L. L. Thurstone

Louis Leon Thurstone (29 May 1887–29 September 1955) was a U.S. pioneer in the fields of psychometrics and psychophysics. He conceived the approach to measurement known as the law of comparative judgment, and is well known for his contributions to factor analysis.

Life

Thurstone originally received a masters in Mechanical Engineering from Cornell University in 1912, before undertaking a PhD in Psychology at the University of Chicago, which he completed in 1917. He later returned to that university, and he taught and conducted research there between 1924 and 1952. He is responsible for the standardized mean and standard deviation of IQ scores used today, as opposed to the Intelligence Test system originally used by Alfred Binet. He is also known for the development of the Thurstone scale.

Thurstone was also an environmentalist, and suggested an early system for generating hydroelectric power from rivers and waterfalls.

Work

Factor analysis and work on intelligence

Thurstone's work in factor analysis led him to formulate a model of intelligence center around "Primary Mental Abilities" (PMAs), which were independent group factors of intelligence that different individuals possessed in varying degrees. He opposed the notion of a singular general intelligence that factored into the scores of all psychometric tests and was expressed as a mental age. This idea was unpopular at the time due to its obvious conflicts with Spearman's "mental energy" model, and is today still largely discredited. Nonetheless, Thurstone's contributions to methods of factor analysis have proved invaluable in establishing and verifying later psychometric factor structures, and has influenced the hierarchical models of intelligence in use in intelligence tests such as WAIS and the modern Stanford-Binet IQ test.

The seven primary mental abilities in Thurstone's model were verbal comprehension, word fluency, number facility, spatial visualization, associative memory, perceptual speed and reasoning.

Thurstone Scale

In psychology, the Thurstone scale was the first formal technique for measuring an attitude. It was developed by Louis Leon Thurstone in 1928, as a means of measuring attitudes towards religion. It is made up of statements about a particular issue, and each statement has a numerical value indicating how favorable or unfavorable it is judged to be. People check each of the statements to which they agree, and a mean score is computed, indicating their attitude.

Thurstone's method of pair comparisons can be considered a prototype of a normal distribution-based method for scaling-dominance matrices. Even though the theory behind this method is quite complex (Thurstone, 1927a), the algorithm itself is straightforward. For the basic Case V, the frequency dominance matrix is translated into proportions and interfaced with the standard scores. The scale is then obtained as a left-adjusted column marginal average of this standard score matrix (Thurstone, 1927b). The underlying rationale for the method and basis for the measurement of the "psychological scale separation between any two stimuli" derives from Thurstone's Law of comparative judgment (Thurstone, 1928).

The principal difficulty with this algorithm is its indeterminacy with respect to one-zero proportions, which return z values as plus or minus infinity, respectively. The inability of the pair comparisons algorithm to handle these cases imposes considerable limits on the applicability of the method.

The most frequent recourse when the 1.00-0.00 frequencies are encountered is their omission. Thus, e.g., Guilford (1954, p. 163) has recommended not using proportions more extreme than .977 or .023, and Edwards (1957, pp. 41-42) has suggested that “if the number of judges is large, say 200 or more, then we might use pij values of .99 and .01, but with less than 200 judges, it is probably better to disregard all comparative judgments for which pij is greater than .98 or less than .02."’ Since the omission of such extreme values leaves empty cells in the Z matrix, the averaging procedure for arriving at the scale values cannot be applied, and an elaborate procedure for the estimation of unknown parameters is usually employed (Edwards, 1957, pp. 42-46). An alternative solution of this problem was suggested by Krus and Kennedy (1977).

With later developments in psychometric theory, it has become possible to employ direct methods of scaling such as application of the Rasch model or unfolding models such as the Hyperbolic Cosine Model (HCM) (Andrich & Luo, 1993). The Rasch model has a close conceptual relationship to Thurstone's law of comparative judgment (Andrich, 1978), the principal difference being that it directly incoroporates a person parameter. Also, the Rasch model takes the form of a logistic function rather than a cumulative normal function.

Contributions to measurement

Despite his contributions to factor analysis, Thurstone (1959, p. 267) cautioned: "When a problem is so involved that no rational formulation is available, then some quantification is still possible by the coefficients of correlation of contingency and the like. But such statistical procedures constitute an acknowledgement of failure to rationalize the problem and to establish functions that underlie the data. We want to measure the separation between the two opinions on the attitude continuum and we want to test the validity of the assumed continuum by means of its internal consistency". Thurstone's approach to measurement was termed the law of comparative judgment. He applied the approach in psychophysics, and later to the measurement of psychological values. The so-called 'Law', which can be regarded as a measurement model, involves subjects making a comparison between each of a number of pairs of stimuli with respect to magnitude of a property, attribute, or attitude. Methods based on the approach to measurement can be used to estimate such scale values.

Thurstone's Law of comparative judgment has important links to modern approaches to social and psychological measurement. In particular, the approach bears a close conceptual relation to the Rasch model (Andrich, 1978), although Thurstone typically employed the normal distribution in applications of the Law of comparative judgment whereas the Rasch model is a simple logistic function. Thurstone anticipated a key epistemological requirement of measurement later articulated by Rasch, which is that relative scale locations must 'transcend' the group measured; i.e. scale locations must be invariant to (or independent of) the particular group of persons instrumental to comparisons between the stimui. Thurstone (1929) also articulated what he referred to as the additivity criterion for scale differences, a criterion which must be satisfied in order to obtain interval-level measurments.

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.

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 handwrighting 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 persons 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.

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 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.

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.

References

ISBN links support NWE through referral fees

• Andrich, D. (1978b) Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 449-460.

• Andrich, D. & Luo, G. (1993) A hyperbolic cosine model for unfolding dichotomous single-stimulus responses. Applied Psychological Measurement, 17, 253-276.

• Babbie, E., 'The Practice of Social Research', 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0534620299

• Edwards, A. L. Techniques of attitude scale construction. New York: Appleton-Century- Crofts, 1957.

• Guilford, J. P. Psychometric methods. New York: McGraw-Hill, 1954.

• Krus, D.J., & Kennedy, P.H. (1977) Normal scaling of dominance matrices: The domain-referenced model. Educational and Psychological Measurement, 37, 189-193 (Request reprint).

• Krus, D.J., Sherman, J.L., & Kennedy, P.H. (1977) Changing values over the last half-century: the story of Thurstone's crime scales. Psychological Reports, 40, 207-211 (Request reprint).

• Thurstone, L. L. (1927a) A Law of comparative judgment. Psychological Review, 34, 273-286.

• Thurstone, L. L. (1927b) The method of paired comparisons for social values. Journal of Abnormal and Social Psychology, 21, 384-400.

• Thurstone, L. L. (1928). Attitudes can be measured. American Journal of Sociology, 33, 529-54.

• Andrich, D. (1978b). Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 449-460.

• Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 278-286.

• Thurstone, L.L. (1929). The Measurement of Psychological Value. In T.V. Smith and W.K. Wright (Eds.), Essays in Philosophy by Seventeen Doctors of Philosophy of the University of Chicago. Chicago: Open Court.

• Thurstone, L.L. (1959). The Measurement of Values. Chicago: The University of Chicago Press.

• Andrich, D. (1978b). Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 449-460.

• Bradley, R.A. and Terry, M.E. (1952). Rank analysis of incomplete block designs, I. the method of paired comparisons. Biometrika, 39, 324-345.

• Luce, R.D. (1959). Individual Choice Behaviours: A Theoretical Analysis. New York: J. Wiley.

• Michell, J. (1997). Quantitative science and the definition of measurement in psychology. British Journal of Psychology, 88, 355-383.

• Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests. (Copenhagen, Danish Institute for Educational Research), expanded edition (1980) with foreword and afterword by B.D. Wright. Chicago: The University of Chicago Press.

• Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 278-286.

• Thurstone, L.L. (1929). The Measurement of Psychological Value. In T.V. Smith and W.K. Wright (Eds.), Essays in Philosophy by Seventeen Doctors of Philosophy of the University of Chicago. Chicago: Open Court.

• Thurstone, L.L. (1959). The Measurement of Values. Chicago: The University of Chicago Press.

External links

• "The Measurement of Psychological Value." by L.L. Thurstone
• L.L. Thurstone psychometric laboratory
• Thurstone's law

• The Vectors of Mind 1934
• Human intelligence: Thurstone
• Biographical information
• History of Thurstone's psychometric lab

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