Table 1.

Type I error for testing the global hypothesis at level 0.05

	Scenario	n	P-FL	adonis2	VE-freq	VE-arcsin	VE-omni
Independent samples, one-way case–control design
Adjusting for confounder	S1	20	0.040	0.052	0.045	0.031	0.038
		100	0.053	0.052	0.050	0.054	0.053
	S2	20	0.039	0.054	0.043	0.026	0.034
		100	0.050	0.050	0.052	0.047	0.053
Not adjusting for confounder	S1	20	0.299	0.296	0.215	0.408	0.362
		100	0.987	0.987	0.913	0.998	0.997
	S2	20	0.065	0.064	0.056	0.076	0.067
		100	0.151	0.151	0.083	0.245	0.194
Independent samples, two-way design
Testing for U1	S1	100	0.046	0.013	0.048	0.048	0.050
	S2	100	0.049	0.048	0.051	0.051	0.050
Testing for U2	S1	100	0.049	0.036	0.049	0.046	0.049
	S2	100	0.053	0.048	0.053	0.054	0.054
Clustered samples, one-way case–control design
Accounting for clustering	S1	100	0.050	0.050	0.048	0.051	0.053
	S2	100	0.044	0.047	0.047	0.044	0.045
Not accounting for clustering	S1	100	1	1	1	1	1
	S2	100	1	1	0.999	1	1
Independent samples, continuous trait
Adjusting for confounder		100	0.049	0.051	0.055	0.049	0.051
Not adjusting for confounder		100	0.095	0.095	0.090	0.093	0.091

Note: P-FL, PERMANOVA-FL. Results for PERMANOVA-FL and adonis2 are based on the Bray–Curtis distance. n is the number of samples. When testing U₁, we set ${\beta }_2=0.5$; when testing U₂, we set ${\beta }_1=0.5$. All analyses of clustered data adjust for the confounder.