Variations on a Theme by Thurstone

Abstract

A test model based on the Thurstone judgmental model is described. By restricting various parameters of the model, 3 Rasch models, 2 pseudoRasch models, 3 two-parameter ICC models, and a Weber's Law model are derived.The thematic model for latent trait approaches to test scaling was developed by Thurstone (1927) from his law of comparative judgment. The model was variously termed the method of successive intervals (Saffir, 1937), the method of graded dichotomies (Attneave, 1949), and the law of categorical judgment (Torgerson, 1958). In the model, stimuli were conceived as having a discriminal dispersion around a central location on an attribute continuum and judgmental categories (or category boundaries) as having a similar distribution on the same continuum. A variant of this model, which will be called Thurstone Model A, is produced by substituting items for stimuli and persons for judgmental categories, as illustrated in Figure 1.Thurstone Model A locates items and persons on the same attribute continuum. Each item and each person has a normal distribution of values on the attribute resulting from moment to moment fluctuation. The standard deviations of APPLIED the distributions may differ from item to item or from person to person. Item and person param_eters are assumed to be independent of each other (ri, = 0). The correlation between location and dispersion parameters is unspecified for either items or persons. A person answers an item correctly if, at the moment of attempting it, his/her momentary attribute value is higher than the momentary attribute value of the item; otherwise he/she answers it incorrectly. Torgerson (1958) suggested certain simplifications of the thematic model in order to overcome the formidable estimation problems. He set the standard deviations for categories (subjects) equal and called the result Condition B; he set standard deviations for stimuli (items) equal and called the result Condition C; finally, he set both the standard deviations of categories and the standard deviations of stimuli equal and called the result Condition D. It is the purpose of this paper to examine more systematically a wider range of simplifying possibilities and to relate these to ancient and modern approaches to test scaling. Restrictions on Both Item and Person Dispersions Torgerson did not consider the cases where item and person dispersions are set to zero. It

A Pearson-Type-VII item response model for assessing person fluctuation

“…However, 26 PSYCHOMETRIKA most of them converge essentially to the same basic model: a threshold model for binary items in which the individual fluctuation is assumed to be Gaussian. In a set of three papers, Lumsden (1977Lumsden ( , 1978Lumsden ( , 1980 gave what is probably the best-known systematization of this type of model. Furthermore, he provided a rationale based on Thurstone scaling and also related the different variations of the basic model to item response theory (IRT).…”

“…The basic scaling model from which the IRT model will be derived is Lumsden's (1980) 'Pseudo-Rasch 1 model', which is formally equivalent to Torgerson's (1958) condition C of Thurstone's law of categorical judgement. The model locates items and respondents in the same trait continuum.…”

Commentary

“…His colleagues resisted adopting such stringent data requirements to the point that Thurstone later dropped the entire effort, and devoted himself to factor analysis [23, p. 214; 24]. One critic sees Thurstone's life work as tragic on a Promethean scale, saying that, in his measurement work, Thurstone stole fire from the gods, and in retribution, they chained him to factor analysis [25]. Thurstone [23, p. 321] himself admitted that he had ''tried to avoid controversy when it would have been better to ignore it.…”

A Generalized Logistic Item Response Model Parameterizing Test Score Inappropriateness

“…Based on ideas suggested by Weiss (1973;Trabin & Weiss, 1979Vale & '~~~ss9 1975), Lumsden proposed the concept of a person response curve as an alternative to the item response curve. to Lumsden (1977Lumsden ( , 1978Lumsden ( , 1980, it is much more realistic to localize measurement unrellability within the person rather than within the test item, as is customary. The model he favored, then, is that of a two-parameter person response curve of an ~~~~~~~~~9s expected item scores regressed onto item difficulty, which is the sole item parameter.…”

“…Lumsden rejected the assumption of differential item variances implied by a model that includes an item slope parameter because, in his opinion, it precludes the existence of test homogeneity, which is the primary assumption of all IRT models and is necessary to ensure local independence. Lumsden (1980) and others (e.g., Andrich, 1978) have found it useful to conceptualize a person-item encounter using Thurstone's law of comparative judgment (see Torgerson, 1958). When the variances of both person and item distributions are possibly nonzero and are not necessarily identical, this formulation leads to a model that is similar in spirit to the one introduced in this paper.…”