The usual 2PL model has only one random effect but the model is bilinear. Let t_i be the random trait of subject i. Then for item j
logit(p_ij) = a_j * t_i + b_j
Usually specify
T ~ N(0,1).
There are other specifications. Estimation is then MML with integration over T.
The Rasch model restricts a_j = a, or, equivalently, estimates the variance of T. This model can be easily fit by MML using -xtlogit- or -xtmelogit- by stacking the data long and using item dummies as fixed effects. There is a nice example out there showing how to do this with -clogit- by Phil Ender on ATS web page (Google for it, I can't dig it out right now).
JV
-----Original Message-----
From: "Joseph Coveney" <[email protected]>
To: [email protected]
Sent: 3/5/2009 10:07 AM
Subject: RE: st: IRT with GLLAMM
I'm not sure what kind of convergence problems you're experiencing with
-gllamm-. Is it just slowness? With the two-parameter model, my understanding
is that you'd be fitting 30 random effects--something that would require a great
deal of patience with -gllamm- at least with more than a few integration points
and without multiple processors.
There are some examples of these kinds of models fitted with -gllamm- in Xiaohui
Zheng & Sophia Rabe-Hesketh. (2007) Estimating parameters of dichotomous and
ordinal item response models with gllamm. _The Stata Journal_ 7(3):313-33. They
limit themselves to a relative few test items, nowhere near 30.
As far as fitting an analogous model with -xtmelogit-, couldn't you set up an
equation on the random effects side of the double-pipe for student-by-test item
interaction terms (the 30 random effects)? It would seem that the common tactic
of omitting the first test item in the random effects equation (omitting it from
the equation as the constant) identifies the model by fixing the first test
item's loading factor (allowing the variance for the random effect for students
to be free).
I think that traditionally with IRT models, the random effects for students
would be constrained to unit variance, which allows for all of the item factor
loadings to be estimated (free)--they're held to be equal for the Rasch model (a
single random effect, fitted with -xtlogit- as Jay mentions and as you show
below) and allowed to be independently estimated in the two-parameter model
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