Determining the Change in Mean Duration Due to a Shift in the Hazard Rate Function

From a major worm event (such as a military action) to a seemingly minor detail (such as the use o f a new plastic washer in a faucet design) change must be accounted for when collecting, interpreting and analyzing data. Indeed the intervention itself may be the focus o f the study. Theoretically, the best way to model some interventions, especially time-dependent ones, is via the hazard function. On the other hand, it may be necessary to translate into simpler concepts in order to answer practical questions. The average duration, for example, may have well-understood relationships with costs, making it the best choice for presenting the result. For example, Shaun Wang [3] discusses deforming the hazard function by a constant multiplicative factor--proportional hazard transform--as a way to price risk load, with the mean playing the role of the pure loss premium. This paper investigates how a shift in the hazard rate impacts the mean. The primary focus of the discussion is the case o f bounded hazard rate functions of finite support. A formal framework is defined for that case and a practical calculation is described for measuring the impact on the mean duration of any deformation of the hazard function. The primary tool is the Cox Proportional Hazard model Several formal results are derived and concrete illustrations o f the calculation are provided in an Appendix, using the SAS implementation. The paper establishes that the method can be applied in a very general context and, in particular, to deformations which are not globally proportional shifts. Indeed the method demands no assumed form for either the survival distribution or the deformation. The discussion begins with a case study that illustrates the application o f these ideas to assess the cost impact o f a TPA referral program.