Analysis of Multifactor Classifications with Unequal Numbers of Observations

Abstract

The sums of squares in the general unequal numbers analysis of variance for an n-way or n-factor classification may be obtained in general terms from standard regression theory. However, the computing formulas using the usual normal equation and solution procedures become unmanageable even with the largest computers whenever the number of factors and/or the number of levels become moderately large. With the use of the calculus of factorials as developed by Kurkjian and Zelen in 1962, and with the additional theoretical results obtained by the authors in a recent paper, it has been possible to set forth relatively simple computational procedures for obtaining sums of squares in the analysis of variance for any effect eliminating all other effects. For sums of squares of interaction to order two, the computations may be performed on an ordinary desk calculator. Higher order interactions may require the use of high speed computers. In order to ease the computational difficulty in computing sums of squares still further, an upper bound sum of squares and a sharp upper bound sum of squares were obtained. Both forms are computationally simple and appear to be closer approximations than present approximations in the literature. A numerical example of a 4 X 3 X 2 factorial with unequal numbers of observations in the subclasses is used to illustrate the application of various formulae for computing sums of squares. A convenient computational form for computing interaction sums of squares has been set forth. The results are easily extended to four factor and larger factorials with unequal numbers of observations.