Table 3.

Parameter values for data generation

C	Item(s)	$\lambda$	${\theta }_r$	$\tau$	Proportion ($\%$)	Skewness
Negatively skewed
2	1-10	0.6	0.64	-0.59	(28, 72)	-1.00
5	1-10	0.6	0.64	-1.55, -1.08, -0.55, 0.15	(6, 8, 15, 27, 44)	-1.00
7	1-10	0.6	0.64	-1.65, -1.23, -0.92, -0.61, -0.28, 0.3	(5, 6, 7, 9, 12, 23, 36)	-1.00
Positively skewed
2	1-10	0.6	0.64	0.59	(72, 28)	1.00
5	1-10	0.6	0.64	-0.151, 0.553, 1.08, 1.555	(44, 27, 15, 8, 6)	1.00
7	1-10	0.6	0.64	-0.305, 0.279, 0.613, 0.915, 1.227, 1.645	(36, 23, 12, 9, 7, 6, 5)	1.00
BACS
	1	2.68	1	-2.92	(0.2, 99.8)	-1.92
	2	2.16	1	-2.66	(0.4, 99.6)	-2.18
	3	2.21	1	-2.13	(1.7, 98.3)	-1.58
	4	1.83	1	-2.01	(1.8, 98.2)	-1.89
2	5	1.43	1	-2.37	(0.9, 99.1)	-2.93
	6	1.38	1	-1.41	(7.9, 92.1)	-1.46
	7	1.31	1	-1.41	(7.9, 92.1)	-1.52
	1	2.68	1	-3.34, -1.84, -0.22, 2.27	(0, 3.2, 38.2, 57.4, 1.2)	-0.53
	2	2.16	1	-3.29, -1.99, -0.69, 0.99	(0.1, 2.3, 22.1, 59.4, 16.2)	-0.68
	3	2.21	1	-3.58, -1.90, -0.33, 1.32	(0, 2.8, 34.1, 53.7, 9.3)	-0.46
	4	1.83	1	-3.31, -1.94, -0.41, 1.37	(0, 2.6, 31.6, 57.3, 8.5)	-0.53
5	5	1.43	1	-3.47, -1.88, -0.74, 0.57	(0, 3, 19.9, 48.8, 28.3)	-0.64
	6	1.38	1	-2.09, -1.15, -0.34, 1.25	(1.8, 10.8, 24.1, 52.7, 10.6)	-0.58
	7	1.31	1	-2.36, -1.10, -0.53, 0.77	(0.9, 12.6, 16.2, 48.4, 21.9)	-0.62

Note. Parameter values used to generate data with negatively skewed distributions, positively skewed distributions, and based on the BACS example. C = number of response categories. $\lambda$ = loadings. ${\theta }_r$ = unique variances for the reference group. Unique variances for the focal groups may change depending on the simulation conditions. $\tau$ = thresholds. Proportion ($\%$) = proportions of endorsing response categories of 0 and 1 for $C=2$, 0 to 4 for $C=5$, and 0 to 6 for $C=7$