We have collected data and analyzed it. We believe that the our findings are a reasonable basis for predicting the future. Now we want to make that prediction.
Here is what our analysis has told us about a particular partial-credit item of our instrument:
|Observed Partial Credit Category||Observed frequency||Observed sample average measure||Expected sample average measure||Mean Expected rating measure||Median Rasch-Thurstone threshold||Modal (Rasch-Andrich?) threshold|
Our example uses a partial-credit item, but this discussion is equally applicable to predicting from "rating scale" data, and much of it applies to dichotomous data.
A. The conventional descriptive-statistical approach of, for instance, Generalizability Theory, is to assume that the next sample will exactly resemble the current one. In which case, the first three columns will suffice. For persons rated in category 1, we would predict a measure of -.51, which was the average measure of those observed in category 1 of this item in the earlier sample.
B. The earlier sample performed largely as the Rasch model predicts, but not exactly. We assume that the next sample will have the same measure distribution and exhibit the same Rasch-coherent behavior as the earlier sample, but the next sample's idiosyncratic non-Rasch behavior is unpredictable. In which case, the fourth column, the "expected sample average measure" is our prediction. It reflects only the Rasch-coherent aspect of the current sample. We expect that the next sample will exhibit small, but different, idiosyncratic departures from these measures, but, since we don't know what these idiosyncrasies will be, for persons rated in category 1, we would predict a measure of -.42, which would have been the average measure of those observed in category 1 of this item in the earlier sample, if that sample had followed exact Rasch-model predictions.
We expect the next person to behave in the same Rasch-conforming way as the previous sample, but we can make no distributional assumptions relevant to the next person. This is sometimes called a "non-informative Bayesian prior" and parallels our use of a tape measure or bathroom scale.
C. For the next individual who receives a rating of 2, we predict the measure corresponding to the point on the latent variable where a rating of 2 is most probable to be observed. This is -.61, its "expected rating measure". From this perspective, the measures corresponding to extreme categories, 1 and 3, are infinite, so the "expected rating measure" reported in the Table for category 1 corresponds to an expected rating of 1.25 (at measure -2.22), conceptually half-way between 1 and 1.5 (at measure -1.50), a boundary between category 1 and category 2. Similarly for category 3, the reported expected rating is for 2.75 (at measure 1.00).
D. Our prediction for a person in a particular category is the range of measures for which there is a 50% or greater chance that the person would be observed in this category or above, and also a 50% or greater chance that the person would be observed in this category or below. For this, the range boundaries are the Rasch-Thurstone thresholds. In our Table, the range of measures for category 1 would be from -infinity to -1.18. These values appear on cumulative probability plots as the points where a .5 probability line intercepts the cumulative curves.
E. Our prediction is the range of measures for which the observed category is the most likely category to be observed. These are the modal thresholds, and correspond to the Rasch-Andrich thresholds (when those are ordered). When the Rasch-Andrich thresholds are disordered, some categories will never be the ones most likely to be observed. In our Table, the range of measures for category 1 would be from -infinity to -.79. These values appear on category probability plots as the abscissae of the points where the probability curves for modal categories meet.
F. Our prediction is the range of measures for which the average rating is in the neighborhood of this category. The Rasch model predicts the probability of any category being observed anywhere along the latent variable. From these probabilities, the average value of the ratings at any point along the variable can be predicted. For any intermediate category, its neighborhood can be defined as the interval from "category value - 0.5" to "category value + 0.5". For extreme categories, the outer ends of the intervals are infinite. These "neighborhoods" are shown as transition values in the Mean column. . Here, the range of measures for category 1 would be from -infinity to -1.50. These values can be seen on the "expected score ogive" (the "model" item characteristic curve).
John Michael Linacre
Predicting Measures from Rating Scale or Partial Credit Categories for Samples and Individuals, Linacre J.M., Rasch Measurement Transactions, 2004, 18:1 p.972
|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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