# Thurstonian model

A Thurstonian model is a latent variable model for describing the mapping of some continuous scale onto discrete, possibly ordered categories of response. In the model, each of these categories of response corresponds to a latent variable whose value is drawn from a normal distribution, independently of the other response variables and with constant variance. Thurstonian models have been used as an alternative to generalized linear models in analysis of sensory discrimination tasks.[1] They have also been used to model long-term memory in ranking tasks of ordered alternatives, such as the order of the amendments to the US Constitution.[2] Their main advantage over other models ranking tasks is that they account for non-independence of alternatives.[3]

## Definition

Consider a set of m options to be ranked by n independent judges. Such a ranking can be represented by the ordering vector rn = (rn1, rn2,...,rnm).

Rankings are assumed to be derived from real-valued latent variables zij, representing the evaluation of option j by judge i. Rankings ri are derived deterministically from zi such that zi(ri1) < zi(ri2) < ... < zi(rim).

The zij are assumed to be derived from an underlying ground truth value μ for each option. In the most general case, they are multivariate-normally distributed:

$z_{ij} = \mu_j + \epsilon_{ij}$

where εj is multivariate-normally distributed around 0 with covariance matrix Σ. In a simpler case, there is a single standard deviation parameter σi for each judge:

$z_{ij}\ \sim\ \mathcal{N}(\beta_j,\, \sigma_i^2).$

## Inference

The Gibbs-sampler based approach to estimating model parameters is due to Yao and Bockenholt (1999).[3]

• Step 1: Given β, Σ, and r_i, sample z_i.

The zij must be sampled from a truncated multivariate normal distribution to preserve their rank ordering. Hajivassiliou's Truncated Multivariate Normal Gibbs sampler can be used to sample efficiently.[4][5]

• Step 2: Given Σ, z_i, sample β.

β is sampled from a normal distribution:

$\beta\ \sim\ \mathcal{N}(\beta^*, \Sigma^*).$

where β* and Σ* are the current estimates for the means and covariance matrices.

• Step 3: Given β, z_i, sample Σ.

Σ−1 is sampled from a Wishart posterior, combining a Wishart prior with the data likelihood from the samples εi =zi - β.

## History

Thurstonian models were introduced by Louis Leon Thurstone to describe the law of comparative judgment.[6] Prior to 1999, Thurstonian models were rarely used for modeling tasks involving more than 4 options because of the high-dimensional integration required to estimate parameters of the model. In 1999, Yao and Bockenholt introduced their Gibbs-sampler based approach to estimating model parameters.[3]

## Applications to sensory discrimination

Thurstonian models have been applied to a range of sensory discrimination tasks, including auditory, taste, and olfactory discrimination, to estimate sensory distance between stimuli that range along some sensory continuum.[7][8][9]

The Thurstonian approach motivated Frijter (1979)'s explanation of Gridgeman's Paradox, also known as the paradox of discriminatory nondiscriminators:[1][8][10][11] People perform better in a three-alternative forced choice task when told in advance which dimension of the stimulus to attend to. (For example, people are better at identifying which of one three drinks is different from the other two when told in advance that the difference will be in degree of sweetness.) This result is accounted for by differing cognitive strategies: when the relevant dimension is known in advance, people can estimate values along that particular dimension. When the relevant dimension is not known in advance, they must rely on a more general, multi-dimensional measure of sensory distance.

## References

1. ^ a b Lundahl, David (1997). "Thurstonian Models — an Answer to Gridgeman's Paradox?". CAMO Software Statistical Methods.
2. ^ Lee, Michael; Steyvers, Mark; de Young, Mindy; Miller, Brent (2011). "A Model-Based Approach to Measuring Expertise in Ranking Tasks". CogSci 2011 Proceedings (PDF). ISBN 978-0-9768318-7-7.
3. ^ a b c Yao, G.; Bockenholt, U. (1999). "Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler". British Journal of Mathematical and Statistical Psychology 52: 19–92. doi:10.1348/000711099158973.
4. ^ Hajivassiliou, V.A. (1993). "Simulation estimation methods for limited dependent variable models". In Maddala,, G.S.; Rao, C.R.; Vinod, H.D. Econometrics. Handbook of statistics 11. Amsterdam: Elsevier. ISBN 0444895779.
5. ^ V.A., Hajivassiliou; D., McFadden; P., Ruud (1996). "Simulation of multivariate normal rectangle probabilities and their derivatives. Theoretical and computational results". Journal of Econometrics 72: 85–134. doi:10.1016/0304-4076(94)01716-6.
6. ^ Thurstone, Louis Leon (1927). "A Law of Comparative Judgment". Psychological Review 34 (4): 273–286. doi:10.1037/h0070288. Reprinted: Thurstone, L. L. (1994). "A law of comparative judgment". Psychological Review 101 (2): 266–270. doi:10.1037/0033-295X.101.2.266.
7. ^ Durlach, N.I.; Braida, L.D. (1969). "Intensity Perception. I. Preliminary Theory of Intensity Resolution". Journal of the Acoustical Society of America 46 (2): 372–383. doi:10.1121/1.1911699.
8. ^ a b Dessirier, Jean-Marc; O’Mahony, Michael (9 October 1998). "Comparison of d′ values for the 2-AFC (paired comparison) and 3-AFC discrimination methods: Thurstonian models, sequential sensitivity analysis and power". Food Quality and Preference 10 (1): 51–58. doi:10.1016/S0950-3293(98)00037-8.
9. ^ Frijter, J.E.R. (1980). "Three-stimulus procedures in olfactory psychophysics: an experimental comparison of Thurstone-Ura and three-alternative forced choice models of signal detection theory". Perception & Psychophysics 28 (5): 390–7. doi:10.3758/BF03204882.
10. ^ Gridgement, N.T. (1970). "A Reexamination of the Two-Stage Triangle Test for the Perception of Sensory Differences". Journal of Food Science 35 (1).
11. ^ Frijters, J.E.R. (1979). "The paradox of discriminatory nondiscriminators resolved". Chemical Senses & Flavor 4 (4): 355–8. doi:10.1093/chemse/4.4.355.