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 method	Software	Item difficulties = Columns			Person abilities (thetas)= Rows			Andrich Thresholds			Analyst
		Mean	Popn. S.D.	Sample S.D.	Mean	Popn. S.D.	Sample S.D.	1	2	3
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.

No estimates received yet.

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

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