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Say I have a $K$ arm experiment that generates survival (time-to-event) endpoints. There are $K-1$ experimental arms and a single control arm.

Say I compute a log rank test statistic comparing the hazards for each arm to control. There will be $K-1$ standardized test statistics, $Z_1, \dots, Z_{k-1}$. What is the correlation between test statistics?

I've found course-notes that say when $K=2$ the correlation is 0.5. Under the null they would be distributed as $(Z_1, Z_2) \sim \mathcal{N}_2(0, \Sigma)$, where $$\Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}.$$ This seems plausible but I don’t see why it’s true. If it is true does it expand to multiple log rank test statistics?

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The correlation of 0.5 is an assumption, it is not derived. I will not accept this as an answer to my question because I do not have a source. This was based on a conversation with an expert, for whatever that's worth.

The assumption is made when there are equal sample sizes in the experimental arm and the control arm. The numerator of the hazard ratio is calculated using the experimental arms and the denominator is calculated using the control arm. So the numerators of the hazard ratios are independent while the denominator is perfectly correlated. This heuristic is why the correlation is assumed to be 1/2

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