Type 1 error rate inflation in multi-armed clinical trials

In recent years design modifications in already ongoing clinical trials have attracted more and more attention. Several publications have shown that applying naively conventional fixed sample size tests in adaptive trials with sample size reassessment based on un-blinded interim data may result in a considerable inflation of the type 1 error rate. The methodological research has considered complicated trial designs to achieve type 1 error control allowing for various options of flexibility for an interim trial re-design. The underlying principles used to achieve type 1 error rate control have been discussed controversially as in general non-standard test statistics, violating the sufficiency principle, are used. Although lot of research has been done investigating methods controlling the type 1 error for adaptive design modifications, less is known about the maximum type 1 error inflation when conventional fixed sample size tests are applied for the analysis of the trial data simply ignoring the adaptive character of the study. The specific question then is how much the actual type 1 error rate may be inflated by such a naive analysis following adaptations. To put it in a practical context, an experimenter may decide not to go through all that adaptive test machinery if for his type of potential design modification the maximum possible type 1 error rate inflation can be quantified and remains negligible. For the comparison of a single treatment with a single control in parallel groups it can be shown that the maximum type 1 error rate can become substantially larger when allowing for unbalanced sample size modifications (unequal sample size for both groups) as compared to the case of balanced sample size modifications (equal sample size for both groups). The goal of this research project is to investigate the maximum type 1 error inflation when more than one treatment is compared to a single control allowing different sample size adaptations in the groups. The maximum inflation of the type 1 error rate for such types of designs can be calculated by searching for "worst case" scenarios, i.e. sample size adaptation rules that lead to the largest conditional type 1 error rate (the probability that under the global null hypotheses at least one treatmant-control comparison rejects in the final analysis, given the interim data) in any point of the sample space. To show the most extrem inflation of the type 1 error rate, it is first assumed that the second-stage-sample-sizes are unconstrained. To see how the numbers will change in more realistic scenarios, in a second step, constraints are put on the second-stage-sample-size, which may lead to scenarios not inflating the type 1 error rate.