Statistical Methods in Medical Research 4ed

Table 10.1 Some properties of three equivalent two-sample distribution-free tests.
Bounds Sampling distribution
Variance
All xi < all yj All yj < all xi Mean No ties Ties
Mann±Whitney U test
UXY ˆ No: of pairs with xi < yj n1n2 0
9
>=
1
UYX ˆ No: of pairs with yj < xi 0
>;
n1n2
Wilcoxon rank sum test
T1 ˆ Sum of ranks for xis
T2 ˆ Sum of ranks for y js n1n2 ‡ 1
2 n2…n2 ‡ 1†
Kendall's S test
1
2 n1…n1 ‡ 1† n1n2 ‡ 1
2 n1…n1 ‡ 1†
1
2 n2…n2 ‡ 1†
2 n1n2
1
2 n1…n ‡ 1†
1
2 n2…n ‡ 1†
S ˆ UXY UYX n1n2 n1n2 0
Notation: n1 ˆ Sample size of xis.
n2 ˆ Sample size of yjs.
n ˆ n1 ‡ n2.
n1n2…n ‡ 1†
12
As above
n1n2…n ‡ 1†
3
n1n2
12n…n 1† n3
n1n2
3n…n 1† n3
n P
…t3 t†
t
n P
…t3 t†
t
280 Analysing non-normal data