Table 2. Type l error rates for different methods with the estimated Volume trait correlation matrix $\widehat{\mathit{R}}\left({\text{10}}^{-\text{4}}\right)$.

α	0.05	0.01	1 × 104	1 × 106	5 × 107	5 × 108
SUM	5.0 × 10−2	1.0 × 10−2	1.0 × 10−4	1.0 × 10−6	5.0 × 10−7	4.7 × 10−8
SSU	4.6 × 10−2	1.0 × 10−2	1.9 × 10−4	4.1 × 10−6	2.4 × 10−6	3.9 × 10−7
Chi-squared	6.4 × 10−2	2.8 × 10−2	1.1 × 10−2	7.4 × 10−3	7.1 × 10−3	6.2 × 10−3
Hom	1.7 × 10−1	9.9 × 10−2	5.0 × 10−2	3.6 × 10−2	3.4 × 10−2	3.1 × 10−2
MAT(1)	5.0 × 10−2	1.0 × 10−2	1.0 × 10−4	1.0 × 10−6	5.2 × 10−7	4.9 × 10−8
MAT(10)	5.0 × 10−2	1.0 × 10−2	9.9 × 10−5	1.0 × 10−6	5.0 × 10−7	5.7 × 10−8
MAT(30)	5.0 × 10−2	1.0 × 10−2	1.0 × 10−4	1.0 × 10−6	5.2 × 10−7	4.5 × 10−8
MAT(50)	5.0 × 10−2	1.0 × 10−2	9.9 × 10−5	1.0 × 10−6	5.0 × 10−7	4.5 × 10−8
aMAT	4.7 × 10−2	9.4 × 10−3	9.9 × 10−5	1.2 × 10−6	6.4 × 10−7	4.7 × 10−8

We simulated 500 million (5 × 10⁸) replications with true Volume trait correlation matrix R under the null and constructed test statistics with $\widehat{\mathit{R}}\left({10}^{-4}\right)$. Type 1 error rates were estimated as the proportions of P-values less than significance level α.