High-Order Hazard Functions and Treatment Choice

Hazard function applications in medical research are problematic for two reasons. First, they are not cast within a decision theory framework. Second, they often adopt severe self-imposed restrictive structures (e.g., in the constant hazard ratio models). The disadvantage of an excessively restrictive structure is self-evident. The disadvantage of the lack of a theoretical basis, which is more subtle, is that treatment choice itself becomes unnecessarily restrictive because decision theory insights remain untapped. This paper uses a decision-theory-based framework for treatment choice, thus addressing the two issues above. It shows that high-order stochastic dominance tools, in conjunction with risk preference attributes, can often be used to compare treatments under weaker conditions than the ones currently used. The paper compares treatments by using what we call high-order hazard functions. These high-order hazard functions are obtained by calculating areas under low-order hazard functions (such as standard and cumulative hazard functions). The paper provides necessary and sufficient conditions for treatment comparisons based on these high-order hazard functions. These conditions are shown to be weaker than the ones currently used because we are able to exploit theoretical tools that are otherwise unavailable. Thus, for example, it shows that our framework often allows treatment comparisons even when hazard functions cross. An example using real-world data shows that the use of high-order stochastic dominance and risk preference attributes allows us to identify a preferred treatment even if low-order hazard functions cross.