Clinical Trials

Clinical Trials: A Practical Guide ■❚❙❘Proportional hazards modelModel descriptionAs described earlier, the hazard function h(t) is the risk that a subject experiencesan event at time t given that the subject has not experienced the event up to t. Theproportional hazards model relates the hazard function to a number of covariates(such as the patient’s characteristics at randomization and the treatment receivedin a clinical trial) as follows [1,2,5]:h i(t) = h 0(t)exp(b 1x 1i+ b 2x 2i+ … + b px pi) (1)where x kiis the value of the covariate x k(k=1, 2,..., p) for an individual i (i=1, 2,..., n).The equation shows that the hazard for individual i at time t is the product oftwo factors:• A baseline hazard function h 0(t) that is left unspecified, except that itcannot be negative.• A linear function of a set of p fixed covariates, which is then exponentiated.The baseline hazard function can be regarded as the hazard function for individualswhose covariates all have value 0 and changes according to time t.Two basic types of model are available for us to use, depending on whether wespecify h 0(t). In general, if we specify a parametric function for h 0(t) in equation (1),we will have a parametric hazards regression model. So, if we specify h 0(t) = λt α ,we get the Weibull hazards regression model. The most widely used hazardsregression model, however, is the Cox regression model, in which such choices ofh 0(t) are unnecessary [1,2]. In the Cox model, the baseline hazard function h 0(t)can take any form. Therefore, the Cox regression model is sometimes calleda semi-parametric hazards regression model and is commonly referred to as theCox proportional hazards regression model.Proportional hazards assumptionWhy is this called a proportional hazards model? The reason is that while thebaseline hazard can constantly change over time, the hazard for any individual isassumed to be proportional to the hazard for any other individual for all times t,and will depend on the covariate values. To illustrate this, let us assume that themodel has only one covariate (treatment, x 1i, x 1i= 0 for standard treatment and1 for new treatment). We first calculate the hazards for two individuals 1 and 2according to equation (1) and then take the ratio of the two hazards:245