Statistical algorithms for semi-parametric variance regression with application to biomarkers

Variance regression allows for heterogeneous variance, or heteroscedasticity, by incorporating a regression model into the variance. This thesis uses a variant of the Expectation-Maximisation (EM) algorithm to develop a new method for fitting additive variance regression models that allow for regression in both the mean and the variance. The algorithm is easily extended to allow for B-spline bases, thus allowing for the incorporation of a semi-parametric model in both the mean and variance. Although here are existing methods to fit these types of models, this new algorithm provides a reliable approach that is not susceptible to numerical instability that can be seen with other approaches. We utilise the developed algorithm with a series of simulation studies and analysis of biomarker datasets. Various simulation studies show that the algorithm is capable of recovering the true model for a variety of scenarios. We also study automatic selection of model complexity based on various information criteria, and show that the Akaike information criterion (AIC) is useful for choosing the optimal number of knots in a B-spline model. It is also found that the ability to estimate the model complexity automatically is greatly improved with a larger sample size. The algorithm is extended to allow for censored outcome data, and to allow for nonnormal data with the incorporation of a skew regression model, using the skew-normal distribution. The algorithms developed in this thesis are available through an R package called VarReg, and a demonstration of the package is given using a biomarker dataset.This algorithm has wide capabilities for analysis of biomarker data, and provides a useful and stable additional tool for fitting variance regression models.