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The present paper gives a general introduction to the linear logistic test model (Fischer, 1973), an extension of the Rasch model with linear constraints on item parameters, along with eRm (an R package to estimate different types of Rasch models; Mair, Hatzinger, & Mair, 2014) functions to estimate the model and interpret its parameters. The applications of the model in test validation, hypothesis testing, cross-cultural studies of test bias, rule-based item generation, and investigating construct irrelevant factors which contribute to item difficulty are explained. The model is applied to an English as a foreign language reading comprehension test and the results are discussed.

Though calibration of an achievement test within psychological and educational context is very often carried out by the Rasch model, data sampling is hardly designed according to statistical foundations. However, Kubinger, Rasch, and Yanagida (2009) recently suggested an approach for the determination of sample size according to a given Type I and Type II risk, and a certain effect of model misfit when testing the Rasch model is supported by some new results. The approach uses a three-way analysis of variance design [image omitted] x "C" with mixed classification. There is a (fixed) group factor "A", a (random) factor "B" of testees within "A", and a (fixed) factor "C" of items cross-classified with [image omitted]. The simulation study in this article deals with further item parameter ranges and ability parameter distributions, and with larger sample sizes and item numbers than the original paper. The results are: The approach works given several restrictions, and its main aim, the determination of the sample size, is attained. (Contains 2 notes, 6 tables, and 1 figure.)

The linear logistic test model (LLTM) breaks down the item parameter of the Rasch model as a linear combination of some hypothesized elementary parameters. Although the original purpose of applying the LLTM was primarily to generate test items with specified item difficulty, there are still many other potential applications, which may be of use for psychometric research on various testing conditions. This article provides some examples of such applications. The examples include (a) position effect of item presentation (in particular, learning and fatigue effects); (b) content-specific learning effect; (c) effect of speeded item presentation; and (d) effect of item response format. (Contains 3 figures.)

In this article, we emphasize that the Rasch model is not only very useful for psychological test calibration but is also necessary if the number of solved items is to be used as an examinee's score. Simplified proof that the Rasch model implies specific objective parameter comparisons is given. Consequently, a model check per se is possible. For data and item pools that fail to fit the Rasch model, various reasons are listed. For instance, the two-parameter logistic or three-parameter logistic models would probably be more suitable. Several suggestions are given for controlling the overall Type I risk, for including a power analysis (i.e., taking the Type II risk into account), for disclosing artificial model check results, and for the deletion of Rasch model misfitting examinees. These suggestions are empirically founded and may serve in the establishment of certain rough state-of-the-art standards. However, a degree of statistical elaboration is needed; and forthcoming test authors will still suffer from the fact that no standard software exists that offers all of the given approaches as a package.