Sign Test (nonparametric)
Purpose: to test for statistically significant differences between two population means,
paired-difference design.
H0: the population means are equal.
Ha: the population means are not equal (2-sided). 1-sided tests are also possible.
Rejection of H0 means that there is a statistically significant difference between
the sample means, and therefore the population means are probably not equal.
Sign test is based on the binomial distribution where n is the number of pairs.
If H0 is true, the number of times out of n that one group will exceed the other is
binomially distributed with p = q = 0.5.
Tied observations are discarded.
Useful to prepare a table with the headings:
"Number of +s" & "p" for determining rejection regions and p-values.
Table is based on the binomial distribution.
Reject H0 if the number of +s is in the rejection region. Reject H0 if p  α.
Sign test is less powerful than the (parametric) t-test because much information is not
used. The sign test uses only the sign of the difference between the items in a pair.
The size of the difference is ignored.
H0 rejected:
The number of pluses in the sample is statistically significant at the 0.05 level.
The population means are probably not equal.
H0 not rejected:
The number of pluses in the sample is not statistically significant at the 0.05 level.
The population means could be equal.
Wilcoxon Signed-Rank Test (nonparametric)
Purpose: to test for statistically significant differences between two population means,
paired-difference design.
H0: the population means are equal.
Ha: the population means are not equal (2-sided). 1-sided tests are also possible.
Rejection of H0 means that there is a statistically significant difference between the
sample means, and therefore the population means are probably not equal.
Procedure:
1. Absolute values of the differences between the observations of each pair are
ranked from lowest (rank 1) to highest (rank n).
Tied observations are discarded.
Tied absolute values of differences are not discarded, but are assigned the
2.
3.
4.
5.
average rank of the tied items.
The sum of the rank numbers for each of the two groups is computed.
If H0 is true, the two rank sums will tend to be equal.
The test statistic, Tc, is the lower of the two rank sums.
Tables give the critical values, Tt.
Reject H0 if Tc  Tt. (!) Because the lower the Tc the stronger the evidence
against H0.
This test is more powerful than the sign test because the sizes of the differences are
Not totally ignored. Yet the differences are not used as fully as they are in the
Paired-difference t-test.
H0 rejected:
The Tc computed from the sample is statistically significant at the 0.05 level.
The population means are probably not equal.
H0 not rejected:
The Tc computed from the sample is not statistically significant at the 0.05 level.
The population means could be equal.
Mann-Whitney "U" Test (nonparametric)
Purpose: to test for statistically significant differences between two population means,
unpaired design.
H0: the population means are equal.
Ha: the population means are not equal (2-sided). 1-sided tests are also possible.
Rejection of H0 means that there is a statistically significant difference between the
sample means, and therefore the population means are probably not equal.
Procedure:
1. All of the (n1 + n2) observations are ranked from lowest (rank 1) to
highest (rank n).
Tied values are assigned the average rank of the tied items.
2. The sum of the rank numbers for each of the two groups is computed.
If H0 is true, the two rank sums will tend to be equal.
3. Compute the following:
U1 = (n1n2) + n1(n1+ 1) / 2 - group 1 rank sum
U2 = (n1n2) + n2(n2+ 1) / 2 - group 2 rank sum
Arithmetic check : U1 + U2 must = (n1*n2).
If H0 is true, U1 & U2 will tend to be equal.
4. The test statistic Uc, is the lower of U1 & U2.
5. Tables give the critical values, Ut.
6. Reject H0 if Uc  Ut. Because the lower the Uc the stronger the evidence
against H0.
This test is less powerful than the (parametric) unpaired t-test because much
information is not used. The test uses only the ranks of the observations, not the
observation values themselves.
H0 rejected:
The Uc computed from the sample is statistically significant at the 0.05 level.
The population means are probably not equal.
H0 not rejected:
The Uc computed from the sample is not statistically significant at the 0.05 level.
The population means could be equal.
=====================================================
Power of a statistical hypothesis test:
Power is a test's ability to detect when H0 is false, and give the correct conclusion
(rejection of H0).
Power is a test's ability to avoid a Type II error.
The power of any test can be increased by increasing the sample size.
The lesser power of nonparametric tests can be offset by using larger samples.
If two different tests are applied to the same data, the more powerful test will
generally produce the lower p-value.
==================================================================
Runs Test (nonparametric)
Purpose: to test for randomness (independence) in a series of binomial-type events.
H0:
Ha:
in the population, events are independent (occur in a random sequence).
in the population, events are dependent (do not occur in a random sequence).
One- or two-sided tests are possible (for positive or negative dependence).
Rejection of H0 means that there is a statistically significant dependency, which may
make better-than-chance forecasts possible.
Procedure:
1. n1 is the number of occurrences of one type of event.
n2 is the number of occurrences of the other type of event.
If n1  n2, n1 must be the lower number.
2. A run is a string of like events.
3. The test statistic, rc, is the number of runs in the series.
A very high number of runs is evidence for negative dependency.
A very low number of runs is evidence for positive dependency (streaks).
4. Tables give probabilities of various numbers of runs under independence.
Useful to prepare a table with the headings:
"Number of runs" & "p" for determining rejection regions and p-values.
Reject H0 if the number of runs is in the rejection region. Reject H0 if p  α.
H0 rejected:
The number of runs in the sample is statistically significant at the 0.05 level.
The sequence of events in the population probably has positive (or negative)
dependence.
H0 not rejected:
The number of runs in the sample is not statistically significant at the 0.05 level.
The sequence of events in the population could be independent.
====================================================
A:
B:
C:
h h x h x h x h x h x h
h x x x h h x x h h h h
h h h h h x x x x x h h
r
2
3
4
5
6
7
8
9
10
11
p (r)
.003
.012
.061
.121
.227
.228
.202
.101
.037
.008
|
|
|
|
|
|
|
|
|
|
|
r
2
2,3
2,3,4
2,3,4,5
2,3,4,5,6
2, . . . ,7
2, . . . ,8
2, . . . ,9
2, . . . ,10
2, . . . ,11
p pos
0.003
0.015
0.076
0.197
0.424
0.652
0.854
0.955
0.992
1.000
rc = 11
rc = 5
rc = 3
|
|
|
|
|
|
|
|
|
|
|
n1= 5; n2= 7
r
11
10,11
9,10,11
8,9,10,11
7, . . . ,11
6, . . . ,11
5, . . . ,11
4, . . . ,11
3, . . . ,11
2, . . . ,11
p neg
0.008
0.045
0.146
0.348
0.576
0.803
0.924
0.985
0.997
1.000
Kruskal-Wallis Test (nonparametric)
Purpose: the nonparametric version of one-way analysis of variance (ANOVA).
H0: all of the population medians are equal.
Ha: not all of the population medians are equal (2-sided).
Rejection of H0 means that there is a statistically significant difference between the
sample medians, and therefore the population medians are probably not equal.
Procedure:
1. All of the observations are ranked from lowest (rank 1) to highest (rank n).
Tied values are assigned the average rank of the tied items.
2. The sum of the rank numbers for each group is computed.
If H0 is true, the rank sums will tend to be equal.
3. The chi-square distribution is used for this test:
c = 12 / (n(n+1)) * (R2 i / n i ) - 3(n+1)
where:
n = the total number of observations
Ri = rank sum for group i
ni = number of observations in group i
t is based on k-1 degrees of freedom where k is the number of groups.
Reject H0 if c >= t.
All sample sizes should equal or exceed 5.
This test is less powerful than the (parametric) ANOVA because much information
is not used. The test uses only the ranks of the observations, not the observation
values themselves.
H0 rejected:
The c computed from the sample is statistically significant at the 0.05 level.
The population medians are probably not all equal.
H0 not rejected:
The c computed from the sample is statistically significant at the 0.05 level.
The population medians could all be equal.