Basic Statistics

THE WILCOXON SIGNED RANKSTEST 193

The assumptions made when performing this test include:

1. The distribution of the Di values is symmetric. Note that when this assumption
   is true, the mean and the median are expected to be equal but the Di values do
   not have to be normally distributed.


2. The Di all have the same median or mean.


3. The Di are independent,


4. The Di are measured with an interval scale (see Conover [1999]).

Because the distribution is assumed to be symmetric, the sum of the positive ranks
should equal the sum of the negative ranks when the two-tail null hypothesis that
Ho : E(D) = 0 is true.
The data set in Table 13.1 consists of hypothetical values of systolic blood pressure
from 15 patients before and after treatment. The difference of each observation is
computed by subtracting the before measurement from the after measurement. Then
the absolute value of the difference is obtained and is ranked. Finally, the original
sign of the difference is applied to the ranks.

Table 13.1 Systolic Blood Pressures and Differences in Ranks

Before X, After Y, Difference D, Absolute ID,I Rank ID,l Sign Differences

135

137

140

140

145

148

150

154

155

158

159

160

162

163

164

125

136

140

142

137

151

154

147

151

153

148

151

159

157

150

• 10
  -1
  0
  +2
  -8
  +3
  +4
  -7
  -4
  -5
  -11
  -9
  -3
  -6


• 14

10

1 0 2 8 3 4 7 4 5

11

9

3

6

14

12

1

2

10

.9

.5

9

.5

7

13

11

3.5

8

14

-12

-1

+2

• 10
  +3.5
  +5.5
  -9
  -5.5
  -7


• 13
  -11
  -3.5
  -8


• 
  

  14

  
  

Note that there is one zero difference that is omitted, so the sample size is now 14.

There are two threes, so the average rank is 3.5, the average of 3 and 4. Likewise,

there are two fours, so the average rank is the average of 5 and 6 or 5.5. The sum of

the ranks of the positive values is

T+ = 2 + 3.5 + 5.5 = 11