Inter-rater Reliability

The relationship between Rasch separation reliability statistics and conventional inter-rater reliability (IRR) statistics is often unclear. One reason is because numerous IRR coefficients have been used to quantify the effect of rater behavior on examinee scores.

The usual IRR approach mistreats ordinal ratings as interval measures and subjects them to analysis of variance (Ebel R L, 1951, Estimation of the reliability of ratings. Psychometrika, 16, 4:407-424). The variance components are combined to produce an IRR formula similar to KR-20.

IRR  =  True Variance in Examinee Score
        -------------------------------
        Observed Variance in Examinee Score

     =  True Variance
        -------------
        True Variance + Error Variance

There is, however, considerable latitude as to what is included in True Variance and what is included in Error Variance. When all examinees are rated by all judges on one task, the observed score is modelled to consist of a grand mean, an examinee mean, a rater mean and residual error. Then True Variance may be the variance of examinee means, and Error Variance may be the residual error variance. This is equivalent to the Rasch separation reliability for examinees. On the other hand, particularly when there are missing observations, the IRR Error Variance may be instead the residual error variance plus the variance of rater means. Further, when the examinees are rated on several items, True Variance may be the variance of examinee means plus the variance of item means.

In practice, there is usually little numerical difference between Rasch and IRR reliabilities when they are computed from equivalent variance terms. If the Rasch person separation reliability is noticeably higher, then the reported IRR probably adds the rater variance to the residual error variance. If the examinees are rated on several items, then the IRR probably adds the item variance to the examinee variance. If neither of these adjustments explains the differences, the reported IRR may be based on an entirely different approach, such as Cohen's kappa of inter-rater agreement on assigning nominal categories. In this case, reconciliation of the Rasch and IRR statistics may be impossible.

The Table gives an example of Rasch inter-rater reliability calculations for data based on the three facets of examinee, rater and item, described in J P Guilford, 1954, Psychometric Methods. 2nd Ed. New York: McGraw-Hill.

Rasch Analysis Results:
      Examinee Rater  Item
S.D.  0.43     0.14   0.31
RMSE  0.18     0.12   0.15
S.D.^2 = Observed Variance
     (0.18)   (0.02) (0.09)
RMSE^2 = Error variance
      0.03     0.01   0.02
Observed - Error = True variance
     (0.15)    0.01  (0.07)

  Formula  Values  Result
Rasch person separation reliability=
   (Examinee True)/(Examinee Obs) = 0.15/0.18 = 0.83
Rasch person separation reliability attenuated by rater variance =
   (Examinee True)/(Examinee Obs + Rater Obs) = 0.15/(0.18+0.02) = 0.75
Rasch person separation reliability adjusted for item variance =
   (Examinee True + Item True)/(Examinee Obs + Item Obs) =
   (0.15+0.07)/(0.18+0.09) =  0.81

Inter-rater Reliability, J Linacre … Rasch Measurement Transactions, 1991, 5:3 p. 166

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	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
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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
Other 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

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