# Estimating Rasch Models with Stata Conditional (fixed-effects) Logistic Regression

For short tests of i=1,L dichotomous items taken by n=1,N subjects, we can easily obtain conditional-maximum-likelihood estimates of the item difficulty parameters, Di, of the Rasch model using the logit fixed-effects estimator in the clogit and xtlogit, fe commands of Stata. Data may be unbalanced.

These commands require that all responses (usually coded, 0/1) are stored in separate observations, while a group-variable is used to identify the observations that belong to the same subject. The reshape command may be useful to convert data into this format.

We describe the Rasch model to Stata as a "conditional logit model" with L covariates x(nik), k=1,L, so that x(nik) = -1 if i=k, and 0 otherwise. The regression coefficient of x(..k) becomes Dk. In order to set the origin of the Rasch measurement scale, Stata will automatically withdraw the first predictor variable D1 and fix it at zero. Alternatively, and better, is to withdraw the most stable of the Di yourself.

Conditional (fixed-effects) logistic regression
Number of observations = 1050
mathEst.S.E.
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
0
.08
.96
1.47
1.80
2.08
2.83
3.46
4.03
4.18
- (fixed)
.40
.38
.38
.38
.38
.39
.40
.42
.43

Here is the procedure for obtaining item calibrations:

(1) if necessary, transform the data into long-format
reshape item*, n(subject_id) j(ItemID)

(2) create explanatory variables for the Di
for num 1/L : gen D = -(X==ItemID)

(3) estimate the Di parameters
clogit item D*, group(subject_id)
or
xtlogit item D*, n(subject_id) fe

 Subject Score Distribution Score Freq. % 0 1 2 3 4 5 6 7 8 9 10 7 5 11 18 8 12 17 12 13 9 8 6 4 9 15 7 10 14 10 11 7 7 dropped as extreme                   dropped as extreme Total: 120 100

Example: The responses of 120 subjects to 10 math problems are coded as 1 (correct) and 0 (incorrect) - see Subject Score Distribution Table. I want to know whether the 10 problems maintain their difficulties, regardless of the subjects' abilities.

Apply clogit - the box above shows selected output.

To examine item difficulty invariance, we compare the item calibrations obtained from the full sample with those obtained from the low-performing (scores<=5) and high-performing (scores>=5) subjects. The box below shows selected results.

Item Calibrations
Math All
Scores
Low
Scores
Difference
Low-All
High
Scores
Difference
High-All
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
0
.08
.96
1.47
1.81
2.08
2.83
3.46
4.04
4.18
0
-.09
.87
1.17
1.57
2.03
3.69
3.39
3.39
3.39
0
-.17
-.09
-.30
-.24
-.05
.86*
-.07
-.65*
-.79*
0
1.43
1.67
2.76
2.94
2.94
3.63
4.37
4.86
5.09
0
1.35
.71
1.28
1.14
.86
.79
.91
.83
.91

We are pleased to see that for low scorers, the items are exhibiting statistical invariance according to the hausman test. Nevertheless, items 7, 9 and 10 (the harder items) are somewhat problematic, meriting further investigation. For high scorers, it appears that almost all items have become about 1 logit more difficult. In fact, it is item 1 that has become about 1 logit easier. Item 1 is the easiest item and is most off target for the high scorers. This illustrates the need to choose the fixed (i.e., withdrawn) item carefully in order to avoid confusion in interpreting results. After an adjustment is made for item 1, the item difficulties are seen to be statistically invariant also for the high scorers.

If the wording of the items is known, then producing an item map of the construct based on these item difficulty calibrations is straightforward via the Stata graphical plotting command graph.

Once item difficulties have been computed, the subject measures may be estimated (via maximum-likelihood, conditional on the estimates of the item parameter) via logit (combined with byvar).

With the Stata command xtlogit, re, it is easy to estimate the Rasch model in which the person-parameters are treated as random effects, conditional on person-level variables.

For complete details of this analysis, see www.stata.com/support/faqs/stat/rasch.html

Jeroen Weesie
Department of Sociology/ICS
Utrecht University

Estimating Rasch Models with Stata: Conditional (fixed-effects) Logistic Regression, Weesie J. … Rasch Measurement Transactions, 2000, 13:4 p. 724

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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