How to carry out a Wilcoxon test (Gehan)
The uncensored case
Suppose we have two samples with observations
X
1
, . . . , X
m
;
Y
1
, . . . , Y
n
.
Order the combined sample and find the ranks of each observation within the combined
sample. We will assume for now that there are no ties. Let
R
i
be the rank of
X
i
within
the combined sample and let
R
+
=
m
i
=1
R
i
.
We would reject the null hypothesis that the two samples are drawn from an identical
underlying distribution in favor of the alternative that one distribution is “shifted” up
or down if
R
is either too large or too small. It can be shown that
R
is asymptotically
normally distributed under the null hypothesis with
E
0
(
R
) =
m
(
m
+
n
+ 1)
2
and
V
0
(
R
) =
mn
(
m
+
n
+ 1)
12
.
Thus we could construct a test based on the standardized value of
R
,
Z
=
R

E
0
(
R
)
V
0
(
R
)
.
The MannWhitney version of the Wilcoxon is useful.
Consider all
m
×
n
possible
pairs of observations, (
X
i
, Y
j
, and define
U
ij
=
+1
if
X
i
> Y
j
;
0
if
X
i
=
Y
j
;

1
if
X
i
< Y
j
.
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 Winter '06
 Tsodikov
 Normal Distribution, Null hypothesis, Xi > Yj, Censored data case

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