[Rasch] Sparse matrices of results

Fraser Rew Fraser.Rew at nzqa.govt.nz
Mon Jan 28 08:09:25 EST 2008

I work for the New Zealand Qualifications Authority which, amongst other things, is responsible for awarding qualifications to secondary school (high school) students in New Zealand. As these can be attained in any combination of subjects, we would like to know the comparative difficulties of each of the subjects. As a background, each subject (about 30 in total) is split into standards (about 5 per subject at each of three levels), each relating to a different topic. 

We are trying to find the best statistical model for this and are considering Rasch as one possibility, with each standard as one item. However a model of the type typically used on exams will not be particularly efficient, as there are around 40,000 students, and most students attempt around 25 of the approximately 150 available standards. The results matrix will therefore be very sparse which leads to difficulties both technical (how do we treat the many empty squares?) and practical (having 40,000 rows and 150 columns will take some time to compute).

For each standard there are three grades of achievement and one of non achievement, however to begin with we'll just be looking at the Achieved/Not Achieved grade boundary as this is the most significant.

So we're trying to work out 1) what is the best way of finding which standards, and by implication which subjects (if any), are more difficult than others?, and 2) how can we do this as efficiently as possible given the large numbers of students and standards available? If anyone is able to help with this, or even advise of some places to start looking for information, that would be much appreciated.

Fraser Rew

Fraser Rew
Researcher/Data Analyst
Qualificiations Division
DDI: 04 463 4368
Extn: 4368


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