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The logistic regression model has become a standard model for binary outcomes in many areas of application and is widely used in medical statistics. Much work has been carried out to examine the asymptotic behaviour of the distribution of Maximum Likelihood Estimates (MLE) for the logistic regression model, although the most widely known properties apply only if the assumed model is correct. There has been much work on goodness-of- t tests to address the last point. The rst part of this thesis investigates the behaviour of the asymptotic distribution of the (MLE) under a form of model mis-speci cation, namely when covariates from the true model are omitted from the tted model. When the incorrect model is tted the maximum likelihood estimates converge to the least false values. In this work, key integrals cannot be evaluated explicitly but we use properties of the skew-Normal distribution and the approximation of the Logit by a suitable Probit function to obtain a good approximation for the least false values. The second part of the thesis investigates the assessment of a particular goodness-of- t test namely the information matrix test (IM) test as applied to binary data models. Kuss (2002), claimed that the IM test has reasonable power compared with other statistics. In this part of the thesis we investigate this claim, consider the distribution of the moments of the IM statistic and the asymptotic distribution of the IM test (IMT) statistic. We had di culty in reproducing the results claimed by Kuss (2002) and considered that this was probably due to the near singularity of the variance of IMT. We de ne a new form of the IMT statistic, IMTR, which addresses this issue.

Incomplete data analysis is often considered with other problems such as model uncertainty or non-identi ability. In this thesis I will use the idea of the local sensitivity analysis to address problems under both ignorable and non-ignorable missing data assumptions. One problem with ignorable missing data is the uncertainty for covariate density. At the mean time, the misspeci cation for the missing data mechanism may happen as well. Incomplete data biases are then caused by di erent sources and we aim to evaluate these biases and interpret them via bias parameters. Under non-ignorable missing data, the bias analysis can also be applied to analyse the di erence from ignorability, and the missing data mechanism misspeci cation will be our primary interest in this case. Monte Carlo sensitivity analysis is proposed and developed to make bias model selection. This method combines the idea of conventional sensitivity analysis and Bayesian sensitivity analysis, with the imputation procedure and the bootstrap method used to simulate the incomplete dataset. The selection of bias models is based on the measure of the observation dataset and the simulated incomplete dataset by using K nearest neighbour distance. We further discuss the non-ignorable missing data problem under a selection model, with our developed sensitivity analysis method used to identify the bias parameters in the missing data mechanism. Finally, we discuss robust con dence intervals in meta-regression models with publication bias and missing confounder.

Identifying genetic interactions for a given microorganism, such as yeast, is difficult. Quantitative Fitness Analysis (QFA) is a high-throughput experimental and computa tional methodology for quantifying the fitness of microbial cultures. QFA can be used to compare between fitness observations for different genotypes and thereby infer genetic interaction strengths. Current “naive” frequentist statistical approaches used in QFA do not model between-genotype variation or difference in genotype variation under differ ent conditions. In this thesis, a Bayesian approach is introduced to evaluate hierarchical models that better reflect the structure or design of QFA experiments. First, a two-stage approach is presented: a hierarchical logistic model is fitted to microbial culture growth curves and then a hierarchical interaction model is fitted to fitness summaries inferred for each genotype. Next, a one-stage Bayesian approach is presented: a joint hierarchi cal model which simultaneously models fitness and genetic interaction, thereby avoiding passing information between models via a univariate fitness summary. The new hierarchical approaches are then compared using a dataset examining the effect of telomere defects on yeast. By better describing the experimental structure, new evidence is found for genes and complexes which interact with the telomere cap. Various extensions of these models, including models for data transformation, batch effects and intrinsically stochastic growth models are also considered.

The PolyQ model is a large stochastic kinetic model that describes protein aggregation within human cells as they undergo ageing. The presence of protein aggregates in cells is a known feature in many age-related diseases, such as Huntington's. Experimental data are available consisting of the proportions of cell death over time. This thesis is motivated by the need to make inference for the rate parameters of the PolyQ model. Ideally observations would be obtained on all chemical species, observed continuously in time. More realistically, it would be hoped that partial observations were available on the chemical species observed discretely in time. However, current experimental techniques only allow noisy observations on the proportions of cell death at a few discrete time points. This presents an ambitious inference problem. The model has a large state space and it is not possible to evaluate the data likelihood analytically. However, realisations from the model can be obtained using a stochastic simulator such as the Gillespie algorithm. The time evolution of a cell can be repeatedly simulated, giving an estimate of the proportion of cell death. Various MCMC schemes can be constructed targeting the posterior distribution of the rate parameters. Although evaluating the marginal likelihood is challenging, a pseudo-marginal approach can be used to replace the marginal likelihood with an easy to construct unbiased estimate. Another alternative which allows for the sampling error in the simulated proportions is also considered. Unfortunately, in practice, simulation from the model is too slow to be used in an MCMC inference scheme. A fast Gaussian process emulator is used to approximate the simulator. This emulator produces fully probabilistic predictions of the simulator output and can be embedded into inference schemes for the rate parameters. The methods developed are illustrated in two smaller models; the birth-death model and a medium sized model of mitochondrial DNA. Finally, inference on the large PolyQ model is considered.

Bayesian statistics has a tendency to produce objects that are of many more than three dimensions, typically of the same dimensionality as the parameter set of the problem. This thesis takes the idea of visual, exploratory data analysis and attempts to apply it to those objects. In order to do this it examines several areas, Monte Carlo Markov Chains (MCMC), display graphics and methods – especially Projection Pursuit methods – and Kernel Density Estimation (KDE). During the course of this work acceptable prior technology was found for MCMC and, once the decision for it had been made, Projection Pursuit. However, the current state of KDE gave rise to several objections. Not least among these came from the Bayesian background of the researcher, KDE had not been put in a suitable Bayesian framework and so clashed with the other technology. In addition it was felt that KDE needed too much user input and that is was somewhat ad hoc. This led to reformulating KDE in a Bayesian framework which had the added advantage of removing the need for a user to provide a bandwidth for each application. Chapter 6 of this thesis considers Bayesian theory and how it can be applied to KDE to produce a form more usable and satisfying in terms of Bayesian mathematics. This is shown to provide a powerful and flexible statistical tool without the need for the ad hoc choices often associated with these methods. This formulation of the KDE as a Bayesian problem is believed to be unique. As part of this work, software was produced in R to provide a usable visualisation of BKDE. A large number of examples is provided to demonstrate how this software can allow easy visualisation of a variety of types of dataset both with and without Kernel Density Estimation.

The classical multivariate theory has been largely based on the multivariate normal distribution (MVN): the scarcity of alternative models for the meaningful and consistent analysis of multiresponse data is a well recognised problem. Further, the complexity of generalising many non-normal univariate distributions makes it undesirable or impossible to use their multivariate versions. Hence, it seems reasonable to inquire about ways of transforming the data so as to enable the use of more familiar statistical techniques that are based implicitly or explicitly on the normal distribution. Techniques for developing data-based transformations of univariate observations have been proposed by several authors. However, there is only one major technique in the multivariate (p-variable) case by Andrews et. al. [1971]. Their approach extended the power transformations proposed by Box & Cox [1964] to the problem of estimating power transformations of multiresponse data so as to enhance joint normality. The approach estimates the vector of transformation parameters by numerically maximising the log-likelihood function. However, since there are several parameters to be estimated, p(p+5)/2 for multivariate data without regression, the resulting maximisation is of high dimension, even with modest values of p and sample size n. The purpose of the thesis is to develop computationally simpler and more informative statistical procedures which are incorporated in a package. The thesis is in three main parts: - A proposed complementary procedure to the log-likelihood approach which attempts to reduce the size of the computational requirements for obtaining the estimates. Though computational simplicity is the main factor, the statistical qualities of the estimates are not compromised, indeed the estimated values are numerically identical to those of the log-likelihood. Further, the procedure implicitly produces diagnostic statistics and some useful statistical quantities describing the structure of the data. The technique is a generalisation of the constructed variables method of obtaining quick estimates for transformation parameters [Atkinson 1985]. To take into account the multiresponse nature of the data and, hence, joint estimates, a seemingly unrelated regression is carried out. The algorithm is iterative. However, there is considerable savings in the number of iterations required to converge to the maximum likelihood (MLE) estimates compared to those using the log-likelihood function. The technique is refered to as the Seemingly Unrelated Regressions/Constructed Variable (SURCON) analysis, and the estimates obtained are the Surcon estimates. - The influence of individual observations on the need for transformations is quite crucial and, hence, it is necessary to investigate the data for any spurious or suspicious observations, outliers. The thesis also proposes an iterative technique for detecting and identifying outliers based on Mahalanobis distances computed from sub-samples of the observations. The results of the analysis are displayed in a graphical summary called the Stalactite Chart, hence, the analysis is refered to as the Stalactite Analysis. - The development of a userfriendly microcomputer-based statistical package which incorporates the above techniques. The package is written in the C programming language.