Table 3.

Empirical type I error rates using different meta-analysis methods (Unit: %)

Methods	Asymptotic method (Het. under H0)		Permutation method (Het. under H0)		Permutation method (No het. under H0)
	DL	SJ	DL	SJ	DL	SJ
K = 10
Richness	1.49	0.72	4.91	4.89	4.94	4.85
Shannon	1.63	0.80	5.15	5.08	5.17	5.10
Simpson	1.61	0.86	4.86	4.89	4.87	4.86
PD	1.68	0.76	5.09	4.99	5.11	5.03
PE	1.64	0.86	4.93	4.96	4.97	5.00
PQE	1.53	0.74	4.95	4.95	4.91	4.87
mMeta	-	-	1.54	1.60	1.54	1.57
aMeta	-	-	3.83	3.84	3.87	3.68
K = 20
Richness	1.67	0.80	4.87	4.95	4.83	4.93
Shannon	1.50	0.64	4.86	4.84	4.88	4.88
Simpson	1.60	0.67	4.97	4.94	4.96	4.97
PD	1.62	0.68	4.98	5.06	4.94	5.08
PE	1.66	0.76	4.87	4.85	4.88	4.89
PQE	1.74	0.74	4.98	4.97	5.02	4.96
mMeta	-	-	1.89	1.93	1.89	1.94
aMeta	-	-	3.66	3.71	3.72	3.77
K = 30
Richness	1.52	0.62	4.85	4.93	4.85	4.95
Shannon	1.51	0.67	4.65	4.71	4.64	4.78
Simpson	1.63	0.74	4.81	4.82	4.85	4.86
PD	1.68	0.65	4.94	4.96	4.96	4.98
PE	1.59	0.74	4.93	4.93	4.96	5.00
PQE	1.66	0.70	4.88	4.97	4.95	4.94
mMeta	-	-	1.92	1.89	1.92	1.90
aMeta	-	-	3.38	3.44	3.41	3.48

^*[Asymptotic method (Het. under H₀)] represents the p-value calculation method based on the asymptotic normality and for the traditional random effects meta-analysis for the existence of heterogeneity under H₀; [Permutation method (Het. under H₀)] represents the group permutation method and for the traditional random effects meta-analysis for the existence of heterogeneity under H₀; [Permutation method (No het. under H₀)] represents the group permutation method and for the Han and Eskin’s modified random effects meta-analysis for the assumption of no heterogeneity under H₀.

^*DL and SJ represents the DerSimonian Laird (method of moments) estimator and the Sidik and Jonkman (robust variance or sandwich variance) estimator, respectively.

^*Richness, Shannon, Simpson, PD, PE and PQE represent the Species richness, Shannon, Simpson, PD, PE and PQE indices, respectively.