# Rasch Estimates for Standard Datasets

Let's build a library of standard datasets and their Rasch estimates. These can be used to confirm that Rasch software is functioning correctly and also for teaching about Rasch estimation.

Estimation method:
AMLE = Anchored Maximum Likelihood Estimation (MLE for estimating person abilities with known item difficulties)
CMLE = Conditional Maximum Likelihood Estimation (R- eRm, WINMIRA)
JMLE = Joint Maximum Likelihood Estimation (R-mixRasch, Winsteps) - no correction for estimation bias
MMLE = Marginal Maximum Likelihood Estimation (R-ltm, ConQuest)
PMLE = Pairwise Maximum Likelihood Estimation (R-pairwise, RUMM2030)
WMLE = Warm's Mean Likelihood Estimation (applied to MLE estimates)

All estimates are in logits. The estimate for column 1 (item 1) is set to 0.0 logits.

Standard dataset 1:
Complete dichotomous dataset of 2 columns (items) and 2 rows (persons):
0,1
1,0

All Rasch estimation methods: column estimates: 0.0, 0.0 ; row estimates: 0.0, 0.0.

Standard dataset 2:
Complete dichotomous dataset of 2 columns (items) and 3 rows (persons):
0,1
1,0
0,1

CMLE column estimates: 0.00000, -0.69315
AMLE row estimates: -0.34658, -0.34658, -0.34658

JMLE column estimates: 0.00000, -1.38629
JMLE row estimates: -0.69315, -0.69315, -0.69315

MMLE column estimates: 0.00000, -1.38629
AMLE row estimates: -0.69315, -0.69315, -0.69315

PMLE column estimates: 0.00000, -0.69315
AMLE row estimates: -0.34658, -0.34658, -0.34658

Standard dataset 3:
Complete dichotomous dataset of 3 columns (items) and 3 rows (persons):
1,0,0
0,1,1
0,1,1

CMLE column estimates: 0.00000, -1.00505, -1.00505
AMLE row estimates: -1.40449, 0.05635, 0.05635
WMLE row estimates: -1.17098, -0.18498, -0.18498

JMLE column estimates: 0.00000, -1.56593, -1.56593
JMLE row estimates: -1.84142, -0.27549, -0.27549
WMLE row estimates: -1.63506, -0.50850, -0.50850

MMLE column estimates: 0.00000, -1.38629, -1.38629
AMLE row estimates: -1.69820, -0.17070, -0.17070
WMLE row estimates: -1.48169, -0.40644, -0.40644

PMLE column estimates: 0.00000, -0.69315, -0.69315
AMLE row estimates: -1.17436, 0.24746, 0.24746
WMLE row estimates: -0.93166, 0.00210, 0.00210

Standard Dataset 4: Rating Scale
8 persons (rows) respond to 8 items (columns) on a 0-3 rating scale:

```10000000
00210203
02001113
01012123
00122033
02110333
00123333
03333332
```
or
```1,0,0,0,0,0,0,0
0,0,2,1,0,2,0,3
0,2,0,0,1,1,1,3
0,1,0,1,2,1,2,3
0,0,1,2,2,0,3,3
0,2,1,1,0,3,3,3
0,0,1,2,3,3,3,3
0,3,3,3,3,3,3,2
```

The row (column) totals are 1, 8, 8, 10, 11, 13, 15, 20

This data matrix is symmetric. In principle, person (row) and item (column) standard deviations (S.D.) are the same.

Rasch logit estimates: Andrich Rating Scale Model, "Restricted" Model
Estimation methodSoftware Item difficulties = Columns Person abilities (thetas)= Rows Andrich ThresholdsAnalyst
MeanPopn. S.D.Sample S.D. MeanPopn. S.D.Sample S.D. 123
JMLE Winsteps .00 1.33 1.42 -.39 1.33 1.42 -.39 .14 .25 J.M. Linacre
JMLE jMetrik .00 1.3793 -.3831 1.3801 -.35 .14 .21 Dr. Bill Plummer
Priyanka
JMLE mixRasch (R) .00 1.382 -.3837 1.3821 -.35 .14 .21 Vernon Mogol

Standard Dataset 5: The LSAT Dichotomous Data
1000 persons (rows) respond to 5 items (columns) scored 0-1:

The data matrix is here.

Comments, corrections and suggestions for more standard datasets are welcome.

John Michael Linacre
mike \at/ winsteps.com

Rasch Estimates for Standard Datasets. Linacre JM. … Rasch Measurement Transactions, 2014, 28:1 p. 1453-4

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

 Forum Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com