Optimal group sequential tests
Sequential procedures were originally designed for use in an industrial
context. However the flexibility and efficiency of sequential methods made them
attractive to those involved in medical experimentation.
The earliest sequential designs for clinical trials were fully sequential, that is
they required an analysis to be conducted after every patient response. More
recently the emphasis has been on group sequential designs, where analyses are
carried out after groups of patient responses.
One of the distinguishing features of sequential procedures is that the
required sample size is a random variable. For fixed group sizes, a given
maximum number of analyses and given error constraints, group sequential tests
can be designed which minimize a given function of expected sample size. We
term such procedures optimal group sequential tests.
In this thesis we introduce a computationally efficient and numerically stable
method for the computation of optimal group sequential tests. Although we
approach this problem from a frequentist perspective, our method makes use of
both Bayesian decision theory and dynamic programming.
In Chapters 3 and 4 we consider computing optimal one-sided and two-sided
tests respectively. The two-sided tests permit the rejection of the null hypothesis,
H0, at any analysis, but they only allow H0 to be accepted at the final analysis.
In Chapter 5 we consider computing optimal wedge tests which, like two-sided
tests, test Hq against a two-sided alternative, but, unlike two-sided tests, allow H0
to be accepted or rejected at each analysis.
In Chapter 6 we consider some of the Bayesian and Bayes decision theoretic
procedures proposed in the literature. Finally, in Chapter 7, we look at a number
of ideas for future research as well as some relevant topics which have not been
considered elsewhere in the thesis.