Paired comparison is one of the oldest and most reliable methods in psychological scaling. Each trial presents exactly two stimuli, and the observer judges which is greater on the attribute of interest. Over many trials and observers, the data form a proportion matrix P where each entry p(i,j) gives the proportion of times stimulus i was preferred to stimulus j.
The Bradley-Terry Model
Equivalently in log-odds form:
log[P(i,j)/P(j,i)] = sᵢ − sⱼ
The Bradley-Terry model (1952) assigns each stimulus a positive scale value v, with choice probabilities proportional to these values. This is mathematically equivalent to Luce's Choice Axiom applied to paired comparisons. The log-linear form shows that the log-odds of choosing i over j equals the difference in scale values on a logistic scale.
Relationship to Thurstone
Thurstone's Case V model produces similar results but assumes a different generative process: each stimulus evokes a normally distributed percept, and the observer chooses whichever percept is larger on a given trial. Converting the observed proportions to z-scores yields scale values directly. For most practical purposes, Thurstonian and Bradley-Terry models produce very similar scale values.
Paired comparisons have been applied to psychophysical scaling, preference measurement, sports rankings, and any domain where direct numerical judgment is difficult but pairwise choices are natural. Modern extensions handle incomplete designs (not all pairs observed), ties, and individual differences in scale values.