Bias in randomised factorial trials

Factorial trials are an efficient method of assessing multiple treatments in a single trial, saving both time and resources. However, they rely on the assumption of no interaction between treatment arms. Ignoring the possibility of an interaction in the analysis can lead to bias and potentially misleading conclusions. Therefore, it is often recommended that the size of the interaction be assessed during analysis. This approach can be formalised as a two-stage analysis; if the interaction test is not significant, a factorial analysis (where all patients receiving treatment A are compared with all not receiving A, and similarly for treatment B) is performed. If the interaction is significant, the analysis reverts to that of a four-arm trial (where each treatment combination is regarded as a separate treatment arm). We show that estimated treatment effects from the two-stage analysis can be biased, even in the absence of a true interaction. This occurs because the interaction estimate is highly correlated with treatment effect estimates from a four-arm analysis. Simulations show that bias can be severe (over 100% in some cases), leading to inflated type I error rates. Therefore, the two-stage analysis should not be used in factorial trials. A preferable approach may be to design multi-arm trials (i.e. four separate treatment groups) instead. This approach leads to straightforward interpretation of results, is unbiased regardless of the presence of an interaction, and allows investigators to ensure adequate power by basing sample size requirements on a four-arm analysis.