Gideon Keren Free University of Amsterdam
A problem encountered by many researchers in psychology and other disciplines is the one of an "unbalanced design" (also referred to as "unequal frequencies" or "non-orthogonal" design). This is the case in which the number of observations in different cells of an experimental design are not necessarily equal. The question of how to treat such designs has become a major controversial issue in recent years ( Appelbaum & Cramer, 1974; Cramer & Appelbaum, 1980; Herr & Gaebelein, 1978; Keren & Lewis, 1976, 1977; Overall & Spiegel, 1960; Overall, Spiegel, & Cohen, 1975; Speed & Hocking, 1976, to mention just a few). The purpose of the present chapter is to discuss briefly several methods that were recently proposed and to provide the reader with some guidelines of how to deal with a nonorthogonal design.
The approach presented in the chapter employs least square regression methods as a substitute for the traditional analysis of variance (ANOVA). Several investigators ( Cohen, 1968; Cohen & Cohen, 1975; Darlington, 1968; Kerlinger & Pedhazur, 1973) have pointed out the equivalence of analysis of variance (ANOVA) and multiple regression (MR) and have advocated the use of the latter on grounds of greater flexibility. It is unfortunate that only a few researchers have indeed applied the MR approach and it is still the case that over 80% of the articles in psychological journals are employing the traditional ANOVA, even under circumstances where MR is the more appropriate tool at least in terms of flexibility and elegance. In the case of nonorthogonal designs it seems to the present author that the use of MR is almost unavoidable. The only treatment of unequal n's proposed within the traditional ANOVA is the so-called "unweighted