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 t hat have the potential for an interaction with treatment may be suspected o n the basis of treatment mechanism or may be known risk factors, as it is o ften 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 cova riate in turn for interaction may be inadequate. We propose a procedure whe reby treatment interaction is examined along a single severity index that i s a linear combination of baseline covariates. Formally, we derive a likeli hood ratio test based on the null beta(0) = beta(1) versus the alternative a beta(0) = beta(1), 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 explici t test requires a Gaussian response, it can be readily implemented whenever the estimates of beta(0), beta(1) are approximately multivariate normal. F or example, it is appropriate for large clinical trials where <(beta)over c ap>(k) is based on a logisitic or Cox regression of response on X.