Fixed-Effects Modeling of Cohen's Weighted Kappa for Bivariate Multinomial Data: A Perspective
of Generalized Inverse
Table 2
Simulation results.
(a)
Case
1
80
2
0.495
0.490
−0.005
0.467
0.458
−0.009
0.493
0.487
−0.007
0.488
0.482
−0.007
1
160
2
0.495
0.495
0.000
0.467
0.466
−0.002
0.493
0.492
−0.001
0.488
0.488
−0.001
2
80
2
−0.128
−0.127
0.001
−0.030
−0.036
−0.006
−0.122
−0.121
0.002
−0.097
−0.096
0.001
2
160
2
−0.128
−0.126
0.001
−0.030
−0.031
−0.002
−0.122
−0.120
0.002
−0.097
−0.095
0.002
3
80
2
0.851
0.850
−0.001
0.841
0.834
−0.007
0.848
0.845
−0.002
0.845
0.843
−0.003
3
160
2
0.851
0.851
0.000
0.841
0.838
−0.003
0.848
0.847
−0.001
0.845
0.844
−0.001
4
80
2
0.538
0.535
−0.004
0.556
0.554
−0.002
0.583
0.580
−0.004
0.606
0.602
−0.004
4
160
2
0.538
0.537
−0.002
0.556
0.556
0.000
0.583
0.581
−0.002
0.606
0.604
−0.002
5
80
3
0.056
0.053
−0.003
−0.016
−0.019
−0.003
0.141
0.138
−0.003
0.171
0.167
−0.004
5
160
3
0.056
0.053
−0.003
−0.016
−0.020
−0.004
0.141
0.138
−0.003
0.171
0.168
−0.003
6
80
3
0.398
0.395
−0.003
0.305
0.299
−0.006
0.392
0.387
−0.005
0.378
0.371
−0.007
6
160
3
0.398
0.397
−0.001
0.305
0.303
−0.002
0.392
0.390
−0.002
0.378
0.374
−0.003
7
80
3
−0.174
−0.172
0.002
−0.254
−0.253
0.001
−0.166
−0.165
0.001
−0.190
−0.188
0.002
7
160
3
−0.174
−0.174
0.000
−0.254
−0.254
0.000
−0.166
−0.166
0.000
−0.190
−0.189
0.000
8
80
3
0.148
0.150
0.002
−0.058
−0.053
0.005
0.018
0.021
0.004
−0.013
−0.009
0.004
8
160
3
0.148
0.148
0.000
−0.058
−0.056
0.002
0.018
0.019
0.001
−0.013
−0.011
0.002
9
80
3
−0.319
−0.313
0.007
−0.411
−0.404
0.007
−0.296
−0.290
0.006
−0.322
−0.316
0.006
9
160
3
−0.319
−0.316
0.003
−0.411
−0.408
0.003
−0.296
−0.293
0.003
−0.322
−0.319
0.003
10
80
4
0.564
0.561
−0.003
0.543
0.532
−0.010
0.515
0.505
−0.009
0.522
0.511
−0.011
10
160
4
0.564
0.562
−0.001
0.543
0.538
−0.005
0.515
0.510
−0.005
0.522
0.516
−0.005
11
80
4
−0.075
−0.075
0.000
−0.087
−0.089
−0.002
−0.075
−0.076
−0.001
−0.087
−0.087
0.000
11
160
4
−0.075
−0.076
−0.001
−0.087
−0.090
−0.003
−0.075
−0.076
−0.001
−0.087
−0.088
−0.001
12
80
4
0.791
0.790
0.000
0.769
0.767
−0.002
0.703
0.699
−0.004
0.701
0.684
−0.017
12
160
4
0.791
0.792
0.001
0.769
0.770
0.000
0.703
0.702
−0.001
0.701
0.694
−0.007
Note: is the sample size, is equal to the number of categories in the response. is the true coefficient value for Cohen’s weighted kappa, stands for , stands for , and stands for . , , , and are their estimates, respectively. , , , and are the biases, respectively.
(b)
Case
1
80
2
0.450
0.482
0.477
0.007
0.011
0.008
0.009
0.011
0.011
0.008
0.008
0.009
0.009
1
160
2
0.462
0.490
0.485
0.004
0.006
0.004
0.004
0.006
0.006
0.004
0.004
0.004
0.004
2
80
2
−0.041
−0.119
−0.094
0.004
0.007
0.004
0.005
0.007
0.007
0.004
0.004
0.005
0.004
2
160
2
−0.034
−0.119
−0.094
0.002
0.004
0.002
0.002
0.004
0.003
0.002
0.002
0.002
0.002
3
80
2
0.827
0.843
0.839
0.003
0.005
0.004
0.004
0.005
0.005
0.004
0.004
0.004
0.004
3
160
2
0.835
0.845
0.843
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
4
80
2
0.547
0.576
0.598
0.005
0.007
0.004
0.004
0.007
0.007
0.004
0.004
0.004
0.004
4
160
2
0.554
0.579
0.602
0.002
0.004
0.002
0.002
0.004
0.004
0.002
0.002
0.002
0.002
5
80
3
−0.022
0.135
0.163
0.008
0.016
0.007
0.008
0.016
0.015
0.007
0.007
0.008
0.008
5
160
3
−0.022
0.137
0.166
0.004
0.008
0.004
0.004
0.008
0.008
0.004
0.003
0.004
0.004
6
80
3
0.295
0.382
0.365
0.008
0.015
0.008
0.010
0.015
0.014
0.008
0.007
0.010
0.009
6
160
3
0.301
0.387
0.371
0.004
0.007
0.004
0.005
0.007
0.007
0.004
0.004
0.005
0.005
7
80
3
−0.252
−0.163
−0.186
0.004
0.008
0.003
0.004
0.008
0.008
0.003
0.003
0.004
0.004
7
160
3
−0.254
−0.166
−0.188
0.002
0.004
0.002
0.002
0.004
0.004
0.002
0.002
0.002
0.002
8
80
3
−0.049
0.025
−0.006
0.006
0.010
0.003
0.004
0.010
0.010
0.003
0.003
0.004
0.004
8
160
3
−0.054
0.021
−0.009
0.003
0.005
0.001
0.002
0.005
0.005
0.001
0.001
0.002
0.002
9
80
3
−0.399
−0.284
−0.311
0.003
0.008
0.004
0.006
0.008
0.008
0.004
0.005
0.006
0.007
9
160
3
−0.406
−0.291
−0.317
0.002
0.004
0.002
0.003
0.004
0.004
0.002
0.002
0.003
0.003
10
80
4
0.522
0.496
0.500
0.006
0.009
0.008
0.008
0.009
0.009
0.008
0.008
0.008
0.008
10
160
4
0.533
0.505
0.511
0.003
0.004
0.004
0.004
0.004
0.005
0.004
0.004
0.004
0.004
11
80
4
−0.091
−0.075
−0.087
0.003
0.007
0.003
0.004
0.007
0.007
0.003
0.003
0.004
0.004
11
160
4
−0.091
−0.076
−0.088
0.001
0.004
0.001
0.002
0.004
0.004
0.001
0.001
0.002
0.002
12
80
4
0.763
0.696
0.671
0.004
0.007
0.005
0.007
0.007
0.007
0.005
0.005
0.007
0.008
12
160
4
0.768
0.700
0.686
0.002
0.004
0.002
0.003
0.004
0.003
0.002
0.002
0.003
0.003
Note: is the sample size, is equal to the number of categories in the response. , , and are the mean bootstrap estimates for , , and , respectively. , , , and are the MSE for the estimates of the four coefficients, respectively. and are the empirical variance and the bootstrap variance for , respectively. and are the empirical variance and the bootstrap variance for , respectively. and are the empirical variance and the bootstrap variance for , respectively.
