Table 1.

Empirical Type I Errors for MiRKAT and Optimal MiRKAT with Continuous Outcome

Simulation Setup	n	Empirical Type I Errors
Kw	Ku	KBC	K0	K0.25	K0.5	K0.75	Koptimal	${\mathit{K}}_{{\mathbf{p}}_{\mathbf{min}}}$
Simulation Scenario 1: Clustered OTUs
$X\perp Z$, no adjustment for X	100	0.053	0.050	0.050	0.046	0.047	0.048	0.052	0.050	0.023
200	0.052	0.047	0.051	0.053	0.049	0.048	0.051	0.051	0.026
$X\perp Z$, adjustment for X	100	0.056	0.048	0.047	0.049	0.045	0.050	0.048	0.046	0.024
200	0.051	0.050	0.053	0.048	0.047	0.052	0.049	0.050	0.027
XZ, no adjustment for X	100	0.389∗	0.062∗	0.172∗	0.268∗	0.345∗	0.384∗	0.182∗	0.268∗	0.183∗
200	0.790∗	0.080∗	0.398∗	0.587∗	0.732∗	0.791∗	0.387∗	0.651∗	0.547∗
XZ, adjustment for X	100	0.055	0.047	0.047	0.049	0.046	0.049	0.046	0.049	0.024
200	0.052	0.049	0.051	0.047	0.047	0.052	0.050	0.049	0.026
Simulation Scenario 2: Top Ten OTUs
$X\perp Z$, no adjustment for X	100	0.053	0.050	0.050	0.045	0.048	0.049	0.053	0.050	0.025
200	0.051	0.047	0.050	0.053	0.050	0.047	0.051	0.050	0.026
$X\perp Z$, adjustment for X	100	0.056	0.048	0.047	0.050	0.046	0.051	0.047	0.049	0.021
200	0.051	0.049	0.053	0.047	0.047	0.052	0.050	0.051	0.023
XZ, no adjustment for X	100	0.153∗	0.048∗	0.669∗	0.105∗	0.124∗	0.147∗	0.157∗	0.516∗	0.067∗
200	0.307∗	0.048∗	0.976∗	0.194∗	0.239∗	0.293∗	0.320∗	0.932∗	0.151∗
XZ, adjustment for X	100	0.056	0.048	0.047	0.049	0.046	0.050	0.047	0.049	0.020
200	0.052	0.049	0.051	0.048	0.048	0.051	0.049	0.049	0.024

Type I error was evaluated for scenarios in which additional covariates were independent of the OTUs $\left(X\perp Z\right)$ or related to the OTUs (XZ) with the use of 5,000 simulated datasets. K_w, K_u, K_BC, K₀, K_0.25, K_0.5, and K_0.75 represent MiRKAT results for the weighted UniFrac kernel, unweighted UniFrac kernel, Bray-Curtis kernel, and generalized UniFrac kernels with α = 0, 0.25, 0.5, and 0.75, respectively. K_optimal represents the simulation results for optimal MiRKAT considering all seven kernels, and $K_{{\mathrm{p}}_{\min }}$ shows the results for a naive Bonferroni-adjusted test. The p values for optimal MiRKAT were obtained by 1,000 permutations. ^∗Inflated type I error.