Does Acquiescence Disagree with Measurement Invariance Testing?

Figures & data

Table 1. Identification and MI constraints Wu and Estabrook (2016) for MI testing with MG-CCFA.

	MI constraints for MG-CCFA with two groups
	Identification constraints
	LV				LRV
	$\mathit{\kappa }$		$\varphi$		$\mathit{\nu }$		${\mathit{\sigma }}^{\mathit{2}}$
	G1	G2	G1	G2	G1	G2	G1	G2	MI constraints	Model comparison
1 – Configural	0	0	1	1	0	0	1	1	${\mathit{\mu }}^{(\mathit{g})}={\mathit{\nu }}^{(\mathit{g})}+{\mathbf{\Lambda }}^{(\mathit{g})}{\mathit{\kappa }}^{(\mathit{g})},$ ${\mathbf{\Sigma }}^{(\mathit{g})}={\mathbf{\Lambda }}^{(\mathit{g})}{\mathbf{\Phi }}^{(\mathit{g})}{\mathbf{\Lambda }}^{(\mathit{g})}+{\mathbf{\Psi }}^{(\mathit{g})},{\mathit{T}}^{(\mathit{g})}={\mathit{T}}^{(\mathit{g})}$
2 – Thresholds	0	0	1	1	0	Free	1	Free	${\mathit{\mu }}^{(\mathit{g})}={\mathit{\nu }}^{(\mathit{g})}+{\mathbf{\Lambda }}^{(\mathit{g})}{\mathit{\kappa }}^{(\mathit{g})},$ ${\mathbf{\Sigma }}^{(\mathit{g})}={\mathbf{\Lambda }}^{(\mathit{g})}{\mathbf{\Phi }}^{(\mathit{g})}{\mathbf{\Lambda }}^{(\mathit{g})}+{\mathbf{\Psi }}^{(\mathit{g})},{\mathit{T}}^{(\mathit{g})}=\mathit{T}$	2 vs. 1
3 – Loadings	0	0	1	Free	0	Free	1	Free	${\mathit{\mu }}^{(\mathit{g})}={\mathit{\nu }}^{(\mathit{g})}+\mathbf{\Lambda }{\mathit{\kappa }}^{(\mathit{g})},$ ${\mathbf{\Sigma }}^{(\mathit{g})}=\mathbf{\Lambda }{\mathbf{\Phi }}^{(\mathit{g})}\mathbf{\Lambda }+{\mathbf{\Psi }}^{(\mathit{g})},$ ${\mathit{T}}^{(\mathit{g})}=\mathit{T}$	3 vs. 2
4 – Intercepts	0	free	1	Free	0	0	1	Free	${\mathit{\mu }}^{(\mathit{g})}=\mathit{\nu }+\mathbf{\Lambda }{\mathit{\kappa }}^{(\mathit{g})},$ ${\mathbf{\Sigma }}^{(\mathit{g})}=\mathbf{\Lambda }{\mathbf{\Phi }}^{(\mathit{g})}\mathbf{\Lambda }+{\mathbf{\Psi }}^{(\mathit{g})},$ ${\mathit{T}}^{(\mathit{g})}=\mathit{T}$	4 vs. 3

Note. LV: latent variable; LRV: latent response variable.

Table 2. Population values for the simulation study.

	Loadings content factor(s)			Loadings ARS factor			Thresholds
	One factor	Two factors		Equal ARS	Unequal ARS (un) balanced scales	Unequal ARS semi-balanced scales	τ
	λ1	λ1	λ2	λARS	λARS	λARS	τ1	τ2	τ3	τ4
X1	0.6	0.6	0	0.175/0.350	0.1/0.275	0.125/0.3	−0.250	−0.750	−1.250	−1.750
X2	0.6	0	0.6	0.175/0.350	0.1/0.275	0.125/0.3	−0.068	−0.568	−1.068	−1.568
X3	0.6	(−)0.6	0	0.175/0.350	0.1/0.275	0.125/0.3	0.114	−0.386	−0.886	−1.386
X4	0.6	0	(−)0.6	0.175/0.350	0.1/0.275	0.125/0.3	0.295	−0.205	−0.705	−1.205
X5	0.6	0.6	0	0.175/0.350	0.1/0.275	0.125/0.3	0.477	−0.023	−0.523	−1.023
X6	0.6	0	0.6	0.175/0.350	0.1/0.275	0.125/0.3	0.659	0.159	−0.341	−0.841
X7	(−)0.6	0.6	0	0.175/0.350	0.25/0.425	0.125/0.3	0.841	0.341	−0.159	−0.659
X8	(−)0.6	0	0.6	0.175/0.350	0.25/0.425	0.125/0.3	1.023	0.523	0.023	−0.477
X9	(−)0.6	(−)0.6	0	0.175/0.350	0.25/0.425	0.275/0.45	1.205	0.705	0.205	−0.295
X10	(−)0.6	0	(−)0.6	0.175/0.350	0.25/0.425	0.275/0.45	1.386	0.886	0.386	−0.114
X11	(−)0.6	(−)0.6	0	0.175/0.350	0.25/0.425	0.275/0.45	1.568	1.068	0.568	0.068
X12	(−)0.6	0	(−)0.6	0.175/0.350	0.25/0.425	0.275/0.45	1.750	1.250	0.750	0.250

Note. Loadings in bold indicates those that were negative for semi-balanced scales. The original ARS factor loadings values were 0.3/0.6 but here we display the rescaled ARS loadings obtained by multiplying the original ARS loadings values by the standard deviation of a right-censored normal distribution, which was used to generate the ARS factor scores. For ease of readability, the unequal ARS factor columns refer to the unidimensional factor model only.

Table 3. Average fit value for MI testing when the ARS factor is not included for unidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for unidimensional scales without “equal” ARS factor
				Configural				Thresholds
ARS	Scale	N	J	${\chi }^2$ (SD)	df	RMSEA (SD)	CFI (SD)	${\chi }_{\tau }^2$ (SD)	$d{f}_{\tau }$	$\mathit{RMSE}{A}_{\tau }$ (SD)	$CF{I}_{\tau }$ (SD)
0.175	Bal	250	12	131.4 (16.0)	108	0.027 (0.012)	0.990 (0.007)	157.6 (16.9)	132	0.026 (0.010)	0.989 (0.007)
			24	587.0 (41.5)	504	0.025 (0.007)	0.988 (0.006)	636.6 (42.1)	552	0.024 (0.006)	0.988 (0.006)
		1000	12	202.7 (31.2)	108	0.029 (0.005)	0.990 (0.004)	234.3 (34.4)	132	0.027 (0.005)	0.989 (0.004)
			24	861.3 (74.0)	504	0.026 (0.003)	0.987 (0.003)	920.8 (76.0)	552	0.026 (0.003)	0.986 (0.003)
	Semi	250	12	129.9 (15.8)	108	0.026 (0.011)	0.991 (0.006)	155.2 (17.4)	132	0.024 (0.011)	0.990 (0.007)
			24	581.0 (35.6)	504	0.024 (0.006)	0.989 (0.006)	631.0 (36.5)	552	0.023 (0.006)	0.988 (0.006)
		1000	12	186.2 (26.7)	108	0.027 (0.005)	0.992 (0.003)	216.5 (29.3)	132	0.025 (0.005)	0.991 (0.003)
			24	852.5 (68.7)	504	0.026 (0.003)	0.987 (0.003)	911.6 (70.9)	552	0.025 (0.003)	0.987 (0.003)
	Unbal	250	12	109.7 (12.7)	108	0.010 (0.011)	0.998 (0.003)	133.6 (14.3)	132	0.010 (0.011)	0.998 (0.003)
			24	516.8 (22.6)	504	0.009 (0.007)	0.998 (0.002)	565.3 (23.4)	552	0.009 (0.007)	0.998 (0.002)
		1000	12	110.0 (14.8)	108	0.005 (0.006)	0.999 (0.001)	134.2 (16.1)	132	0.005 (0.006)	0.999 (0.001)
			24	512.8 (28.1)	504	0.004 (0.004)	0.999 (0.001)	560.8 (29.7)	552	0.004 (0.004)	0.999 (0.001)
0.350	Bal	250	12	539.7 (80.1)	108	0.126 (0.012)	0.895 (0.021)	597.4 (87.2)	132	0.118 (0.011)	0.886 (0.023)
			24	1863.9 (243.8)	504	0.104 (0.009)	0.875 (0.026)	1952.9 (252.4)	552	0.101 (0.009)	0.871 (0.027)
		100	12	1927.0 (170.4)	108	0.130 (0.006)	0.885 (0.013)	2110.4 (186.8)	132	0.122 (0.006)	0.875 (0.014)
			24	6523.6 (564.5)	504	0.109 (0.005)	0.861 (0.016)	6806.0 (589.2)	552	0.106 (0.005)	0.855 (0.016)
	Semi	250	12	538.1 (107.0)	108	0.126 (0.015)	0.898 (0.027)	594.5 (115.8)	132	0.118 (0.015)	0.890 (0.029)
			24	1876.6 (232.0)	504	0.104 (0.009)	0.877 (0.024)	1966.3 (240.6)	552	0.101 (0.009)	0.873 (0.024)
		1000	12	1953.5 (211.9)	108	0.131 (0.008)	0.885 (0.015)	2136.3 (232.0)	132	0.123 (0.007)	0.875 (0.016)
			24	6497.7 (473.9)	504	0.109 (0.004)	0.860 (0.014)	6779.1 (494.4)	552	0.106 (0.004)	0.855 (0.015)
	Unbal	250	12	113.4 (14.1)	108	0.012 (0.013)	0.999 (0.002)	137.3 (15.5)	132	0.012 (0.012)	0.999 (0.002)
			24	508.1 (20.0)	504	0.006 (0.007)	0.999 (0.001)	556.0 (20.5)	552	0.006 (0.006)	0.999 (0.001)
		1000	12	105.9 (13.0)	108	0.004 (0.005)	1.000 (0.000)	130.5 (14.5)	132	0.004 (0.005)	1.000 (0.000)
			24	508.2 (24.4)	504	0.003 (0.004)	1.000 (0.000)	557.5 (26.2)	552	0.003 (0.004)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items. Values in bold indicate conditions where the average model fit results were below commonly accepted cut-off values for “good” fit.

Table 4. Average fit value for MI testing when the ARS factor is not included for unidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for unidimensional scales without “equal” ARS factor
				Loadings				Intercepts
ARS	Scale	N	J	${\chi }_{\lambda }^2$ (SD)	$d{f}_{\lambda }$	$\mathit{RMSE}{A}_{\lambda }$ (SD)	$CF{I}_{\lambda }$ (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	167.4 (18.3)	143	0.024 (0.011)	0.989 (0.007)	220.8 (32.2)	154	0.040 (0.011)	0.972 (0.014)
			24	656.0 (42.6)	575	0.023 (0.007)	0.988 (0.006)	749.4 (62.0)	598	0.031 (0.007)	0.978 (0.009)
		1000	12	237.7 (32.8)	143	0.025 (0.005)	0.990 (0.004)	449.4 (56.2)	154	0.044 (0.004)	0.969 (0.006)
			24	925.0 (75.0)	575	0.025 (0.003)	0.987 (0.003)	1305.4 (108.0)	598	0.034 (0.003)	0.974 (0.004)
	Semi	250	12	166.1 (18.5)	143	0.023 (0.011)	0.990 (0.007)	202.0 (27.1)	154	0.034 (0.011)	0.980 (0.011)
			24	651.2 (37.0)	575	0.022 (0.006)	0.989 (0.006)	735.3 (45.3)	598	0.030 (0.005)	0.980 (0.007)
		1000	12	221.6 (30.5)	143	0.023 (0.005)	0.992 (0.003)	374.8 (51.5)	154	0.038 (0.004)	0.977 (0.005)
			24	915.7 (71.9)	575	0.024 (0.003)	0.987 (0.003)	1294.8 (100.9)	598	0.034 (0.002)	0.974 (0.004)
	Unbal	250	12	143.3 (15.3)	143	0.009 (0.010)	0.998 (0.004)	154.8 (16.7)	154	0.009 (0.010)	0.997 (0.004)
			24	587.7 (25.0)	575	0.008 (0.008)	0.998 (0.003)	608.8 (24.9)	598	0.007 (0.007)	0.998 (0.003)
		1000	12	144.0 (18.3)	143	0.005 (0.006)	0.999 (0.001)	154.0 (19.0)	154	0.004 (0.005)	0.999 (0.001)
			24	579.4 (30.4)	575	0.004 (0.004)	0.999 (0.001)	597.5 (33)	598	0.003 (0.003)	1.000 (0.001)
0.350	Bal	250	12	585.4 (85.4)	143	0.111 (0.011)	0.892 (0.022)	711.4 (103.7)	154	0.120 (0.011)	0.864 (0.026)
			24	1939.6 (249.7)	575	0.097 (0.009)	0.875 (0.027)	2112.6 (264.2)	598	0.100 (0.009)	0.861 (0.029)
		100	12	2032.2 (184.4)	143	0.115 (0.006)	0.881 (0.014)	2600.3 (221.8)	154	0.126 (0.006)	0.846 (0.016)
			24	6729.6 (588.4)	575	0.103 (0.005)	0.858 (0.016)	7548.7 (594.0)	598	0.108 (0.005)	0.839 (0.017)
	Semi	250	12	581.3 (111.9)	143	0.110 (0.014)	0.896 (0.028)	664.2 (116.9)	154	0.115 (0.013)	0.879 (0.029)
			24	1955.3 (240.5)	575	0.098 (0.009)	0.876 (0.024)	2129.0 (250.8)	598	0.101 (0.008)	0.863 (0.025)
		1000	12	2046.6 (225.3)	143	0.115 (0.007)	0.881 (0.016)	2413.6 (248.6)	154	0.121 (0.007)	0.859 (0.017)
			24	6702.4 (492.9)	575	0.103 (0.004)	0.857 (0.014)	7517.4 (509.4)	598	0.108 (0.004)	0.839 (0.015)
	Unbal	250	12	149.3 (17.6)	143	0.012 (0.012)	0.998 (0.002)	159.0 (18.7)	154	0.011 (0.012)	0.998 (0.002)
			24	581.7 (21.9)	575	0.007 (0.007)	0.999 (0.001)	603.9 (22.4)	598	0.006 (0.006)	0.999 (0.001)
		1000	12	145.5 (18.1)	143	0.005 (0.006)	1.000 (0.001)	156.3 (18.9)	154	0.005 (0.006)	1.000 (0.001)
			24	589.1 (30.4)	575	0.004 (0.004)	1.000 (0.000)	611.7 (34.7)	598	0.004 (0.004)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items. Values in bold indicate conditions where the average model fit results were below commonly accepted cut-off values for “good” fit.

Table 5. Average fit value for MI testing when the ARS factor is not included for multidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for multidimensional scales without “equal” ARS factor
				Configural				Thresholds
ARS	Scale	N	J	${\chi }^2$ (SD)	df	RMSEA (SD)	CFI (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	123.4 (17.1)	106	0.023 (0.013)	0.986 (0.011)	148.8 (19.0)	130	0.021 (0.013)	0.985 (0.013)
			24	558.2 (32.2)	502	0.020 (0.007)	0.986 (0.008)	608.2 (33.4)	550	0.019 (0.007)	0.985 (0.008)
		1000	12	163.4 (22.1)	106	0.023 (0.004)	0.989 (0.004)	192.7 (24.4)	130	0.022 (0.004)	0.988 (0.005)
			24	705.0 (58.7)	502	0.020 (0.003)	0.988 (0.004)	761.1 (60.9)	550	0.019 (0.003)	0.987 (0.004)
	Semi	250	12	122.4 (14.4)	106	0.022 (0.012)	0.987 (0.010)	147.9 (15.9)	130	0.021 (0.012)	0.986 (0.011)
			24	553.3 (32.6)	502	0.019 (0.007)	0.987 (0.008)	602.9 (34.0)	550	0.018 (0.007)	0.987 (0.008)
		1000	12	151.2 (22.1)	106	0.020 (0.006)	0.991 (0.004)	179.7 (24.3)	130	0.019 (0.006)	0.991 (0.005)
			24	669.5 (35.6)	502	0.018 (0.002)	0.990 (0.002)	723.9 (37.6)	550	0.018 (0.002)	0.990 (0.002)
	Unbal	250	12	109.8 (16.2)	106	0.012 (0.013)	0.994 (0.008)	134.2 (17.8)	130	0.011 (0.012)	0.994 (0.009)
			24	520.7 (28.2)	502	0.011 (0.008)	0.995 (0.005)	569.3 (29.1)	550	0.010 (0.008)	0.995 (0.006)
		1000	12	106.3 (12.9)	106	0.004 (0.005)	0.999 (0.001)	130.7 (14.5)	130	0.004 (0.005)	0.999 (0.002)
			24	507.5 (32.0)	502	0.004 (0.004)	0.999 (0.001)	556.0 (33.6)	550	0.004 (0.004)	0.999 (0.001)
0.350	Bal	250	12	392.1 (65.8)	106	0.103 (0.012)	0.866 (0.031)	441.7 (71.9)	130	0.097 (0.011)	0.854 (0.034)
			24	1415.5 (175.1)	502	0.085 (0.008)	0.862 (0.026)	1494.6 (181.5)	550	0.083 (0.008)	0.857 (0.027)
		100	12	1315.6 (148.4)	106	0.107 (0.007)	0.857 (0.019)	1456.5 (162.8)	130	0.101 (0.006)	0.843 (0.021)
			24	4378.3 (357.6)	502	0.088 (0.004)	0.853 (0.014)	4586.4 (372.3)	550	0.086 (0.004)	0.847 (0.014)
	Semi	250	12	377.6 (66.3)	106	0.101 (0.012)	0.879 (0.030)	425.2 (71.9)	130	0.095 (0.011)	0.868 (0.033)
			24	1382.4 (176.7)	502	0.083 (0.009)	0.872 (0.028)	1459.6 (182.4)	550	0.081 (0.008)	0.868 (0.028)
		1000	12	1188.6 (111.9)	106	0.101 (0.005)	0.877 (0.013)	1316.3 (122.7)	130	0.095 (0.005)	0.866 (0.014)
			24	4192.1 (371.8)	502	0.086 (0.004)	0.864 (0.014)	4390.7 (387.3)	550	0.083 (0.004)	0.858 (0.015)
	Unbal	250	12	110.3 (15.4)	106	0.012 (0.013)	0.998 (0.003)	134.5 (16.7)	130	0.012 (0.012)	0.997 (0.004)
			24	515.6 (29.1)	502	0.009 (0.008)	0.998 (0.002)	564.4 (30.0)	550	0.009 (0.008)	0.998 (0.002)
		1000	12	114.8 (14.3)	106	0.008 (0.006)	0.999 (0.001)	139.9 (16.4)	130	0.007 (0.006)	0.999 (0.001)
			24	531.9 (37.8)	502	0.007 (0.005)	0.999 (0.001)	581.7 (39.9)	550	0.007 (0.005)	0.999 (0.001)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items. Values in bold indicate conditions where the average model fit results were below commonly accepted cut-off values for “good” fit.

Table 6. Average fit value for MI testing when the ARS factor is not included for multidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for multidimensional scales without “equal” ARS factor
				Loadings				Intercepts
ARS	Scale	N	J	${\chi }_{\lambda }^2$ (SD)	$d{f}_{\lambda }$	$\mathit{RMSE}{A}_{\lambda }$ (SD)	$CF{I}_{\lambda }$ (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	159.3 (18.3)	140	0.021 (0.012)	0.984 (0.013)	203.4 (25.6)	150	0.036 (0.011)	0.959 (0.020)
			24	628.3 (33.4)	572	0.019 (0.007)	0.986 (0.008)	697.4 (36.6)	594	0.026 (0.005)	0.974 (0.009)
		1000	12	201.4 (25.8)	140	0.020 (0.004)	0.988 (0.005)	353.2 (48.4)	150	0.037 (0.004)	0.961 (0.009)
			24	781.2 (60.6)	572	0.019 (0.003)	0.988 (0.004)	1054.7 (80.7)	594	0.028 (0.003)	0.973 (0.005)
	Semi	250	12	158.3 (16.1)	140	0.020 (0.012)	0.985 (0.011)	195.1 (22.7)	150	0.033 (0.010)	0.966 (0.017)
			24	624.8 (35.8)	572	0.018 (0.007)	0.987 (0.009)	686.9 (41.2)	594	0.024 (0.006)	0.977 (0.011)
		1000	12	191.7 (24.3)	140	0.019 (0.005)	0.990 (0.004)	323.8 (39.0)	150	0.034 (0.004)	0.967 (0.007)
			24	747.1 (37.5)	572	0.017 (0.002)	0.990 (0.002)	971.3 (60.7)	594	0.025 (0.002)	0.977 (0.004)
	Unbal	250	12	144.3 (18.1)	140	0.011 (0.012)	0.993 (0.009)	154.2 (19.1)	150	0.011 (0.012)	0.993 (0.009)
			24	592.1 (31.1)	572	0.010 (0.008)	0.994 (0.006)	612.8 (32.0)	594	0.010 (0.008)	0.994 (0.006)
		1000	12	141.1 (16.1)	140	0.005 (0.005)	0.999 (0.002)	150.9 (17.4)	150	0.004 (0.005)	0.999 (0.002)
			24	576.3 (34.6)	572	0.004 (0.004)	0.999 (0.001)	596.3 (34.2)	594	0.003 (0.004)	0.999 (0.001)
0.350	Bal	250	12	447.9 (71.7)	140	0.093 (0.011)	0.855 (0.034)	574.9 (81.9)	150	0.106 (0.010)	0.800 (0.039)
			24	1501.6 (179.5)	572	0.080 (0.008)	0.860 (0.027)	1686.1 (200.0)	594	0.086 (0.008)	0.835 (0.030)
		100	12	1461.1 (162.6)	140	0.097 (0.006)	0.844 (0.021)	1985.9 (204.7)	150	0.111 (0.006)	0.783 (0.027)
			24	4584.4 (370.9)	572	0.084 (0.004)	0.848 (0.014)	5392.3 (414.1)	594	0.090 (0.004)	0.818 (0.016)
	Semi	250	12	437.7 (71.3)	140	0.092 (0.011)	0.867 (0.032)	538.1 (87.1)	150	0.101 (0.011)	0.827 (0.040)
			24	1468.6 (180.1)	572	0.079 (0.008)	0.870 (0.028)	1616.4 (193.9)	594	0.083 (0.008)	0.851 (0.030)
		1000	12	1338.0 (122.4)	140	0.092 (0.005)	0.864 (0.014)	1771.5 (145.7)	150	0.104 (0.005)	0.816 (0.017)
			24	4402.5 (387.1)	572	0.082 (0.004)	0.858 (0.014)	5084.9 (425.6)	594	0.087 (0.004)	0.834 (0.016)
	Unbal	250	12	143.9 (18.2)	140	0.011 (0.012)	0.997 (0.004)	154.0 (18.4)	150	0.011 (0.011)	0.997 (0.004)
			24	587.7 (29.8)	572	0.009 (0.008)	0.998 (0.003)	608.3 (29.9)	594	0.009 (0.008)	0.998 (0.003)
		1000	12	152.6 (17.8)	140	0.008 (0.006)	0.999 (0.001)	161.4 (18.9)	150	0.007 (0.006)	0.999 (0.001)
			24	608.4 (40.7)	572	0.007 (0.005)	0.999 (0.001)	630.0 (40.6)	594	0.007 (0.005)	0.999 (0.001)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items. Values in bold indicate conditions where the average model fit results were below commonly accepted cut-off values for “good” fit.

Table 7. Latent mean bias in the intercept invariance model in function of the simulated conditions when the ARS factor is not included.

				Latent mean bias
				Equal ARS		Unequal ARS
ARS	scale	sample	j	1 Factor	2 Factors	1 Factor	2 Factors
0.175	Bal	250	12	−0.013	−0.007	−0.118	−0.093
		250	24	0.001	0.013	−0.082	−0.098
		1000	12	−0.023	−0.003	−0.101	−0.090
		1000	24	0.001	0.001	−0.090	−0.095
	Semi	250	12	0.105	0.072	−0.004	−0.020
		250	24	0.077	0.067	−0.029	−0.019
		1000	12	0.094	0.067	−0.011	−0.024
		1000	24	0.058	0.072	−0.026	−0.026
	Unbal	250	12	0.184	0.187	0.210	0.204
		250	24	0.183	0.188	0.193	0.205
		1000	12	0.184	0.192	0.202	0.194
		1000	24	0.193	0.187	0.189	0.201
0.350	Bal	250	12	−0.078	−0.007	−0.216	−0.163
		250	24	0.001	−0.014	−0.109	−0.143
		1000	12	−0.085	−0.014	−0.206	−0.161
		1000	24	−0.009	−0.013	−0.130	−0.152
	Semi	250	12	0.163	0.135	−0.018	−0.027
		250	24	0.118	0.108	−0.028	−0.041
		1000	12	0.162	0.134	−0.024	−0.027
		1000	24	0.114	0.132	−0.038	−0.046
	Unbal	250	12	0.339	0.330	0.367	0.352
		250	24	0.346	0.335	0.342	0.340
		1000	12	0.343	0.340	0.358	0.347
		1000	24	0.336	0.344	0.349	0.346

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 8. Convergence rate in function of the simulated conditions when the ARS factor is included.

Convergence rate MI testing with ARS factor
				1. Content factor				2. Content factors
ARS	Scale	N	J	Configural	Thresholds	Loadings	Intercepts	Configural	Thresholds	Loadings	Intercepts
0.175	Bal	250	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
		1000	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
	Semi	250	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
		1000	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
	Unbal	250	12	0.410	0.690	0.990	0.990	0.430	0.760	0.970	0.960
			24	0.690	0.780	1	0.980	0.350	0.620	0.890	0.940
		1000	12	0.480	0.690	0.970	0.990	0.430	0.790	1	0.990
			24	0.600	0.630	1	1	0.400	0.870	0.930	0.950
0.350	Bal	250	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
		100	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
	Semi	250	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
		1000	12	1	1	1	1	1	1	1	1
			24	1	1	1	1	1	1	1	1
	Unbal	250	12	0.410	0.820	0.990	0.990	0.290	0.730	0.860	0.920
			24	0.640	0.870	0.990	1	0.170	0.560	0.860	0.830
		1000	12	0.420	0.790	1	0.990	0.250	0.780	0.920	0.930
			24	0.550	0.720	0.970	1	0.120	0.750	0.850	0.890

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 9. Average fit value for MI testing when the ARS factor is included for unidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for unidimensional scales with “equal” ARS factor
				Configural				Thresholds
ARS	Scale	N	J	${\chi }^2$ (SD)	df	RMSEA (SD)	CFI (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	109.9 (13.4)	106	0.011 (0.012)	0.997 (0.004)	134.7 (14.1)	130	0.011 (0.011)	0.997 (0.005)
			24	515.0 (22.1)	502	0.009 (0.007)	0.997 (0.002)	562.7 (22.4)	550	0.009 (0.007)	0.997 (0.002)
		1000	12	106.2 (13.4)	106	0.004 (0.006)	0.999 (0.001)	129.8 (14.8)	130	0.004 (0.005)	0.999 (0.001)
			24	507.1 (25.3)	502	0.004 (0.004)	1.000 (0.001)	555.4 (26.5)	550	0.004 (0.003)	1.000 (0.001)
	Semi	250	12	111.3 (12.4)	106	0.013 (0.012)	0.997 (0.004)	135.3 (13.6)	130	0.012 (0.011)	0.997 (0.004)
			24	514.4 (23.5)	502	0.009 (0.008)	0.998 (0.003)	563.0 (24.7)	550	0.009 (0.007)	0.997 (0.003)
		1000	12	106.6 (15.1)	106	0.005 (0.006)	0.999 (0.001)	130.4 (16.7)	130	0.005 (0.006)	0.999 (0.001)
			24	510.7 (24.6)	502	0.004 (0.004)	0.999 (0.001)	559.1 (25.0)	550	0.004 (0.003)	0.999 (0.001)
	Unbal	250	12	99.1 (12.3)	106	0.005 (0.008)	0.999 (0.002)	118.8 (13.3)	130	0.003 (0.007)	0.999 (0.002)
			24	510.5 (24.4)	502	0.008 (0.008)	0.998 (0.002)	548.8 (23.4)	550	0.005 (0.006)	0.999 (0.002)
		1000	12	98.8 (15.5)	106	0.003 (0.005)	1.000 (0.001)	121.1 (17.4)	130	0.003 (0.005)	1.000 (0.001)
			24	502.9 (29.2)	502	0.003 (0.004)	1.000 (0.001)	542.1 (31.9)	550	0.002 (0.003)	1.000 (0.000)
0.350	Bal	250	12	116.1 (14.8)	106	0.017 (0.013)	0.997 (0.003)	140.2 (16.4)	130	0.015 (0.012)	0.997 (0.003)
			24	568.6 (34.6)	502	0.022 (0.007)	0.994 (0.003)	584.2 (27.8)	550	0.014 (0.007)	0.997 (0.002)
		100	12	124.5 (18.3)	106	0.011 (0.007)	0.999 (0.001)	147.7 (21.3)	130	0.010 (0.007)	0.999 (0.001)
			24	778.6 (81.5)	502	0.023 (0.004)	0.994 (0.002)	696.4 (56.3)	550	0.016 (0.003)	0.997 (0.001)
	Semi	250	12	115.3 (15.1)	106	0.016 (0.013)	0.997 (0.003)	137.6 (16.4)	130	0.014 (0.012)	0.997 (0.003)
			24	572.8 (37.4)	502	0.023 (0.007)	0.994 (0.004)	584.8 (29.9)	550	0.015 (0.007)	0.997 (0.003)
		1000	12	132.7 (23.6)	106	0.014 (0.008)	0.998 (0.001)	154.1 (23.8)	130	0.012 (0.007)	0.998 (0.001)
			24	776.7 (66.4)	502	0.023 (0.003)	0.994 (0.002)	695.1 (49.7)	550	0.016 (0.003)	0.997 (0.001)
	Unbal	250	12	99.5 (13.6)	106	0.004 (0.009)	1.000 (0.001)	123.7 (15.8)	130	0.005 (0.010)	0.999 (0.001)
			24	501.1 (23.4)	502	0.005 (0.006)	0.999 (0.001)	540.8 (21.1)	550	0.003 (0.005)	1.000 (0.001)
		1000	12	95.8 (14.5)	106	0.002 (0.004)	1.000 (0.000)	117.8 (15.1)	130	0.002 (0.003)	1.000 (0.000)
			24	499.9 (27.6)	502	0.003 (0.003)	1.000 (0.000)	543.3 (25.0)	550	0.002 (0.003)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 10. Average fit value for MI testing when the ARS factor is included for unidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for unidimensional scales with “equal” ARS factor
				Loadings				Intercepts
ARS	Scale	N	J	${\chi }_{\lambda }^2$ (SD)	$d{f}_{\lambda }$	$\mathit{RMSE}{A}_{\lambda }$ (SD)	$CF{I}_{\lambda }$ (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	146.2 (16.0)	141	0.012 (0.011)	0.996 (0.005)	155.8 (18.6)	151	0.011 (0.012)	0.996 (0.006)
			24	586.0 (23.9)	573	0.009 (0.007)	0.997 (0.003)	607.4 (26.4)	595	0.008 (0.007)	0.997 (0.003)
		1000	12	142.1 (17.4)	141	0.005 (0.006)	0.999 (0.001)	152.1 (20.4)	151	0.004 (0.006)	0.999 (0.002)
			24	578.9 (30.7)	573	0.004 (0.004)	0.999 (0.001)	597.3 (34.0)	595	0.003 (0.004)	0.999 (0.001)
	Semi	250	12	147.2 (14.6)	141	0.012 (0.011)	0.996 (0.004)	156.2 (16.2)	151	0.012 (0.011)	0.996 (0.005)
			24	586.5 (26.3)	573	0.009 (0.007)	0.997 (0.003)	606.5 (28.0)	595	0.008 (0.007)	0.997 (0.003)
		1000	12	141.2 (20.0)	141	0.005 (0.006)	0.999 (0.001)	153.2 (21.5)	151	0.005 (0.006)	0.999 (0.002)
			24	581.7 (30.8)	573	0.004 (0.004)	0.999 (0.001)	603.5 (36.8)	595	0.004 (0.004)	0.999 (0.001)
	Unbal	250	12	134.9 (14.0)	141	0.005 (0.008)	0.999 (0.003)	141.7 (13.4)	151	0.003 (0.007)	0.999 (0.002)
			24	580.2 (24.5)	573	0.007 (0.007)	0.998 (0.002)	594.7 (23.7)	595	0.005 (0.006)	0.999 (0.002)
		1000	12	133.1 (17.6)	141	0.003 (0.005)	1.000 (0.001)	140.0 (16.8)	151	0.002 (0.004)	1.000 (0.001)
			24	573.7 (30.7)	573	0.003 (0.004)	1.000 (0.001)	583.2 (31.0)	595	0.002 (0.003)	1.000 (0.000)
0.350	Bal	250	12	152.6 (18.2)	141	0.016 (0.012)	0.997 (0.004)	162.0 (18.0)	151	0.015 (0.012)	0.997 (0.003)
			24	639.0 (40.5)	573	0.020 (0.008)	0.994 (0.004)	658.8 (39.5)	595	0.020 (0.007)	0.994 (0.003)
		100	12	163.7 (22.5)	141	0.011 (0.007)	0.998 (0.001)	172.2 (24.5)	151	0.010 (0.007)	0.999 (0.001)
			24	844.6 (83.5)	573	0.021 (0.004)	0.994 (0.002)	852.1 (79.2)	595	0.021 (0.003)	0.994 (0.002)
	Semi	250	12	150.9 (17.2)	141	0.015 (0.012)	0.997 (0.003)	160.3 (17.7)	151	0.014 (0.011)	0.997 (0.003)
			24	645.9 (40.1)	573	0.022 (0.007)	0.993 (0.004)	663.9 (39.5)	595	0.020 (0.007)	0.994 (0.004)
		1000	12	174.3 (27.5)	141	0.014 (0.007)	0.998 (0.002)	182.6 (27.3)	151	0.013 (0.007)	0.998 (0.002)
			24	843.8 (69.4)	573	0.022 (0.003)	0.994 (0.002)	851.2 (67.6)	595	0.021 (0.003)	0.994 (0.002)
	Unbal	250	12	139.5 (16.4)	141	0.007 (0.011)	0.999 (0.002)	144.4 (16.3)	151	0.005 (0.009)	0.999 (0.001)
			24	573.4 (22.3)	573	0.005 (0.006)	0.999 (0.001)	591.3 (22.9)	595	0.004 (0.006)	1.000 (0.001)
		1000	12	134.1 (15.8)	141	0.003 (0.004)	1.000 (0.000)	140.6 (16.3)	151	0.002 (0.004)	1.000 (0.000)
			24	582.1 (28.1)	573	0.004 (0.004)	1.000 (0.000)	597.5 (30.0)	595	0.003 (0.004)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 11. Average fit value for MI testing when the ARS factor is included for multidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for multidimensional scales with “equal” ARS factor
				Configural				Thresholds
ARS	Scale	N	J	${\chi }^2$ (SD)	df	RMSEA (SD)	CFI (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	108.2 (15.2)	104	0.013 (0.012)	0.994 (0.008)	132.5 (17.0)	128	0.012 (0.012)	0.993 (0.009)
			24	514.7 (26.8)	500	0.010 (0.008)	0.995 (0.005)	562.9 (27.9)	548	0.010 (0.007)	0.995 (0.005)
		1000	12	104.2 (14.4)	104	0.005 (0.006)	0.999 (0.002)	127.8 (16.0)	128	0.005 (0.005)	0.999 (0.002)
			24	507.0 (30.9)	500	0.004 (0.004)	0.999 (0.001)	555.1 (31.9)	548	0.004 (0.004)	0.999 (0.001)
	Semi	250	12	109.4 (13.4)	104	0.013 (0.012)	0.994 (0.008)	134.0 (15.1)	128	0.013 (0.012)	0.993 (0.008)
			24	512.1 (29.1)	500	0.009 (0.008)	0.996 (0.006)	560.5 (30.6)	548	0.008 (0.008)	0.995 (0.006)
		1000	12	103.9 (14.5)	104	0.005 (0.006)	0.999 (0.002)	128.0 (16.0)	128	0.005 (0.005)	0.999 (0.002)
			24	504.9 (27.2)	500	0.004 (0.004)	0.999 (0.001)	552.8 (29.1)	548	0.004 (0.004)	0.999 (0.001)
	Unbal	250	12	99.0 (13.0)	104	0.006 (0.009)	0.998 (0.005)	120.5 (12.0)	128	0.004 (0.007)	0.999 (0.003)
			24	494.5 (24.1)	500	0.004 (0.006)	0.998 (0.003)	550.3 (22.4)	548	0.006 (0.006)	0.998 (0.003)
		1000	12	95.8 (10.2)	104	0.002 (0.003)	1.000 (0.001)	117.9 (11.9)	128	0.001 (0.003)	1.000 (0.001)
			24	480.4 (30.3)	500	0.001 (0.003)	1.000 (0.001)	538.8 (31.5)	548	0.002 (0.003)	1.000 (0.001)
0.350	Bal	250	12	113.5 (14.2)	104	0.017 (0.013)	0.995 (0.006)	136.9 (15.0)	128	0.014 (0.012)	0.995 (0.006)
			24	539.8 (27.6)	500	0.017 (0.007)	0.994 (0.004)	577.0 (29.3)	548	0.013 (0.008)	0.995 (0.004)
		100	12	134.8 (18.5)	104	0.016 (0.006)	0.996 (0.002)	158.1 (20.9)	128	0.014 (0.006)	0.996 (0.002)
			24	620.7 (39.9)	500	0.015 (0.003)	0.995 (0.001)	623.9 (39.1)	548	0.011 (0.003)	0.997 (0.001)
	Semi	250	12	111.8 (15.7)	104	0.016 (0.013)	0.995 (0.006)	135.2 (17.6)	128	0.014 (0.013)	0.995 (0.006)
			24	531.6 (28.7)	500	0.014 (0.008)	0.995 (0.004)	570.6 (28.8)	548	0.011 (0.008)	0.996 (0.004)
		1000	12	128.5 (20.4)	104	0.014 (0.007)	0.997 (0.002)	152.7 (20.3)	128	0.013 (0.006)	0.997 (0.002)
			24	586.1 (36.3)	500	0.013 (0.003)	0.997 (0.001)	604.8 (34.9)	548	0.010 (0.004)	0.998 (0.001)
	Unbal	250	12	93.9 (11.9)	104	0.003 (0.007)	1.000 (0.001)	118.5 (14.0)	128	0.004 (0.007)	0.999 (0.001)
			24	494.5 (25.4)	500	0.004 (0.006)	0.999 (0.001)	541.7 (25.0)	548	0.004 (0.006)	0.999 (0.001)
		1000	12	98.0 (10.7)	104	0.002 (0.004)	1.000 (0.000)	122.6 (13.7)	128	0.003 (0.004)	1.000 (0.000)
			24	474.2 (18.7)	500	0.000 (0.000)	1.000 (0.000)	541.9 (24.6)	548	0.002 (0.003)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 12. Average fit value for MI testing when the ARS factor is included for multidimensional scales in function of the simulated conditions with equal ARS.

Average fit value in MI testing for multidimensional scales with “equal” ARS factor
				Loadings				Intercepts
ARS	Scale	N	J	${\chi }_{\lambda }^2$ (SD)	$d{f}_{\lambda }$	$\mathit{RMSE}{A}_{\lambda }$ (SD)	$CF{I}_{\lambda }$ (SD)	${\chi }_{\nu }^2$ (SD)	$d{f}_{\nu }$	$\mathit{RMSE}{A}_{\nu }$ (SD)	$CF{I}_{\nu }$ (SD)
0.175	Bal	250	12	143.0 (16.4)	138	0.012 (0.011)	0.993 (0.009)	152.7 (16.8)	147	0.012 (0.011)	0.992 (0.009)
			24	584.2 (28.3)	570	0.010 (0.007)	0.995 (0.005)	603.8 (27.5)	591	0.009 (0.007)	0.995 (0.005)
		1000	12	137.0 (17.3)	138	0.004 (0.005)	0.999 (0.002)	145.6 (17.9)	147	0.004 (0.005)	0.999 (0.002)
			24	577.5 (31.2)	570	0.004 (0.004)	0.999 (0.001)	596.3 (32.9)	591	0.004 (0.004)	0.999 (0.001)
	Semi	250	12	143.9 (15.5)	138	0.012 (0.011)	0.993 (0.009)	152.2 (17.2)	147	0.012 (0.012)	0.993 (0.009)
			24	582.7 (31.7)	570	0.008 (0.008)	0.995 (0.006)	604.3 (33.8)	591	0.008 (0.008)	0.995 (0.007)
		1000	12	137.7 (15.7)	138	0.004 (0.005)	0.999 (0.002)	146.6 (16.7)	147	0.004 (0.005)	0.999 (0.002)
			24	574.7 (28.8)	570	0.003 (0.004)	0.999 (0.001)	595.0 (29.1)	591	0.003 (0.004)	0.999 (0.001)
	Unbal	250	12	135.2 (15.1)	138	0.007 (0.010)	0.997 (0.006)	141.4 (15.8)	147	0.005 (0.009)	0.997 (0.006)
			24	575.3 (29.9)	570	0.007 (0.007)	0.997 (0.005)	592.6 (29.0)	591	0.006 (0.007)	0.997 (0.004)
		1000	12	130.4 (13.6)	138	0.002 (0.004)	1.000 (0.001)	136.4 (13.8)	147	0.001 (0.003)	1.000 (0.001)
			24	558.3 (34.9)	570	0.002 (0.003)	0.999 (0.001)	574.5 (34.9)	591	0.002 (0.003)	1.000 (0.001)
0.350	Bal	250	12	150.1 (15.9)	138	0.016 (0.012)	0.993 (0.007)	158.5 (17.0)	147	0.016 (0.012)	0.993 (0.007)
			24	610.5 (31.7)	570	0.016 (0.007)	0.994 (0.004)	630.5 (33.6)	591	0.015 (0.008)	0.994 (0.004)
		100	12	178.0 (23.0)	138	0.016 (0.006)	0.995 (0.003)	187.7 (23.7)	147	0.016 (0.006)	0.995 (0.003)
			24	696.2 (43.7)	570	0.015 (0.003)	0.995 (0.002)	715.2 (44.8)	591	0.014 (0.003)	0.995 (0.002)
	Semi	250	12	147.7 (18.6)	138	0.015 (0.012)	0.994 (0.007)	156.0 (18.0)	147	0.015 (0.012)	0.994 (0.006)
			24	601.7 (29.4)	570	0.013 (0.007)	0.995 (0.004)	621.0 (29.5)	591	0.013 (0.007)	0.995 (0.004)
		1000	12	168.3 (22.6)	138	0.014 (0.006)	0.997 (0.003)	175.5 (23.3)	147	0.013 (0.006)	0.997 (0.002)
			24	656.2 (40.1)	570	0.012 (0.003)	0.997 (0.002)	676.6 (42.4)	591	0.012 (0.003)	0.997 (0.002)
	Unbal	250	12	130.8 (16.1)	138	0.005 (0.009)	0.999 (0.002)	138.6 (16.2)	147	0.004 (0.008)	0.999 (0.002)
			24	564.6 (24.7)	570	0.004 (0.006)	0.999 (0.001)	582.6 (24.0)	591	0.004 (0.005)	0.999 (0.001)
		1000	12	135.3 (15.8)	138	0.004 (0.005)	1.000 (0.001)	140.7 (16.9)	147	0.003 (0.004)	1.000 (0.001)
			24	563.8 (31.7)	570	0.002 (0.003)	1.000 (0.000)	583.2 (31.0)	591	0.002 (0.003)	1.000 (0.000)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items.

Table 13. Latent mean bias in the intercept invariance model in function of the simulated conditions when the ARS factor is included.

Latent mean bias
				Equal ARS		Unequal ARS
ARS	Scale	Sample	J	1 Factor	2 Factors	1 Factor	2 Factors
0.175	Bal	250	12	0.012 (100%)	−0.004 (100%)	−0.080 (100%)	−0.079 (100%)
		250	24	−0.005 (100%)	0.016 (100%)	−0.072 (100%)	−0.084 (100%)
		1000	12	−0.005 (100%)	0.000 (100%)	−0.072 (100%)	−0.075 (100%)
		1000	24	0.004 (100%)	0.004 (100%)	−0.075 (100%)	−0.080 (100%)
	Semi	250	12	0.015 (100%)	0.005 (100%)	−0.094 (100%)	−0.079 (100%)
		250	24	0.029 (100%)	−0.002 (100%)	−0.068 (100%)	−0.079 (100%)
		1000	12	−0.001 (100%)	0.000 (100%)	−0.095 (100%)	−0.081 (100%)
		1000	24	0.001 (100%)	0.002 (100%)	−0.077 (100%)	−0.085 (100%)
	Unbal	250	12	0.161 (30%)	−0.098 (32.5%)	0.094 (20%)	0.264 (14.5%)
		250	24	−0.400 (23%)	−0.283 (24.5%)	−0.065 (27%)	0.207 (14.5%)
		1000	12	−0.078 (20%)	0.240 (23.5%)	0.436 (20%)	0.705 (8%)
		1000	24	−0.158 (24%)	0.066 (26.5%)	0.923 (26%)	−0.228 (2%)
0.350	Bal	250	12	0.002 (100%)	0.006 (100%)	−0.064 (100%)	−0.077 (100%)
		250	24	0.003 (100%)	−0.001 (100%)	−0.060 (99%)	−0.068 (100%)
		1000	12	−0.007 (100%)	0.001 (100%)	−0.053 (100%)	−0.075 (100%)
		1000	24	−0.013 (100%)	0.000 (100%)	−0.084 (100%)	−0.077 (100%)
	Semi	250	12	−0.005 (100%)	0.006 (100%)	−0.073 (100%)	−0.060 (100%)
		250	24	0.013 (100%)	−0.016 (100%)	−0.068 (100%)	−0.078 (100%)
		1000	12	0.002 (100%)	0.004 (100%)	−0.078 (100%)	−0.06 (100%)
		1000	24	0.008 (100%)	0.005 (100%)	−0.084 (100%)	−0.079 (100%)
	Unbal	250	12	0.108 (26%)	0.509 (25.5%)	0.116 (28%)	0.678 (8.5%)
		250	24	−0.072 (23%)	0.013 (14%)	0.526 (23%)	−0.064 (4%)
		1000	12	0.139 (25%)	0.234 (16.5%)	0.658 (26%)	2.404 (1%)
		1000	24	0.057 (20%)	−0.018 (11%)	0.835 (38%)	2.445 (1%)

Note. ARS: ARS factor loadings; N: sample size within each group; Bal: balanced scale; Semi: semi-balanced scale; Unbal: unbalanced scale; J: number of items; (X%): proportion of undistorted data sets used to calculate the latent mean bias.

Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81, 1014–1045. https://doi.org/10.1007/s11336-016-9506-0

 PubMed Web of Science ®Google Scholar