## Survival data

Survival data - or more generally, time-to-event data - in two or more groups will most often be compared using the logrank test (see Section 9.3.4). The method for sample size calculation given here makes certain assumptions and approximations, as have the earlier methods, in particular it assumes 'proportional hazards.' This is discussed further in Chapter 9, but briefly, the hazard is the risk of experiencing an event at a given instant in time. The hazard ratio (HR) is the ratio of this risk in one group divided by the risk in the comparison group. Thus a hazard ratio of 0.8 indicates that the risk of the event in the treatment group is 0.8 of the risk in the control group at a given point in time (i.e. the risk is 20 per cent lower). If we assume proportional hazards, then we assume the hazard ratio is constant over time.

With a and ยก3 defined as before, and again assuming equal numbers of patients being allocated to each treatment group, the total number of events (E) needed is given by

In this situation, the proportion of patients who are event-free at a given time in the treatment group (P2) and control group (P1) are related to the hazard ratio as follows:

Thus, it is only necessary to have estimates of the survival proportions rather than the HR, and indeed most sample size tables offer this option. The total number of patients (N) required can be derived from this, and is given by

It is important to note that these formulae assume that the trial analysis will take place at a minimum time T after the last patient has entered the trial, and also assume that any information obtained after this time is not used in the analysis. Consequently, it slightly over-estimates the total number of patients required, but in general such conservatism is no bad thing. Alternative methods (e.g. [15]) allow one to incorporate estimates of the accrual rate into the sample size calculation hence permitting estimation of the total duration of the trial (accrual and follow-up) required before the required number of events is likely to be observed.

In general, one would of course wish to carry out the definitive analysis of the trial when the majority of events have occurred. The survival rates used to calculate the sample size should be taken from this time T, and should therefore be chosen such that the event rate after this time is low. Graphically, this is generally the point at which survival curves 'level off.'

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