Currently I am reading Multidimensional Item Response Theory by one of my mentors Mark Reckase and I realize how very cool item reponse theory (IRT) is.
The underlying purpose of item response theory is measure two things simultaneously.
1. Measure the item's parameters. This means measure the characteristics of the test item such as difficulty of the item, likelihood that someone will correctly guess the item, and typically the discriminatory power of the item, ie. how much power the item has to identify the difference in ability between one student and another student.
2. While attempting to measure the items' parameters on a test, IRT methods must also estimate the ability of the students taking the test simultaneously. That is to say that each student has a different level of ability when entering the test and generally that ability level is unknown a-priori. Of course after taking the test the ability level is still not "known" but at least some reasonable approximation of the ability level of the student should be known at that time.
So why is this cool? Well, imagine having a data set with K items (variables) and N students (observations). Now, only knowing that either the students got the answers correct or they failed do an estimation which is largely similar to a logit model except that your exogenous variables (Xs) also need to be estimated.
At this point many of you will probably say, well, this is clearly possible if difficult you are willing to make some identifying assumptions.
So far, the only identifying assumptions that I see needed are 1. The probability that a student answers a question correctly is greater the greater the ability of the student, dP(Y=1|X,T,D)/dX>0. And 2. The probability that the question is answered incorrectly is greater the greater the difficulty of the item dP(Y=1|X,T,D)/dD<0 .="." and="and" assumption="assumption" form="form" know="know" make="make" model.="model." nbsp="nbsp" of="of" p="p" structural="structural" that="that" the="the" underlying="underlying" we="we" well="well">
I will be posting more on IRT as I learn.0>