Repeated tests on serially correlated data
Suppose two treatments are being compared in a clinical trial with a fixed number of subjects. Half the subjects receive treatment 1, while the other half receive treatment 2. Measurements of some relevant response variable are made on each subject at equally spaced time points. This thesis proposes a procedure for testing at each time point the null hypothesis H0 of no treatment differences, while controlling the overall probability of type I error. Chapter 1 describes alternative methods for estimating the unknown parameters of models generating short series of measurements. Maximum likelihood estimation, and modified variance based estimation, are considered in 1.1 and 1.2 respectively. A 4 parameter model is described in 1.3. This allows for increasing or decreasing variances of measurements within subjects. In 1.4, maximum likelihood estimation is used in fitting the 4 parameter model to two data sets. In Chapter 2, attention focuses on sequential testing, with data arising from the 4 parameter model. The possibility of using stopping rules based on posterior probabilities, rather than sequentially adjusted P values, is considered briefly in 2.1. Two methods for determining appropriate nominal significance levels for successive tests, when the model for data is assumed known, are described in 2.2. An algorithm for integrating quickly a special case of the multivariate normal density, in high dimensions, presented in 2.3, facilitates implementation of the two methods. In 2.4, test procedures based on the methods of 2.2 are described, in which model parameters are (re-) estimated at each time point, as data accumulate. Test statistics and z-values are adjusted for each test, according to the updated parameter estimates. Simulations show that these procedures give overall significance levels close to those required. One of the two procedures is preferred on the basis of power to detect deviations from H0.