# Item discrimination of poorly targeted items

In test theory, it appears to be widely recognized that very "easy" or "very difficult" items, relative to the ability of the sample of respondents, will have deflated correlations with the rest of the test (item-rest correlations). One way of explaining this, is that floor and ceiling effects will limit the variability of the responses, and hence any correlations with these. Item response theory would seem to take this into account. Looking at a sigma shaped item characteristic curve of e.g. a very easy item, data points are distributed at the upper right corner of the curve, where the slope is much less steep than at the middle of the curve. The above graph only illustrates this to some degree, but imagine that the dots were skewed even further to the right. It appears to me that the Rasch model (as well as other IRT models, I would assume) extrapolates the item response function across to the lower ability/person location levels, and estimates the item discrimination at the midpoint (the steepest part of the curve, where p=.5), where indeed no actual data points may exist. Would it be correct to say then, that the IRT item discrimination in this case offers a sample independent alternative to an item-rest correlation, an alternative that isn't affected by low variance? I've found plenty of references to the Rasch model being sample independent, but can't find anyone touching specifically on the above issue. The conclusion I read from the above ICC is that the model predicts the item to work well in a sample with lower trait level (θ). However, I would like to make sure this is an accurate conclusion, and preferably be able to back this statement up with a reference. Any thoughts?

• This is pretty hard to read, at least on my phone. Consider some line breaks? The graph is much appreciated and this is a good question – shadowtalker Mar 26 '15 at 4:38
• @ssdecontrol - Reads fine on my galaxy s4. – Erik Mar 26 '15 at 9:20
• I think @ssdecontrol is referring to the flow of the text, in that it's hard to visually discern where and what your questions are, and what is important to read. Something more like this might be helpful to break up the text flow stats.stackexchange.com/questions/125287/… – philchalmers Mar 29 '15 at 16:06

• @Erik what do you mean you have range restriction only on a few items, how would you know that? All you should know about those items is the item's slope and intercept. Your person parameters ($\theta$) should be distributed around N(0,1) if you parametrized the 2PL in the usual way, so I'm not sure the graph you are referring to above can happen. Can you elaborate or demonstrate what you are seeing? (might require an edit to your original question) – philchalmers Apr 6 '15 at 23:08