Some Scheffé-Type Tests for Some Behrens-Fisher-Type Regression Problems

In educational and psychological applications as well as in other applications, it may be necessary to make certain comparisons of two regression lines when the variances are unequal. Such problems arise, for example, in studies comparing two alternative curriculums or two different teaching methods. By generalizing an idea which Scheffé used to obtain a test for the Behrens-Fisher problem, this paper develops some tests for comparing two regression lines when the two sets of error terms are normally distributed but with two different variances. Scheffé's test itself is a randomized test, but in this paper we present both randomized and non-randomized tests. Both simple and multiple regression are considered, but the simple regression tests are computationally easier than the multiple regression tests. The basic test statistic which is used is the ordinary t-statistic. Essentially two types of problems are dealt with: (A) determining whether the two regression lines are identical when they are known to be parallel; and (B) determining whether the two regression lines are parallel. Confidence bounds as well as tests of hypotheses are available. For Problem A, a minimax estimator of the distance between the two lines is obtained. In addition to the Scheffé-type tests, we also consider some tests based on an approach of Welch and Hájek. A numerical example is presented.