3PL IRT model. Could someone point to a nice explanation and/or tutorial? I'm very new to IRT. Even so, have just been asked to try and learn to use 3PL model on the data that we have. I'm completely out of my depth, and what's worse is we'll have to run our own parameter estimations -- I can't just find appropriate app that will run all the necessary calculations and give me the end result.
I realize the model is not exactly easy, and to run it smoothly will most likely require its in-depth studying. Still, I would like to have somewhat quick and easy explanation/tutorial with the help of which I could at least superficially acquaint myself with what I will be dealing with. Ideally, it should not only have closed formulas for how parameters are estimated, but one or two examples. 
Could someone please point me to anything like this? Could be a book, a paper, anything...
 A: I know there are a lot of R users here, but in my experience, Stata's tutorials for the models it fits are clear and comprehensible, and their technical writing is generally better than the R packages I've inspected. Moreover, you can import Stata's data into R with the haven library, as documented here.
Stata's general IRT reference is here. I would advise that you familiarize yourself with the 2-PL model first. In a 2-PL model, you're basically assuming that one latent trait, $\theta$, drives logistic responses to a number of questions. Below, for the 2-PL let i index items/questions, j index persons, \$theta_j\ be the j-th person's trait level. Let a be the item discrimination, and b be the item difficulty.
$P(y_{ij} = 1 | \theta_j) = logit^-1[a_i(\theta_j - b_i) ]$
$\theta$ ~ $N(0,1)$
$logit^-1(x) = \frac{exp(x)}{1 + exp(x)}$
The graph below demonstrates discrimination and difficulty:
This stems from real data on the Patient Health Questionnaire (a depression screener), to which I fit a 2-PL model. The items represented by the blue and red lines have pretty similar discrimination, but their difficulty parameters vary quite a bit. You have to have $\theta = 0.71$ to have a 50% chance of endorsing the dyspohria (i.e. sad mood) question. You need $\theta_j = 2.57$, i.e. 2.57 whole SD above the mean level of depression, to have a 50% chance of saying that you're better off dead. So, the suicide ideation item is consistent with a much more severe level of depression than the dysphoria question; if this were an educational test, we'd say the latter question is much more difficult than the former.
Discrimination is the slope of the logistic curve. See how for a given change in depression, the probability of endorsing the appetite change item increases a lot more slowly than for the other two items? The appetite question is less able to discriminate between people who are low vs. high in depression.
Now, with the 3-PL model, you're adding a lower asymptote in. In education, this basically says that we assume everyone will guess at the correct answer. I remember the GRE was a 4-option multiple choice test, so the lower asymptote corresponds to a probability of 0.25. Math and more detail in the Stata documentation. I'm less familiar with this type of model, because I don't work in education.
