Thurstone Model

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The Thurstone model for paired comparisons is a way to analyze people’s choice of action. For example, automobile parking choices, career choices, or product preferences.

The Thurstone model, which assumes that the preferences for two stimuli are normally distributed in the population [2], is characterized by the following assumptions:

• When a subject is presented with a pair of stimuli, it will generate a continuous preference — referred to as a utility function or, in Thurstone’s terminology, a “discriminal process” — for each stimulus. Continuous preference means that they can take on a value of any real number.
• The subject will prefer the stimulus with the larger value at the time of the comparison. The subject will never be indifferent between two stimuli.
• These preferences, which are not directly observed, follow a normal distribution in the population.

The Thurstone model and the Mallows distribution are two different models that can be used to measure distance and order, respectively. Stimuli measured using the Thurstone model can be modeled with the Mallows distribution. If we assume that the latent variables of the stimuli are normally distributed, the Mallows distribution provides a way to calculate the probability of a particular ranking of the stimuli.

Thurstone model and SEM

Thurstone’s model is simply a multivariate normal density with an structured mean vector and covariance matrix that has been dichotomized. Thus, it is somewhat natural to consider its estimation using existing procedures for structural equation modeling (SEM) for dichotomous variables [3]. To put this another way, each ability in the model is a hidden variable that can be assessed through a collection of yes-or-no questions. The model assumes that the hidden variables follow a normal distribution, while the questions are binary versions of these variables.

SEM is a statistical technique that can be used to estimate the parameters of a model. It can estimate the parameters of Thurstone’s model by treating the dichotomous items as observed variables. The reason why it is somewhat natural to consider estimating Thurstone’s model using SEM is that SEM can be used to estimate the parameters of a wide variety of models. SEM is also a fairly easy technique to use, and there are many software packages available that can be used to implement it.

References

1. Parked cars by Bob Harvey, CC BY-SA 2.0 <https://creativecommons.org/licenses/by-sa/2.0>, via Wikimedia Commons
2. Thurstone, L.L. (1927). A law of comparative judgment. Psychological Review, 79, 281-299
3. Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551-560.