## Building Variables with Measure vs. Fit Plots

How can we construct robust definitions for variables such as "abiding belief in corporate quality," or "empathy for therapy clients" when, initially, we hardly know what the variable is, or even whether the concept constitutes a variable at all?

Let us start with a vaguely understood variable, "Intensity of Classroom Power Harassment (CPH)" - the manipulation of the power differential between teachers and students in the traditional educational setting. An "intensive definition" specifying criteria for inclusion and exclusion for this concept is not available. We must retreat to "ostensive", "point to", definitions. We do this by collecting anecdotal vignettes of CPH, and assembling a "CPH test" in which each story becomes a test item. Then we ask people to rate each item on a rating scale with categories ordered from trivial to blatant CPH. We don't need a large sample of items or of respondents, but a good spread of CPH blatancy and person sensitivity gives better separation and the potential for a clearer depiction of the variable.

Here is where the "qualiquantal map" is useful. "Qualiquantal" refers to a variable's quality, the fit statistics, and its quantity, the measures. Rasch analysis gives us a measure for each respondent, quantifying sensitivity to CPH, and a calibration for each item, quantifying intensity of CPH. We also obtain fit statistics, "infits", which quantify coherence with the core variable underlying the interactions between respondents and items. Plotting fit statistics against measures produces a map of the relationship between the items (or respondents) and the core variable.

It is this qualiquantal, fit vs. measure, map which clarifies the variable. Write each item on a card and position it on a large sheet by its calibration and infit statistic. Do the same thing for each person with cards containing demographics. To provide greater clarity as to what the placement of respondents on the map implies, focus groups of respondents with similar measures and fits can be asked to explore the concept of interest. Brief excerpts of their testimony can then be displayed in position on the map.

How are we to understand what the map is telling us? A "similarities and differences" approach is useful. Scan vertically for similarities and differences between high infit and low infit items (or respondents) with similar measures. Scan horizontally for similarities and differences between high and low measure items (or respondents) with similar fits.

An item/person can be high measure/high misfit, low measure/low misfit, or either of the two high/low combinations. A high measure/low misfit CPH item is a blatant CPH episode which is coherent with the core concept of CPH extracted from these data. High misfit items or people are "marching to a different drummer." Such an item might portray a dramatic or poignant example of some facet of CPH. But because it has high misfit, it must be down-played as only marginally germane to the current core (even though it might be central to some other expression of the CPH concept.) As we refine our ideas, we selectively delete items and respondents, recalculate the measures and fit statistics, and see how the coherence and construct validity of the variable changes. Thus we jiggle the data until we are satisfied with the empirical definition.

Examining the scatter-plot leads to inferences about how the concept works. In the CPH map, low misfit items/people tend to be harassment tolerant while high misfit items/people tend to be harassment militant. If this trend had been opposite, it could mark CPH as a scourge on the educational system because the more relevant items/people are more militant. Even though CPH is too frequently practiced and too often unnoticed by its perpetrators, the observed trend implies that it can be treated with forbearance and viewed as a hazard of the unequal power of student and professor, to be dealt with gently but firmly.

High               CRITICIZE
Misfit
SHIFT BLAME
COMPLAIN         PUNITIVE
BELITTLE
HARSH           SEXIST
....................................................
NOSY
SHAMING

PLAY FAVORITES
DOMINEER
INDIFFERENT           PREJUDICED
Low
Misfit                                             CUSS
-----------------------------------------------------
Trivial        Intensity of Vignette          Blatant

Building variables with measure vs. fit plots. Durkin JE. … Rasch Measurement Transactions, 1992, 5:4 p.173

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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