Thurstone, L. L.

Revision as of 03:57, 19 November 2007

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

Louis Leon Thurston was born in Chicago Illinois on May 29,1887 to two Swedish immigrants Conrad and Sophia Thunstrom. The family eventually changed the last name to Thurston to avoid confusion of pronunciation and spelling. The first 14 years of Louis’s life was transient as his father changed careers several times. The career changes took the family first from Chicago to a small town in Illinois (Berwyn), then to Mississippi. From Mississippi, the family moved to Sweden where they stayed for almost six years. When he was fourteen, the family settled in Jamestown, New York.

He later reported that moving around had a positive effect on him as he received a multicultural education. By going to different schools in different countries, he could compare the goals of education that each country offered. Through this comparison, he noticed that the heroes of the stories taught in school were always of the same nationality as the teacher. From this experience, he reflected on whether the social sciences could be and should be studied from a more objective point of view.

Young Thurston was very adept at academics. He published for the first time at the age of sixteen in the journal Scientific America. This journal article explained how the hydroelectric plants at the Niagara Falls could be constructed so that they did not interfere with the natural beauty of the falls. His second article was published at age eighteen, again in the Scientific America. It was based on work he had done as a high school student.

Thurston studied engineering at Cornell University beginning in 1908. Studying machine design lead Thurston to a fascination with the human factor implied in all design. This was the beginning of his interest in psychology. Another experience encouraging his interest in psychology was working with Thomas A. Edison. Edison had heard about Thurston inventing an innovative motion picture projector and offered him an internship.

Two of Thurston’s biographers agreed that working with Edison was the beginning of Thurston’s interest in psychology. According to A.R. Jensen, it was at Edison laboratory that Thurston became interested in audio perception. According to Dorothy Adkins Wood, Thurston was very much interested in Edison’s unique problem solving ability. Did Edison’s problem solving stem from his genius or did his genius stem from his problem solving? Thurston’s interest in Psychology lead him to graduate school where he earned his Master’s degree in Psychology at age 24.

For 18 years, Thurston worked at the Carnegie Institute of Technology in the Division of Applied Psychology. He was there at the onset of World War I. Although he tried to enlist in the Army, he was not accepted due to a physical problem. However, he did help the Army by adapting the intelligence tests to help in recruiting and training.

In 1923, Thurston went to Washington DC to work on government contracts. The year and a half was not very fruitful, but while he was there he met and married Thelma Gwinn. After accepting the Associate Professor of Psychology position at the University of Chicago, he moved with his wife to Chicago where they lived for the next 28 years. The last three years of his life were spent at the University of North Carolina Intelligence Laboratory named after him. He died in September 1955 from heart failure.

Work

Thurstone's main contributions to psychology and psychometrics are his method of factor analysis, his theory of intellingence, and his comparative judgment scale.

Factor Analysis

Although Charles Spearman is credited with inventing factor analysis, Thurstone is the one who first coined the term. In addition, Thurstone is recognized as the inventor of exploratory factor analysis, a more practical variation than the confirmatory factor analysis of Spearman. The goal of Thurstone ’s model is to determine the number of meaningful common factors in a correlation matrix. This produces simple structures that accounts for many of the correlations observed among the factors. Exploratory factor analysis determines the number and the nature of latent constructs within a set of observed variables. Analyzing the correlated factors can rank the factors in order of importance to the correlation. Thus, exploratory factor analysis is important tool in determining hierarchy of factors such as the contributors to intelligence.

Theory of Intelligence

Thurstone’s theory of intelligence centered on the existence of Primary Mental Abilities (PMA). His approach was in direct contrast with Spearman’s theory of general intelligence. Thurstone felt that differences in the results of intellectual tasks could be attributed to one or more of seven independent abilities. These seven abilities were named Space, Verbal Comprehension, Word Fluency, Number Facility, Induction, Perceptual Speed, Deduction, Rote Memory and Arithmetic Reasoning.

The Space PMA represents the ability to recognize that two shapes are the same when one has been rotated. Perceptual Speed is the ability to recognize similarities and differences between pairs of stimuli. Verbal Comprehension involves recognizing synonyms and antonyms. Induction requires establishing a rule or pattern within a given set. Deduction involves drawing a logical inference from a set of facts or premises.

Thurstone ’s theory was well supported by his early research when the subjects were University of Chicago undergraduates. It did not hold up when he tested school aged children. Apparently, the more intellectually elite subjects at the University of Chicago did not differ very much on their general intelligence. Their observable differences were noted among the PMAs. The grade school children were more diverse in their general intelligence. Therefore, the differences among their PMAs were not as notable as the differences among their general intelligence.

Comparative Judgment 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.

This methodological contribution of Thurstone has been noted as one of the first attempts at developing a comparative judgment scaling technique. This method of measuring attitudes on an interval scale allowed statements related to an attitude to be ranked in reference to each other. The extreme opposites of the attitude and the opinions representing the equally-distanced steps in between the opposites could be established.

This rank scale can be used to rank all possible feelings related to an issue and to categorize people expressing an opinion based on the rank of that opinion. It is used today mainly in basic research. Most researchers acknowledge that, while it is very accurate, it is too complex for applied settings.

Legacy

Thurstone’s theory of intelligence was a major influence on later theories of multiple intelligences, such as those of Guilford, Gardner, and Sternberg. Guilford developed a three-dimensional model of intelligence composed of contents, operations, and processes. This model relied on the interactions of various factors similar to the interactions of the correlation of factors in Thurstone’s theory. Although Gardner’s multiple intelligences did not perfectly intersect with Thurstone’s PMAs, both theories support a practical definition of intelligence. Sternberg emphasized speed of perception and the practical application of inductive reasoning as an important part of his triarchic theory of intelligence.

Thustone's attitudinal scale was very influential in encouraging others, such as Guttman and Coombs, to develop practicdal scaling proceduers in the social sciences.

Major publications

Thurstone, L.L. (1927a). "A law of Comparative Judgement" Psychological Review, 34, 278-286.

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.

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.

References
ISBN links support NWE through referral fees

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

Dawes, R. M., (1994). Psychological measurement. Psychological Review, 101, 278-281.

Edwards, A. L. Techniques of attitude scale construction. New York: Appleton-Century- Crofts, 1957.
Glover, J. A., & Ronning, R. R. (1987). Historical foundations of educational psychology. New York: Plenum Press.
Guilford, J. P. Psychometric methods. New York: McGraw-Hill, 1954.
Kail, R., & Pelligrino, J. W. (1985). Human intelligence: Perspectives and prospects. New York: W. H. Freeman and Company.
Kerlinger, F. N. (1986). Foundations of behavioral research. Chicago: Holt, Rhinehart, and Winston, Inc.
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).
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.
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

Credits

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@@ Line 22: / Line 22: @@
 ==Work==
+Thurstone's main contributions to psychology and psychometrics are his method of factor analysis, his theory of intellingence, and his comparative judgment scale.
-===Factor analysis===
+===Factor Analysis===
 Although [[Charles Spearman]] is credited with inventing [[factor analysis]], Thurstone is the one who first coined the term.  In addition, Thurstone is recognized as the inventor of exploratory [[factor analysis]], a more practical variation than the confirmatory [[factor analysis]] of Spearman. The goal of Thurstone ’s model is to determine the number of meaningful common factors in a correlation matrix.  This produces simple structures that accounts for many of the correlations observed among the factors.  Exploratory factor analysis determines the number and the nature of latent constructs within a set of observed variables.  Analyzing the correlated factors can rank the factors in order of importance to the correlation.  Thus, exploratory factor analysis is important tool in determining hierarchy of factors such as the contributors to [[intelligence]].
-===Theory of intelligence===
+===Theory of Intelligence===
 Thurstone’s theory of [[intelligence]] centered on the existence of Primary Mental Abilities (PMA). His approach was in direct contrast with Spearman’s theory of general intelligence. Thurstone felt that differences in the results of intellectual tasks could be attributed to one or more of seven independent abilities.  These seven abilities were named Space, Verbal Comprehension, Word Fluency, Number Facility, Induction, Perceptual Speed, Deduction, Rote Memory and Arithmetic Reasoning.
@@ Line 35: / Line 36: @@
-===Thurstone Scale===
+===Comparative Judgment Scale===
-In [[psychology]], the '''Thurstone scale''' was the first formal technique for measuring an [[attitude (psychology)|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.
+In [[psychology]], the 'Thurstone scale' was the first formal technique for measuring an [[attitude (psychology)|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).
+This methodological contribution of Thurstone has been noted as one of the first attempts at developing a comparative judgment scaling technique.  This method of measuring attitudes on an interval scale allowed statements related to an attitude to be ranked in reference to each other.  The extreme opposites of the attitude and the opinions representing the equally-distanced steps in between the opposites could be established.
-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.
+This rank scale can be used to rank all possible feelings related to an issue and to categorize people expressing an opinion based on the rank of that opinion.  It is used today mainly in basic research.  Most researchers acknowledge that, while it is very accurate, it is too complex for applied settings.
-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 <i> “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."’ </i> 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).
+==Legacy==
-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.
-==Legacy==
 Thurstone’s theory of intelligence was a major influence on later theories of multiple intelligences, such as those of [[Guilford]], [[Gardner]], and [[Sternberg]].  Guilford developed a three-dimensional model of intelligence composed of contents, operations, and processes.  This model relied on the interactions of various factors similar to the interactions of the correlation of factors in Thurstone’s theory.  Although Gardner’s multiple intelligences did not perfectly intersect with Thurstone’s PMAs, both theories support a practical definition of intelligence.  Sternberg emphasized speed of perception and the practical application of inductive reasoning as an important part of his triarchic theory of intelligence.
-===Contributions to measurement===
+Thustone's attitudinal scale was very influential in encouraging others, such as Guttman and Coombs, to develop practicdal scaling proceduers in the social sciences.
-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 (general)|process]] in which a [[comparison]] is made between pairs of a collection of entities with respect to magnitudes of an [[attribute]], trait, [[attitude (psychology)|attitude]], and so on.  Examples of such processes are the comparison of perceived intensity of physical [[stimulus (physiology)|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 comparison|''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 (abstract)|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 [[Units of measurement|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
-:<math>
-S_i-S_j=x_{ij} \sqrt {\sigma_i^2 + \sigma_j^2- 2r_{ij}\sigma_i\sigma_j} ,
-</math>
-in which:
-*<math>S_i</math> is the psychological scale value of stimuli ''i''
-*<math>x_{ij}</math> 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''
-*<math>\sigma_i</math> is the discriminal dispersion of a stimulus <math>R_i</math>
-*<math>r_{ij}</math> 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 <math>R_i</math>, where <math>S_i</math> 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:
-:<math>
-x_{ij}= \frac {S_i-S_j}{ \sigma} \,
-</math>
-where
-:<math>{\sigma}= \sqrt {\sigma_i^2 + \sigma_j^2}.\,</math>
-In this case of the model, the difference <math>{S_i-S_j}</math> can be inferred directly from the proportion of instances in which ''j'' is judged greater than ''i'' if it is hypothesised that <math>x_{ij}</math> 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 <math>\sigma = 1</math>, which is in effect an arbitrary choice of the [[Units of measurement|unit]] of measurement.  Letting <math>P_{ij}</math> be the proportion of occasions on which ''i'' is judged greater than ''j'', if, for example, <math>P_{ij}=0.84</math> and it is hypothesised that <math>x_{ij}</math> is normally distributed, then it would be inferred that <math>S_i-S_j \cong 1</math>.
-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.
 ==Major publications==
@@ Line 116: / Line 64: @@
-===References===
+==References==
-* Andrich, D. (1978b) Relationships between the Thurstone and Rasch approaches to item scaling.  ''Applied Psychological Measurement'', 2, 449-460.
+* Babbie, E., 'The Practice of Social Research', 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0534620299
-* Andrich, D. & Luo, G. (1993) A hyperbolic cosine model for unfolding dichotomous single-stimulus responses. ''Applied Psychological Measurement'', 17, 253-276.
+* Dawes, R. M., (1994). Psychological measurement. <i>Psychological Review, 101<i>, 278-281.
-* Babbie, E., 'The Practice of Social Research', 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0534620299
+* Edwards, A. L. <i>Techniques of attitude scale construction.</i> New York: Appleton-Century- Crofts, 1957.
-* Edwards, A. L. <i>Techniques of attitude scale construction.</i> New York: Appleton-Century- Crofts, 1957.
+* Glover, J. A., & Ronning, R. R. (1987). <i>Historical foundations of educational psychology.<i> New York: Plenum Press.
 * Guilford, J. P. <i>Psychometric methods.</i> New York: McGraw-Hill, 1954.
+* Kail, R., & Pelligrino, J. W. (1985). <i>Human intelligence: Perspectives and prospects.<i> New York: W. H. Freeman and Company.
+* Kerlinger, F. N. (1986). <i>Foundations of behavioral research.<i> Chicago: Holt, Rhinehart, and Winston, Inc.
 * Krus, D.J., & Kennedy, P.H. (1977) Normal scaling of dominance matrices: The domain-referenced model. <i>Educational and Psychological Measurement,</i> 37, 189-193 [http://www.visualstatistics.net/Scaling/Domain%20Referenced%20Scaling/Domain-Referenced%20Scaling.htm (Request reprint).]
@@ Line 131: / Line 83: @@
 * 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 [http://www.visualstatistics.net/Readings/Thurstone%20Crimes%20Scale/Thurstone%20Crimes%20Scale.htm  (Request reprint).]
-*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.
 ==External links==