In rating scale analysis, logit measures are a better basis for statistical inference than the original rating scale categories, but the rating scale categories may be a better basis for communication. Content experts and end-users have often internalized the meaning of each rating scale category, so that they know immediately what a "2" or a "6"implies to them in terms of performance. With experience, they can even recognize a "2.5" or a "3.3". Consequently, users may request that a logit measure be converted back onto the rating scale metric for interpretability.
When all items employ the same rating scale, and the mean item difficulty is set to zero (and also judge severity, task challenge, etc.), then a rating scale value corresponding to any person measure can be obtained directly from the generic expected score curve. The Facets computer program provides this, reporting it is as the "Fair Average"rating. This Fair Average can also be obtained by inspection directly from graphical output. In Figure 1, the Fair Average for student 5 (with a measure of 1.5 logits) is at 3.0 on the rating scale metric.
Approximating the rating scale expected score curve (model item characteristic curve) with a logistic ogive aids understanding and simplifies arithmetical operations when performing logit-measure to rating-metric conversions.
A typical Rasch performance assessment model is:
where Bn is the ability of person n; Di the difficulty of item i; Cj the severity of judge j; Fk the impediment to being observed in category k relative to category k-1; Pnijk is the probability of being observed in category k; and Pnij(k-1) in category k-1.
Exact computation of the rating scale characteristic curve is arduous and error-prone. The curve, however, can be usefully approximated with a simple logistic curve with two parameters, one for location and one for slope. The close relationship between the Rasch characteristic curve and a logistic approximation is shown in Figure 2.
Inspect the output of your Rasch analysis program.
Identify the rating scale parameters (e.g., Table 3 in BIGSTEPS, Table 8 in Facets). Figure 3 is a typical example. Plotting the category scores against the expectation measures produces the rating scale characteristic curve shown in Figure 2.
A logistic approximation is:
where B is the examinee's measure, s is the corresponding value on the rating scale metric; l is the bottom category(1 in the example) and h is the top category (4 in the example); X is a slope parameter and M is a location parameter. The corresponding explicit form for s is:
A serviceable value for X can be estimated from the output of a Rasch calibration program. Decide the range of the expected score curve you wish to match. In this example, I've decided to match the curve along as much of its useful range as possible. In particular, I've chosen to have exact agreement at the points corresponding to expected scores of 1.5 and 3.5. The measure corresponding to 1.5 is -2.9, labeled L, that corresponding to 3.5 is 3.1 labeled H. Then, we have two simultaneous equations using the two points:
yielding M = 0.1, X = 6/(2*ln(5)) = 1.9.
What Measure corresponds to an Expected Rating of 3?
In equation 2,
B = 0.1 + 1.9log[(3-1)/(4-3)]) = .1 + 1.9log(2) = 1.4, which approximates the reported estimate of 1.5.
What Expected Rating corresponds to a Measure of -1.5?
In equation 3,
which approximates the reported value of 2.0.
The difference between the expected rating and the average observed rating can be used to compute the effect of the item difficulties and judge severities on the examinee measure. The reported measure has been adjusted for item difficulty and judge severity and so corresponds to the expected rating. The observed average rating (Obs Avge)includes a context effect due to the particular items and judges encountered by the examinee. An observed measure corresponding to the observed average rating can be computed using the logistic approximation:
where BObs is the measure based on the Observed Average. Then the impact of the context on the measure is
How much impact has context had on a examinee's measure?
Here is one examinee from a Facets report, using the same 4 category rating scale as before:
|Obsvd Obsvd Obsvd Fair Logit |
|Score Count Average Average Measure|
| 87 15 2.8 3.0 1.51 |
This examinee's reported measure is 1.51 logits. The Fair Average (expected rating) is 3.0 rating points. According to the logistic approximation, the measure corresponding to 3.0 is 1.4, which is close to the reported measure of 1.51, as expected.
The observed score is 87 from 15 observations, producing an observed average of 2.8 as shown. The observed measure, BObs, is, by the logistic approximation,
Thus, the effect of the examination context (of more severe than average judges and/or more challenging than usual items) is to make this examinee appear to perform 1.51 - 0.9 = .6 logits worse than in a standard situation.
John Michael Linacre
Communicating Examinee Measures as Expected Ratings. Linacre J. M. Rasch Measurement Transactions, 1997, 11:1 p. 550-551.
|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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