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Journal of Probability and Statistics
Volume 2015, Article ID 751803, 8 pages
Research Article

Measurement of Interobserver Disagreement: Correction of Cohen’s Kappa for Negative Values

Departments of Mechanical Engineering and Industrial & Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA

Received 24 June 2015; Accepted 6 September 2015

Academic Editor: Z. D. Bai

Copyright © 2015 Tarald O. Kvålseth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


As measures of interobserver agreement for both nominal and ordinal categories, Cohen’s kappa coefficients appear to be the most widely used with simple and meaningful interpretations. However, for negative coefficient values when (the probability of) observed disagreement exceeds chance-expected disagreement, no fixed lower bounds exist for the kappa coefficients and their interpretations are no longer meaningful and may be entirely misleading. In this paper, alternative measures of disagreement (or negative agreement) are proposed as simple corrections or modifications of Cohen’s kappa coefficients. The new coefficients have a fixed lower bound of −1 that can be attained irrespective of the marginal distributions. A coefficient is formulated for the case when the classification categories are nominal and a weighted coefficient is proposed for ordinal categories. Besides coefficients for the overall disagreement across categories, disagreement coefficients for individual categories are presented. Statistical inference procedures are developed and numerical examples are provided.