Table 2.

Ratio of empirical type I errors divided by the significance level α estimated over 10⁷ simulations. Listed within parenthesis are the standard errors divided by α. Permutation p-values (denoted “Perm”) are obtained adaptively as follows. We first used 10⁵ Monte Carlo simulations to estimate p-values. For those p-values less than 10⁻³, we then used 10⁶ Monte Carlo permutations to obtain more accurate estimates.

	Kw					Ku
α	0.05	10−2	10−3	10−4	10−5	0.05	10−2	10−3	10−4	10−5
KRV	0.94 (0.001)	1.04 (0.003)	1.36 (0.01)	1.86 (0.04)	2.92 (0.17)	0.92 (0.001)	1.05 (0.003)	1.49 (0.01)	2.41 (0.05)	4.12 (0.20)
RRV	0.93 (0.001)	0.93 (0.003)	1.07 (0.01)	1.25 (0.04)	1.68 (0.13)	0.90 (0.001)	0.91 (0.003)	1.06 (0.01)	1.38 (0.04)	1.66 (0.13)
Perm	1.00 (0.001)	1.00 (0.003)	1.02 (0.01)	0.99 (0.03)	0.99 (0.10)	1.00 (0.001)	1.00 (0.003)	1.00 (0.01)	1.02 (0.03)	0.99 (0.10)
	K0.5					KBC
α	0.05	10−2	10−3	10−4	10−5	0.05	10−2	10−3	10−4	10−5
KRV	0.93 (0.001)	1.05 (0.003)	1.44 (0.01)	2.15 (0.05)	3.50 (0.19)	0.95 (0.001)	1.05 (0.003)	1.39 (0.01)	2.01 (0.04)	3.10 (0.18)
RRV	0.92 (0.001)	0.92 (0.003)	1.06 (0.01)	1.34 (0.04)	1.86 (0.14)	0.93 (0.001)	0.92 (0.003)	1.02 (0.01)	1.23 (0.04)	1.52 (0.12)
Perm	1.00 (0.001)	1.00 (0.003)	1.01 (0.01)	1.01 (0.03)	0.98 (0.10)	1.00 (0.001)	1.00 (0.003)	1.00 (0.01)	1.02 (0.03)	0.98 (0.10)
	Kv					K0
α	0.05	10−2	10−3	10−4	10−5	0.05	10−2	10−3	10−4	10−5
KRV	0.92 (0.001)	1.03 (0.003)	1.44 (0.01)	2.23 (0.05)	3.62 (0.19)	0.92 (0.001)	1.06 (0.003)	1.60 (0.01)	2.80 (0.05)	5.10 (0.23)
RRV	0.90 (0.001)	0.91 (0.003)	1.06 (0.01)	1.33 (0.04)	1.83 (0.14)	0.89 (0.001)	0.90 (0.003)	1.06 (0.01)	1.46 (0.04)	2.09 (0.14)
Perm	1.00 (0.001)	1.00 (0.003)	0.99 (0.01)	0.97 (0.03)	1.01 (0.10)	1.00 (0.001)	1.00 (0.003)	1.00 (0.01)	1.01 (0.03)	1.00 (0.10)