In analyzing clinical or educational data, persons (or items) may already be characterized with "numbers" that are asserted to be linear measures. But it is not known what is the linear transformation between these measures and the local logits of the current data set.
One simple approach to this problem is to analyze the data, and plot the resulting person measures (or item difficulties) against their "numbers" to identify the transformation. This technique, though generally successful, overlooks misfit and sample-item targeting.
Another approach is to optimize the fit statistics. In this technique, one chooses an initial number-to-logit conversion, uscale, that spreads the sample (or items) conspicuously wider along the measurement variable than is reasonable. Perform a data analysis with the persons (or items) anchored at their converted numbers. All summary mean-squares (average infit and outfit mean-squares summarizing all persons and items) are expected to be noticeably above 1.0. If not, double the number-to-logit conversion factor, uscale, and repeat this step of the procedure.
Once all four summary mean-squares are greater than 1.0, a useful number-to-logit conversion factor has proved to be uscale = uscale divided by the average of the four summary mean- squares. Reanalyze the data, and, if the average of the four summary mean-squares is still noticeably greater than 1.0, repeat this step of the procedure.
In preliminary investigations with clinical indicators, this has proved to be a fast and easy way to build useful measurement systems.
Example: When the persons were anchored at their clinical indicator values, the mean person infit statistic was 1.93, outfit 2.19. The mean item infit statistic was 1.59, outfit 2.27. The average of these four numbers is 2.0. On reanchoring the persons at their clinical indicator values, divided by 2.0, the mean-squares became, respectively, .98, 1.08, 1.02, 1.08. This suggests that a probabilistically interpretable measurement system has been constructed.
John Michael Linacre
Using Rasch Fit statistics to Rescale Linear External Numbers or Measures into Rasch Anchor Values. Linacre, J.M. Rasch Measurement Transactions, 2000, 14:2 p. 750.
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