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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 [[discrimination testing|sensory discrimination tasks]].<ref name=L>{{cite web |last=Lundahl |first=David |title=Thurstonian Models — an Answer to Gridgeman's Paradox? |date=1997 |publisher=CAMO Software Statistical Methods |url=http://www.camo.com/resources/infodoc/thurstonian_models.html}}</ref> 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.<ref>{{cite book |last1=Lee |first1=Michael |first2=Mark |last2=Steyvers |first3=Mindy |last3=de Young |first4=Brent |last4=Miller |chapter=A Model-Based Approach to Measuring Expertise in Ranking Tasks |chapterurl=http://mindmodeling.org/cogsci2011/papers/0305/paper0305.pdf |format=PDF |title=CogSci 2011 Proceedings |year=2011 |isbn=978-0-9768318-7-7 |url=http://mindmodeling.org/cogsci2011/}}</ref> Their main advantage over other models ranking tasks is that they account for non-independence of alternatives.<ref name=Yao>{{cite journal |last1=Yao |first1=G. |last2=Bockenholt |first2=U. |title=Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler |journal=British Journal of Mathematical and Statistical Psychology |volume=52 |pages=19–92 |year=1999 |doi=10.1348/000711099158973 }}</ref>

==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 '''z'''i such that ''z''i(ri1)'' < ''zi(ri2)'' < ... < ''zi(rim)''.

The '''z'''ij 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:
:<math> z_{ij} = \mu_j + \epsilon_{ij} </math>
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:
: <math>
z_{ij}\ \sim\ \mathcal{N}(\beta_j,\, \sigma_i^2).
</math>

==Inference==
The [[Gibbs sampling|Gibbs-sampler]] based approach to estimating model parameters is due to Yao and Bockenholt (1999).<ref name=Yao/>

* 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.<ref>{{cite book |last=Hajivassiliou |first=V.A. |chapter=Simulation estimation methods for limited dependent variable models |editor1-first=G.S. |editor1-last=Maddala, |editor2-first=C.R. |editor2-last=Rao |editor3-first=H.D. |editor3-last=Vinod |title=Econometrics |series=Handbook of statistics |publisher=Elsevier |location=Amsterdam |year=1993 |volume=11 |isbn=0444895779}}</ref><ref>{{cite journal |first1=Hajivassiliou |last1=V.A. |first2=McFadden |last2=D. |first3=Ruud |last3=P. |title=Simulation of multivariate normal rectangle probabilities and their derivatives. Theoretical and computational results |journal=Journal of Econometrics |volume=72 |pages=85–134 |year=1996 |doi=10.1016/0304-4076(94)01716-6 }}</ref>

* Step 2: Given Σ, '''z'''_i, sample β.

β is sampled from a [[normal distribution]]:
: <math>
\beta\ \sim\ \mathcal{N}(\beta^*, \Sigma^*).
</math>

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 distribution|Wishart]] posterior, combining a [[Wishart distribution|Wishart]] prior with the data likelihood from the samples '''ε'''i ='''z'''i - β.

==History==
Thurstonian models were introduced by [[Louis Leon Thurstone]] to describe the [[law of comparative judgment]].<ref>{{cite journal |last1=Thurstone |first1=Louis Leon |title=A Law of Comparative Judgment |journal=Psychological Review |volume=34 |pages=273–286 |year=1927 |doi=10.1037/h0070288 |issue=4 }} Reprinted: {{cite journal |journal=Psychological Review |volume=101 |issue=2 |pages=266–270 |year=1994 |doi=10.1037/0033-295X.101.2.266 |title=A law of comparative judgment |last1=Thurstone |first1=L. L. }}</ref> 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 sampling|Gibbs-sampler]] based approach to estimating model parameters.<ref name=Yao/>

==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.<ref>{{cite journal |last1=Durlach |first1=N.I. |last2=Braida |first2=L.D. |journal=[[Journal of the Acoustical Society of America]] |volume=46 |issue=2 |pages=372–383 |year=1969 |doi=10.1121/1.1911699 |title=Intensity Perception. I. Preliminary Theory of Intensity Resolution }}</ref><ref name=D>{{cite journal |first1=Jean-Marc |last1=Dessirier |first2=Michael |last2=O’Mahony |title=Comparison of d′ values for the 2-AFC (paired comparison) and 3-AFC discrimination methods: Thurstonian models, sequential sensitivity analysis and power |journal=Food Quality and Preference |volume=10 |issue=1 |pages=51–58 |date=9 October 1998 |doi=10.1016/S0950-3293(98)00037-8 }}</ref><ref>{{cite journal |first1=J.E.R. |last1=Frijter |title=Three-stimulus procedures in olfactory psychophysics: an experimental comparison of Thurstone-Ura and three-alternative forced choice models of signal detection theory |journal=Perception & Psychophysics |volume=28 |pages=390–7 |year=1980 |doi=10.3758/BF03204882 |issue=5 }}</ref>

The Thurstonian approach motivated Frijter (1979)'s explanation of Gridgeman's Paradox, also known as the paradox of discriminatory nondiscriminators:<ref>{{cite journal |last1=Gridgement |first1=N.T. |title=A Reexamination of the Two-Stage Triangle Test for the Perception of Sensory Differences |journal=Journal of Food Science |volume=35 |issue=1 |year=1970 }}</ref><ref>{{cite journal |first1=J.E.R. |last1=Frijters |title=The paradox of discriminatory nondiscriminators resolved |journal=Chemical Senses & Flavor |volume=4 |pages=355–8 |year=1979 |doi=10.1093/chemse/4.4.355 |issue=4 }}</ref><ref name=D/><ref name=L/> 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.

==See also==
*[[Thurstone scale]]

== References ==
{{reflist}}

[[Category:Latent variable models]]
[[Category:Psychometrics]]

Reinforced solid - Revision history

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