Handling Correlated Variables

Many situations exist where multiple responses are measured during an experiment, e.g. parent and drug metabolite concentration and drug toxicity and efficacy. Typically the responses will be mixed (continuous and discrete) and display varying dependence structures. When standard multivariate distributions do not exist or are not applicable, a more flexible approach to the construction of the multivariate distribution is required. This talk will introduce a class of functions, called Copulas, that enable the joint distribution of the multiple responses to be expressed as a function of the marginal distributions and the Copula, which “joins” the marginal distributions together. This approach can prove useful as an experimenter will typically have knowledge of the marginal distribution of the responses but may not know the joint distribution. Design of experiments for studies with multiple responses is an important issue. Poor or incorrect experimental design can lead to poorly estimable models. Historically, experiments have either been designed for only one of the responses or where independence of responses has been assumed. These approaches are flawed as they ignore the inherent relationship between the responses. However, Copula functions allow the dependence between the responses to be incorporated into the design of experiment. We shall consider a bivariate binary response model for the efficacy and toxicity of a drug where the marginal distribution of each binary response is assumed to be logistic. The use of Copula functions will be illustrated, as well as the effect of accounting for the dependence between the responses.