The relative efficiency of two tests is a measure of the relative power of two tests. Suppose tests 1 and 2 are tests for the same null-hypothesis and at the same significance level “alpha” (probability of type I error). The relative efficiency of test 2 with respect to test 1 is the ratio N1 / N2, where N2 is the sample size of test 2 required to achieve the same power for a given alternative as is achieved by test 1 using a sample of size M N_1 .
This measure usually depends on the significance level “alpha”, the alternative hypothesis, and the sample size N1 of test 1.
Note: the relative efficiency of tests should not be confused with the relative efficiency of estimators.