Estimating Item Discriminations with Rasch Models

Plots of empirical item characteristic curves (ICCs) enable one to estimate the empirical item discrimination, at least as well as a 2-PL IRT computer program. This is because 2-PL discrimination estimation is degraded by the imputation of a person distribution and by arbitrary constraints placed on discrimination values. It is also skewed by accidental outliers which the eye can disregard.

Figure 1. Empirical and model ICCs.

On a plot such as Figure 1 draw in the line that, to your eye, matches the central slope of the empirical item characteristic curve (ICC). In Figure 1, the dots indicate the Rasch-model-predicted ICC. The points marked `x' indicate the empirical values, obtained by averaging the observed responses, `0' and `1', for persons whose measures are estimated to be in the .1 logit interval centered on the `x'. When a dot and an `x' coincide, a `*' is shown. The line follows the trail of `x' and `*'.

Estimate the logit (x-axis) distance from where the line intercepts the .0 score value to where it intercepts the 1.0 score value (for dichotomies). The logit distance here is about 4.0 logits.

Use the nomogram in Figure 2 to estimate empirical item discrimination. In this nomogram, looking at the middle traceline, a logit distance of 4.0 logits, corresponds to a logit discrimination of 1.0, in accordance with model prediction. Shorter distances, i.e., steeper slopes, correspond to higher discriminations.

Figure 2. Nomogram for item discrimination.

In Figure 2, there are 3 lines in order to simplify comparison and reporting of results in an IRT context. 2-PL programs generally report their results in probits. So the probit discrimination for logits distances is given by the upper line. That corresponding to a 4-logit distance is 1.7. On the other hand, if one is looking at plot scaled in probits, instead of logits, one would look at the lower line to find the logit discrimination.

John M. Linacre

Estimating item discriminations with Rasch models. Linacre JM. … 16:1 p.868

Estimating item discriminations. Linacre JM. … Rasch Measurement Transactions, 2002, 16:1 p.868

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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