Let us say, I observe 3 variables in a control condition and a treatment condition. I would like to find out whether the treatment has some effect on all 3 variables at the same time. I should mention that I cannot observe the 3 variables simultaneously. In a first experiment, I would measure variable A in control vs. treatment, in a second experiment, I would measure variable B in control vs. treatment and so on.
For each of the variables, the null hypothesis that the treatment has no effect was tested. So, I have 3 p-values, one for each variable. (and I only have access to these data, not the raw data).
Let us see whether I understand this correctly:
this would mean, I test, whether the treatment has the same effect on all 3 variables.
and this would mean, the treatment has an effect on at least one variable.
Is there also a way, to test whether there is some effect (but not necessarily the same effect) on all of the variables? Intuitively, this should be the case, if all 3 p-values are small and it would not be the case, if the maximum p-value would be large, I would say.