## Thursday, January 13, 2011

### Rasch Estimated Measures

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Deep within the Rasch model is the mystery of how person and item normal scales are combined into one logit scale. Ministep starts with a typical data matrix, such as PUP Table 3.

Question difficulty is converted from the number of right marks to the number of wrong marks in Step 1.  The average difficulty of 84% right is now expressed as 16% wrong.

The cells recording right and wrong marks in PUP Table 3 are converted to probabilities. The marginal cells, in Table 3, for student score and question difficulty are converted from normal to logit values in Step 2.

The initial location of student scores and question difficulty ranges from  nearly -5 logits to nearly +5 logits for the student nurse test results on PUP Table 3.

In Step 3, the final location for estimated student ability and item difficulty measures results when the item difficulty measure of zero (0) logits rests below the normal average student score (84% right). The normal average test item difficulty (16% wrong) rests below the student ability measure of zero (0) logits.

--------0--------------84%------- person ability
----------16%-------------0-----       item difficulty

Equivalent means are in registration. They mark off equivalent lengths (1.66 measures) of the logit scale on this test.

This is not just a case of shifting the item tally past the ability tally, but a re-plotting of the item values onto a single logit scale. Re-plotting compressed the negative item measure locations by 0.84 and expanded the positive item measure locations by 1.16 to create a close fit to the Winsteps Person-Item Bar Chart.

There are many ways to set person and item final locations, the basis for the test characteristic curve (TCC). Winsteps uses two methods in series, normal approximation algorithm (PROX) and joint maximum likelihood estimation (JMLE). For this test it cycled through PROX twice and then through JMLE twice.