Conventional statistics and examinee raw scores may be all that remain of the data from a previous testing. These statistics provide the mean of the N examinee raw scores, their standard deviation and, for each item i, the rate of success of the sample of examinees, a percent-correct-value, "p-value", P_{i}, for i = 1 to L items.
In order to equate examinee performances on this earlier test with those on other tests, or to add these earlier items to an item bank, conversion from the raw score metric to a linear metric is needed. This can be achieved with a simple, usefully accurate technique, provided that it is reasonable to think of the examinees as randomly selected from a normal distribution.
Here's how to construct logit measures from raw score statistics:
1. Check the raw score statistics for consistency:
Is the examinee mean raw score | ? |
Are there typographical errors?
Even if p-values were obtained from one sample and mean examinee score from another, they may still be close enough for this computation.
2. Compute a raw score-to-ability conversion factor, C_{b}:
3. Compute a logit examinee ability variance from SD, the examinee raw score standard deviation:
4. Obtain an item calibration expansion factor, X_{f}, to adjust item difficulties for examinee ability variance:
5. Compute a logit difficulty calibration for each item, d_{i}:
6. The standard error, SE_{i}, of calibration d_{i} for item i is:
7. Compute an initial logit ability estimate br^{o} corresponding to each raw score r from 1 to L-1:
8. Compute a final ability estimate br corresponding to each raw score r from 1 to L-1 by iterating the equation:
where
replacing b_{r}^{o} by b_{r} and recomputing, until
9. For examinees with extreme scores of 0, compute b_{0} using steps 7 and 8 with r = 0.25.
10. For examinees with extreme scores of L, compute b_{L} using steps 7 and 8 with r = L - 0.25.
11. The standard error, SE_{r}, of person ability measure b_{r}, corresponding to raw score r, including r = 0.25 and r = L - .25, is:
This algorithm has been applied successfully to empirical data.
A. J. Stenner, B. D. Wright & J. M. Linacre 1994 RMT 8:1 p. 338
From P-values and raw score statistics to logits. Stenner AJ., Wright BD., Linacre JM. … Rasch Measurement Transactions, 1994, 8:1 p.338
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