New Mathematical Model for the Law of Comparative Judgment

Abstract

The law of comparative judgment provides a useful approach for psychophysical scaling in subjective printing quality measurement and evaluation. It links the scales, through a set of equations, to the proportion of times that one stimulus is judged greater than another in terms of perceived strengths of stimuli. The equations can be further simplified with additional assumptions on the perceptual dispersion of each stimulus and on their correlations to various forms that can be solved, via least squares method, for practical applications. However, the normal deviates can approach the value of (plus or minus) infinity. Further, because the equations for the law of comparative judgment deal with the normal deviates, solutions are generally not optimized for the proportions of choice. The method of modeling the correlation between two stimuli can also be improved to reflect the underlying mechanism of perception. To overcome these problems, we propose to directly model the discriminal dispersion variation and use a maximum likelihood model to describe the law of comparative judgment. The model describes the law of comparative judgment directly in the proportion of choice domain. Solutions are also given for actual paired comparison data. Other useful information such as the standard error of predictions can also be obtained easily based on the simplified forms of the model.