Statistical Methods in Medical Research 4ed

Table 10.2 Percentage change in area of gastric ulcer after 3 months' treatment (Doll & Pygott, 1952).
Number
X: In-patients 32 100 …12† , 93, 92 91 …2† , 90, 85, 83, 81,
80, 78, 46, 40, 34, 0, 29, 62, 75, 106, 147, 1321
Y: Out-patients 32 100 …5† , 93, 89, 80, 78, 75, 74, 72, 71,
66, 59, 41, 30, 29, 26, 20, 15, 20, 25,
37, 55, 68, 73, 75, 145, 146, 220, 1044
Notation: 100 …5† indicates 5 observations at 100, etc.
xi 100 …12† 93 92 91 …2† 90 85 ...
Number of yj > xi 27 26 26 26 26 25 ...
Thus,
and
P ˆ 12…27†‡5…26†‡3…25†‡24 ‡ 23 ‡ 17 ‡ 2…16†‡11 ‡ 9 ‡ 7 ‡ 2…4†‡2 ‡ 0
ˆ 662,
Q ˆ 5…20†‡19 ‡ 15 ‡ 11 ‡ 7…10†‡9 ‡ 5…7†‡2…6†‡2…5†‡2…4†‡3 ‡ 2…2†‡2…1†
ˆ 298,
From Table 10.1,
10.3 Comparison of two independent groups 281
S ˆ 662 298 ˆ 364:
var…S† ˆ …32†…32†
3…64†…63† 643 64 …17 3
17† ... ,
where the terms arising from the small groups of ties (like two observations at 93) have
been omitted as they can easily be seen to be very much smaller than the other terms.
var…S† ˆ…0 084656†…257 184†
ˆ 21 772
SE…S† ˆ 2
p 1 772 ˆ 147 5:
Using the normal approximation, the standardized normal deviate is 364=147 5
ˆ 2 47…P ˆ 0 014†.
For the Mann±Whitney version, each tie contributes 1
2 to UXY and UYX . Thus, for
example,
UYX ˆ 5…26†‡19 1
2
ˆ 330,
1 ‡ 15 ‡ 11 2 ‡ ...
E…UYX †ˆ1 2 …32†…32† ˆ512,
UYX E…UYX †ˆ 182…ˆ 1
2 S†,
var…UYX †ˆ5443…ˆ 1
4 var…S††,
and the standardized normal deviate is 2 47, as before apart from the sign.