# 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 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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Coming Rasch-related Events
March 21, 2019, Thur. 13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019
April 4 - 8, 2019, Thur.-Mon. NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual
April 5 - 9, 2019, Fri.-Tue. AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting
April 12, 2019, Fri. On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch
May 24 - June 21, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - 30, 2019, Wed.-Thu. Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/
June 4 - 7, 2019, Tue.-Fri.In-Person Italian Rasch Analysis Workshop based on RUMM (Fabio La Porta and Serena Caselli; entirely in Italian). Prof David Andrich from Western Australia University will be hosted by the workshop. For enquiries and registration email to workshop.rasch@gmail.com
June 17-19, 2019, Mon.-Wed. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement
June 20-21, 2019, Thurs.-Fri. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 2-5, 2019, Tue.-Fri. 2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org
July 11-12 & 15-19, 2019, Thu.-Fri. A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses
Aug 5 - 10, 2019, Mon.-Sat. 6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html
Aug. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
August 25-30, 2019, Sun.-Fri. Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/
Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Nov. 3 - Nov. 4, 2019, Sun.-Mon. International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm
Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - June 19, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 26 - July 24, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 7 - Sept. 4, 2020, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com