A **Thurstonian model** is a stochastic transitivity model with latent variables 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. Developments over the last two decades, however, have led to Thurstonian models that allow unequal variance and non zero covariance terms. 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] } Ennis ^{ [4] } provides a comprehensive account of the derivation of Thurstonian models for a wide variety of behavioral tasks including preferential choice, ratings, triads, tetrads, dual pair, same-different and degree of difference, ranks, first-last choice, and applicability scoring. In Chapter 7 of this book, a closed form expression, derived in 1988, is given for a Euclidean-Gaussian similarity model that provides a solution to the well-known problem that many Thurstonian models are computationally complex often involving multiple integration. In Chapter 10, a simple form for ranking tasks is presented that only involves the product of univariate normal distribution functions and includes rank-induced dependency parameters. A theorem is proven that shows that the particular form of the dependency parameters provides the only way that this simplification is possible. Chapter 6 links discrimination, identification and preferential choice through a common multivariate model in the form of weighted sums of central F distribution functions and allows a general variance-covariance matrix for the items.

Consider a set of *m* options to be ranked by *n* independent judges. Such a ranking can be represented by the ordering vector **r _{n}** = (r

Rankings are assumed to be derived from real-valued latent variables *z _{ij}*, representing the evaluation of option

The **z**_{i} are assumed to be derived from an underlying ground truth value *μ* for each option. In the most general case, they are multivariate-normal:

One common simplification is to assume an isotropic Gaussian distribution, with a single standard deviation parameter for each judge:

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

- Step 1: Given β, Σ, and
**r**, sample_{i}**z**._{i}

The *z _{ij}* 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.

- Step 2: Given Σ,
**z**, sample β._{i}

β is sampled from a normal distribution:

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

- Step 3: Given β,
**z**, sample Σ._{i}

Σ^{−1} is sampled from a Wishart posterior, combining a Wishart prior with the data likelihood from the samples **ε**_{i} =**z**_{i} - β.

Thurstonian models were introduced by Louis Leon Thurstone to describe the law of comparative judgment.^{ [7] } 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] } This comment, however, only applies to ranking and Thurstonian models with a much broader range of applications were developed prior to 1999. For instance, a multivariate Thurstonian model for preferential choice with a general variance-covariance structure is discussed in chapter 6 of Ennis (2016) that was based on papers published in 1993 and 1994. Even earlier, a closed form for a Thurstonian multivariate model of similarity with arbitrary covariance matrices was published in 1988 as discussed in Chapter 7 of Ennis (2016). This model has numerous applications and is not limited to any particular number of items or individuals.

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.^{ [8] }^{ [9] }^{ [10] }

The Thurstonian approach motivated Frijter (1979)'s explanation of Gridgeman's Paradox, also known as the paradox of discriminatory nondiscriminators:^{ [1] }^{ [9] }^{ [11] }^{ [12] } 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.

The above paragraph contains a common misunderstanding of the Thurstonian resolution of Gridgeman's paradox. Although it is true that different decision rules (cognitive strategies) are used in making a choice among three alternatives, the mere fact of knowing an attribute in advance does not explain the paradox, nor are subjects required to rely on a more general, multidimensional measure of sensory difference. In the triangular method, for instance, the subject is instructed to choose the most different of three items, two of which are putatively identical. The items may differ on a unidimensional scale and the subject may be made aware of the nature of the scale in advance. Gridgeman's paradox will still be observed. This occurs because of the sampling process combined with a distance-based decision rule as opposed to a magnitude-based decision rule assumed to model the results of the 3-alternative forced choice task.

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

In probability theory and statistics, a **Gaussian process** is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

In statistics, the **Wishart distribution** is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.

In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.

In statistics, **multivariate analysis of variance** (**MANOVA**) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests involving individual dependent variables separately.

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In statistics, **ordinary least squares** (**OLS**) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function.

**Weighted least squares** (**WLS**), also known as **weighted linear regression**, is a generalization of ordinary least squares and linear regression in which the errors covariance matrix is allowed to be different from an identity matrix. WLS is also a specialization of generalized least squares in which the above matrix is diagonal.

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In probability theory and statistics, the **normal-inverse-gamma distribution** is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

In probability and statistics, an **elliptical distribution** is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots.

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- 1 2 Lundahl, David (1997). "Thurstonian Models — an Answer to Gridgeman's Paradox?". CAMO Software Statistical Methods.
- ↑ Lee, Michael; Steyvers, Mark; de Young, Mindy; Miller, Brent (2011). "A Model-Based Approach to Measuring Expertise in Ranking Tasks" (PDF).
*CogSci 2011 Proceedings*(PDF). ISBN 978-0-9768318-7-7. - 1 2 3 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. - ↑ Ennis, Daniel (2016).
*Thurstonian Models — Categorical Decision Making in the Presence of Noise*. Richmond: The Institute for Perception. ISBN 978-0-9906446-0-6. - ↑ Hajivassiliou, V.A. (1993). "Simulation estimation methods for limited dependent variable models". In Maddala, G.S.; Rao, C.R.; Vinod, H.D. (eds.).
*Econometrics*. Handbook of statistics.**11**. Amsterdam: Elsevier. ISBN 0444895779. - ↑ 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**(1–2): 85–134. doi:10.1016/0304-4076(94)01716-6. - ↑ 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. - ↑ 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. Bibcode:1969ASAJ...46..372D. doi:10.1121/1.1911699. PMID 5804107. - 1 2 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. - ↑ 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 . PMID 7208248. - ↑ Gridgement, N.T. (1970). "A Reexamination of the Two-Stage Triangle Test for the Perception of Sensory Differences".
*Journal of Food Science*.**35**(1): 87–91. doi:10.1111/j.1365-2621.1970.tb12376.x. - ↑ 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.

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