Residual patterns should be studied in order to understand when and why data deviate from a model. This paper illustrates some techniques for exploring residual patterns resulting from the difference between observed and expected scores as predicted by a Rasch model for polychotomous data. The score residual divided by its standard deviation is…

Approximately 1200 Danish school children aged 8, 11, and 14 participated in a questionnaire study under the caption, "Why I like to read." Data from Denmark formed part of an international study, in which 12 other countries used the same questionnaire to explore various functions of reading and to compare them over different countries.…

Necessary and sufficient conditions for the existence and uniqueness of a solution of the so-called "unconditional" and the "conditional" maximum-likelihood estimation equations in the dichotomous Rasch model are given. It is shown how to apply the results in practical uses of the Rasch model. (Author/JKS)

Recent advances in item analysis have provided greater capabilities for the analysis of tests, but have also significantly increased the gap between the theory and practice of item analysis. This paper traces the lines of development in item analysis under latent trait theory. (MV)

Deciding whether sets of test data are consistent with any of a large class of item response models is considered. The assumption of local independence is weakened to a new condition, local nonnegative dependence (LND). Necessary and sufficient conditions are derived for a LND item response model. (Author/JKS)

Six statistical methods for identifying possible sources of bias in standardized test items are presented. The relationship between chi-squared methods and item-response theory methods are also discussed. (Author/BW)

Item response theory (IRT) has proven to be a very powerful and useful measurement tool. However, most of the IRT models that have been proposed, and all of the models commonly used, require the assumption of unidimensionality, which prevents their application to a wide range of tests. The few models that have been proposed for use with…

In previous research, no relationship was found between test factor structure and test hierarchical structure. This study found some correspondence between test factor structure and test inter-item dependency structure, as measured by a log-linear model. There was an inconsistency, however, which warrants further study: more significant two-item…

The relatively hard problem of transforming a given set of curves to curves with the same shape can sometimes be reduced to the easier problem of rendering curves parallel. In this paper a group is associated with the given curves, and it is shown that the reduction from the hard problem to the easy problem is valid whenever the group is…

Fisher's information measure for the item difficulty parameter in the Rasch model and its marginal and conditional formulations are investigated. It is shown that expected item information in the unconditional model equals information in the marginal model, provided the assumption of sampling examinees from an ability distribution is made. For the…

A procedure for the sequential optimization of the calibration of an item bank is given. The procedure is based on an empirical Bayes approach to a reformulation of the Rasch model as a model for paired comparisons between the difficulties of test items in which ties are allowed to occur. First, it is indicated how a paired-comparisons design…

The Pearson and likelihood ratio statistics are frequently used for assessing the absolute fit of probability models. Researchers are often interested in comparing fits provided by different models which may have a subsuming or non-subsuming relation. A subsuming relation exists when the parameters of the reduced model form a subset of those…

A mathematical setting based on a statistical sampling probability mechanism is described. In this setting, a mathematical meaning is given for the information function; and it becomes possible to study the relative merits of various ability-estimating procedures. The maximum likelihood estimation procedure under the one-, two-, three-parameter…

Latent trait test score theory is discussed primarily in terms of Birnbaum's three-parameter logistic model, and with some reference to the Rasch model. Equations and graphic illustrations are given for item characteristic curves and item information curves. An example is given for a hypothetical 20-item adaptive test, showing cumulative results…

An approximate relationship is devised between the unidimensional model used in data analysis and a multidimensional model hypothesized to be generating the item responses. Scale shrinkage is successfully predicted for several sets of simulated data. (Author/LMO)