Designing Factorial Clinical Trials: An Examination of Power

Abstract

Investigators will sometimes consider clinical trials involving more than one treatment or intervention, as these trials allow for the simultaneous evaluation of the individual efficacy of multiple treatments. The most common design choice is a factorial trial, in which patients are randomized to all possible combinations of treatments, including control. Factorial trials are an attractive choice in examining individual treatment effects provided certain conditions are met, including the important assumption that no interaction exists between the treatments of interest.

However, even without interaction, the statistical power for a treatment can be substantially influenced by the effectiveness of the other treatment in the trial, an issue that has not been widely recognized. This issue is compounded by the fact that the impact on power depends on the scale on which interactions are defined.

In the current work, we evaluate how the power for a treatment in a binary outcome 2x2 factorial trial changes as a function of the effectiveness of a second treatment in the same trial, under a range of possible parameter conditions. We provide analytical results to describe the behavior of these functions on the additive, risk ratio, and odds ratio scales and attempt to determine where the maximum power occurs for each scale.

Sets of numerical evaluations were also implemented to support these analytic results, as well to evaluate how the minimum required sample size for the trial changes as a function of the first and second treatment effects. Controllable parameters within the evaluations include the event rate in the control group, sample size, treatment effect sizes, and Type-I error thresholds. Separate evaluations were created for scenarios where the treatments are assumed to not have an interaction on either the additive, risk ratio, or odds ratio scales. We also provide two examples of factorial trials using real data to illustrate our findings.

In general, we find that power for an individual treatment decreases as a function of the effectiveness of the other treatment if they do not interact on the risk ratio scale. A similar pattern is observed in the odds ratio case at low base rates, but at high base rates, power increases may occur if the first treatment is moderately more effective than its planned value. When treatments do not interact on the additive scale, power may either increase or decrease depending on the response rate in the control group.

Results from these analyses may benefit investigators in planning clinical trials. Assumptions about the anticipated effects of each treatment under study, even if there is no interaction between them, are critical in calculating a valid sample size that will yield sufficient power for individual treatments in the context of factorial studies.