# 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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May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
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