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In a 3-arms clinical trial, with time to event data, either A is superior to C or B is superior to C it will be considered significant. No need to compare A and B. I know how to calculate the sample size in two arms, A vs C; or B vs C, separately.

In the 3-arm case (A vs C or B vs C), how should I calculate the sample size?

any suggestions or any reference that I can take a look? Thanks in advance.

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Since you say you already know how to choose a sample size to control the 2-arm false positive rate, you can use that same technique, but shoot for false positive rate which is one-half of the desired $\alpha$ for the 3-arm test.

Let's see why. Your 3-arm case is not some separate omnibus test requiring a separate analysis, but simply the disjunction of two "2-arm" cases: your alternative hypothesis is confirmed if $A > C$ or $B > C$ or both. We do not know that these two tests are independent - indeed, there is a common case where the sample size of the control group C is much smaller than either treatment group where most of the variance in each test comes from the imprecise measurement of C, and in this case the outcomes of the two tests are highly correlated. However, let's say the the false positive rate for A vs. C and B vs. C are the same, say $\alpha_2$ for "2-arm." Then the false positive rate for the 3-arm test $\alpha_3$ is bounded

$$ \alpha_2 < \alpha_3 < 2\alpha_2 \tag{1} $$

For completeness I should mention that if we do know that the 2-arm tests are independent, then we can take:

$$ \alpha_3 = 2\alpha_2 - (\alpha_2)^2 \tag{2} $$

If A vs. C and B vs. C have different false positive rates $\alpha_A$ and $\alpha_B$, then formula (1) generalizes to:

$$ \max(\alpha_A, \alpha_B) < \alpha_3 < \alpha_A + \alpha_B \tag{3} $$

Let's say for concreteness's sake that we want the false positive rate of 3-arm test to be .05. Then we can solve (1) for the required 2-arm false positive rate:

$$ \begin{align} 0.05 & = \alpha_3 < 2 \alpha_2 \\ 0.0025 & = \frac{\alpha_3}{2} = \alpha_2 \end{align} \tag{4} $$

This is an example of a Bonferroni correction. If you believe in independence, or have different rates for A and B, you could instead solve equations (2) and (3) in much the same way.

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  • $\begingroup$ Thank you very much for your detailed explanation. We plan for a 1:1:1 allocation in the 3 arms. A arm and B arm have dome similarities (difference is treatment sequence or order of treatment), so probably the two comparisons (A vs C & B vs C) are not independent. $\endgroup$
    – ziweiguan
    May 3, 2019 at 18:03

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