If two statistical tests have different sizes then is it reasonable to compare their powers at an alternative away from the null? if not, would multiplying the two tests to have same desired level (size) by constants qualify tehm by comparison? In such cases if the multipliers are functions of nuisance parameters and are replaced by their estimates what can happen?
Thanks in advance.
What do you mean by "tests of different sizes"? Do you mean tests on different-sized samples?
Sir, by 'size' I mean type-I errors of tests. I have simulated power functions of several tests for the same hypothesis. their type-I errors are different and one of them has very high type-I errors though it has highest powers. In such cicumstances does the power comparison make sense?
I have suggested some dada based ad-hoc corrective factors to the tests which fairly control the type-I errors but then the previous pattern of pwer functions for uncorrected tests substancially changes.
Comparioson of which power functions is more reasonable? Corrected ones with desirable and almost equal type-I errors Or the un-corrected ones with substancially different type-I errors?
Thanks a lot in advance.
It is not reasonable.
Consider two test procedures that are the same test performed at the 0.05 level and also at the 0.10 level. The latter will always be more powerful, but the comparison is not meaningful.
A result may be significant at the 0.1 level but not the 0.05 level. Thus it is more likely to be rejected at the 10% level, i.e. higher power.
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