. Author manuscript; available in PMC: 2012 May 1.

Published in final edited form as: J Math Imaging Vis. 2011 May;40(1):20–35. doi: 10.1007/s10851-010-0240-4

Table 5.

Estimated significance levels and proportions of agreement between the tests based on Monte Carlo simulations of distances with three groups, $X$ , $Y$ , and $Z$ with sizes n_x , n_y , and n_z , respectively, with N_mc =10000 Monte Carlo replicates. ${\hat{α}}_{K W}$ is the empirical size estimate for K-W test, ${\hat{α}}_{F_{1}}$ , ${\hat{α}}_{F_{2}}$ are for ANOVA F-tests with and without HOV, respectively; ${\hat{α}}_{K W, F_{1}}$ , ${\hat{α}}_{K W, F_{2}}$ , and ${\hat{α}}_{F_{1}, F_{2}}$ are the values of proportion of agreement between the indicated tests in the subscripts. The empirical sizes in the same row with the same superscript are not significantly different from each other. (^>(^<) Empirical size is significantly larger (smaller) than 0.05; i.e. method is liberal (conservative); (^ℓ (^≈) The proportion of agreement (not) significantly less than the minimum of the empirical sizes. )

	Empirical size			Prop. of agreement
(n_x,n_y,n_z)	${\hat{α}}_{K W}$	${\hat{α}}_{F_{1}}$	${\hat{α}}_{F_{2}}$	${\hat{α}}_{K W, F_{1}}$	${\hat{α}}_{K W, F_{2}}$	${\hat{α}}_{F_{1}, F_{2}}$
(1000,1000,1000)	.0511 ^a	.0508^a	.0506^a	.0417^ℓ	.0419^ℓ	.0499^≈
(5000,5000,10000)	.0495 ^a	.0498^a	.0497^a	.0386^ℓ	.0386^ℓ	.0491^≈
(5000,7500,10000)	.0480 ^a	.0451^a, ^<	.0449^a, ^<	.0368^ℓ	.0369^ℓ	.0446^≈
(10000,10000,10000)	.0483 ^a	.0483^a	.0480^a	.0392^ℓ	.0392^ℓ	.0477^≈