Wilcoxon

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NON- PARAMETRIC TESTS

• M.Prasad Naidu
• MSc Medical Biochemistry, Ph.D,.

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Non-parametric tests

• Note When valid use parametric
• Commonly used
• Wilcoxon
• Chi square etc.
• Performance comparable to parametric
• Useful for non-normal data
• If normalization not possible
• Note CI derivation-difficult/impossible

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Wilcoxon signed rank test

• To test difference between paired data

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STEP 1

• Exclude any differences which are zero
• Put the rest of differences in ascending order
• Ignore their signs
• Assign them ranks
• If any differences are equal, average their ranks

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STEP 2

• Count up the ranks of ives as T
• Count up the ranks of ives as T-

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STEP 3

• If there is no difference between drug (T) and
  placebo (T-), then T T- would be similar
• If there were a difference
• one sum would be much smaller and
• the other much larger than expected
• The smaller sum is denoted as T
• T smaller of T and T-

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STEP 4

• Compare the value obtained with the critical
  values (5, 2 and 1 ) in table
• N is the number of differences that were ranked
  (not the total number of differences)
• So the zero differences are excluded

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Patient Hours of sleep Hours of sleep Difference Rank Ignoring sign
Patient Drug Placebo Difference Rank Ignoring sign
1 6.1 5.2 0.9 3.5
2 7.0 7.9 -0.9 3.5
3 8.2 3.9 4.3 10
4 7.6 4.7 2.9 7
5 6.5 5.3 1.2 5
6 8.4 5.4 3.0 8
7 6.9 4.2 2.7 6
8 6.7 6.1 0.6 2
9 7.4 3.8 3.6 9
10 5.8 6.3 -0.5 1
3rd 4th ranks are tied hence averaged T
smaller of T (50.5) and T- (4.5) Here T4.5
significant at 2 level indicating the drug
(hypnotic) is more effective than placebo
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Wilcoxon rank sum test

• To compare two groups
• Consists of 3 basic steps

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Non-parametric equivalent of t test
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Step 1

• Rank the data of both the groups in ascending
  order
• If any values are equal average their ranks

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Step 2

• Add up the ranks in group with smaller sample
  size
• If the two groups are of the same size either one
  may be picked
• T sum of ranks in group with smaller sample size

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Step 3

• Compare this sum with the critical ranges given
  in table
• Look up the rows corresponding to the sample
  sizes of the two groups
• A range will be shown for the 5 significance
  level

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Non-smokers (n15) Non-smokers (n15) Heavy smokers (n14) Heavy smokers (n14)
Birth wt (Kg) Rank Birth wt (Kg) Rank
3.99 27 3.18 7
3.79 24 2.84 5
3.60 18 2.90 6
3.73 22 3.27 11
3.21 8 3.85 26
3.60 18 3.52 14
4.08 28 3.23 9
3.61 20 2.76 4
3.83 25 3.60 18
3.31 12 3.75 23
4.13 29 3.59 16
3.26 10 3.63 21
3.54 15 2.38 2
3.51 13 2.34 1
2.71 3
Sum272 Sum163
17, 18 19are tied hence the ranks are averaged
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