# PCA: Variance in Data Explained by Rasch Measures

A Rasch model predicts that there will be a random aspect to the data. This is well understood. But what does sometimes surprise us is how large the random fraction is.

The Figure shows the proportion of raw-observation randomness predicted to exist in dichotomous observations under various conditions. Note (2015): Practical experience indicates that these curves are useable approximations for polytomous items.

The x-axis is the absolute difference between the mean of the person and item distributions, from 0 logits to 5 logits. The y-axis is the percent of variance in the data explained by the Rasch measures.

Each plotted line corresponds to one combination of standard deviations. The lesser of the person S.D. and the item S.D. is first, 0 to 5 logits, followed by "~". Then the greater of the person S.D. and the item S.D.

Thus, the arrows indicate the line labeled "0-3". This corresponds to a person S.D. of 0 logits and an item S.D. of 3 logits, or a person S.D. of 0 logits and an item S.D. of 3 logits. The Figure indicates that, with these measure distributions about 50% of the variance in the data is explained by the Rasch measures.

When the person and item S.D.s, are around 1 logit, then only 25% of the variance in the data is explained by the Rasch measures, but when the S.D.s are around 4 logits, then 75% of the variance is explained. Even with very wide person and item distributions with S.D.s of 5 logits only 80% of the variance in the data is explained.

For the unexplained variance, see Critical Eigenvalue Sizes (Variances) in Standardized Residual Principal Components Analysis (PCA).

In early versions of Winsteps, specify PRCOMP=R

How the Table was computed

This Table was produced with Excel:

1. Normal distributions of logit measures for the items and persons were simulated using specified S.D.s and targeting (= mean difference between person measures and item measures).
2. For each person-item combination were computed:
the Rasch-dichotomous expected-score: Eni = 1/(1+exp(Di-Bn)
and model-variance of the expected-score: Wni = Eni(1-Eni)
3. The expected scores were averaged
4. Sum-of-squares of explained variance is: sum[(expected score - average expected score)²]
5. Sum-of-squares of unexplained variance is: sum( model-variances of expected scores )
6. Explained variance = (Sum-of-squares of explained variance) / (Sum-of-squares of explained variance + Sum-of-squares of unexplained variance)

Here are some percentages for empirical datasets:

 % Variance Explained Dataset Winsteps File name 71.1% Knox Cube Test exam1.txt 29.5% CAT test exam5.txt 0.0% coin tossing - 25.8% Liking for Science(3 categories) example0.txt 37.5% NSF survey(3 categories) interest.txt 30.0% NSF survey(4 categories) agree.txt 78.7% FIM® (7 categories) exam12.txt

John Michael Linacre

Editor, Rasch Measurement Transactions

Variance in Data Explained by Rasch Measures. Linacre, J.M. … Rasch Measurement Transactions, 2008, 22:1 p. 1164

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 (entirely in Italian). 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