An Analysis of Variance for Paired Comparisons

Abstract In a paired comparison test of m brands of a product each of the ½m(m – 1) pairs is presented to 2r judges: to r in one order, and to r in the other. An analysis of variance is developed for the case in which the judges’ preferences are expressed on a 7 or 9-point scale. Account is taken of the effects of order of presentation. Main effects are defined for the brands. The hypothesis of subtractivity, analogous to the hypothesis of additivity in a two-way layout, states roughly that the results for any pair, after order effects are eliminated, can be attributed entirely to the difference of the main effects of the two brands in the pair. Significance tests for the main effects, for the order effects, and for the hypothesis of subtractivity are given, as well as estimates of various parameters and their standard errors. The main effects are analyzed by considering all possible comparisons. A numerical example illustrates the method.