Wilcoxon Signed Rank Test
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)
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).
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:
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]
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.
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.