Vertical Equating and Item Selection

George Ingebo believes that bank calibrations need to be all-purpose values. I have a specific purpose in mind with vertical test equating -- to evaluate a particular sequence of curriculum material. I have no intention of accounting for third graders' performance on eighth grade tests. If the scale is used to place new students in the curriculum sequence, I would only use the students' answers to items that were within a reasonable range of each student's estimated position on the scale.

All models engage data uncritically. Evaluation of model fit, or appropriateness of items for linking, comes after the model has been applied. Poor linking items are identified after an MFORMS analysis through the study of residuals and item fit. The method differs from looking at a plot of paired link-item calibrations, but the result is the same. Either way, the "noisy" items can be detected, deleted, and the analysis rerun. More work needs to be done to find the relationship between noise, as revealed by a plot of paired item calibrations (multistep procedure) and noise as revealed through the study of (one-step, concurrent) residuals.

In one sense, the removal of noisy items from the test linking data may be immaterial. Consider the plot of paired item calibrations. One tends to reject outlying items from both sides of the identity line, so the mean difference in difficulty between tests is hardly changed -- neither does the resulting scale except for the absence of the outlying items. If there is no "right" item for linking tests at different grade levels -- which is what some might argue -- we should not be much comforted by the removal of a few noisy items.

There are problems with vertical test equating, but I think it is too soon to give up. A vertically equated scale has a great deal of potential. How else are we to compare the difficulty of curricular material across grade levels and track growth in student ability? The extent to which the scale can be linear and unidimensional across grade levels is a matter for empirical research -- we have to construct scales in order to evaluate them.

Vertical equating and item selection. Schulz EM. … Rasch Measurement Transactions 2:2 p.18

Vertical equating and item selection. Schulz EM. … Rasch Measurement Transactions, 1988, 2:2 p.18

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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
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