. Author manuscript; available in PMC: 2025 Jan 1.

Published in final edited form as: Commun Stat Simul Comput. 2022 Feb 23;53(2):1048–1067. doi: 10.1080/03610918.2022.2039396

Table 2:

Performance of WJK-OLS and WJK-GLS approaches compared to the other methods in the estimation of the conditional means, and the hypothesis testing of the treatment effect when the cluster-level heterogeneity varies.

Cluster-level unequal variance^a	Measure^b		Method
Cluster-level unequal variance^a	Measure^b		LMM-KR	LMM-Permu.	OLS-CRVE	FGLS-CRVE	WJK-OLS	WJK-GLS

25	MSE	Est. μ₁	1.53	1.53	2.06	1.53	1.34	1.17
	MSE	Est. μ₂	1.06	1.06	1.21	1.06	1.08	0.996

	Rejection % (δ = 0)	Sig. 1	0.011	0.025	0.026	0.024	0.037	0.038
	Rejection % (δ = 0)	Sig. 2	0.050	0.083	0.082	0.078	0.084	0.093

	Rejection % (δ ≠ 0)	Sig. 1	0.608	0.651	0.614	0.651	0.650	0.691
	Rejection % (δ ≠ 0)	Sig. 2	0.767	0.784	0.709	0.760	0.751	0.849

36	MSE	Est. μ₁	1.71	1.71	2.43	1.71	1.38	1.17
	MSE	Est. μ₂	1.06	1.06	1.21	1.06	1.08	0.996

	Rejection % (δ = 0)	Sig. 1	0.008	0.025	0.024	0.025	0.038	0.036
	Rejection % (δ = 0)	Sig. 2	0.049	0.077	0.079	0.072	0.087	0.088

	Rejection % (δ ≠ 0)	Sig. 1	0.567	0.617	0.589	0.621	0.618	0.661
	Rejection % (δ ≠ 0)	Sig. 2	0.741	0.749	0.674	0.723	0.712	0.822

49	MSE	Est. μ₁	1.88	1.88	2.81	1.88	1.39	1.197
	MSE	Est. μ₂	1.06	1.06	1.21	1.06	1.08	0.996

	Rejection % (δ = 0)	Sig. 1	0.008	0.024	0.023	0.025	0.036	0.041
	Rejection % (δ = 0)	Sig. 2	0.049	0.075	0.076	0.07	0.088	0.087

	Rejection % (δ ≠ 0)	Sig. 1	0.532	0.593	0.568	0.595	0.595	0.647
	Rejection % (δ ≠ 0)	Sig. 2	0.706	0.718	0.642	0.689	0.689	0.814

Variances all equal to 1	MSE	Est. μ₁	0.79	0.79	0.79	0.79	0.84	0.89
	MSE	Est. μ₂	1.13	1.13	1.21	1.13	1.08	0.996

	Rejection % (δ = 0)	Sig. 1	0.008	0.044	0.046	0.05	0.05	0.034
	Rejection % (δ = 0)	Sig. 2	0.033	0.112	0.1	0.103	0.104	0.078

	Rejection % (δ ≠ 0)	Sig. 1	0.779	0.839	0.818	0.832	0.842	0.841
	Rejection % (δ ≠ 0)	Sig. 2	0.904	0.919	0.889	0.915	0.921	0.938

There are 5 clusters/condition, and the cluster sizes are: 80, 50, 120, 25, 30, 30, 60, 75, 50, 5. The variance of cluster-level random effect is 1 if no unequal variances exit or an unequal variance is associated with the cluster of size 120.

Since the same 1000 random seeds are used for the simulations with the true null and the true effect situations, the MSEs are the same when the other settings are identical. Thus the MSE is presented only once.

Est. is an abbreviation for Estimate of; Sig. is for nominal significance level. Due to the limited number of permutations, Sig. 1=0.048 (12/252) and Sig. 2=0.103 (26/252).