Policy implications resulting from connecting survival models to the underlying biological processes
*Sidney Klawansky, Harvard School of Public Health 

Keywords: survival, Cox proportional hazards model, accelerated failure time model, non-proportional hazards, postponement, cost-effectiveness

The Cox proportional hazards model is widely used to compare groups in survival analysis. For some diseases, the proportionality assumption seems to be consistent with both the data and the underlying biological process. However, in a number of major disease areas, the hazard ratio changes over time. In these cases, the non-proportional hazards behavior over time may be understood as the natural biological consequence of diseases, such as heart disease and cancer, that progress over time to a sentinel event such as heart attack, stroke or death. The time scaled proportional hazards model, aka the joint hazard and time scaling model, was derived to transform a distribution of disease progression rates into a distribution of arrival times at the sentinel event as reflected in the hazard. Because the model independently scales the hazard and time dimensions, it has far greater flexibility to accommodate a wide range of data, especially non-proportional data, than do standard models. In fact, standard models such as the proportional hazards (PH) and the accelerated failure time (AFT) model are simply special cases of the joint model.

Because of the biodynamic behavior with time, the hazards comparing a successfully treated group having a better prognosis with a control group having a less favorable one will likely show the smallest hazard ratio early, indicating the greatest survival benefit. However, the hazard ratio will probably increase over time, indicating decreased survival benefit at later times, due to the increased occurrence of deaths that were initially postponed in the more favorable group. This increasing hazard ratio over time was observed in NCI registry data for 22,488 local v. 21,968 regional breast cancer cases, where local breast cancer carries a better prognosis than does regional disease. The hazard ratio increased from 0.16 to 0.45 over a follow-up period of 20 years.

The joint model suggests important features for cost-effectiveness analyses and for health policy. Clinical trials in heart disease and cancer that typically end at 3-5 years may cite hazard ratios that are favorable, but that may well increase at later times. With survival benefits decreasing and side effects accumulating over time, the best cost-effectiveness ratios could occur early and less favorable ratios could occur later. The policy implications of this changing benefit over time due to the underlying biological behavior need to be considered.