Fitting polynomials to main effects in multi factor experiments: unequal spacing and unbalanced data with no missing cells

Abstract

In multi factor experiments, when the factor of interest arises from distinct levels of a quantitative variable, the main effects for that factor, which can be arbitrary weighted averages of the cell means, may be expressed as a polynomial function of the corresponding levels of the quantitative variable. The main effect sum of squares may be partitioned into individual one degree of freedom sums of squares, which may be combined into numerator sums of squares for F-tests of significance of the polynomial coefficients. The purpose of this paper is to show how these sums may be obtained as sequential sums of squares from a no-intercept regression of the observation vector on specially constructed independent variables when sample sizes are not all the same and consecutive values of the quantitative variable are not equally spaced. The construction of the variables and subsequent no-intercept regression may be performed easily with the aid of any statistical com-computing package. No special software is necessary.

Keywords:

• regression
• Gram-Schmidt
• orthogonal decomposition
• sequential sums of squares
• cell means
• weighted averages