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NON-PARAMETRIC
STATISTICS
Definition
Nonparametric satistics, also known as
distribution-free statistics, are methods of testing
hypotheses when the nature of the distributions
are unknown.
Some of nonparametric statistics: Sign test,
Wilcoxon signed rank test, Wilcoxon rank sum
test, Kruskal-Wallis test, Friedman test, Rank
correlation
Sign Test
Sign test is probably the simplest of the nonparametric tests.
This test is used for paired data.
Step of analysis:
- Evaluate the difference between each paired data and
take the sign (positive or negative)
- Calculate the number of positive signs and negative
signs
- Take the larger number of sign and compare to the
corresponding table (Bolton: Table IV.12).
- If the calculated number (larger number) is greater than
the critical number in the table, then the difference
between two means is significant.
Example: Time to peak plasma concentration
Subject
Drug A
Drug B
Difference
(B-A)
Rank
1
2
3
4
5
6
7
8
9
10
11
12
2.5
3.0
1.25
1.75
3.5
2.5
1.75
2.25
3.5
2.5
2.0
3.5
3.5
4.0
2.5
2.0
3.5
4.0
1.5
2.5
3.0
3.0
3.5
4.0
+ 1.0
+ 1.0
+ 1.25
+ 0.25
0
+ 1.5
- 0.25
+ 0.25
- 0.5
+ 0.5
+ 1.5
+ 0.5
7.5
7.5
9
2
10.5
2
2
5
5
10.5
5
Evaluation
 Number of positive signs: 9
 Number of negative sign: 2
 No difference: 1
 New sample size: 11
 Critical value: 10 for 5% level and 11 for 1%
level
 Conclusion: No significant difference
Wilcoxon Signed Rank Test
 In Wilcoxon sign rank test, the magnitude of difference is taken

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




into consideration
The differences are then ranked in order of magnitude,
disregarding the sign
Differences of equal magnitude are given average rank
The signs corresponding to the signs of the original differences
are reassigned to the ranks
Ranks of the same sign are summed
Take smaller sum
Compared to critical value in corresponding Table (Bolton, Table
IV.13)
If the smaller rank sum is less than the critical value, then the
mean difference is significant.
For larger sample size, normal approximation is
available to compare two population means using the
Wilcoxon signed rank test:
z
R  N ( N  1) / 4
 N ( N  1/ 2)( N  1) /12
Application to the above example:
z
z
R  N ( N  1) / 4
 N ( N  1/ 2)( N  1) /12
59  11(11  1) / 4
11(11  1/ 2)(11  1) /12
 2.31
From Table IV.2 (Bolton), Z=2.31 corresponds to tail
area of 0.01 or P=0.02 for two sided test
Wilcoxon Rank Sum Test (test for
differences between two independent groups)
 In WRST, data are ranked in order of
magnitude
 Ranks of each group are summed
 For moderate sample size, the statistical test
for equality of the distribution means may be
approximated using the normal distribution
(calculation of z value). This approximation
works well if the smaller sample size is equal
to or greater than 10. For samples less than
size 10, refer to Table IV.16 (Bolton).
Calculation of z for WRST
z
T  N1 ( N1  N2  1) / 2
N1 N 2 ( N1  N2  1) /12
T is the sum of ranks for the smaller sample size, N1 is the
smaller sample size, N2 is the larger sample size.
If z is greater than or equal to 1.96, the twotreatments can be
said to be significantly different at the 5% level (two-sided
test)
Example:
Calculation of z
105.5  11(11  12  1) / 2
26.5
z

 1.63
(11)(12)(11  12  1) /12 16.25
From Table of cummulative area for normal distribution,
value (Table IV.2 Bolton), z=1.63 corresponds to tail area of
about 0.052 or P=0.104 (two-sided test). Therefore, these data
do not provide sufficient evidence to show that the two
different peices of apparatus give different dissolution
results (for 5% significance level).
 The Friedman test is a non-parametric
statistical test developed by the U.S.
economist 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, it is thus a special case of the
Durbin test.
 The Friedman test is used for two-way
repeated measures analysis of variance by
ranks. In its use of ranks it is similar to the
Kruskal-Wallis one-way analysis of variance
by ranks.
1.
Given data , that is, a tableau with n rows (the
blocks), k columns (the treatments) and a single
observation at the intersection of each block and
treatment, calculate the ranks within each block. If
there are tied values, assign to each tied value the
average of the ranks that would have been
assigned without ties. Replace the data with a new
tableau where the entry rij is the rank of xij within
block i.
 The test statistic is given by
 . Note that the value of Q as computed
above does not need to be adjusted for tied
values in the data.
 Finally, when n or k is large (i.e. n > 15 or k >
4), the probability distribution of Q can be
approximated by that of a chi-square
distribution. In this case the p-value is given
by . If n or k is small, the approximation to
chi-square becomes poor and the p-value
should be obtained from tables of Q specially
prepared for the Friedman test. If the p-value
is significant, appropriate post-hoc multiple
comparisons tests would be performed.
 In statistics, the Kruskal-Wallis one-way analysis of
variance by ranks (named after William Kruskal and
W. Allen Wallis) is a non-parametric method for
testing equality of population medians among groups.
Intuitively, it is identical to a one-way analysis of
variance with the data replaced by their ranks. It is an
extension of the Mann-Whitney U test to 3 or more
groups.
 Since it is a non-parametric method, the KruskalWallis test does not assume a normal population,
unlike the analogous one-way analysis of variance.
However, the test does assume an identically-shaped
distribution for each group, except for any difference
in medians.
1. Rank all data from all groups together; i.e.,
rank the data from 1 to N ignoring group
membership. Assign any tied values the
average of the ranks they would have
received had they not been tied.
2. The test statistic is given by:
 ni is the number of observations in group i
 rij is the rank (among all observations) of
observation j from group i
 N is the total number of observations across
all groups
3. A correction for ties can be made by dividing
K by
where G is the number of groupings of different
tied ranks, and ti is the number of tied
values within group i that are tied at a
particular value. This correction usually
makes little difference in the value of K
unless there are a large number of ties.
 Finally, the p-value is approximated by
 If some ni's are small (i.e., less than 5) the
probability distribution of K can be quite
different from this chi-square distribution. If a
table of the chi-square probability distribution
is available, the critical value of chi-square,
 can be found by entering the table at g − 1
degrees of freedom and looking under the
desired significance or alpha level. The null
hypothesis of equal population medians
would then be rejected if
 Appropriate multiple comparisons would then
be performed on the group medians.