# Difficulties and Average Category Measures

"Why do the average category measures sometimes say an obviously easy lower rating scale category is more difficult than a much harder higher category?"

Tetsuhito Shizuka

This often perplexes us. First, a definition: "difficulty" is always a combination of "difficulty of observing" and "difficulty of doing". We hope that "difficulty of observing" is much less than "difficulty of doing" so that "doing" dominates the data.

For instance, imagine an observation protocol designed to assess speed. We can easily observe cars driving less than 50 km/hour: we can give them a rating of "1". It is more difficult to observe cars driving faster than 50 km/hour, because we have less time in which to observe each one: we can give them a rating of "3". It is almost impossible to observe cars driving exactly 50 km/hour, because they are rare, but we can give those cars (if we observe any) a rating of "2".

On the intended rating scale of "slow", "medium", "fast", the ratings are ordered 1,2,3. In difficulty of observing, however, the ratings are ordered 1,3,2. These two orderings will be confounded in our data set. Consequently we can expect that the step calibrations, the Rasch rating scale parameters, will be disordered.

If we note down all the cars observed in category 1, and then average their measures on our entire instrument, we obtain an average measure for category 1. This reports "what is the average measure of cars rated 1 in this sample?". Similarly, the average measure for category 2 reports on "what is the average measure of cars rated 2 in this sample?" These averages are measures on the underlying linear metric, but are descriptive of this sample and this use of the rating scale. Since observation of a higher category is supposed to be indicative of an object with more of the variable, we expect that the average measure for category 2 will be noticeably higher than that for category 1, and 3 higher than 2. If not, this use of the rating scale has produced a blurred or contradictory description of this sample, so that we have good reason to examine our measuring instrument for flaws. Do the items cooperate to form one variable? Do higher categories indicate more of the variable in a uniform way? Are raters using the instrument in the manner intended?

John Michael Linacre

Difficulties and Average Category Measures Shizuka T., Linacre J.M. … Rasch Measurement Transactions, 2000, 13:4 p. 717

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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
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