Table 4.

Simulation Results: Bifactor model, Null Condition

K	N	reps	M2			RMSEA		rejection rates
			df	Mean	Var	Mean	(90% CI)	.150	.100	.050	.010
test statistic based on full one- and two-way marginal tables (M2)
2	750	935	75	74.4	147.1	.006	(.000,.019)	.144	.106	.049	.006
	1500	922	75	74.7	147.8	.004	(.000,.013)	.153	.104	.044	.007
3	750	995	390	390.1	754.1	.004	(.000,.012)	.153	.098	.047	.009
	1500	998	390	389.8	706.4	.003	(.000,.009)	.140	.093	.039	.007
4	750	998	915	888.7	1838.3	.002	(.000,.008)	.057	.030	.009	.002
	1500	1000	915	891.2	1703.6	.001	(.000,.006)	.059	.035	.010	.003
5	750	989	1650	1601.5	3149.9	.001	(.000,.006)	.029	.015	.008	.001
	1500	989	1650	1604.8	3318.4	.001	(.000,.004)	.033	.014	.007	.001
test statistic based on reduced one- and two-way marginal tables ( $M_2^{\ast }$)
3	750	995	60	59.4	133.5	.006	(.000,.021)	.140	.096	.052	.014
	1500	998	60	59.4	116.5	.004	(.000,.014)	.128	.091	.042	.009
4	750	998	45	45.2	96.6	.007	(.000,.023)	.154	.106	.059	.017
	1500	1000	45	45.1	86.3	.005	(.000,.015)	.157	.097	.042	.007
5	750	989	30	29.8	62.3	.006	(.000,.024)	.140	.092	.055	.017
	1500	989	30	30.1	60.2	.004	(.000,.017)	.151	.095	.047	.013

Notes. K is the number of categories. N is the sample size. The number of valid (converged and stable) solutions are reported under the column heading “reps.” The reference distribution of M₂ and $M_2^{\ast }$ statistics is central chi-square with df degrees of freedom. Note that M₂ is equivalent to $M_2^{\ast }$ when K is 2.