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Large sample standard errors of kappa and weighted kappa.
Abstract
The statistics kappa (Cohen, 1960) and weighted kappa (Cohen, 1968) were introduced to provide coefficients of agreement between two raters for nominal scales. Kappa is appropriate when all disagreements may be considered equally serious, and weighted kappa is appropriate when the relative seriousness of the different possible disagreements can be specified. The papers describing these two statistics also present expressions for their standard errors. These expressions are incorrect, having been derived from the contradictory assumptions of fixed marginal totals and binomial variation of cell frequencies. Everitt (1968) derived the exact variances of weighted and unweighted kappa when the parameters are zero by assuming a generalized hypergeometric distribution. He found these expressions to be far too complicated for routine use, and offered, as alternatives, expressions derived by assuming binomial distributions. These alternative expressions are incorrect, essentially for the same reason as above. Assume that N subjects are distributed into k* cells by each of them being assigned to one of k categories by one rater and, independently, to one of the same k categories by a second
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“At the balanced matched point (p 11 , p 10 , p 01 , p 00 ) = (0.4,0.1, 0.1, 0.4)which adds π = 0.5 on top of matched marginals κ = ϕ = 0.6, and the multinomial delta method using the closed-form κ variance of Fleiss et al. (1969) gives Var(κ) = Var( φ) = 16/(25N ) = 0.640/N -not the 0.41/N that the bivariatenormal approximation (1 -ϕ 2 ) 2 /N would yield for continuous Pearson, which does not”
From Agreement Metrics for LLM-as-Judge Evaluation: What to Report and Why (2026) · cites this paper as Fleiss et al. (1969)
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Metadata
- Authors
- Fleiss, J. L., Cohen, J., & Everitt, B. S.
- Venue
- Psychological Bulletin
- Published
- 1969-11
- Volume
- 72
- Issue
- 5
- Pages
- 323-327
- ISSN
- 0033-2909 (ISSN-L)
- 1939-1455
- OpenAlex
- W1979773093
- Cited by
- 1,520
- References
- 1
- FWCI
- 8.67
- Type
- journal article
- Language
- en
- Open access
- closed
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