## Abstract

### Background

Interaction in clinical trials presents challenges for design and appropriate sample size estimation. Here we considered interaction between treatment assignment and a dichotomous prognostic factor with a continuous outcome. Our objectives were to describe differences in power and sample size requirements across alternative distributions of a prognostic factor and magnitudes of the interaction effect, describe the effect of misspecification of the distribution of the prognostic factor on the power to detect an interaction effect, and discuss and compare three methods of handling the misspecification of the prognostic factor distribution.

### Methods

We examined the impact of the distribution of the dichotomous prognostic factor on power and sample size for the interaction effect using traditional one-stage sample size calculation. We varied the magnitude of the interaction effect, the distribution of the prognostic factor, and the magnitude and direction of the misspecification of the distribution of the prognostic factor. We compared quota sampling, modified quota sampling, and sample size re-estimation using conditional power as three strategies for ensuring adequate power and type I error in the presence of a misspecification of the prognostic factor distribution.

### Results

The sample size required to detect an interaction effect with 80% power increases as the distribution of the prognostic factor becomes less balanced. Misspecification such that the actual distribution of the prognostic factor was more skewed than planned led to a decrease in power with the greatest loss in power seen as the distribution of the prognostic factor became less balanced. Quota sampling was able to maintain the empirical power at 80% and the empirical type I error at 5%. The performance of the modified quota sampling procedure was related to the percentage of trials switching the quota sampling scheme. Sample size re-estimation using conditional power was able to improve the empirical power under negative misspecifications (i.e. skewed distributions) but it was not able to reach the target of 80% in all situations.

### Conclusions

Misspecifying the distribution of a dichotomous prognostic factor can greatly impact power to detect an interaction effect. Modified quota sampling and sample size re-estimation using conditional power improve the power when the distribution of the prognostic factor is misspecified. Quota sampling is simple and can prevent misspecification of the prognostic factor, while maintaining power and type I error.