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

Potthoff, Richard F.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
NATO Research Grants Programme, United States Air Force Office of Scientific Research, Behrens-Fisher Problem, Regression (Statistics), Scheffe Test, Statistical Significance


In psychological and other applications, it may be necessary to make certain comparisons of two regression lines when the variances are unequal. Such problems arise rather frequently, for example, in studies comparing two alternative curriculums or two different teaching methods. By generalizing an idea which Scheffe 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. Scheffe'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. This paper is on a theoretical level; a more practically-oriented discussion, with numerical examples, is given in Research Bulletin 63-36.

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