From STAT-L Mon Oct 1 00:00:00 1993
Date: Sat, 2 Oct 1993 16:47:00 EST
Reply-To: Stat-l Discussion List
Sender: Stat-l Discussion List
From: John Uebersax
Subject: Rasch model software and FAQ
This message has three parts: (1) some frequently asked questions
about the Rasch model, (2) a list of current software, and (3)
references.
The information here is mainly oriented to consulting
statisticians, rather than people who want do delve into the
theoretical aspects. Attention is restricted to models for
dichotomous data.
Thank you to the following people who provided valuable
assistance with this material: Harold Ayabe, David Delong,
Travis Gee, Joop Hox, Brian Junker, Dennis Roberts, Magnus
Stenbeck, and Robert Terry. (Thank you also to anyone whose name
I may have inadvertently left out).
***NOTE: This file is 411 lines long***
FREQUENTLY ASKED QUESTIONS (FAQ) ABOUT THE RASCH MODEL
1. What is the Rasch model?
The Rasch model is a data analysis tool for creating multi-item
scales. One view is that it jointly locate items and subjects on
a continuum. For example, in the figure below, i1, i2, and i3
denote three dichotomous (scored right or wrong) items, and s1,
s2, and s3 denote three subjects. The location of subjects on
the continuum corresponds to their ability level. The item
locations correspond to their difficulty levels. If a person's
ability level exceeds an item's difficulty level, the person will
(generally) pass the item.
i1 i2 i3
<-----+---*---+----*----+---*----> trait level
s1 s2 s3
2. What is the Rasch model used for?
The Rasch model can be used for measurement (i.e., locating a
person on the latent continuum) or exploratory data analysis
(e.g., understanding the structure of items or selecting a useful
subset). It permits identification of items or behaviors that
are ordered (e.g., what is the sequence of skills one attains in
becoming a computer programmer). In that sense, it is similar to
Guttman scaling; however, unlike a Guttman scale, a Rasch scale
allows that (with reference to the figure above), there is a
certain probability that, for example, subject s2 might answer
item i1 incorrectly or i2 correctly. A Rasch scale is thus
similar to a Guttman scale with allowance for measurement error.
An article by Safrit, Cohen and Costa (1989) gives an excellent
introduction showing practical applications of the Rasch and
related models.
3. What are related methods?
The Rasch model is subsumed under the larger heading of item
response theory (IRT); other essentially synonymous terms for IRT
are latent trait models and modern psychometric theory.
Various models of this larger set differ according to whether the
response function (the probability of a positive response given
various ability levels) follows a logistic or a normal ogive
curve (the two are very similar).
Models also differ in the number of parameters allowed for the
response function of each item. The Rasch model is the same as a
1-parameter logistic (1PL) latent trait model. Other IRT models
include 2 and 3 parameter logistic models (2PL, 3PL) and 1, 2,
and 3 parameter normal ogive (1PN, 2PN, and 3PN) models.
The normal ogive models follow from a simple model that involves
response thresholds and normally distributed measurement error.
The logistic models are generally justified as close and
computationally convenient approximations to the normal ogive
model. The Rasch or 1PL model, however, can be viewed as
deriving directly from a different theoretical model.
4. When do you use the Rasch model?
The Rasch model is fairly simple IRT model. That is an advantage
and a disadvantage. The disadvantage is that such a simple model
may not fit the data. The advantage is that, if you only have a
small number of subjects, it may be better to use a simpler model
with fewer parameters.
5. Are there limiting assumptions?
The first limiting assumption with the usual Rasch model is that
there is only one latent dimension underlying the items; that is
not a severe limitation, since one can easily eliminate items
that appear to violate the assumption. Second, because it is a
"one parameter" model, the Rasch model makes an assumption
analogous to equal measurement error for each item.
6. Are there variations on the Rasch model?
The Rasch model has been extended to items with ordered response
levels, and to multidimensional latent traits.
7. Are there different estimation methods?
Most programs use one of three estimation methods: joint maximum
likelihood (ML), conditional ML, and marginal ML.
Joint ML (also sometimes called unconditional ML) jointly
estimates the locations of items and subjects on the trait
continuum. Conditional ML estimates item locations by making
"distribution free" assumptions. Marginal ML estimation assumes
a prior distribution of subject abilities (usually a normal
distribution), and estimates item locations based on this
distribution. The consensus is that all three methods usually
give comparable results. Some are more computationally demanding
than others, however, and the number of items may affect which is
used. Any will produce satisfactory results in most applied
situations.
Conditional ML estimation is sometimes favored by statisticians,
because of interesting technical properties. For example, it is
possible to obtain the conditional ML estimates by a variety of
methods, including loglinear analysis (e.g., Kelderman, 1984) and
special forms of latent class analysis (Lindsay, Clogg & Grego,
1990).
It is also possible to closely approximate the Rasch model with
so-called heuristic estimation procedures. This is briefly
discussed at the end of the software list below.
8. What are some good texts or references for more information?
Hambleton, Swamination & Rogers (1991) give a good overview of
logistic IRT methods, including the Rasch model. Andrich (1988)
provides a good and affordable (part of the Sage "Quantitative
Applications" series) introduction to the Rasch model. Wright and
Stone (1979) also give a good introduction to the Rasch model.
Rasch (1960/1980), of course, is an obvious source to consult.
Lord & Novick (1968) remains an important source for IRT methods
in general.
9. What software should I use or recommend?
Any of the programs listed below should be adequate to estimate
the Rasch model. Some estimate more complex 2-parameter and
3-parameter models as well, which may be a consideration for some
applications.
Perhaps the best recommendation is to use what is most easily
available. Rasch and IRT models are most common in educational
and psychological research. At a large university, there is
a good chance that someone in the Educational Psychology or
Psychology departments already has one of these programs.
RASCH MODEL SOFTWARE
Much of the following list can be found in Hambleton, Swaminathan
and Rogers (1985), pp. 48-50 and 159-160. Some of that
information has been updated here, and a few other programs
noted.
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PC-BILOG $345
PC-MULTILOG $200
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Scientific Software
1525 East 53rd St
Suite 906
Chicago, IL 60615
phone: 800-247-6113
312-684-4920
fax: 312-684-4979
Comments: 1, 2, and 3PL models; marginal ML estimation.
-----------------------------------------------------------------
LOGIMO
-----------------------------------------------------------------
Henk Kelderman
Department of Educational Measurement
University of Twente
Postbus 217
7500 AE Enschede
The Netherlands
Comments: Loglinear formulation of Rasch model; conditional
ML estimation.
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RASCAL $175
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Assessment Systems Corporation
2233 University Avenue
Suite 200
St. Paul, MN 55114
Phone: 612-647-9220
Fax: 612-647-0412
email: asc@mr.net
Comments: Joint ML estimation; another program, ASCAL
($400) estimates 1, 2, and 3PL models.
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BIGSTEPS $550
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MESA Press
5835 S. Kimbark Ave
Chicago, IL 60637-1609
Phone: 312-702-1596
Fax: 312-702-0248
email: MESA@uchicago.edu
Comments: Replaces BICAL, BIGSCALE; joint ML estimation.
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PML $250
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Jan-Eric Gustafsson
University of Goteborg
Institute of Education
S-431 20 Molndal
SWEDEN
Also available from the dutch company ProGamma (email:
gamma.post@gamma.rug.nl).
Comment: Conditional ML estimation.
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RASCH 1.0 free
-----------------------------------------------------------------
Germano Rossi
Universita degli Studi di Verona
Centro di Informatica e Calcolo Automatico
via dell'Artigliere 19
I-37129 Verona
ITALY
email: Germano@irvuniv.bitnet
The program can be obtained by ftp to rigel.oakland.edu. It is
in the directory \pub\msdos\statistics and is named RASCH10.ZIP
(self-extracting archive).
Comment: The program appears good, but, with one data set
it seemed to give incorrect results (perhaps
I did not use the program correctly). My
suggestion would be to contact the author and
see if a new version is available, or else
test it yourself to verify the results.
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MIRTE
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Mark Reckase
American College Testing Program
P.O. Box 168
Iowa City, IA 52243
Comments: 1, 2, and 3PL models; joint ML estimation.
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LOGIST
-----------------------------------------------------------------
Educational Testing Service
Rosedale Road
Princeton, NJ 08541
Comments: 1, 2, and 3PL models; joint ML estimation.
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NOHARM
-----------------------------------------------------------------
Colin Fraser
Centre for Behavioral Studies
University of New England
Armidale, NSW
AUSTRALIA 2351
Comment: Uses McDonald's nonlinear factor analysis model.
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OTD (Optimal Test Design) $550
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Norman Verhelst
National Institute for Educational Measurement
P.O. Box 1034
6801 MG Arnhem
The Netherlands
FAX: 01131 85 52 13 56
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RIDA
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Cees Glas
National Institute for Educational Measurement
P. O. Box 1034
6801 MG Arnhem
The Netherlands
Comment: Conditional or marginal ML estimation.
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MIRA
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Jurgen Rost
Institute for Science Education
Kiel University
Olshausenstrasse 62
D-2300 Kiel 1
Germany
Comment: Also estimates Rasch mixture model.
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TESTAT (add-on module) $110
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Available from various
commercial dealers.
Comments: Requires SYSTAT
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Other Alternatives
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Frank Baker's recent book on IRT includes many programs. You can
consult either the book or perhaps contact him directly to check
on the programs' availability. His address is
Department of Educational Psychology
Univ. of Wisconsin
Madison, WI 53706
Because the Rasch model can be set up as a loglinear
(quasisymmetry) model, it is possible to estimate it with general
software such as GLIM or SAS PROC CATMOD.
Lindsay, Clogg & Grego (1991) recently described how certain
latent class models could be used to estimate item parameters of
the Rasch model, and that these are the same as the conditional
ML estimates. In the article, they refer to a computer program
they have written for this. You can contact Bruce Lindsay or
Clifford Clogg at
Department of Statistics
Pennsylvania State University
University Park, PA 16802
email (Clogg): ccc@psuvm.psu.edu
PC-LISCOMP, distributed by Scientific Software (see address above
for PC-BILOG), although generally thought of as a tool for
structural equation modeling, can probably be used to estimate
normal-ogive IRT models.
Lord and Novick (1968; pp. 377-378) noted how factor analysis can
be used to approximate results of 1PN and 2PN IRT models. Bock &
Lieberman (1970), applying this "heuristic" approach, factor
analyzed matrices of tetrachoric correlations and obtained
results that closely approximated results of marginal ML
estimation. This alternative would require software that
calculates tetrachoric correlations, and probably a reasonably
flexible factor analysis program (e.g., LISREL or EQS).
REFERENCES
Andrich D. (1988). Rasch models for measurement. Newbury Park,
CA: Sage.
Bock RD, Lieberman M. (1970). Fitting a response curve model for
dichotomously scored items. Psychometrika, 35, 179-198.
Hambleton RK, Swaminathan H. (1985). Item response theory.
Boston: Kluwer-Nijohoff (pp. 144-147, 7.8 Approximate Estimation
Procedures)
Hambleton RK, Swaminathan H, Rogers HJ. (1991). Fundamentals of
Item Response Theory Newbury Park: Sage.
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika
49, 223-245.
Lindsay B, Clogg CC, Grego J. (1991). Semiparametric estimation
in the Rasch model and related exponential response models,
including a simple latent class model for item analysis. JASA,
86, 96-107.
Lord FM, Novick MR. (1968). Statistical theories of mental
test scores. Reading, Massachusetts: Addison-Wesley.
Rasch G. (1960; reprinted 1980). Probabilistic models for some
intelligence and attainment tests. Chicago: University of
Chicago Press.
Safrit MJ, Cohen AS, Costa MG. (1989). Item response theory and
the measurement of motor behavior. Research Quarterly For
Exercise and Sport, 60, 325-335.
Wright BD, Stone MH. (1979). Best test design: Rasch measurement.
Chicago: MESA Press.
--John Uebersax
Bowman Gray Medical School
Winston-Salem, NC 27157
uebersax@phs.bgsm.wfu.edu
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