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).
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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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