Calculation of p-value for Wilcoxon Signed-Rank test
 The null hypothesis for the Signed-Rank test is that
the distributions of eye-itch scores are equivalent for
each treatment.
 The probabilities of a certain magnitude being positive
or negative are equivalent (p = ½).
 Under H0 the probability for any arrangement of signs
of the ranks is (½ )n
 The p-value is just k∙(½)n, where k is the number of
arrangements of signs that yield a statistic as or more
extreme than the observed statistic
ranks of magnitudes of observations
obs data
Other
arrangements
of signs that
result in SPR
as or more
extreme
3 3
- +
- + - - - -
3
+
-
3
+
-
3 6.5 6.5
+ -
8
-
sum of
positive
ranks
3
0
3
3
3
3
In our eye-itch data, there are 6 possible arrangements of
the signs that would yield a sum of the positive ranks equal
to or less than that observed (note we include the observed
arrangement).
The one-sided p-value would be 6∙(½)8 = 0.0234.
The two-sided p-value is 2 × 0.0234 = 0.0468.
P-value for Wilcoxon Rank-Sum test
 The null hypothesis the distributions of each group is
the same
 Under H0 both samples come from the same
distribution so the probability of any n1 of the n1+n2
ranks is equally likely.
 Can compute the distribution of all possible sums of
the ranks in group 1, then compare the observed
statistic to this distribution
distribution of possible sums of ranks of controls
p = 0.009
36.5
p = 0.008
50
102.5
116
rank sums
 This distribution was created by computer sampling.
 If there are no ties in the data, then distribution
calculation is easier (and tables are available in some
texts)
 Note distribution looks normal. See coursepack for
computation of Normal approximation p-value.