Moments When Utilities Are Functional

Abstract

Individual patients’ values and preferences are critical to patient-centered decisions. When dealing with health outcomes on an interval scale, where many intermediate outcomes must be considered, it can be helpful to encourage the patient decision maker to consider what functional form may be specified and parameterized for a given outcome attribute. This assessment problem is described in Keeney and Raiffa’s classic book. 1 For simplicity, let us restrict ourselves to unidimensional settings. We are asked first to specify an attitude toward risk (risk neutral, risk averse, or risk seeking) and then subclassify attitudes depending on whether the risk attitude is constant, absolutely decreasing, or relatively decreasing, in the case of the most common attitude, risk aversion. These questions then often specifyafunctionalformforthepatient’sutilityfunction,whichcanbeparameterizedonthebasisofalimited number of responses from the patient decision maker. This approach tends to consider just risk aversion, the curvature of the utility function as determined by the ratio of its second to its first derivative. That curvature can be quantified by the Pratt-Arrow measure of risk aversion, 2 u 2 /u 1 , although other measures, such as the stronger Ross measure of risk aversion, 3–6 are sometimes used. We use the notation u i to denote the ith derivative of u(x), d i u(x)/dx i . For utilitydistribution functions,onecanconsider either utility density functions (udfs) or their integrals, cumulative utility functions (cufs), analogous toprobabilitydensity functions (pdfs) and cumulative distribution functions (cdfs). For density functions, shape can be described by the function’s moments (e.g., 0-mean, 1-slope, 2-variance, 3-skewness, 4-kurotosis,andhigherunnamedmoments).Normatively, utilities with positive skewness are preferred by individuals whose utility function has u 3 . 0, whereas low kurtosis should be preferred by individuals whose utility function has u 4 \0. 7