Special Seminar by Dr. Steve Coad

A seamless phase II/III clinical trial is conducted in two stages. It is assumed that the treatment responses have normal distributions with unequal known variances and that there is unequal allocation of patients. The first stage studies all of the treatments and selects the one with the largest sample mean to continue to the second stage. However, the sample mean for the selected treatment is a positively biased estimator for the corresponding population mean. Also, the maximum likelihood estimator is a biased estimator of this population mean, due to combining data from both stages and ignoring the selection rule.

An unbiased estimator based on second-stage data only can be found, but this estimator is inefficient. Therefore, a uniformly minimum variance conditionally unbiased estimator for the mean of the selected treatment is derived to correct for the selection bias. The derivation starts with the unbiased estimator based on the second-stage data only. Then the Rao-Blackwell theorem is used to improve this estimator, by taking conditional expectations with respect to sufficient statistics based on the data from both stages and the selection rule.

For the case of two treatments, using the theorem of total expectations, exact expressions are obtained for the bias and mean square error of the two estimators for the mean of the selected treatment. Although the bias and mean square error formulae for the maximum likelihood estimator can be easily computed, the expression for the mean square error of the uniformly minimum variance conditionally unbiased estimator includes a term involving the expectation of the squared inverse Mills’ ratio and needs to be evaluated using numerical integration.

A comparison of the properties and distributions of the estimators is made, both analytically and by using simulation. For example, it is shown that the bias of the maximum likelihood estimator is maximised when there is no difference between the treatment means. On the other hand, when the treatment mean difference is large, the mean square errors of the two estimators converge. In general, both estimators have approximate normal distributions, but the maximum likelihood estimator is closer to normal, especially in the tails.