## Coxs proportional hazards model

Cox's proportional hazards model is used extensively in the analysis of survival data. The use of the model includes comparing treatment effects in trials and adjusting these comparisons for baseline characteristics. A further use of these models is to assess variables for prognostic significance. The basic assumption for this model is that the ratio of hazards of an event at a given time in the groups being considered is proportional over time. It is beyond the scope of this book to consider how one might test this assumption, and the reader is referred to more specific books on survival analysis for these (such as Ref. [9]). Despite this apparently rather limiting assumption, the proportional hazards model has been extensively used, partly because experience suggests that it works well in practice in many situations, but also because it is available in nearly all major statistical packages.

In studying a group of patients we might wish to relate the duration of their survival to various clinical characteristics that they have at diagnosis, for example age, stage of disease, grade of disease, performance status, etc. In survival analysis terms we model the probability of dying at a given time, t, given the patient has survived to that time. This is called the instantaneous death rate or the hazard rate and denoted as X(t). The proportional hazards model relates this to an overall average hazard for the whole group X0(t), using the following expression:

where X is the group of variables representing the characteristics of the patient, for example their age, stage of disease, grade of disease, etc., and 0 are a set of parameters which need to be estimated from the data. For example if we are trying to evaluate the risk of dying at a particular time, given survival to that time, and given the patient's age, stage of disease and grade of disease:

risk of dying at time t = average risk of dying at time t x exp (01 x age + 02 x stage + 03 x grade). (9.25)

The parameters 01,02,... (together with their standard errors) give an indication of the relative importance of each of the variables. In most situations when comparing two treatments in a randomized trial, the Cox model with the treatment group as the only variable in the model will give very similar results to the logrank test described above, and the hazard ratio will be given by exp (0).

It is clear from the above that the logrank test and the Cox model can both be used for the same purposes, usually with little difference between the results. We discuss the pros and cons of both approaches in Box 9.1.

It can be seen from Box 9.1 that, except for the simplest analyses, the Cox model although more complicated, provides a more general framework for the analysis of time-to-event data.

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