An Adaptive Location-Scale Test

A new location-scale test based on a combination of the ideas of Levene and Lepage

Lepage's test combines the Wilcoxon rank-sum and the Ansari-Bradley statistics. We propose to replace the latter statistic by a Wilcoxon rank-sum calculated after Levene's transformation. We use the medians for this transformation, i.e. absolute deviations from sample medians are calculated. The new location-scale test can be carried out as a permutation test based on permutations of the original observations, the Levene transformation has to be applied for each permutation in an intermediate step to calculate the test statistic. Simulations indicate that the new test can be more powerful than an O'Brien-type test and Lepage's test, the latter is the standard nonparametric location-scale test. The new test is illustrated using real data about colony sizes of yellow-eyed penguins and an SAS program to perform the test is freely available.

Estimation in flexible two stage designs

Adaptive test designs for clinical trials allow for a wide range of data driven design adaptations using all information gathered until an interim analysis. The basic principle is to use a test statistics which is invariant with respect to the design adaptations under the null hypothesis. This allows for a control of the type I error rate for the primary hypothesis even for adaptations not specified a priori in the study protocol. Estimation is usually another important part of a clinical trial, however, is more difficult in adaptive designs. In this research paper we give an overview of point and interval estimates for flexible designs and compare methods for typical sample size rules. We also make some proposals for confidence intervals which have nominal coverage probability also after an unforeseen design adaptation and which contain the maximum likelihood estimate and the usual unadjusted confidence interval.

Optimal Conditional Error Functions for the Control of Conditional Power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.

The advantages and disadvantages of adaptive designs for clinical trials

Flexible designs for clinical trials permit mid-trial design modifications, which are based on interim information from inside or outside the trial while meeting (regulatory) requirements for the control of the type I error rate. The basic principle is to combine stage standardized test statistics such as p-values or z-scores in a pre-specified way. The flexibility covers changes of sample sizes, treatment allocation ratios and the number of interim analyses, as well as the selection of treatments, doses and end points. The price to be paid is that non-standard test statistics must be used after an adaptation.

Issues in designing flexible trials

We outline the general framework of adaptive combination tests and discuss their relationship to flexible group sequential designs. An important field of applications is sample size reassessment. We discuss reassessment rules based on conditional power arguments using either the observed or the prefixed effect size. These rules tend to lead to large expected sample sizes for small actual effects. However, the application of a maximal bound for the second stage sample size leads to more favourable properties. Additionally, we consider an optimized reassessment rule in terms of expected sample sizes. Since the adaptive design does not use the classical test statistics for some types of sample size reassessments, the adaptive test may reject the null hypothesis while the classical one-sample test does not. We characterize sample size reassessment rules, where such inconsistencies are avoided. Finally, the extension of flexibility to the number of stages is explored. In the first interim analysis a second interim analysis is only planned if the chance to achieve a decision there is high. This leads to savings in the average number of interim analysis performed, without paying a noticeable price in terms of expected sample size.