Wilcoxon Signed Rank Test
Last Updated :
23 Aug, 2021
Prerequisites: Parametric and Non-Parametric Methods
Hypothesis Testing
Wilcoxon signed-rank test, also known as Wilcoxon matched pair test is a non-parametric hypothesis test that compares the median of two paired groups and tells if they are identically distributed or not.
We can use this when:
Differences between the pairs of data are non-normally distributed.
Independent pairs of data are identical. (or matched) Eg. (Math, English: Both subjects) ; (June, July: Both months)
Steps involved:
Step 1 - Determine the null (h0) and alternate (ha) hypothesis.
Step 2 - Find the difference (D) between the two columns. [D = B-A]
Step 3 - Find absolute difference (Abs-D). [Abs-D = |D|]
Step 4 - Assign ranks to Abs-D from lowest (1) to highest (n).
Assigning Ranks:
If any two or more Abs-D values are same, then assign them consecutive ranks, then find the average of the ranks for each set of duplicate value. Consider the following scenario:
Abs-D |
1 |
2 |
3 |
3 |
4 |
4 |
4 |
5 |
Rank |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
Modified Ranks |
1 |
2 |
3.5 |
3.5 |
6 |
6 |
6 |
7 |
Case I - For Abs-D = 3
-> Assign them consecutive possible ranks. (3,4)
-> Find average of 3,4 => (3+4)/2 = 3.5
-> Assign the rank = 3.5 to both the 3's present in the table.
Case II - For Abs-D = 4
-> Assign them consecutive possible ranks. (5,6,7)
-> Find average of 5,6,7 => (5+6+7)/3 = 6
-> Assign the rank = 6 to all the 4's present in the table.
Step 5 - Find the sum of the ranks assigned to positive (T+) and negative (T-) Abs-D values.
Step 6 - Find the Wilcoxon Rank. (Wcalc = minimum(T+,T-))
Step 7 - Use the value of n and α and find Wtable in two-tailed section of
'Critical values of wilcoxon signed rank test'.
(take α = 0.05, if not given)
Step 8 - Interpretation of result.
NOTE : We use two-tailed test when we are dealing with two hypothesis. (null and alternate)
Interpretation of result
When Wcalc < Wtable :
-> Reject H0 (null hypothesis)
-> The two groups are not identically distributed.
When Wcalc > Wtable :
-> Accept H0 (null hypothesis)
-> The two groups are identically distributed.
Example Problem (Step by Step):
Consider the following example. The smog concentration data of 13 states of India were measured. Perform the Wilcoxon signed rank test and determine if there’s a significant difference in the concentrations recorded in May to that in December. [take α = 0.05]
States
|
Smog concentration in May (A)
|
Smog concentration in December (B)
|
Difference [D] (B-A)
(Step-2)
|
Absolute Difference [Abs-D]
(Step-3)
|
Rank
(Step-4)
|
Delhi
|
13.3
|
11.1
|
-2.2
|
2.2
|
5
|
Mumbai
|
10.0
|
16.2
|
6.2
|
6.2
|
9
|
Chennai
|
16.5
|
15.3
|
-1.2
|
1.2
|
3
|
Kerala
|
7.9
|
19.9
|
12.0
|
12.0
|
11
|
Karnataka
|
9.5
|
10.5
|
1.0
|
1.0
|
2
|
Tamil Nadu
|
8.3
|
15.5
|
7.2
|
7.2
|
10
|
Orissa
|
12.6
|
12.7
|
0.1
|
0.1
|
1
|
UP
|
8.9
|
14.2
|
5.3
|
5.3
|
7
|
MP
|
13.6
|
15.6
|
2.0
|
2.0
|
4
|
Rajasthan
|
8.1
|
20.4
|
12.3
|
12.3
|
12
|
Gujarat
|
18.3
|
12.7
|
-5.6
|
5.6
|
8
|
West Bengal
|
8.1
|
11.2
|
3.1
|
3.1
|
6
|
Jammu
|
13.4
|
36.8
|
23.4
|
23.4
|
13
|
n = 13
α = 0.05
Step 1 - h0 : Cmay = Cdecember (no change in the smog concentration)
h1 : Cmay ≠Cdecember (smog concentration changed)
Step 2,3,4 - Refer the table given above.
Step 5 - T+ marked as [ ] in table.
T- marked as [ ] in table.
∑T+ = 75
∑T- = 16
Step 6 - Wcalc = minimum(75,16)
= 16
Step 7 - Using n = 13 and α = 0.05 in table (click here)
Wtable = 17
Step 8 - Wcalc < Wtable :
Rejecting H0.
i.e smog concentration have changed from before.
Conclusion:
Wilcoxon signed-rank test is a very common test in the fields of pharmaceuticals, especially amongst drug researchers, to find out the dominant symptoms of various drugs on humans. Being a non-parametric test, it works as an alternative to T-test which is parametric in nature. For any doubt/query, comment below.
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