Hypothesis Testing Incorporating a Preliminary Test of Significance

Abstract

Suppose that independent samples are available from two normal populations with means μ₁ and μ₂ and known variances. It is desired to test H: μ₁ =0. A possible preliminary test procedure in this case consists of first testing μ₁ = μ₂ and then testing H using either the sample mean from the first population or the pooled sample mean depending on the outcome of the test regarding equality of μ₁ and μ₂. This procedure is shown to be biased. A power comparison of this procedure with the generalized likelihood ratio test of H indicates that the latter test is more powerful in a fairly extensive parametric region. It is noted that biasedness is frequently encountered in preliminary test procedures. In light of the example, it is recommended that power comparisons be made between such procedures and any available unbiased tests to ascertain whether extensive power loss regions exist or not.