The Use of Loglinear Models for Assessing Differential Item Functioning Across Manifest and Latent Examinee Groups

Loglinear latent class models are used to detect differential item functioning (DIF). These models are formulated in such a manner that the attribute to be assessed may be continuous, as in a Rasch model, or categorical, as in Latent Class Mastery models. Further, an item may exhibit DIF with respect to a manifest grouping variable, a latent grouping variable, or both. Likelihood-ratio tests for assessing the presence of various types of DIF are described, and these methods are illustrated through the analysis of a "real world" data set

Item bias detection using the loglinear Rasch model:Observed and unobserved subgroups

A method is proposed for the detection of item bias with respect to observed or unobserved subgroups. The method uses quasi-loglinear models for the incomplete subgroup x test score x item 1 x ... x item k contingency table. If the subgroup membership is unknown, the models are the incomplete-latent-class models of S. J. Haberman (1979). The (conditional) Rasch model is formulated as a quasi-loglinear model. The parameters in this model that correspond to the main effects of the item responses are the conditional estimates of the parameters in the Rasch model. Item bias can then be tested by comparing the quasi-loglinear-Rasch model with models that contain parameters for the interaction of item responses and the subgroups. An example uses data from a test taken by 286 Dutch undergraduates who took a multiplication test using Roman numerals and numbers written out in Dutch. Some of the examinees had received training in multiplying Roman numerals. It was expected that Roman items would be biased, and the procedure confirmed this bias. Five tables present the models and study data

Estimating quasi-loglinear models for a Rasch table if the numbers of items is large

The Rasch Model and various extensions of this model can be formulated as a quasi loglinear model for the incomplete subgroup x score x item response 1 x ... x item response k contingency table. By comparing various loglinear models, specific deviations of the Rasch model can be tested. Parameter estimates can be computed using programs such as GLIM, ECTA, and MULTIQUAL, but this becomes impractical if the number of items is large. In that case, the tables of observed and expected counts become too large for computer storage. In this paper, a method of parameter estimation is described that does not require the internal representation of all observed and expected counts, but rather uses only the observed and expected sufficient statistics of the parameter estimates, which are the marginal tables corresponding to the model terms only. The computational problem boils down to computation of the expected sufficient statistics which, in its raw form, amounts to summation of a very large number of expected counts. However, it is shown that, depending on the structure of the model, the number of computations can be reduced considerably by making use of the distributive law. As a result, simpler models may be computed much more efficiently in terms of both storage and processing times

Loglinear multidimensional IRT models for polytomously scired Items

A loglinear item response theory (IRT) model is proposed that relates polytomously scored item responses to a multidimensional latent space. Each item may have a different response function where each item response may be explained by one or more latent traits. Item response functions may follow a partial credit model (D. Andrich, 1978; and G. N. Masters, 1982), a multidimensional Rasch model (G. Rasch, 1961; and E. B. Andersen, 1973, 1983), or other forms of response functions to be defined by the user. Conditional maximum likelihood estimates are derived, and the models may be tested generally or against alternative loglinear models. The latter tests are sensitive to deviations from local independence subgroup invariance or assumptions about the form of the operating characteristic curves. The model was illustrated through application to data from a test to identify learning problems in Dutch children from 4 to 6.5 years of age. Fifteen items were administered to 66 children aged 4 to 5 years, 132 children aged 5 to 5.5 years, and 65 children aged 5.5 to 6 years. Three appendices illustrate the dichotomous Rasch model, the partial credit model, and Rasch's multidimensional model

Loglinear-latent-class models for detecting item bias

The use of loglinear latent class models to detect item bias was studied. Purposes of the study were to: (1) develop procedures for use in assessing item bias when the grouping variable with respect to which bias occurs is not observed; (2) develop bias detection procedures that relate to a conceptually different assessed trait--a categorical attribute; and (3) exemplify the use of these developed procedures with real world data. Models are formulated so that the attribute to be measured may be continuous, as in a Rasch model, or categorical, as in latent class models. The item bias to be studied may correspond to a manifest grouping variable, a latent grouping variable, or both. Likelihood-ratio tests for assessing the presence of various types of bias are described. These methods are illustrated through analysis of a "real world" data set from a study of multiplication items administered to 286 Dutch undergraduates. Bias was related to a manifest grouping variable by giving 143 of the subjects some training in Roman numerals, in which some of the multiplication problems were written. Results indicate that it was possible to explain item bias through differences in item difficulties or error rates across levels of grouping variables. The model represented can be extended to include several observed and unobserved variables. Ten tables present information about the models and findings of the study

IRT-based test construction

Four discussions of test construction based on item response theory (IRT) are presented. The first discussion, "Test Design as Model Building in Mathematical Programming" (T.J.J.M. Theunissen), presents test design as a decision process under certainty. A natural way of modeling this process leads to mathematical programming. General models of test construction are discussed, with information about algorithms and heuristics; ideas about the analysis and refinement of test constraints are also considered. The second paper, "Methods for Simultaneous Test Construction" (Ellen Boekkooi-Timminga), gives an overview of simultaneous test construction using zero-one programming. The item selection process is based on IRT. Some objective functions and practical constraints are presented, the construction of parallel tests is considered, and two tables are provided. The third paper, "Automated Test Construction Using Minimax Programming" (Wim J. van der Linden), proposes the use of the minimax principle in IRT test construction and indicates how this results in test information functions deviating less systematically from the target function than for the usual criterion of minimal test length. An alternative approach and some practical constraints are considered. The final paper, "A Procedure To Assess Target Information Functions" (Henk Kelderman), discusses the concept of an information function and its properties. An interpretable function of information is chosen: the probability of a wrong order of the ability estimates of two subjects

Differences in quality-of-life dimensions of Adult Strabismus Quality of Life and Amblyopia & Strabismus Questionnaires

Purpose: The Adult Strabismus Quality of Life Questionnaire (AS-20) and the Amblyopia & Strabismus Questionnaire (A&SQ) both measure health-related quality of life in strabismus patients. We evaluated to what extent these instruments cover similar domains by identifying the underlying quality-of-life factors of the combined questionnaires. Methods: Participants were adults from a historic cohort with available orthoptic childhood data documenting strabismus and/or amblyopia. They had previously completed the A&SQ and were now asked to complete the AS-20. Factor analysis was performed on the correlation-matrix of the combined AS-20 and A&SQ data to identify common underlying factors. The identified factors were correlated with the clinical variables of angle of strabismus, degree of binocular vision, and visual acuity of the worse eye. Results: One hundred ten patients completed both questionnaires (mean age, 44 years; range, 38–51 years). Six factors were found that together explained 78% of the total variance. The factor structure was dominated by the first four factors. One factor contained psychosocial and social-contact items, and another factor depth-perception items from both questionnaires. A third factor contained seven items—only from the AS-20—on eye strain, stress, and difficulties with reading and with concentrating. A fourth factor contained seven items—only from the A&SQ—on fear of losing the better eye and visual disorientation, specific for amblyopia. Current visual acuity of the worse eye correlated with depth-perception items and vision-related items, whereas current binocular vision correlated with psychosocial and social-contact items, in 93 patients. Conclusions: Factor analysis suggests that the AS-20 and A&SQ measure a similar psychosocial quality-of-life domain. However, functional problems like avoidance of reading, difficulty in concentrating, eye stress, reading problems, inability to enjoy hobbies, and need for frequent breaks when reading are represented only in the AS-20. During the development of the A&SQ, asthenopia items were considered insufficiently specific for strabismus and were excluded a priori. The patients who generated the items for the AS-20 had, in majority, adulthood-onset strabismus and diplopia and were, hence, more likely to develop such complaints than our adult patients with childhood-onset strabismus and/or amblyopia

Multidimensional rasch models for partial credit scoring

Rasch models for partial-credit scoring are discussed and a multidimensional version of the model is formulated. A model may be specified in which consecutive item responses depend on an underlying latent trait. In the multidimensional partial-credit model, different responses may be explained by different latent traits. Data from van Kuyk’s (1988) size concept test and the Raven Progressive Matrices test were analyzed. Maximum likelihood estimation and goodness-of-fit testing are discussed and applied to these datasets. Goodness-of-fit statistics show that for both tests, multidimensional partial-credit models were more appropriate than the unidimensional partial-credit model. Index terms: X2 testing, exponential family model, multidimensional item response theory, multidimensional Rasch model, partial-credit models, Progressive Matrices test, Rasch model