Say you have a drug of which you want to test wether it increases the number of immune cells in your blood. You divide your sample in two groups where one group receives the treatment and the other group doesn't. In each group, you measure the effect both before treatment initialization and after 2 months of treatment. One can now test the effect of the drug in two ways:
Test the number of immune cells at two months in both groups
Test the increase in immune cells (=effect after 2 months minus baseline) in each group.
If there would be no effect of the drug at all, but you would do both tests at the 5% level and pick the smallest p-value, what would then be the family-wise error rate (FWE)? As you can see, both tests are not independent, however, they do generate different p-values, so I already reasoned the FWE should be somewhere between 5 and 9.75%.
Thanks a lot!