Functional data analysis with applications in biostatistics

Functional data analysis is concerned with the analysis of data for which the observed responses for each subject are continuous curves. In practice measurements are taken at discrete time points but estimates are required over the entire time interval. Traditional techniques for analysis of multiple curves such as longitudinal data analysis or time series methods are unsuitable for this type of data since there are generally more measurements per subject than subjects and stationarity assumptions do not necessarily hold. With a technology induced growth in data of this kind research into techniques for functional data analysis has become an emerging area in recent years. This thesis aims to develop new techniques for functional data analysis focusing on three problems: logistic regression with a functional regressor, linear and logistic regression for a repeatedly stimulated functional regressor, and a functional mixed-effects type model for joint mean and covariance modelling. For each of the problems, we develop solutions using a basis function approach, that is, expressing the data for each subject as a linear combination of known basis functions. Using this approach we are able to overcome singularity problems associated with having more measurements than subjects. As well as calculating maximum likelihood or least squares parameter estimates model diagnostic and smoothing parameter selection issues are addressed. The techniques developed in this thesis are applied to novel biostatistical data sets electroencephalographic data and fetal heart rate data. Of main interest is the fetal heart rate data which motivated the development of the regression techniques for a repeatedly stimulated functional parameter. It was found that the stimulated fetal heart rates could be used to predict an infant's risk category at birth and psychomotor development at 18 months of age. Most of the material presented in the thesis is my own work. The exception is: the work described in Section 6.3 is partly due to Victor Solo.