Clinical Trials

❘❙❚■ Chapter 21 | Analysis of Survival DataFigure 2. Proportional hazards assumption: equal distance between two hazard functions for two individuals.Individual 1Individual 2Log hazard functionTimeh 1(t) = h 0(t)exp(b 1x 11)h 2(t) = h 0(t)exp(b 1x 12)h 1(t) = exp(b1 [x 11– x 12]) (2)h 2(t)What is important about equation (2) is that h 0(t) is canceled out of the numeratorand denominator. As a result, the ratio of hazards, exp(b 1[x 11– x 12]), is constantover time. In this example, exp(b 1[x 11– x 12])= exp(0)= 1 if the two individuals havethe same treatment, or exp(b 1[x 11– x 12]) = exp(b 1) (or exp[–b 1]) if they havedifferent treatments. After performing a logarithmic transformation to both sidesof equation (2), we have the following equation:log(h 1[t]) – log(h 2[t]) = b 1(x 11– x 12)If we plot the log hazards for the two individuals, the proportional hazardsproperty implies that the hazard functions should strictly have the same distanceat any time during the study as shown in Figure 2. If these curves cross each otheror diverge, then the proportional hazards (sometimes called proportionality)assumption may not be met.246