One of the more common objectives in research is to determine if two populations differ, so it is not surprising that a variety of tests of significance have been developed to perform two-sample tests for location. The pooled *t* test and the unequal variance *t* test are the most popular of the parametric tests, but many others have been proposed, including several rank-based tests. One of these rank tests is the Wilcoxon rank-sum test, which is popular because it is reasonably powerful and robust. It was one of the rank-based tests that were described in Section 1.3. In Chapter 1 we also described the adaptive test of Hogg, Fisher, and Randies (1975), which we called the HFR test, and we compared the power of that test to the power of the *t* test.

In this chapter we will describe a modern adaptive two-sample test. Unlike the HFR test, this test can be generalized to perform adaptive tests of significance for any subset of coefficients in a linear model. Hence, the approach used in this chapter will also be used in subsequent chapters to perform many tests, including a test of slope in a linear regression, a test of interaction in a two-way layout, and a test of a main effect in a multifactorial design.

Now suppose we have *n* observations and we want to perform a test of location. Let *y*_{i} be the dependent (or response) variable and let *x*_{i} be the independent (or predictor) variable for the *i*th observation. For a test of location ...

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