## Insights through Graphing

Conventional item analysis presents its results as tables of numbers. The Rasch model provides a framework in which familiar results can be made more meaningful and new insights can emerge. Graphical presentation is the key.

The Figure overleaf summarizes the performance of 7,132 students on Question 1 of the 1990 Australian Higher School Certificate Physics Examination. Overall, 79% of the sample succeeded on it.

The ability distributions of the students who succeeded and failed on this item are shown in the top panel of the Figure. The x-axis shows the variable in terms of percent correct on the whole test (clearly squeezed together around 50%), and also logit abilities relative to the mean sample performance. The y-axis indicates the number of students.

A secondary x-axis, immediately below the ability measures, locates percentiles for this sample. The students meriting special attention are at the tails.

The item calibration (which targets students with 50% probability of success) is shown by the vertical rule. We can confirm this by eye, because it corresponds to the location on the variable where the succeed and fail bars are of equal length.

Based on performance on the test as a whole, we would expect each vertical bar to end at a . Only a few bars are much shorter or longer, indicating generally good fit of the item for this sample.

Another perspective on Question 1 is given by the comparison of its item characteristic curves in the bottom panel. The model curve, , follows the logistic ogive for this right/wrong item. The empirical ICC is shown by . It is somewhat fuzzy, but close to the model line. High performers have done a little better than expected on this item. The few low performers somewhat worse. There is no evidence of a lower "guessing" asymptote. Most intriguing is the slight sag in performance among those about 1 logit above the item (around 70% correct). Perhaps these students have a usually successful, but slightly faulty approach: "A little knowledge is a dangerous thing!"

Andrew Stephanou
Centre for the Study of Higher Education (CSHE)
The University of Melbourne, Australia

Insights through Graphing.Stephanou A. … Rasch Measurement Transactions, 1997, 11:2 p. 571-572.

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
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