Reliability, Separation, Strata Statistics

Reliabilities are often reported as though they were invariable characteristics of tests. Of course, they are not. They depend not only on the construction of the test, but also on the distribution of the examinee sample tested. Conventionally, only person separation reliability is reported, but item separation statistics are also useful indicators. They tell how well this sample of examinees have spread out the items along the measure of the test, and so defined a meaningful variable.

The Table below aids interpreting and predicting reliabilities. Its underlying components are:

Observed SD = the observed standard deviation of reported measures, for examinees or for items.

Root Mean-Square Error (RMSE) = "average" measurement error of reported measures.

True SD = standard deviation of reported measures corrected for measurement error inflation.

Observed SD and RMSE are calculated directly from the reported measures and their standard errors. Then,
(True SD)^2 = (Observed SD)^2 - (RMSE)^2

Separation Ratio:
G = (True SD)/(RMSE) is a ratio scale index comparing the "true" spread of the measures with their measurement error. It indicates the measure of spread of this sample of examinees (or test items) in units of the test error in their measures.

Separation Reliability:
G^2/(1+G^2) = (True SD)^2/(Observed SD)^2 = KR-20 or Alpha. This is a correlation coefficient, the ratio of true measure variance to observed measure variance. When G=1, True SD = RMSE, and reliability is 0.5. A reliability less than 0.5 implies that the differences between measures are mainly due to measurement error.

Discernible Strata: (4G+1)/3
The functional range of measures is around 4 True SD. Inflate this by 1 RMSE to allow for the error in the observed measures. Set a significant difference between two measures at 3 RMSE. Then there are (4 True SD + RMSE)/(3 RMSE) = (4G+1)/3
significantly different levels of measures in the functional range. See discussion at RMT 16:3 p. 888

Separation  KR-20, Alpha:    % Variance:         Distinct Strata:
 Ratio: G    G^2/(1+G^2)   Not Due Error/Due Error   (4G+1)/3
    0          .00          0/100                    1
    1          .50          50/50                    1
    1.5        .70          70/30                    2
    2          .80          80/20                    3
    3          .90          90/10                    4
    4          .94          94/6                     5
    5          .96          96/4                     7
    6          .97          97/3                     8
    7          .98          98/2                     9

William P. Fisher, Jr.

Wright, B. D., & Masters, G. N. (1982, pp. 92, 105-106). Rating scale analysis: Rasch measurement. Chicago, Illinois: MESA Press.

  1. Reliability, separation, strata statistics, Fisher WP Jr. … 6:3 p.238
  2. Reliability and separation nomograms, Linacre JM. … 1995, 9:2 p.421
  3. Reliability and separation, Wright BD. … 1996, 9:4 p.472
  4. Predicting Reliabilities and Separations of Different Length Tests, Linacre, J.M. … 2000, 14:3 p.767
  5. Going beyond Unreliable Reliabilities, Mallinson T., Stelmack J. … 2001, 14:4 p.787-8
  6. Separation, Reliability and Skewed Distributions: Statistically Different Levels of Performance, Wright B.D. … 2001, 14:4 p.786
  7. Number of Person or Item Strata (4G+1)/3, Wright BD, Masters GN. … 2002, 16:3 p.888
  8. Cash value of Reliability, WP Fisher … Rasch Measurement Transactions, 2008, 22:1 p. 1160

Reliability, Separation, Strata Statistics, W Fisher Jr … Rasch Measurement Transactions, 1992, 6:3 p. 238

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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