Incorrect F Formulas in Regression Analysis

Abstract

In regression analysis, an F test can be viewed as a comparison between a full and a restricted model. The most general F formula compares the error sums of squares (SSE's) of these two models. This F formula is always correct because the SSE comparison is meaningful in all tests. Other formulas use the corrected model sum of squares (SSM) or the coefficient of determination (R ²) to compare the full and restricted models. This article gives several examples where the SSM's or R ²'s of the two models cannot be compared, and hence where the use of F formulas based on SSM or R ² would be incorrect. This problem usually arises in tests of nonhomogeneous hypotheses, although it may also appear in other situations.

Key Words:

• Coefficient of determination
• Full model
• Hypothesis test
• Linear model
• Reparametrization
• Restricted model