Friedman's Test is a nonparametric alternative
to twoway analysis of variance. The hypothesis
is:
H_{0} the means of all
the samples are equal
H_{1} 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 F_{r} statistic using
the formula:
Where:
I 
Number of samples (treatments) 
J 
Number of blocks 
R_{i} 
sum of the ranks in row 'i' 
This gives a value for F_{r} of 4.65
The value of F_{r} has an approximately
chisquare distribution with I  1 degrees of
freedom.
This is a distribution free alternative
to ANOVA. It will compare several samples and
test the hypothesis:
H_{0} the means of all the samples
are equal
H_{1} 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:
R_{i} 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 c^{2
} 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:
H_{0} the variance is the same for
all the samples
H_{1} 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 z_{ij} values 

the average of the z_{ij} values
in row 'i' 
This is the nonparametric version of the two
sample ttest; it compares the means of two samples.
It tests the hypothesis:
H_{0} the means are equal
H_{1} 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
m_{1}
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 KruskalWallace
test. The hypothesis is:
H_{0} the medians of all the samples
are equal
H_{1} 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 




 find the expected value for each cell:
 find the chisquare 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 they are not as efficient; for a given
data set the nonparametric test will give a higher
pvalue.
Parametric
Hypothesis Test 

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 

This is the nonparametric equivalent of the one
sample ttest. It tests the hypothesis:
H_{0} the mean equals zero
H_{1} 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 nonzero 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 zstatistic 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 pvalue can be calculated using normal distribution
tables.
