## Friday, January 21, 2011

### Person Item Map

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Ministep Table 1.0 combines the data in Table 13.1 Person, and Table 17.1, Item, into a second visual display of test results (also see bubble graphic). Estimated student ability and item difficulty measures are placed side by side, in one vertical dimension, and in the same sequence as the normal distributions are on PUP Table 3 in two dimensions.
The normally distributed test scores (right marks) and difficulties (wrong marks) have been edited into Table 1.0 to again show the difference between the two distributions (normal and logit). The logit scale suggests that it takes more effort to move from a score of 22 to 23 than from 15 to 16.

Winsteps Table 1.0 shows which students and questions match on an estimated student-ability:question-difficulty measure scale. The most efficient testing is done with items and students that have similar estimated measures.

The Rasch Model is therefore a common method for calibrating test items for use in computerized adaptive testing (CAT). After each examinee’s forced response to a question, the computer quickly calculates the expected range of success for this student and delivers a more difficult one if the response was right and an easier one if the response was wrong. The test ends when the score falls within, or without, preset confidence limits or the maximum number of questions or time is reached.

Healy, Nicho, and Summi, estimated person measure of 1.73, can be expected to earn a score of 85% on items with an estimated difficulty measure of zero (=exp(Ability-Difficulty)/(1+(exp(Ability-Difficulty)) or exp(1.73 - 0)/(1+exp(1.73 - 0)) or exp(1.73)/(1+exp1.73) or 5.65/(1+5.64) or 5.64/6.64 or 0.85 or 85%). This makes sense, as the average test score and item difficulty were 84%.

Insert your choice of ability and difficulty into the Rasch model to predict expected raw scores on future tests. Salto (or a group of students with Salto’s ability), estimated person measure of 0.34, can expect to answer 20% of items correctly that have an estimated difficulty measure of 1.73 (=exp(0.34 – 1.73)/(1+(exp(0.34 – 1.73)). Salto also has a 20% or 0.2 chance of answering correctly any one question with a difficulty measure of 1.73. CAT would use some easy questions to assess Salto. They could be like the questions on this test.