## What do you expect of the control arm

It is important to have an idea of the outcome anticipated in the control group as this forms the baseline from which improvements will be judged, and of course affects the sample size. This is illustrated in Fig. 5.3, which shows the number of patients required in a survival study to detect an 'absolute' difference of 10 per cent according to different control group rates. It is important to distinguish between absolute and relative differences. Given a baseline survival rate of 50 per cent, an absolute difference of 10 per cent, for example, means an increase in survival from 50 to 60 per cent, while a relative or proportionate 10 per cent improvement means an increase from 50 to 55 per cent.

For proportions and time-to-event outcomes, the precision with which the control group rates need to be estimated is least for rates in the region of50 per cent and greatest for very low and very high rates since, as shown in Fig. 5.3, the slopes are greatest at the extremes. However, the number of events, and hence the number of patients, required to detect an absolute difference of a given magnitude is highest when the baseline rate is in the region of 50 per cent. This is because it is the relative difference in event rates that is important (and not the absolute difference). For example, an absolute increase from 50 to 55 per cent is a relative increase of 10 per cent, while an absolute increase from 10 to 15 per cent is a 50 per cent relative increase. For simple proportions, the event rate and non-event rate can be reversed, thus the number of patients required to detect an absolute increase from 10 to 20 per cent is identical to the number required to detect an absolute increase from 80 to 90 per cent; this is because the standard error of a proportion, P, estimated in a group of N patients is ^[P(1 — P)/N]. However, for time to event outcomes, additional information is contained in the time to event; thus in general the number of patients required to detect a given absolute increase in survival proportions is slightly fewer than the number required to detect the same absolute increase in simple proportions when the event rate is high (see Fig. 5.3). However, as

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Survival proportions Simple proportions

20 40 60 80 Control group survival rate (%)

Fig. 5.3 Number of patients required to detect an absolute 10 per cent increase in proportions. Note: Example from a 2-arm trial, absolute difference to detect = 10 per cent, 90 per cent power and 2-sided 5 per cent significance level.

the additional information is effectively added only by the patients with an event, it is important to distinguish between the event rate and the non-event rate with time-to-event data and, as shown in Fig. 5.3, the number of patients required to detect a given absolute increase is not perfectly symmetrical about the 50 per cent rate. More patients are required to detect an increase from 80 to 90 per cent event-free than to detect an increase from 10 to 20 per cent event-free, since in the latter case many more patients have events and thus contribute the additional information.

The best source of data for the control group outcome is a previous trial using the same eligibility criteria. If this is not available, bear in mind that patients in trials tend to have better survival than those not in trials, that selection criteria will almost inevitably identify patients with a better prognosis than basic registry data would suggest, and wherever possible err on the side of caution by tending towards a rate closer to 50 per cent.

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