Table 2. Type I error rates of the G×E test and the joint test for rare-variant (RV) simulations.

Nominal level	SimRega	Burden-based	GESAT
Joint test $(a_G,a_{GE})=(0,0)$
0.05	0.0504 (0.0022)b	0.0511 (0.0022)	NA
0.01	0.0093 (0.0010)	0.0110 (0.0010)	NA
0.005	0.0047 (0.0007)	0.0056 (0.0007)	NA
0.001	0.0010 (0.0003)	0.0011 (0.0003)	NA
G×E test $(a_G,a_{GE})=(0,0)$
0.05	0.0496 (0.0022)	0.0523 (0.0022)	0.05090 (0.0024)
0.01	0.0085 (0.0009)	0.0104 (0.0010)	0.0119 (0.0011)
0.005	0.0038 (0.0006)	0.0044 (0.0007)	0.0050 (0.0007)
0.001	0.0007 (0.0026)	0.0008 (0.0003)	0.0007 (0.0003)
G×E test $(a_G,a_{GE})=\left(0.02,0\right)$c
0.05	0.0473 (0.0021)	0.0482 (0.0021)	0.0602 (0.0024)
0.01	0.0099 (0.0010)	0.0112 (0.0011)	0.0119 (0.0011)
0.005	0.0052 (0.0007)	0.0055 (0.0007)	0.0062 (0.0008)
0.001	0.0014 (0.0004)	0.0010 (0.0003)	0.0009 (0.0003)

Data were generated using a case–control design. The corresponding standard errors (SEs) are shown in parentheses. The values in italics/boldface type are those whose 95% confidence intervals (i.e., rate ± 1.96 × SE) fall below/above the nominal level. $a_G$ and $a_{GE}$ are the group PARs of the genetic main effect and the G×E effect, respectively. The results were obtained based on 10,000 replications.

^aUsing p (the proportion of variation explained by the leading eigenvalues in $S_G$) = 0.99.

^bStandard errors of the type I error rates.

^cAssuming 40 SNPs with causal main (G) effect.