# Probability of rejecting null hypothesis with correlated test statistics

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.