Table I.

Simulation results for the probit generalized linear mixed model and κ_m based upon 100 data sets for I = 50 and J = 50 for two different sets of parameter values $\mathit{\theta }=(\eta ,\beta ,{\sigma }_u^2,{\sigma }_v^2{\sigma }_{v_1}^1,{\rho }_v)$.

Parameter	True value	Model (a)	Model (b)	Model (c)
(i) η = 1, ${\sigma }_{u_0}^2={\sigma }_v^2={\sigma }_{v_1}^2=1$
η	1	0.95 (0.17)	0.99 (0.24)	1.02 (0.36)
β	0.5	—	0.46 (0.25)	0.48 (0.41)
${\sigma }_u^2$	1	0.97 (0.20)	0.94 (0.26)	0.94 (0.21)
${\sigma }_v^2$	1	1.02 (0.23)	1.09 (0.31)	0.97 (0.32)
${\sigma }_{v_1}^2$	1	—	—	1.37 (0.46)
ρv01	0.25	—	—	−0.0004 (0.001)
κ		0.19 (0.04)
κm		0.21 (0.03)
(i) η = 1, ${\sigma }_{u_0}^2={\sigma }_{v_0}^2={\sigma }_{v_1}^2=5$
η	1	0.89 (0.29)	0.71 (0.48)	1.049 (0.50)
β	0.5	—	0.38 (0.67)	0.43 (0.66)
${\sigma }_u^2$	5	4.90 (0.84)	4.65 (0.98)	4.66 (0.92)
${\sigma }_v^2$	5	5.26 (1.26)	5.09 (1.36)	4.28 (0.82)
${\sigma }_{v_1}^2$	5	—	—	6.64 (2.24)
ρv	0.25	—	—	0.0004 (0.005)
κ		0.28 (0.05)
κm		0.29 (0.04)

The three models fitted are: (a) Φ⁻¹(p_ij) = η+u_i+v_j, (b) Φ⁻¹(p_ij) = η+βx_j+u_i+v_j and (c) Φ⁻¹(p_ij) = η+βx_j+u_i+v_j+z₂v_1j; x_i ~ Bin(1,0.5), z₂ ~ Bin(1,0.5). Cohen’s kappa = κ and model-based kappa = κ_m. Mean parameter estimates are presented with associated standard errors in parentheses.