Range-Frequency Theory

Range-frequency theory is a model of the psychological evaluation of stimuli in context developed by psychologist Alan Parducci. It posits that when a stimulus is rated alongside other stimuli, its rating will depend in part on where it ranks among the stimulus set. For example, a patient comparing doctors may provide a different rating to the same doctor when presented in a group of doctors perceived to be superior by the patient than when presented in a group of doctors perceived to be inferior by the patient.

Formally, the rating of a stimulus is described by a weighted average of the utility of the stimulus (relative to the range of utilities present in the context) and the rank of the stimulus (relative to the range of ranks present in the context). The theory is formulated mathematically as

In this formula, Ri is the rating response given to the ith stimulus in a set of N stimuli. The first component of the formula gives some relative weight (w) to the difference between the utility of the stimulus, xi, and the utility of the worst stimulus in the set, xmin, relative to the range of utilities between the best and worst stimuli (xmax – xmin). The second component of the formula assigns the remaining relative weight (1 – w) to a similar comparison between the rank of the stimulus, rank(xi), and the rank of the worst stimulus, relative to the range of ranks between the best and worst stimuli. The highest ranked stimulus receives a rank of N and the lowest ranked receives a rank of 1. The function J is a linear transformation. The weighting parameter, w, establishes the relative weight of the utility component and rank component of the model and is often fixed at .5 in modeling.

Range-frequency theory predicts that the same stimulus will receive different ratings depending on its relative position (rank) among the set of stimuli to be rated. For example, the rating of a health state B in a set of rank-ordered (high to low) health states A, B, C, D, E is predicted to be higher than the rating of the same health state in a set of rank-ordered (high to low) health states A, F, B, G, E, because the relative position of B is higher in the first set, where it is the second-best state, than in the second set, where it is the third-best state.

Figure 1 illustrates this key prediction of range-frequency theory. The curves plot the range-frequency theory predictions of ratings of stimuli against their underlying psychological utilities. In the solid upper curve, four stimuli are rated, with underlying utilities of .5, .6, .7, and .9; and in the lower dashed curve, four stimuli are rated, with underlying utilities of .5, .7, .8, and .9. Range-frequency theory predicts that the same stimulus with utility .7 will receive a higher rating in the first context (when it is second-ranked in the stimulus set) than in the second context (when it is third-ranked in the stimulus set). These predictions have been supported in numerous experimental studies in which stimulus context is varied.

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