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Say I'm working with 2 test statistics and want at least one of them to reject its null hypothesis, so formally:

Statistic $T_1$, null hypothesis $H_{10}$ and Statistic $T_2$, null hypothesis $H_{20}$

  • $H_0$: don't reject $H_{10}$ AND $H_{20}$
  • $H_1$: reject $H_{10}$ OR $H_{20}$

Then for example, if both tests are done at 5% signficance level, the probability of rejecting $H_0$ under the assumption that $H_0$ is true would be $1 - (1- 0.05)^2 $.

Now assume both statistics $T_1$ and $T_2$ are calculated on overlapping data and are therefore positively correlated. Would that increase or decrease the probability of rejecting $H_0$ under the assumption that $H_0$ is true if compared to the probability when both statistics are calculated on independent data?

I would personally think that it would increase this probability, but I don't have an explanation for this and am not sure if this is correct.

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