by identifying and quantifying the practical application of Bayesian methods to such cluster randomised trial designs, as well as existing methodological developments in the area, through a methodological systematic review. The review highlights that whilst there have been some efforts to develop Bayesian methodology for Cluster Randomised Controlled Trials, the practical uptake of such methods remains low. Next, a novel application of an informative class of prior distribution, the power prior, is proposed whereby information is borrowed from continuous, clustered, historical data, such as that from a pilot or feasibility study. The performance of this approach is evaluated, and superiority, in comparison to established methods, is demonstrated for certain performance metrics. The novel application of the power prior methodology is then explored in the context of study design and sample size calculation for a Cluster Randomised Controlled Trial, whereby the impact of the use of these new methods is quantified in the context of the impact on type I error and statistical power. It is demonstrated that the adoption of these methods has the potential to reduce sample size requirements, thereby facilitating more efficient trial design and reducing research waste. However, it is also shown that, under the traditional frequentist interpretation, inflated type I error rates can be expected as a result of borrowing information through the power prior. In order to address the limitation of inflated type I error, an approach is presented in which the degree of information borrowing through the power prior is determined in order to control Bayesian type I error at some nominal level. It is shown that by adopting a Bayesian interpretation of design operating characteristics, information borrowing methods can be used whilst maintaining type I error control. Finally, a newly developed R package, PPCRCT is described which allows for straightforward implementation of the methodology presented within this thesis.