2. • It is a non-parametric statistical method that
compares two groups that are independent of
sample data.
• It is used to test the null hypothesis that the
two samples have similar median or whether
observations in one sample are likely to have
larger values than those in other sample
• The parametric equivalent of Mann-Whitney
U test is t- test of unrelated sample
3. Assumption
• The two samples are random
• Two samples are independent of each other
• Measurement is of ordinal type thus
observations are arranged in ranks
4. Steps to perform
• The null hypothesis and alternative hypothesis
are identified.
• The significance level [alpha] related with null
hypothesis is stated. Usually alpha is set at 5%
and therefore, the confidence level is 95 %
• All of the observations are arranged in terms
of magnitude.
5. • The Ra denotes the sum of the ranks in group
a
• The Rb denotes the sum of ranks in group b
• U statistics is determined by
Verify Ua + Ub = nanb
6. • Evaluate U = min [ Ua,Ub]
• The obtained value is smaller of the two
statistics
• Using table of critics evaluate the possibility of
obtaining value of U or lower
• The critical value is compared with the
obtained value.
• The results are then interpreted to draw
conclusion.
14. We have to choose lowest value
hence U= 2
• Use u table= critical value
• N1=6 n2=6
• U critics from table = 5
• We should get the calculated value as equal to
or greater than table value.
• Here we got lesser value than table value
hence null hypothesis is rejected.
15. Wilcoxon Rank sum test
• It is non-parametric dependent samples t test
that can be performed on ranked or ordinal
data.
• Mann-Whitney Wilcoxon test
• It is used to test null hypothesis
• It is used to assess whether the distribution of
observations obtained between two separate
groups on a dependent variable are
systematically different from one another.
16. • It is used to evaluate the populations that are
equally distributed or not
• A population is set of similar items or data
obtained from experiment
• Rank basically two types of rank given Ra large
and Rb small.
17. It can be used in the place of
• One sample t test
• Paired t test
• For ordered categorical data where a
numerical scale is in appropriate but where it
is possible to rank the observations
18. General way to perform test
• State the null hypothesis Ho and the
alternative hypothesis H1
• Define alpha level
• Define decision rule
• Calculate Z statistics
• Calculate results
• Make conclusion
19. For paired data
• State the null hypothesis
• Calculate each paired difference
• Rank di ignoring signs [ assign rank 1 to the
smallest , rank 2 to the next etc ]
• Designate each rank along with its sign. Based
on the sign of di
20. • Calculate W+ the sum of the ranks of positive
di and W- the sum of the ranks of the negative
di.
• [W+] + [W-] = n [n+1]2
21. Problem
Group A p1 Group B p2
41 66
56 43
64 72
42 62
50 55
70 80
44 74
57 75
63 77
78
N1=9 N2=10
22. Group s = P1 + P2 Group Rank
41 A 1
42 A 2
43 B 3
44 A 4
50 A 5
55 B 6
56 A 7
57 A 8
62 B 9
63
64
A
A
10
11
66 B 12
70 A 13
72 B 14
23. Group s = P1 + P2 Group Rank
74 B 15
75 B 16
77 B 17
78 B 18
80 B 19
24. Group A Rank sum
Group s = P1 + P2 Group Rank
41 A 1
42 A 2
44 A 4
50 A 5
56 A 7
57 A 8
63
64
A
A
10
11
70 A 13
SUM OF Rank a 61
25. Group B Rank sum
Group s = P1 + P2 Group Rank
43 B 3
55 B 6
62 B 9
66 B 12
72 B 14
74 B 15
75 B 16
77 B 17
78 B 18
80 B 19
Sum of Group B 129
29. Krushal –Wallis H-test
• H test
• Non parametric statistical procedure used for
comparing more than two independent
sample
• Parametric equivalent to this test is one way
ANOVA
• H test is for non-normally distributed data.
30. Krushal –Wallis H-test
• It is a generalization of the Mann- Whitney
test which is a test for determining whether
the two samples selected are taken from the
same population.
• The p values in both the Krushal –Wallis and
the Mann-Whitney tests are equal
• It is used for samples to evaluate their degree
of association.
31. Description of sample
• 3 independently drawn sample.
• Data in each sample should be more than 5
• Both distribution and population have same
shape
• Data must be ranked
• Samples must be independent
• K independent sample k> 3 or K=3
32. Characteristics
• Test statistics is applied when data is not normally
distributed
• Test uses k samples of data.
• Test can be used for one nominal and one ranked
variable
• Significance level is denoted with α
• Data is ranked and df is n-1
• The rank of each sample is calculated
• Average rank is applied in case if there is tie
33. Problem
• Null hypothesis
• K independent sample drawn from population
which are identically distributed.
• Alternative hypothesis
• K independent sample drawn from population
which are not identically distributed.
34. Notation
Sampl1 obseravtion
1 Xxx Xxx Xxx Xxx Xxx
2 Xxx Xxx Xxx Xxx Xxx Xxx xxx
3 Xxx Xxx Xxx xxx Xxx Xxx Xxx Xxx
K Xxx Xxx Xxx Xxx Xxx Xxx xxx
Observation more than five
K =3 or K>3
35. Procedure
• Define null H0 and alternative H1 hypothesis.
• Rank the sample observations in the
combined series.
• Compute Ti sum of ranks
36. • Apply chi square variate with K-1 degree of
freedom
• K = number of sample
• Conclusion
• Take the table value from Chi 2 [k-1][α]
• If calculated H value > Chi 2 [k-1][α]
• We reject H0
37. Use krushal wallis H test at 5 % level of
significance if three methods are
equally effective
Method
1
99 64 101 85 79 88 97 95 90 100
Method
2
83 102 125 61 91 96 94 89 93 75
Method
3
89 98 56 105 87 90 87 101 76 89
38. Step I
• Null hypotheis
• H0 : μ a = μ b = μc
• Three methods are equally effective
• Alternative hypothesis H1= at least two of the
μ are different
• Three methods are not equally effective
• n1+n2+n3=30
42. • Df = K-1 =3-1=2
• Table value = 5.99
• Calculated value is 0.196
• Since the calculated H value
• 0.196 < 5.99
• We fail to reject H0
• All the teaching methods are equal
43. Friedman test
• It is a non parametric test developed and
implemented by Milton Friedman.
• It is used for finding differences in treatments
across multiple attempts by comparing three
or more dependent samples.
• It is an alternative to ANOVA when the
assumption of normality is not met
44. Friedman test
• The test is calculated using ranks of data
instead of unprocessed data
• It is used to test for differences between
groups when the dependent variable being
measured is ordinal.
• It can also be used for continuous data that
has marked as deviations from normality with
repeated measures.
45. • It is a repeated measures of ANOVA that can
be performed on the ordinal data.
46. Descriptions and requirements
• Dependent variable should be measured at the
ordinal or continuous level
• Data comes from a single group measured on at
least three different occasions.
• Random sampling method must be used
• All of the pairs are independent.
• Observations are ranked within blocks with no
ties
• Samples need not be normally distributed.
47. Problem ordinal data is given .is there
a difference between weeks 1,2,3
using alpha as 0.05
week1 Week 2 Week 3
27 20 34
2 8 31
4 14 3
18 36 23
7 21 30
9 22 6
48. Steps
• Define null and alternative hypothesis.
• State alpha
• Calculate degree of freedom
• State decision rule
• Calculate the statistic
• State result
• conclusion
49. Step 1
• Null hypothesis
• H0 there is no difference between three
conditions.
• Alternative hypotheis
• H1 there is a difference between three
conditions
55. Step 7.state result
• If chi square is greater than 5.991 than reject
the null hypothesis.
• Calculated chi square value is 2.33
• Calculated value is lesser than table value
hence fail to reject null hypotheis.
• Hence there is no difference among the three
group.
