Law of comparative judgment

Conceived by L. L. Thurstone, the law of comparative judgment (LCJ) is a general mathematical representation of a discriminal process, which is any process in which a comparison is made between pairs of a collection of entities with respect to magnitudes of an attribute, trait, attitude, and so on. Examples of such processes are the comparison of perceived intensity of physical stimuli, and comparisons of the level of extremity of an attitude expressed within statements.

Background
Thurstone published a paper on the law of comparative judgment (LCJ) in 1927. In this paper he introduced the underlying concept of a psychological continuum for a particular 'project in measurement' involving the comparison between a series of stimuli, such as weights and handwriting specimens, in pairs. He soon extended the domain of application of the LCJ to phenomena which have no obvious physical counterpart, such as attitudes and values (Thurstone, 1929).

The essential idea behind the LCJ is that it can be used to scale a collection of stimuli based on simple comparisons between stimuli two at a time: that is, based on a series of pairwise comparisons. For example, say that the perceived weights of a series of five objects of varying masses are to be scaled. By having people compare the weights of the objects in a pairwise fashion, data can be obtained and the LCJ applied in order to estimate scale values of the perceived weights on a continuum. Although Thurstone referred to it as a law, in terms of modern psychometric theory the 'law' of comparative judgment is more aptly described as a measurement model. That is, the LCJ represents a general theoretical model which, applied in a particular empirical context, constitutes a scientific hypothesis regarding the outcomes of comparisons between some collection of objects.

Relationships to pre-existing psychophysical theory
Thurstone showed that in terms of his conceptual framework, Weber's law and the so-called Weber-Fechner law, which are generally regarded as one and the same, are independent, in the sense that one may be applicable but not the other to a given collection of experimental data. In particular, Thurstone showed that if Fechner's law applies and the discriminal dispersions associated with stimuli are constant (as in Case 5 of the LCJ outlined below), then Weber's law will also be verified. He considered that the Weber-Fechner law and the LCJ both involve a linear measurement on a psychological continuum whereas Weber's law does not.

Thurstone stated Weber's law as follows: "The stimulus increase which is correctly discriminated in any specified proportion of attempts (except 0 and 100 per cent) is a constant fraction of the stimulus magnitude" (Thurstone, 1959, p. 61). He considered that Weber's law said nothing directly about sensation intensities at all. In terms of Thurstone's conceptual framework, the association posited between perceived stimulus intensity and the physical magnitude of the stimulus in the Weber-Fechner law will only hold when Weber's law holds and the just noticeable difference (JND) is treated as a unit of measurement. Importantly, this is not simply given a priori (Michell, 1997, p. 355), as is implied by purely mathematical derivations of the one law from the other. It is, rather, an empirical question whether measurements have been obtained; one which requires justification through the process of stating and testing a well-defined hypothesis in order to ascertain whether specific theoretical criteria for measurement have been satisfied. Some of the relevant criteria were articulated by Thurstone, in a preliminary fashion, including what he termed the additivity criterion. Accordingly, from the point of view of Thurstone's approach, treating the JND as a unit is justifiable provided only that the discriminal dispersions are uniform for all stimuli considered in a given experimental context. Similar issues are associated with Stevens' power law.

In addition, Thurstone employed the approach to clarify other similarities and differences between Weber's law, the Weber-Fechner law, and the LCJ. An important clarification is that the LCJ does not necessarily involve a physical stimulus, whereas the other 'laws' do. Another key difference is that Weber's law and the LCJ involve proportions of comparisons in which one stimulus is judged greater than another whereas the so-called Weber-Fechner law does not.

The general form of the law of comparative judgment
The most general form of the LCJ is



S_i-S_j=x_{ij} \sqrt {\sigma_i^2 + \sigma_j^2- 2r_{ij}\sigma_i\sigma_j} , $$

in which:


 * $$S_i$$ is the psychological scale value of stimuli i
 * $$x_{ij}$$ is the sigma corresponding with the proportion of occasions on which the magnitude of stimulus i is judged to exceed the magnitude of stimulus j
 * $$\sigma_i$$ is the discriminal dispersion of a stimulus $$R_i$$
 * $$r_{ij}$$ is the correlation between the discriminal dispersions of stimuli i and j

The discriminal dispersion of a stimulus i is the dispersion of fluctuations of the discriminal process for a uniform repeated stimulus, denoted $$R_i$$, where $$S_i$$ represents the mode of such values. Thurstone (1959, p. 20) used the term discriminal process to refer to the "psychological values of psychophysics"; that is, the values on a psychological continuum associated with a given stimulus.

Case 5 of the law of comparative judgment
Thurstone specified five particular cases of the 'law', or measurement model. An important case of the model is Case 5, in which the discriminal dispersions are specified to be uniform and uncorrelated. This form of the model can be represented as follows:



x_{ij}= \frac {S_i-S_j}{ \sigma} \, $$

where


 * $${\sigma}= \sqrt {\sigma_i^2 + \sigma_j^2}.\,$$

In this case of the model, the difference $${S_i-S_j}$$ can be inferred directly from the proportion of instances in which j is judged greater than i if it is hypothesised that $$x_{ij}$$ is distributed according to some density function, such as the normal distribution or logistic function. In order to do so, it is necessary to let $$\sigma = 1$$, which is in effect an arbitrary choice of the unit of measurement. Letting $$P_{ij}$$ be the proportion of occasions on which i is judged greater than j, if, for example, $$P_{ij}=0.84$$ and it is hypothesised that $$x_{ij}$$ is normally distributed, then it would be inferred that $$S_i-S_j \cong 1$$.

When a simple logistic function is employed instead of the normal density function, then the model has the structure of the Bradley-Terry-Luce model (BTL model) (Bradley & Terry, 1952; Luce, 1959). In turn, the Rasch model for dichotomous data (Rasch, 1960/1980) is identical to the BTL model after the person parameter of the Rasch model has been eliminated, as is achieved through statistical conditioning during the process of Conditional Maximum Likelihood estimation. With this in mind, the specification of uniform discriminal dispersions is equivalent to the requirement of parallel Item Characteristic Curves (ICCs) in the Rasch model. Accordingly, as shown by Andrich (1978), the Rasch model should, in principle, yield essentially the same results as those obtained from a Thurstone scale. Like the Rasch model, when applied in a given empirical context, Case 5 of the LCJ constitutes a mathematized hypothesis which embodies theoretical criteria for measurement.