Question

Suppose that m = 2, and that the p-values for the two tests are positively correlated, so that if one is small then the other will tend to be small as well, and if one is large then the other will tend to be large. How does the family-wise error rate associated with these m = 2 tests qualitatively compare to the answer in (b) with m = 2?

In (b), FWER = 1-(1-alpha)^m

Answer 1

solution,

when the p values are positively correlated Sarkar(2008) developedalternative single step and stepwise k-FWER procedure utilizing kth order joint null distribution of the p values.

If we perform m hypothesis tests , probabilities of at least 1 false posibilty

P(making at least 1 error in m tests)=1-(1-alpha)^m

in ques also it is given FWER =1-(1-alpha)^M

now as we know that if two tests are positively correlated it means if one is small then the other will tend to be small as well and if one is large then the other will tend to be large and in ques m=2 and the p values are positively correlated we are to find family wise error rate associated with these m=2 tests qualitively compare

with the help of Bonferroni correction FWER assosiated.bcz Bonferrooni derived the calculation that the overall alpha of performing m significance test is equal to 1-(1-alpha)^m which is the probability that one of them will result in a statisticaaly significant outcome here to perform a Bonferroni correction divide the critical P value(alpha) by the number of comparision being made

so the new critical P value would be ( alpha/2).the statistical power of the study is then calculated based on this modified P value

so for two hypothesis tests an overall alpha =0.5

then P=1-(1-.5)^2=.75

thankss