# RE: st: IRT with GLLAMM

 From jverkuilen To Subject RE: st: IRT with GLLAMM Date Sat, 7 Mar 2009 11:10:57 -0800

```SAS Mixed has a very broad variety of covariance matrices for random effects, including the ability to fit a factorial structure and also to constrain arbitrary elements in a straightforward manner either to constants or if I recall correctly to impose equality constraints.

We'd need to do some stubby pencil and scratch paper work to see if using a one dimensional factorial structure would do the trick for the 2PL model. My inclination is that it does, but I am too jet lagged to believe much of anything right now.

If so, having a few more covariance matrices allowed in -xtmixed-, -xtmelogit- and -xtmepoisson- would mean they could be tricked into doing a broad variety of confirmatory factor analyses with Gaussian, binary and count variables.

-gllamm- can do all of these of course but the speed, computational robustness and relative ease of the XT series would make this very attractive.

I don't have the XT kung fu to figure out if this is already possible by clever choice of existing covariance matrices. I don't think so, but Bobby Guttierez has shown some surprising examples before....

JV

-----Original Message-----
From: "Joseph Coveney" <jcoveney@bigplanet.com>
To: statalist@hsphsun2.harvard.edu
Sent: 3/7/2009 4:24 AM
Subject: RE: st: IRT with GLLAMM

Jay Verkuilen wrote:

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).

--------------------------------------------------------------------------------

Thanks, Jay; I stand corrected--and reminded that the item dummy variables are
not being used as indicators for separately estimated variances.  The factor
variables on the single random effect.  If there's a way to get -xtmelogit-'s
random effects equation to specify regression of variables on a random effect
(for the two-parameter logistic IRT model), then it escapes me, too.  It seems
that what's needed for -xtmelogit- is something analogous to its
-covariance(identity)-, but with the ability to fit (all but one of) the
diagonal elements of that option's identity matrix as regression coefficients
instead of being fixed at 1.

Joseph Coveney

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