Table 3.

Estimated type I error rates at the significance level of 5% for the prior microbial community-level association tests, OMiRKAT, aMiSPU, OMiAT, and aMiAD, for the clustered microbiome data (Unit: %).

Random intercepts
	n = 20			n = 50
${\rho }_{j{\ne j}^{\prime }}$	L	M	H	L	M	H
Gaussian
OMiRKAT	24.36	79.89	97.44	37.98	96.61	99.96
aMiSPU	14.64	52.5	80.78	20.47	75.69	95.65
OMiAT	22.13	79.27	97.77	40.63	98.65	99.97
aMiAD	5.70	6.79	8.22	6.11	7.39	8.82
Binomial
OMiRKAT	7.12	20.19	41.40	9.35	30.02	62.19
aMiSPU	6.17	12.32	24.13	6.88	16.18	34.86
OMiAT	6.87	18.54	39.62	9.09	33.68	71.1
aMiAD	5.41	5.71	6.31	5.64	5.98	6.62
Random intercepts and slopes
Gaussian
OMiRKAT	81.86	99.27	99.89	97.53	99.92	99.94
aMiSPU	72.20	96.42	98.58	92.87	99.88	99.98
OMiAT	81.31	99.41	99.91	98.70	99.93	99.97
aMiAD	8.59	10.68	11.57	8.51	10.24	10.58
Binomial
OMiRKAT	23.98	63.69	84.53	36.73	86.82	97.98
aMiSPU	15.87	42.33	62.83	21.83	63.68	84.62
OMiAT	22.64	63.08	85.10	40.63	93.27	99.49
aMiAD	6.15	7.30	8.35	6.20	7.45	8.24

L: low within-cluster correlation (${\rho }_{j{\ne j}^{\prime }}$ = $\frac{1}{3}$ for the random intercepts, ${\rho }_{j{\ne j}^{\prime }}$ = $\frac{(1+j^2)}{(j^2+3)}$ for the random intercepts and slopes); M: medium within-cluster correlation (${\rho }_{j{\ne j}^{\prime }}$ = $\frac{1}{2}$ for the random intercepts, ${\rho }_{j{\ne j}^{\prime }}$ = $\frac{(1+j^2)}{(j^2+2)}$ for the random intercepts and slopes); H: high within-cluster correlation (${\rho }_{j{\ne j}^{\prime }}$ = $\frac{3}{5}$ for the random intercepts, ${\rho }_{j{\ne j}^{\prime }}$ = $\frac{(1+j^2)}{(j^2+\frac{5}{3})}$ for the random intercepts and slopes).