2. ā¢ It is a non-parametric statistical method that
compares two groups that are independent of
sample data.
ā¢ 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
ā¢ The parametric equivalent of Mann-Whitney
U test is t- test of unrelated sample
3. Assumption
ā¢ The two samples are random
ā¢ Two samples are independent of each other
ā¢ Measurement is of ordinal type thus
observations are arranged in ranks
4. Steps to perform
ā¢ The null hypothesis and alternative hypothesis
are identified.
ā¢ The significance level [alpha] related with null
hypothesis is stated. Usually alpha is set at 5%
and therefore, the confidence level is 95 %
ā¢ All of the observations are arranged in terms
of magnitude.
5. ā¢ The Ra denotes the sum of the ranks in group
a
ā¢ The Rb denotes the sum of ranks in group b
ā¢ U statistics is determined by
Verify Ua + Ub = nanb
6. ā¢ Evaluate U = min [ Ua,Ub]
ā¢ The obtained value is smaller of the two
statistics
ā¢ Using table of critics evaluate the possibility of
obtaining value of U or lower
ā¢ The critical value is compared with the
obtained value.
ā¢ The results are then interpreted to draw
conclusion.
14. We have to choose lowest value
hence U= 2
ā¢ Use u table= critical value
ā¢ N1=6 n2=6
ā¢ U critics from table = 5
ā¢ We should get the calculated value as equal to
or greater than table value.
ā¢ Here we got lesser value than table value
hence null hypothesis is rejected.
15. Wilcoxon Rank sum test
ā¢ It is non-parametric dependent samples t test
that can be performed on ranked or ordinal
data.
ā¢ Mann-Whitney Wilcoxon test
ā¢ It is used to test null hypothesis
ā¢ 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. ā¢ It is used to evaluate the populations that are
equally distributed or not
ā¢ A population is set of similar items or data
obtained from experiment
ā¢ Rank basically two types of rank given Ra large
and Rb small.
17. It can be used in the place of
ā¢ One sample t test
ā¢ Paired t test
ā¢ 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
ā¢ State the null hypothesis Ho and the
alternative hypothesis H1
ā¢ Define alpha level
ā¢ Define decision rule
ā¢ Calculate Z statistics
ā¢ Calculate results
ā¢ Make conclusion
19. For paired data
ā¢ State the null hypothesis
ā¢ Calculate each paired difference
ā¢ Rank di ignoring signs [ assign rank 1 to the
smallest , rank 2 to the next etc ]
ā¢ Designate each rank along with its sign. Based
on the sign of di
20. ā¢ Calculate W+ the sum of the ranks of positive
di and W- the sum of the ranks of the negative
di.
ā¢ [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 āWallis H-test
ā¢ H test
ā¢ Non parametric statistical procedure used for
comparing more than two independent
sample
ā¢ Parametric equivalent to this test is one way
ANOVA
ā¢ H test is for non-normally distributed data.
30. Krushal āWallis H-test
ā¢ 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.
ā¢ The p values in both the Krushal āWallis and
the Mann-Whitney tests are equal
ā¢ It is used for samples to evaluate their degree
of association.
31. Description of sample
ā¢ 3 independently drawn sample.
ā¢ Data in each sample should be more than 5
ā¢ Both distribution and population have same
shape
ā¢ Data must be ranked
ā¢ Samples must be independent
ā¢ K independent sample k> 3 or K=3
32. Characteristics
ā¢ Test statistics is applied when data is not normally
distributed
ā¢ Test uses k samples of data.
ā¢ Test can be used for one nominal and one ranked
variable
ā¢ Significance level is denoted with Ī±
ā¢ Data is ranked and df is n-1
ā¢ The rank of each sample is calculated
ā¢ Average rank is applied in case if there is tie
33. Problem
ā¢ Null hypothesis
ā¢ K independent sample drawn from population
which are identically distributed.
ā¢ Alternative hypothesis
ā¢ 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
ā¢ Define null H0 and alternative H1 hypothesis.
ā¢ Rank the sample observations in the
combined series.
ā¢ Compute Ti sum of ranks
36. ā¢ Apply chi square variate with K-1 degree of
freedom
ā¢ K = number of sample
ā¢ Conclusion
ā¢ Take the table value from Chi 2 [k-1][Ī±]
ā¢ If calculated H value > Chi 2 [k-1][Ī±]
ā¢ 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
ā¢ Null hypotheis
ā¢ H0 : Ī¼ a = Ī¼ b = Ī¼c
ā¢ Three methods are equally effective
ā¢ Alternative hypothesis H1= at least two of the
Ī¼ are different
ā¢ Three methods are not equally effective
ā¢ n1+n2+n3=30
42. ā¢ Df = K-1 =3-1=2
ā¢ Table value = 5.99
ā¢ Calculated value is 0.196
ā¢ Since the calculated H value
ā¢ 0.196 < 5.99
ā¢ We fail to reject H0
ā¢ All the teaching methods are equal
43. Friedman test
ā¢ It is a non parametric test developed and
implemented by Milton Friedman.
ā¢ It is used for finding differences in treatments
across multiple attempts by comparing three
or more dependent samples.
ā¢ It is an alternative to ANOVA when the
assumption of normality is not met
44. Friedman test
ā¢ The test is calculated using ranks of data
instead of unprocessed data
ā¢ It is used to test for differences between
groups when the dependent variable being
measured is ordinal.
ā¢ It can also be used for continuous data that
has marked as deviations from normality with
repeated measures.
45. ā¢ It is a repeated measures of ANOVA that can
be performed on the ordinal data.
46. Descriptions and requirements
ā¢ Dependent variable should be measured at the
ordinal or continuous level
ā¢ Data comes from a single group measured on at
least three different occasions.
ā¢ Random sampling method must be used
ā¢ All of the pairs are independent.
ā¢ Observations are ranked within blocks with no
ties
ā¢ 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
ā¢ Define null and alternative hypothesis.
ā¢ State alpha
ā¢ Calculate degree of freedom
ā¢ State decision rule
ā¢ Calculate the statistic
ā¢ State result
ā¢ conclusion
49. Step 1
ā¢ Null hypothesis
ā¢ H0 there is no difference between three
conditions.
ā¢ Alternative hypotheis
ā¢ H1 there is a difference between three
conditions
55. Step 7.state result
ā¢ If chi square is greater than 5.991 than reject
the null hypothesis.
ā¢ Calculated chi square value is 2.33
ā¢ Calculated value is lesser than table value
hence fail to reject null hypotheis.
ā¢ Hence there is no difference among the three
group.