Fixed-Effects Modeling of Cohen's Weighted Kappa for Bivariate Multinomial Data: A Perspective of Generalized Inverse

Table 1

The bivariate probability table.

𝑋⧵𝑌	0	1	2	⋯	𝑖	⋯	𝑗	⋯	𝑘	𝑃𝑋(𝑥)
0	𝑝00	𝑝01	𝑝02	⋯	𝑝0𝑖	⋯	𝑝0𝑗	⋯	𝑝0𝑘	𝑝0⋅
1	𝑝10	𝑝11	𝑝12	⋯	𝑝1𝑖	⋯	𝑝1𝑗	⋯	𝑝1𝑘	𝑝1⋅
2	𝑝20	𝑝21	𝑝22	⋯	𝑝2𝑖	⋯	𝑝2𝑗	⋯	𝑝2𝑘	𝑝2⋅
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
𝑖	𝑝𝑖0	𝑝𝑖1	𝑝𝑖2	⋯	𝑝𝑖𝑖	⋯	𝑝𝑖𝑗	⋯	𝑝𝑖𝑘	𝑝𝑖⋅
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
𝑗	𝑝𝑗0	𝑝𝑗1	𝑝𝑗2	⋯	𝑝𝑗𝑖	⋯	𝑝𝑗𝑗	⋯	𝑝𝑗𝑘	𝑝𝑗⋅
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
𝑘	𝑝𝑘0	𝑝𝑘1	𝑝𝑘2	⋯	𝑝𝑘𝑖	⋯	𝑝𝑘𝑗	⋯	𝑝𝑘𝑘	𝑝𝑘⋅
𝑃𝑌(𝑦)	𝑝⋅0	𝑝⋅1	𝑝⋅2	⋯	𝑝⋅𝑖	⋯	𝑝⋅𝑗	⋯	𝑝⋅𝑘	1

Note: there are 𝑘+1 categories in the response. 𝑃_𝑋(𝑥) and 𝑃_𝑌(𝑦) denote the marginal probabilities.