Clinical Trials

❘❙❚■ Chapter 21 | Analysis of Survival DataSurvival function and hazard functionIn survival analysis, two functions are of central interest, namely survival functionand hazard function [1,2]. The survival function, S(t), is the probability that thesurvival time of an individual is greater than or equal to time t. Since S(t) isthe probability of surviving (or remaining event-free) to time t, 1 – S(t) is theprobability of experiencing an event by time t. Put simply, as t gets larger,the probability of an event increases and therefore S(t) decreases. Plotting a graphof probability against time produces a survival curve, which is a useful componentin the analysis of such data. Since S(t) is a probability, its value must be ≥0 but ≤1.When t = 0, S(0) = 1, indicating that all patients are event-free at the start ofstudy. Within these restrictions, the S(t) curve can have a wide variety of shapes.The hazard function, h(t), represents the instantaneous event rate at time t foran individual surviving to time t and, in the case of the pancreatic cancer trial,it represents the instantaneous death rate. With regard to numerical magnitude,the hazard is a quantity that has the form of ‘number of events per time unit’(or ‘per person-time unit’ in an epidemiological study). For this reason, the hazardis sometimes interpreted as an incidence rate. To interpret the value of the hazard,we must know the unit in which time is measured.For the pancreatic cancer trial, suppose that the hazard of death for a patient is0.02, with time measured in months. This means that if the hazard remainsconstant over one month then the death rate will be 0.02 deaths per month (or perperson-months). In reality, the 36 patients contributed a total of 950 person-monthsand 16 deaths. Assuming that the hazard is constant over the 48-month period andacross all patients, an estimate of the overall hazard is 16 / 950 = 0.017 deathsper person-month.The hazard function is a very useful way of describing the probability distributionfor the time of event occurrence. It can be a constant, as illustrated above, or itcan be more complex. For example, if h(t) = λ (λ > 0), we have what is known asan exponential survival distribution. If h(t) = λt α (λ > 0; λ and α are constants),we get the Weibull distribution [1,2].Every hazard function has a corresponding survival function as described by thefollowing equation:tS(t) = exp { – ∫ h(u)du }For the exponential distribution h(t) = λ, we substitute this hazard function intothe above equation, perform the integration, and obtain the survival functionS(t) = e –λt .0238