Hi Lee,
If you go to my web page for Latent Trait and Item Response Theory (IRT)
Models,
http://ourworld.compuserve.com/homepages/jsuebersax/lta.htm
(please let me know if this link doesn't work)
that will point to several other pages that might help.
> Then the IRT curve that I am looking for (something they call a 3-parameter
> logistic, which I think is not a 100% correct name) is described by the
> following function (best viewed in a fixed-width font):
A well-kept secret is that it is just as easy to estimate a probit
(cumulative gaussian) latent trait model. The probit model is
theoretically more appropriate in many applications.
Of course, you will need to decide, if you haven't already, whether to
pursue a 1- 2- or 3-parameter model.
> find a reference that tells me exactly the recipe for finding it, but the
> best I can tell is that the algorithm would start with an initial guess for
> T, fit the curve parameters a, b, and c, then use this curve to re-estimate
> T. The process repeats until some convergence criterion is reached.
That's one approach. Another is "brute force" optimization, where one
uses a general purpose optimization routine to (simultaneously) find the
set of paramter values that maximizes a given criterion--usually the
log-likelihood.
Here's a good book that covers the material without making things more
complicated than necessary:
Hulin, C. L., F. Drasgow, C. K. Parsons, Item Response Theory,
Homewood, Illinois, Dow Jones-Irwin, 1983.
I'd also recommend looking at some of Bock's work, such as:
Bock, R. D., and Aitkin, M. (1981). "Marginal Maximum Likelihood
Estimation of Item Parameters: Application of an EM Algorithm,"
Psychometrika, 46, 443-459.
Of course, the "bibles" are still:
Lazarsfeld, P. F., and Henry, N. W. (1968), Latent Structure
Analysis, 2oston: Houghton Mifflin.
Lord FM, Novick MR. (1968). Statistical theories of mental test
scores. Reading, Massachusetts: Addison-Wesley.
> Does anyone know if SAS will do this?
One of my pages describes how to estimate a 2-parameter latent trait
model by factor-analyzing a matrix of tetrachoric correlations. SAS
(via a macro available on the SAS site) can produce a matrix of
tetrachoric correlations. And the matrix can be supplied to and
factored by PROC FACTOR.
This works pretty well for estimating the item paramters (slopes and
thresholds). However if you also want to score respondents (i.e.,
estimate their latent trait levels) that takes a little more work (a
separate page on my site talks about this).
A 1-parameter Rasch model can be formulated as a loglinear model.
Therefore it might be possible to use say, PROC CATMOD or something like
that to estimate a Rasch model.
> I have found a piece of software
> that claims to fit "Rasch models", but the classical Rasch model is a
> one-parameter version of what I'm looking for (set b and c to zero, and
> you have a Rasch model).
Correct. I prefer 2-parameter models, unless there is some theoretical
reason to expect a 1-parameter model (i.e., that all items have the same
correlation with the latent trait).
I maintain that the choice of logistic IRT vs probit IRT vs Rasch model
should be made based on the theoretical assumptions of each model and
the assumptions about your data. For example, Rasch has a very nice
theory about how people answer test items that justifies use of
Rasch modeling. (I don't necessarily agree with the model, but
it is interesting). On the other hand, if you have a familiar:
manifest trait = latent trait + error
model, where error is (a) normally distributed, and (b) homoscedastic (
error variance not correlated with latent trait level), and where
one assumes discretizing thresholds that convert latent continuous
responses to observed binary responses, then a probit latent trait
model is appropriate.
> Plus, the software costs about $1000, and I don't have that to spare.
> The software (one called "BIGSTEPS" is the only one I can find that will
> deal with the 89,000 students I have to deal with) is not exactly
> "Microsoft Bob" in its ease of use.
Check my web site. One page talks about software for estimating IRT and
Rasch models. Personally, for Rasch models, I use MIRA or WINMIRA; for
IRT models I use my own programs for "discrete latent trait" modeling:
Heinen T. Latent class and discrete latent trait models:
Similarities and differences. Thousand Oaks, California: Sage, 1996.
I also have a FAQ on the Rasch model on the site, including information
specifically on Rasch software.
Hope this helps.
John Uebersax
[EMAIL PROTECTED]
http://ourworld.compuserve.com/homepages/jsuebersax
P.S. The limiting factor on IRT software is usually the number of
items, rather than the number of subjects.
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