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:
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 |
Please email me your own percentages to add to this list.
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
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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 |
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