ORTHOGONALLY DECOMPOSING THE INTERACTION SUM OF SQUARES IN TWO FACTOR EXPERIMENTS WHEN BOTH FACTORS ARE QUANTITATIVE: UNBALANCED DATA AND UNEQUAL SPACING

Abstract

In this article, interaction contrasts are expressed as a polynomial function of both of the quantitative classification variables, the values of which give rise to the levels of the two factors. When all polynomial coefficients are zero, there is no interaction between the two factors. When at least some of these coefficients are not zero, interaction between the two factors exists and structure is imposed on the interaction contrasts. The imposed structures are examined in detail. To test the hypothesis that any particular set of polynomial coefficients is zero, the observation vector may be regressed on a specially constructed set of independent variables, and the numerator sum of squares for the associated F test may be obtained by adding certain of the sequential sums from the regression. The construction of the regression variables and subsequent regression may be performed easily with the aid of any statistical computing package. No special software is necessary.

Keywords:

• Regression
• Gram-Schmidt
• Sequential sums of squares
• No missing cells
• Polynomial
• Cell means

ACKNOWLEDGMENTS

The author would like to thank an anonymous referee, whose comments led to a greatly improved presentation of the results of this article.