A multivariate test of interaction for use in clinical trials

An important issue in clinical trials is whether the effect of treatment is essentially homogeneous as a function of baseline covariates. Covariates that have the potential for an interaction with treatment may be suspected on the basis of treatment mechanism or may be known risk factors, as it is often thought that the sickest patients may benefit most from treatment. If disease severity is more accurately determined by a collection of baseline covariates rather than a single risk factor, methods that examine each covariate in turn for interaction may be inadequate. We propose a procedure whereby treatment interaction is examined along a single severity index that is a linear combination of baseline covariates. Formally, we derive a likelihood ratio test based on the null beta0 = beta1 versus the alternative abeta0 = beta1, where X'beta(k) (k = 0, 1) corresponds to the severity index in arm k and X is a vector of baseline covariates. While our explicit test requires a Gaussian response, it can be readily implemented whenever the estimates of beta0,beta1 are approximately multivariate normal. For example, it is appropriate for large clinical trials where beta(k) is based on a logisitic or Cox regression of response on X.