Unit-III Non Parametric tests: Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal-Wallis test, Friedman Test. BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)

1. Unit-III
Non-Parametric tests
✓Wilcoxon Rank Sum Test
✓Mann-Whitney U test
✓Kruskal-Wallis test
✓Friedman Test
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Ravinandan A P
Assistant Professor,
Department of Pharmacy Practice,
Sree Siddaganga College of Pharmacy
In association with
Siddaganga Hospital,
Tumkur-02

2. Level of significance
(Non-parametric data)-
1. Wilcoxan’s signed rank test
2. Wilcoxan rank sum test
3. Mann Whitney U test
4. Kruskal -Wallis test (one way ANOVA)
5. Friedman Test
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4. Why non-parametric methods?
• •Certain statistical tests like the t-test require assumptions of the
distribution of the study variables in the population
• –t-test requires the underlying assumption of a normal distribution
• –Such tests are known as parametric tests
• •There are situations when it is obvious that the study variable
cannot be normally distributed, e.g.,
# of hospital admissions per person per year
# of surgical operations per person
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5. • The study variable generates data which are scores & so should be
treated as a categorical variable with data measured on ordinal scale
–E.g., scoring system for degree of skin reaction to a chemical agent:
•1: intense skin reaction
•2: less intense reaction
•3: No reaction
• For such type of data, the assumption required for parametric tests seem invalid => non-
parametric methods should be used
• •Aka distribution-free tests, because they make no assumption about the underlying
distribution of the study variables
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7. Advantages of Non-Parametric tests
1. Distribution free & hence no assumption about the population is
required.
2. When sample size is small it is simple to & understand & easy to
apply.
3. It is less time consuming & for significant result no further work is
necessary
4. Applicable to all type of data
5. Helpful to researchers collecting pilot study or the medical
researchers working with a rare disease
6. Make fewer assumptions than classical procedures.
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11. Wilcoxon test.
• Used to test whether or not the difference between two paired
population medians is zero.
• The null assumption is that it is, i.e. the two medians are equal.
• Variables can be either metric or ordinal.
• Distributions any shape, but the differences should be
distributed symmetrically.
• This is the non-parametric equivalent of the matched-pairs t
test.
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12. WILCOXON SIGNED RANK TEST
• For the comparison of two treatments in a paired design, a more
sensitive non-parametric test is Wilcoxon signed rank test
• Ex: paired data obtained from bioavailability experiment: Time to
Peak concentration
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14. WILCOXON SIGNED RANK TEST
Subject Time to Peak Difference
A B B-A
1 2.5 3.5 1
2 3 4 1
3 1.25 2.5 1.25
4 1.75 2 0.25
5 3.5 3.5 0
6 2.5 4 1.5
7 1.75 1.5 -0.25
8 2.25 2.5 0.25
9 3.5 3 -0.5
10 2.5 3 0.5
11 2 3.5 1.5
12 3.5 4 0.5
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15. Ranks with +ve Sign Ranks with -Ve sign
2 2
2 5
5
5
7.5
7.5
9
10.5
10.5
59 7
If the Smaller Rank sum is Less than or Equal to the
table value , the comparative groups are different at the
Indicated level of significance
At 5% and 1%
For N= 11 table Value = 13 at 5% - Level
Cal value < Table Value
Difference is Significant Ravinandan A P

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17. WILCOXON RANK SUM TEST
• The wilcoxon signed rank test for non-parametric test for the
comparison of paired sample
• If two treatments are to compared, where the observations have
been obtained from two independent groups, the non-parametric
wilcoxon rank sum test or Mann – whitney U-test is alternative
for Pooled t-test
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18. Wilcoxon rank sum test
(aka Mann-Whitney U test)
• Non-parametric equivalent of parametric t-test for 2
independent samples (unpaired t-test)
• Suppose the waiting time (in days) for cataract surgery at
two eye clinics are as follows:
Patients at clinic A (nA=18) 1, 5, 15, 7, 42, 13, 8, 35, 21,
12, 12, 22, 3, 14, 4, 2, 7, 2
Patients at clinic B (nB=15) 4, 9, 6, 2, 10, 11, 16, 18, 6,
0, 9, 11, 7, 11, 10
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20. Amount Amount Rank(old) Rank(New)
Dissolved Dissolved
53 58 3 11
61 55 14 5.5
57 67 9 21
50 62 1 15.5
63 55 17 5.5
62 64 15.5 18.5
54 66 4 20
59 59 2 12.5
59 68 12.5 12
57 57 9 9
64 69 18.5 23
56 7
105.5 160.5
Z = (| T - N1(N1+N2+1)/2 |) / (sqrt(N1N2(N1+N2+1)/12)
T= Sum of the ranks for the smaller sample size
N1 = Size of sample1 11
N2 = Size of sample2 12
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21. Mann-Whitney test
• Used to test whether or not the difference between two independent
population medians is zero.
• The null assumption is that it is, i.e. the two medians are equal.
• Variables can be either metric or ordinal.
• No requirement as to shape of the distributions, but they need to be
similar.
• This is the non-parametric equivalent of the two-sample t test.
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22. Kruskal-Wallis test.
• Used to test whether the medians of three of more
independent groups are the same.
• Variables can be either ordinal or metric. Distributions any
shape, but all need to be similar.
• This non-parametric test is an extension of the Mann-
Whitney test.
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23. Friedman Test
• The Friedman test is a non-parametric statistical
test developed by Milton Friedman.
• Similar to the parametric repeated
measures ANOVA, it is used to detect differences
in treatments across multiple test attempts.
• The procedure involves ranking each row
(or block) together, then considering the values of
ranks by columns.
• Applicable to complete block designs.
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24. Friedman Test
• The Friedman test is a non-parametric alternative to ANOVA
with repeated measures.
• No normality assumption is required.
• The test is similar to the Kruskal-Wallis Test.
• We will use the terminology from Kruskal-Wallis Test and Two
Factor ANOVA without Replication.
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28. Non-parametric vs. parametric methods
Advantages:
1. Do not requires the assumption needed for parametric tests.
Therefore useful for data which are markedly skewed
2. Good for data generated from small samples. For such small
samples, parametric tests are not recommended unless the nature
of population distribution is known
3. Good for observations which are scores, i.e. measured on
ordinal scale
4. Quick and easy to apply and yet compare quite well with
parametric methods
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29. Non-parametric vs. parametric methods
•Disadvantages
1.Not suitable for estimation purposes as confidence
intervals are difficult to construct
2.No equivalent methods for more complicated parametric
methods like testing for interactions in ANOVA models
3.Not quite as statistically efficient as parametric methods
if the assumptions needed for the parametric methods
have been met
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31. THANK YOU
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