# 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 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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