Table 1.

Parameter Recovery for the 3P-ML-HIRT Model With a Sample Size of 4,000.

Parameter	True	Bias	RMSE	ARB	Ratio	Skewness
Difficulty
M	0.257	0.016	0.117	0.095	0.988	0.006
SD	1.297	0.089	0.082	0.210	0.163	0.427
Maximum	3.367	0.261	0.461	2.550	1.386	1.043
Minimum	−4.224	−0.229	0.029	0.001	0.655	−1.320
Slope
M	1.001	−0.053	0.090	0.054	0.977	0.058
SD	0.428	0.042	0.048	0.031	0.139	0.418
Maximum	2.371	0.039	0.275	0.108	1.339	1.447
Minimum	0.244	−0.154	0.019	0.000	0.730	−0.628
Pseudo-guessing	0.147	0.001	0.003	0.003	1.242	−0.241
Residual variance at Level 1
M	0.246	0.043	0.066	0.191	0.936	0.280
SD	0.263	0.049	0.056	0.175	0.178	0.436
Maximum	1.069	0.181	0.216	0.693	1.228	0.950
Minimum	0.015	−0.003	0.016	0.012	0.681	−0.646
Residual variance at Level 2
${\mathrm{\psi }}_1$	0.028	0.001	0.009	0.025	0.853	0.163
${\mathrm{\psi }}_2$	0.036	0.002	0.006	0.042	1.148	−0.314
${\mathrm{\psi }}_3$	0.046	0.000	0.012	0.002	0.741	0.231
${\mathrm{\psi }}_4$	0.021	0.007	0.015	0.324	0.759	1.019
Residual variance at Level 3
Intercept (β0)	0.428	0.003	0.034	0.006	0.812	0.174
Slope (β1)	0.039	0.002	0.005	0.046	0.874	0.301
Covariance	0.073	0.003	0.009	0.045	0.756	0.501
Regression weight at Level 3
Intercept of β0	0	NA	NA	NA	NA	NA
Slope of β0	0.190	0.003	0.012	0.015	1.116	−0.151
Intercept of β1	0.396	0.006	0.017	0.015	0.794	−0.356
Slope of β1	0.032	0.002	0.005	0.059	1.116	1.146
Factor loading
λ1	1	NA	NA	NA	NA	NA
λ2	1.129	0.124	0.181	0.110	0.663	0.613
λ3	1.494	0.104	0.135	0.069	0.964	0.856

Note. The number of estimated difficulty parameters and estimated slope parameters and the estimated Level 1 residual variances are 105, 102, and 12, respectively. 3P-ML-HIRT = three-parameter multilevel higher order item response theory; True = true values; RMSE = root mean square error; ARB = absolute value of relative bias; NA = not applicable because of model constraints.