So basically I have the following scenario:
I have developed a new way for conditional independence testing, and am bench-marking is against other methods.
To do this, I test the power in scenarios, where the null-hypothesis of conditional independence should be rejected (because it is false), however the tests are very complex (low signal to noise ratio), and hence for some sample sizes, the power is (very) low. Which made me wonder:
If, say, I fix the confidence (desired type 1 error) at 5%, would it ever make sense to have a power that is below 5%?
Because if the test gives you at least 5% type 1 errors, shouldn't it reject at least 5% of the cases, where the null is actually false (as a lower bound).
Logically, it makes complete sense to me - but I couldn't find any mathematical reasoning supporting this. Does anyone know how to derive this lower bound? (or why its not actually a lower bound)