David Andrich

A rating formulation for ordered response categories

A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.

Application of a psychometric rating model to ordered categories which are scored with successive integers

A latent trait measurement model in which ordered response categories are both parameterized and scored with successive integers is investigated and applied to a summated rating or Likert ques tionnaire. In addition to each category, each item of the questionnaire and each subject are para meterized in the model; and maximum likelihood estimates for these parameters are derived. Among the features of the model which make it attractive for applications to Likert questionnaires is that the total score is a sufficient statistic for a subjects at titude measure. Thus, the model provides a formal ization of a familiar and practical procedure for measuring attitudes.

Controversy and the Rasch model: a characteristic of incompatible paradigms?

The development of Rasch models in educational and psychologic measurement in the 1960s coincided with the introduction of other similar models, now described as models of item response theory (IRT). The application of IRT models has now extended to other social sciences, including health. Originally, there was substantial controversy between those who saw Rasch models as simply special cases of IRT models and those who saw them as essentially different. Because these different perspectives continue to manifest themselves in various ways, it seems relevant to understand the source of the original controversy. This paper attempts to do so by invoking Kuhn’s studies in the history and philosophy of science at 3 levels. First, it suggests that the 2 perspectives reflect Kuhn’s concept of legitimate, incompatible paradigms in which controversy is a typical manifestation. Second, because Kuhn recognizes individual histories in the development of paradigms, Rasch’s own shift in perspective is summarized. Third, because proponents of the Rasch models emphasize the models’ compatibility with fundamental measurement found in physical science, an analogy is made between how Kuhn explains the role of measurement in the physical sciences and how proponents of Rasch models explain the role of these models in the social sciences. In particular, these roles cannot be gleaned from textbooks in science and statistics, respectively.

An Extension of the Rasch Model for Ratings Providing Both Location and Dispersion Parameters.

An elaboration of a psychometric model for rated data, which belongs to the class of Rasch models, is shown to provide a model with two parameters, one characterising location and one characterising dispersion. The later parameter, derived from the idea of a unit of scale, is also shown to reflect the shape of rating distributions, ranging from unimodal, through uniform, and then to U-shaped distributions. A brief case is made that when a rating distribution is treated as a random error distribution, then the distribution should be unimodal.

Is the Sense of coherence-instrument applicable on adolescents? A latent trait analysis using Rasch-modelling

This study examines the construct validity of the 13-items Sense of Coherence (SOC) -instrument applied to a sample of 868 eighteen-year-old adolescents from a city in Sweden by using the Rasch model. All items showed relative invariance across the continuum, and although three items showed statistically significant lack of perfect invariance across genders, it was considered that they nevertheless worked sufficiently well to be retained. One item did show a response format that was incompatible with the correct operation of the categories. The questionnaire could separate the adolescents from this general population and it was concluded that the results were consistent with Antonovskys view that the SOC-scale should be dealt with as a measure of one global factor.

Scaling Attitude Items Constructed and Scored in the Likert Tradition

The statement scaling emphasis associated with the Thurstone tradition for the study of attitude provides no direct consequence for attitude measurement, while conversely, the attitude measurement emphasis in the Likert tradition provides no direct consequence for statement scaling. Both aspects of the study of attitude are unified by a generalisation of Raschs simple logistic model for dichotomously scored achievement items. The generalized model caters for the response category system of attitude questionnaires constructed and scored in the Likert tradition, but because both statements and persons are parameterised, the model has consequences for both statement scaling and person measurement. As with the simple logistic model, the two sets of parameter estimates, in this case those of the statements and of the persons, are independent of each other. An illustration involving attitudes to the professional roles of teachers is provided.

Relationships Between the Thurstone and Rasch Approaches to Item Scaling

When the logistic function is substituted for the normal, Thurstones Case V specialization of the law of comparative judgment for paired comparison responses gives an identical equation for the esti mation of item scale values as does the Rasch formulation for direct responses. The law of com parative judgment must be modified to include a subject parameter; but this parameter, which is eliminated statistically with respect to the direct re sponse design, is eliminated experimentally in the paired comparison design. Some comparisons and contrasts are made between the two approaches to item scaling, and it is shown that greater gen eralizability for item scaling is possible when the two approaches are juxtaposed appropriately.

Rating scales and Rasch measurement

Assessments with ratings in ordered categories have become ubiquitous in health, biological and social sciences. Ratings are used when a measuring instrument of the kind found in the natural sciences is not available to assess some property in terms of degree – for example, greater or smaller, better or worse, or stronger or weaker. The handling of ratings has ranged from the very elementary to the highly sophisticated. In an elementary form, and assumed in classical test theory, the ratings are scored with successive integers and treated as measurements; in a sophisticated form, and used in modern test theory, the ratings are characterized by probabilistic response models with parameters for persons and the rating categories. Within modern test theory, two paradigms, similar in many details but incompatible on crucial points, have emerged. For the purposes of this article, these are termed the statistical modeling and experimental measurement paradigms. Rather than reviewing a compendium of available methods and models for analyzing ratings in detail, the article focuses on the incompatible differences between these two paradigms, with implications for choice of model and inferences. It shows that the differences have implications for different roles for substantive researchers and psychometricians in designing instruments with rating scales. To illustrate these differences, an example is provided.

The General Health Questionnaire: a psychometric analysis using latent trait theory

This study examines the Likert-style successive integer scoring of Goldbergs (1972, 1978) General Health Questionnaire (GHQ) with a psychometric model in which the thresholds between successive categories within each item can be estimated. The model is particularly appropriate because the scoring of the successive categories, which are not named in the same way across items, by successive integers has received substantial discussion in the literature. Results from 1967 teachers in Western Australia who completed the 30-item form of the GHQ show that the items conform reasonably well to the model at a general or macro-level of analysis. In particular, the original ordering of categories is supported. However, as expected, there are systematic differences between distances among threshold within items and systematic differences among thresholds between items. The differences between positively and negatively orientated items confirm a suggestion in the literature that these two classes of items form sufficiently different scales so that they could be treated as separate, though reasonably correlated, scales.

A Hyperbolic Cosine Latent Trait Model For Unfolding Dichotomous Single-Stimulus Responses

Social-psychological variables are typically measured using either cumulative or unfolding response processes. In the former, the greater the location of a person relative to the location of a stimulus on the continuum, the greater the proba bility of a positive response; in the latter, the closer the location of the person to the location of the statement, irrespective of direction, the greater the probability of a positive response. Formal probability models for these processes are, respec tively, monotonically increasing and single-peaked as a function of the location of the person relative to the location of the statement. In general, these models have been considered to be independent of each other. However, if statements constructed on the basis of a cumulative model have three ordered response categories, the response function within the statement for the middle category is in fact single-peaked. Using this observation, a unidimen sional model for responses to statements that have an unfolding structure was constructed from the cumulative Rasch model for ordered response categories. A location and unit of measurement parameter exist for each statement. A joint maxi mum likelihood estimation procedure was inves tigated. Analysis of a small simulation study and a small real dataset showed that the model is readily applicable.