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Friedman's Test is a nonparametric alternative
to two-way analysis of variance. The hypothesis
is:
H0 the means
of all the samples are equal
H1 the mean of at least one
of the samples is different
Consider the example where
three treatments are evaluated on four different
patients:
| Therapy |
Andrew |
Belinda |
Chris |
Dave |
| Relaxed |
110 |
140 |
100 |
130 |
| Normal |
115 |
150 |
105 |
135 |
| High Intensity |
117 |
155 |
100 |
135 |
The test involves:
- impose ranks on each of the columns.
If values are equal, average the ranks
they would have got if they were slightly
different:
| Therapy |
Andrew |
Belinda |
Chris |
Dave |
| Relaxed |
1 |
1 |
1.5 |
1 |
| Normal |
2 |
2 |
3 |
2.5 |
| High Intensity |
3 |
3 |
1.5 |
2.5 |
- calculate the Fr statistic
using the formula:

Where:
| I |
Number of samples (treatments) |
| J |
Number of blocks |
| Ri |
sum of the ranks in row 'i' |
This gives a value for Fr of
4.65
The value of Fr has an approximately
chi-square distribution with I - 1 degrees
of freedom.
This is a distribution free
alternative to ANOVA. It will compare several
samples and test the hypothesis:
H0 the means of all the samples
are equal
H1 the mean of at least one
of the samples is different
The test involves:
- sort the combined results into size
order and allocate ranks
- form a table that contains the ranks,
instead of the values
- calculate the test statistic 'K using
the formula:

Where:
Ri sum of the ranks in row 'i'
J number of values in row 'i'
N the total number of values
I the number of rows
'K' has an approximately c2
distribution with degrees of freedom
I - 1
The ANOVA method relies on
the assumption that the variance is the
same for all the samples. Levene's test
is:
H0 the variance is the same
for all the samples
H1 the variance of at least
one of the samples is different
The test uses the statistic 'W' where:
Where:
| N |
total number of values |
| Ni |
number of values in row 'i' |
| k |
number of levels |
 |
median of level 'i' |
 |
the average of all the zij
values |
 |
the average of the zij
values in row 'i' |
This is the nonparametric version of the
two sample t-test; it compares the means
of two samples. It tests the hypothesis:
H0 the means are equal
H1 the mean of sample 'm' is
[less than/greater than/not equal to]
the mean of sample 'n'
It is similar in concept to the Wilcoxon
Signed Rank Test and is also known as the
Wilcoxon Rank Sum Test. The test involves:
- sort the combined results into size
order and allocate ranks
- find the sum 'w' of the ranks of the
sample with the smallest number of values
(if they both have the same number of
values, select either). This is one of
the critical values.
For small data sets the critical value
is found from published tables. Because
the sum of the ranks must be an integer,
it is not usually possible to find the exact
critical value:
m is the number of values in the sample
with the least number of values
n is the number of values in the other sample
m1
is associated with the sample containing
the smallest number of values
See Mood's Median Test
The Mood's Median Test is a nonparametric
equivalent of ANOVA. It is an alternative
to the Kruskal-Wallace test. The hypothesis
is:
H0 the medians of all the
samples are equal
H1 the median of at least one
of the samples is different
The test involves:
- find the median of the combined data
set
- find the number of values in each sample
greater than the median and form a contingency
table:
| |
A |
B |
C |
Total |
| Greater
than the median |
|
|
|
|
| Less
than or equal to the median |
|
|
|
|
| Total |
|
|
|
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- find the expected value for each cell:

- find the chi-square value from:

Most types of hypothesis tests require
that the population conforms to a particular
distribution, usually the normal distribution.
Where this is not the case a nonparametric
test can be used.
Nonparametric tests make no assumptions
about the type of distribution (although
they may require symmetry, or some other
property). The disadvantage is that nonparametric
tests are not as efficient; for a given
data set the nonparametric test will give
a higher p-value.
Parametric tests include ANOVA, the t test,
the F test and the chi-square test.
| Parametric
Hypothesis Test |
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A Parametric test make assumptions about
the underlying distribution of the population
from which the sample is being drawn, and
which is being investigated. This is typically
that the population conforms to a normal
distribution.
Alternative name for Mood's Median Test
See the Mann Whitney Tests
| Wilcoxon
Signed Rank Test |
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This is the nonparametric equivalent of
the one sample t-test. It tests the hypothesis:
H0 the mean equals zero
H1 the mean [is less than/greater
than/not equal to] zero
The test involves:
- sort the values into order of their
absolute magnitude (ignoring the signs)
and allocate ranks
- calculate the sum the ranks of the
data values are positive (S+)
An example would be:
| Rank |
1 |
2 |
3 |
4 |
5 |
6 |
| Data Value |
-1.0 |
-2.0 |
2.5 |
3.0 |
3.0 |
3.5 |
If the mean is non-zero the positive values
will be clustered at one end or other of
the ordered data set and S+ and
S- will be very different.
For the example S+ = 3 + 4 +
5 + 6 = 18
For small data sets values the critical
value is found from published tables. Because
the sum of the ranks must be an integer,
it is not usually possible to find the exact
critical value:
For larger data sets the z-statistic can
be found from:

This gives the upper tail
test (the mean is less than zero). For the
lower tail test, use the sum S+ and for
the two tailed test use both, remembering
to use the table value twice.
The p-value can be calculated using normal
distribution tables.
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