## Flemings single stage design [16

Here we assume that for any experimental treatment, there will be a level of response, pi, below which the treatment would not be considered for further study, and a higher level, p2, above which the treatment would certainly warrant further investigation. The sample size is determined by the need to minimize the probability of concluding that the response rate is greater than p1 when that is false (a) and to minimize the probability (P) of concluding that the response rate is less than p2 when that too is false. This is analogous to the comparison of proportions described in Section 5.4.2, but here we consider one of the proportions effectively to be fixed. As we do not need to allow for uncertainty in the estimation of one of the proportions, the sample size is considerably smaller (approximately one quarter) than the sample size for comparing two parallel groups.

An example of a trial designed in this way is given in Box 5.1. Gehan's two-stage design [17]

In the first stage, a minimum useful level of efficacy (pm) is determined, for example a 25 per cent response rate, and a maximum acceptable probability (a) of rejecting the treatment if it is truly as, or more, effective than pm, for example 5 per cent. The probability of seeing ni consecutive failures given a true response rate p, where p is greater than or equal to (pm) is (1 — pm)n1. The number of patients studied in the first stage is therefore the value of n which satisfies (1 — Pm )n1 < a. For example, if the true response rate p is 0.25, then the probability of eleven consecutive failures is (1 — 0.25)11 = 0.04. Therefore, if you observed eleven consecutive failures, then the trial could stop at this point with the conclusion that the treatment is unlikely (a four in 100 chance) to be associated with a response rate of 25 per cent or more. If any responses are seen, the study continues to the second stage, in which the aim is to estimate the response rate with a specified precision.

The sample size for the second phase depends on two things; the true response rate P - which is estimated from the response rate in the 1st stage of the study - and the desired precision, £ (defined by the standard error of the overall study response rate) which must be specified.

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