[Rasch] Solving longitudinal puzzles with Rasch?

Iasonas Lamprianou liasonas at cytanet.com.cy
Fri Dec 12 15:29:44 EST 2014

Dear all, 
I need to solve a longitudinal puzzle. I would love to use Rasch (if it is the most appropriate tool). My post is long, but my puzzle is complex! 

I have data from a computerized test. The students were allowed to log in whenever they wanted to take any number of short tests.Each test had 3-7 questions. Each test consists of different questions. There is no cosnistent pattern as to which tests were completed by the students (i.e. some students completed test A first but others would complete test Z first). The tests are not of the same difficulty. The items within a test are not of the same difficulty. The tests/items are not calibrated. There are many thousands of students and tens of tests (=hundreds of items). The teachers have a vague idea of the difficulty of each test, so they tried to match the difficulty of the test with the ability of the students. But of course, as I said, the tests are not calibrated (so the teachers were not really sure how difficult each test was), and they did not really have precise measures of the students ability (but of course they knew their students). This practice lasted for a whole year. Some students were more industrious, so they used log in every week (any time during the month/year) and they used to take a test. Others logged in once a month; and others only logged in once and took only one test. Overall, the students have taken on average 4-5 tests (=15-20 items), at random time points across the year. However, the ability of the students changed across the year. My question is how can I use (if I can) the Rasch model to analyze the data? In effect, my aim is: (a) to calibrate all the tests/items so that I can have an item bank, and (b) estimate student abilities at the start and end of the year (wherever possible) to measure progress. I am ready to assume that item difficulties do not change (we do not alter the items) but student abilities do change (hopefully improve) across time. 

I am not sure if this puzzle can be solved using Rasch models. I thought that I could split the year in intervals of, say, 2 months. Assume that the ability of each person during those two months is more or less the same. Also assume that each person is a different version of itself in the next two months. Then assume that item difficulties are fixed. Then run the analysis with six times the number of students (each two months the student "changes"). This has the problem that the students are not really "different" and there should be a lot of collinearity (dependence). 
Any idea will be values and considered to be significant. 
Thank you very much 
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