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RE: st: IRT with GLLAMM


From   jverkuilen <jverkuilen@gc.cuny.edu>
To   <statalist@hsphsun2.harvard.edu>
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
loadings (item discriminations) are regression coefficients for the item dummy
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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