Clinical Trials

❘❙❚■ Chapter 21 | Analysis of Survival DataAlthough the incidence rate uses the information on censored observations, it isbased on the assumption that the hazard of an event is constant during the studyperiod or has an exponential distribution. In the case of the pancreatic cancertrial, it means that the hazard of death is constant over the 48-month period.However, the risk of an event can change with time. To overcome this problem,the Cox model, which does not require such assumptions, can be used to derivea better measurement for the treatment effect.Hazard ratioThe treatment can be simply measured as a binary covariate (1 for new treatmentA and 0 for standard treatment B in the pancreatic cancer trial) and introducedinto a Cox proportional hazards model. In the pancreatic cancer trial, theestimated hazard ratio of death for patients who received treatment A to thosewho received standard treatment B is 0.31, with 95% CI (0.11, 0.89), P = 0.030.This means that the new treatment is estimated to reduce the hazard of death by69%, with 95% CI (11%, 89%), and the reduction in hazard is statisticallysignificant at the 5% significance level. As there is only one covariate (treatment)in the Cox model, the estimated hazard ratio is called a crude or unadjustedtreatment effect. The adjusted hazard ratio for the treatment will be generatedif other baseline patient characteristics are introduced into the model.The assumption of the Cox proportional hazards model is that the hazard ratiobetween two treatment groups is constant over the entire time interval. As directestimates of a hazard function are difficult, especially when the sample size issmall, the proportionality is often checked visually by plotting two log–log survivalcurves [1,2]. A log–log survival curve or log (–log [survival rate]) is a logarithmictransformation of negative logarithmically transformed survival function.According to statistical theory, if two hazard functions are proportional, the twolog–log survival curves differ by a constant amount [1,2].Figure 3 displays the estimated log–log survival curves for the two treatmentgroups in the pancreatic cancer trial. We see that the two curves are approximatelythe same shape and are nearly parallel over the study period, so the proportionalhazards assumption does not seem to be seriously violated. A formal statisticaltest of proportional hazards assumption yields a chi-squared value of 0.20 witha P-value of 0.655. Since P = 0.655, we accept that the proportional hazardsassumption holds true since there is no evidence against the null hypothesis.250