Mokken Scale Analysis for Dichotomous Items Using Marginal Models

Scalability coefficients play an important role in Mokken scale analysis. For a set of items, scalability coefficients have been defined for each pair of items, for each individual item, and for the entire scale. Hypothesis testing with respect to these scalability coefficients has not been fully developed. This study introduces marginal modelling as a framework to derive the standard errors for the scaling coefficients and test hypotheses about these coefficients. Several examples demonstrate the possibilities of marginal modelling in Mokken scale analysis. These possibilities include testing whether Mokken’s criteria for a scale are satisfied, testing whether scalability coefficients of different items are equal, and testing whether scalability coefficients are equal across different groups.     

Psychometrics in Psychological Research: Role Model or Partner in Science?

This is a reaction to Borsboom's (2006) discussion paper on the issue that psychology takes so little notice of the modern developments in psychometrics, in particular, latent variable methods. Contrary to Borsboom, it is argued that latent variables are summaries of interesting data properties, that construct validation should involve studying nomological networks, that psychological research slowly but definitely will incorporate latent variable methods, and that the role of psychometrics in psychology is that of partner, not role model. Requests for reprints should be sent to Klaas Sijtsma, Department of Methodology and Statistics, FSW, Tilburg University, PO Box 90153, 5000 LE, Tilburg, The Netherlands.     

Global, local, and graphical person-fit analysis using person-response functions

Person-fit statistics test whether the likelihood of a respondent's complete vector of item scores on a test is low given the hypothesized item response theory model. This binary information may be insufficient for diagnosing the cause of a misfitting item-score vector. The authors propose a comprehensive methodology for person-fit analysis in the context of nonparametric item response theory. The methodology (a) includes H. Van der Flier's (1982) global person-fit statistic U3 to make the binary decision about fit or misfit of a person's item-score vector, (b) uses kernel smoothing (J. O. Ramsay, 1991) to estimate the person-response function for the misfitting item-score vectors, and (c) evaluates unexpected trends in the person-response function using a new local person-fit statistic (W. H. M. Emons, 2003). An empirical data example shows how to use the methodology for practical person-fit analysis.     

Measuring the ability of transitive reasoning,using product and strategy information

Cognitive theories disagree about the processes and the number of abilities involved in transitive reasoning. This led to controversies about the influence of task characteristics on individuals' performance and the development of transitive reasoning. In this study, a computer test was constructed containing 16 transitive reasoning tasks having different characteristics with respect to presentation form, task format, and task content. Both product and strategy information were analyzed to measure the performance of 6- to 13-year-old children. Three methods (MSP, DETECT, and Improved DIMTEST) were used to determine the number of abilities involved and to test the assumptions imposed on the data by item response models. Nonparametric IRT models were used to construct a scale for transitive reasoning. Multiple regression was used to determine the influence of task characteristics on the difficulty level of the tasks. It was concluded that: (1) the qualitatively distinct abilities predicted by Piaget's theory could not be distinguished by means of different dimensions in the data structure; (2) transitive reasoning could be described by one ability, and some task characteristics influenced the difficulty of a task; and (3) strategy information provided a stronger scale than product information.     

Latent Class Modeling of Phases in the Development of Transitive Reasoning

Fuzzy trace theory posits that during development the use of verbatim information for solving transitive relationships shifts to the use of gist information. In cognitive developmental research that uses a cross-sectional design, the binomial mixture model is often used to identify such shifts. Because the binomial mixture model assumes equal task difficulty and uses the number of correctly solved tasks for data analysis, it may be too restrictive and the more flexible latent class model is adopted as an alternative. This model allows varying task difficulty and uses the pattern of task scores as input for data analysis. The binomial mixture model and the latent class model are compared theoretically, and applied to transitive reasoning test data obtained from a cross-sectional sample of 615 children. The latent class model is found to be more appropriate for identifying multiple phases. Three phases are distinguished which can be interpreted well by means of fuzzy trace theory. These phases do not encompass fixed age periods.     

On the Usefulness of a Multilevel Logistic Regression Approach to Person-Fit Analysis

The logistic person response function (PRF) models the probability of a correct response as a function of the item locations. Reise (2000) Reise, S. P. 2000. Using multilevel logistic regression to evaluate person-fit in IRT models. Multivariate Behavioral Research, 35: 543–568. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] proposed to use the slope parameter of the logistic PRF as a person-fit measure. He reformulated the logistic PRF model as a multilevel logistic regression model and estimated the PRF parameters from this multilevel framework. An advantage of the multilevel framework is that it allows relating person fit to explanatory variables for person misfit/fit. We critically discuss Reise's approach. First, we argue that often the interpretation of the PRF slope as an indicator of person misfit is incorrect. Second, we show that the multilevel logistic regression model and the logistic PRF model are incompatible, resulting in a multilevel person-fit framework, which grossly violates the bivariate normality assumption for residuals in the multilevel model. Third, we use a Monte Carlo study to show that in the multilevel logistic regression framework estimates of distribution parameters of PRF intercepts and slopes are biased. Finally, we discuss the implications of these results and suggest an alternative multilevel regression approach to explanatory person-fit analysis. We illustrate the alternative approach using empirical data on repeated anxiety measurements of cardiac arrhythmia patients who had a cardioverter-defibrillator implanted.     

Robust Mokken Scale Analysis by Means of the Forward Search Algorithm for Outlier Detection

Exploratory Mokken scale analysis (MSA) is a popular method for identifying scales from larger sets of items. As with any statistical method, in MSA the presence of outliers in the data may result in biased results and wrong conclusions. The forward search algorithm is a robust diagnostic method for outlier detection, which we adapt here to identify outliers in MSA. This adaptation involves choices with respect to the algorithm's objective function, selection of items from samples without outliers, and scalability criteria to be used in the forward search algorithm. The application of the adapted forward search algorithm for MSA is demonstrated using real data. Recommendations are given for its use in practical scale analysis.