Agreement coefficients in misskappa are written in terms of pairwise disagreement. For two raters with categorical ratings in \(\{1,\ldots,C\}\), let \(L\) be a loss matrix with \(\ell_{cc}=0\) and \(0 \le \ell_{cd} \le 1\). The generic coefficient is
\[ \kappa = 1 - \frac{D_o}{D_e}, \]
where \(D_o\) is observed disagreement and \(D_e\) is the disagreement expected under a chance model. Different named coefficients mostly differ in how they estimate these two quantities when there are more than two raters.
For a subject \(i\) and raters \(r,s\), complete-data pairwise disagreement is
\[ d_{irs} = \ell_{X_{ir}, X_{is}}. \]
The Conger-style observed component averages over identified rater pairs. The Fleiss-style component averages over pooled marginal category frequencies. Brennan-Prediger uses a uniform chance distribution over categories.
The C++ side accepts agreement weights \(W\) because that is the convention used by irrCAC. Internally the estimators work with \(L = 1 - W\). The R side exposes the same choice as weight, with "identity" for nominal agreement and "linear" or "quadratic" for ordered or scored categories.
The closed-form quadratic estimators use the special case
\[ \ell_{cd} = \frac{(v_c - v_d)^2}{(\max_j v_j - \min_j v_j)^2}, \]
where \(v\) is the supplied category-score vector. This loss lets the coefficient be expressed through means and covariance matrices, which is why kappa_quadratic() has a separate implementation from the general U-statistic estimator.
References
Brennan, R. L., & Prediger, D. J. (1981). Coefficient kappa: Some uses, misuses, and alternatives. Educational and Psychological Measurement, 41(3), 687–699.
Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37–46.
Cohen, J. (1968). Weighted kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 70(4), 213–220.
Conger, A. J. (1980). Integration and generalization of kappas for multiple raters. Psychological Bulletin, 88(2), 322–328.
Fleiss, J. L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76(5), 378–382.