The Scaling of Paired Comparison Data With one or More Extreme Stimuli: a Comparison of three Models

Abstract

Two models for the scaling of paired comparison data are compared to the Thurstone case III model. One model is based upon parameter estimates derived from the normal approximation of the binomial distribution, and the other from the beta density function defined over the unit interval. Procedures are given for estimating the parameters for each model, and arguments are presented in favor of a skewed characteristic distribution function in cases where the scale includes extreme stimuli.Two indices of goodness of fit are presented for each of five data sets. The results generally illustrate the inability of the Thurstone case III model to adequately account for the data when the scale includes one or more extreme stimuli. The beta model performed well for all five data sets but fit best under just those conditions where the Thurstone model performed poorly. The binomial approximation performed adequately on three of the five data sets and would seem to be a simpler alternative estimation procedure to the Thurstone procedures when the normality assumption is tenable.EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 1976, 36, 657-670. IN recent years, several attempts have been made to revise and/or replace the earlier Thurstone models for the scaling of paired comparison judgments. Notable among these competing models are those proposed by Restle (1961 ), Luce (1959), andBradley andTerry (1952). Like the Thurstone model, each of these newer models posits the existence of an underlying real value scale and a corresponding distriat University of Manitoba Libraries on June 22, 2015 epm.sagepub.com Downloaded from