Thurstonian model
A Thurstonian model is a
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 zi 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:
Inference
The Gibbs-sampler based approach to estimating model parameters is due to Yao and Bockenholt (1999).[3]
- Step 1: Given β, Σ, and ri, sample zi.
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.[5][6]
- Step 2: Given Σ, zi, sample β.
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, zi, 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.[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.
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.[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.
See also
References
- ^ a b Lundahl, David (1997). "Thurstonian Models — an Answer to Gridgeman's Paradox?". CAMO Software Statistical Methods.
- ISBN 978-0-9768318-7-7.
- ^ .
- ISBN 978-0-9906446-0-6.
- ISBN 0444895779.
- .
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- PMID 5804107.
- ^ .
- PMID 7208248.
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