The analysis of experiments involving paired comparisons has received considerable attention in statistical methodology. Thurstone (1927) has considered the problem on the assumptions that a linear variate is involved and that perceptible differences exist among the items presented for comparison. More recently, Mosteller (1951 a, b) has elaborated upon Thurstone's method and, having postulated a sensation continuum over which sensations are jointly normally distributed, has developed ax2 test following transformation of the observed variates.

Kendall & Babington Smith (1940) proposed a method of analysis for paired comparisons which does not depend on assumptions of a linear variate or of normality, and the procedure may be described as a combinatorial type test. They form a coefficient of agreement which essentially measures discrepancies from perfect agreement, although the model used in the test is not explicitly formulated. In subjective tests the consistency of a judge is measured in terms of circular triads. We note that tests of consistency and tests of agreement, when differences are known to exist, may also be considered to be tests of null hypotheses upon the postulation of absence of differences. Guttman (1946) has developed a method of quantifying paired comparisons. His problenm was to determine a numerical value for each of a number of items which will best represent the comparisons in some sense. The problem may be considered to be one of estimation as distinct from the problem of testing hypotheses. When only two items are to be compared in a ranking experiment, a test of the hypothesis of no-difference between them on some characteristic may be based on the binomial distribution. The estimation of the probabilities that the items are stuperior in a given comparison may be accomplished, and these estimates afford a method of rating the items or a method of quantification. In the present paper a generalization of the binomial model and distribution is obtained.