Some geometric methods for constructing decision criteria based on two-dimensional parameters

This paper reviews two types of geometric methods proposed in recent years for defining statistical decision rules based on two-dimensional parameters that characterize treatment effect in a medical setting. A common example is that of making decisions, such as comparing treatments or selecting a best dose, based on both the probability of efficacy and the probability of toxicity. In most applications, the two-dimensional parameter is defined in terms of a model parameter of higher dimension including effects of treatment and possibly covariates. Each method uses a geometric construct in the two-dimensional parameter space based on a set of elicited parameter pairs as a basis for defining decision rules. The first construct is a family of contours that partitions the parameter space, with the contours constructed so that all parameter pairs on a given contour are equally desirable. The partition is used to define statistical decision rules that discriminate between parameter pairs in term of their desirabilities. The second construct is a convex two-dimensional set of desirable parameter pairs, with decisions based on posterior probabilities of this set for given combinations of treatments and covariates under a Bayesian formulation. A general framework for all of these methods is provided, and each method is illustrated by one or more applications.