Chapter 14
Nonparametric Statistics
 Learn
….
About Nonparametric
Statistical Methods
Agresti/Franklin Statistics, 1 of 62
Nonparametric Statistical
Methods

Nonparametric methods are used:
• When the data are ranks for the subjects,
rather than quantitative measurements.
• When it’s inappropriate to assume
normality.
Agresti/Franklin Statistics, 2 of 62
 Section 14.1
How Can We Compare Two
Groups by Ranking?
Agresti/Franklin Statistics, 3 of 62
Example: How to Get A Better
Tan

Experiment: A student wanted to
compare ways of getting a tan without
exposure to the sun.

She decided to investigate which of two
treatments would give a better tan:
• An “instant bronze sunless tanner” lotion
• A tanning studio
Agresti/Franklin Statistics, 4 of 62
Example: How to Get A Better
Tan

Subjects:
• Five female students participated in the
experiment.
• Three of the students were randomly
selected to use the tanning lotion.
• The other two students used the
tanning studio.
Agresti/Franklin Statistics, 5 of 62
Example: How to Get A Better
Tan

Results:
• The girls’ tans were ranked from 1 to 5,
with 1 representing the best tan.

Possible Outcomes:
• Consider all possible rankings of the
girls’ tans.
• A table of possibilities is displayed
on the next page.
Agresti/Franklin Statistics, 6 of 62
Example: How to Get A Better
Tan
Agresti/Franklin Statistics, 7 of 62
Example: How to Get A Better
Tan

For each possible outcome, a mean
rank is calculated for the ‘lotion’
group and for the ‘studio’ group.

The difference in the mean ranks is
then calculated for each outcome.
Agresti/Franklin Statistics, 8 of 62
Example: How to Get A Better
Tan

For this experiment, the samples were
independent random samples – the
responses for the girls using the tanning
lotion were independent of the responses
for the girls using the tanning studio.
Agresti/Franklin Statistics, 9 of 62
Example: How to Get A Better
Tan

Suppose that the two treatments have
identical effects.
• A girl’s tan would be the same regardless
of which treatment she uses.
• Then, each of the ten possible outcomes is
equally likely. So, each outcome has
probability of 1/10.
Agresti/Franklin Statistics, 10 of 62
Example: How to Get A Better
Tan

Using the ten possible outcomes, we
can construct a sampling distribution
for the difference between the sample
mean ranks.

The distribution is displayed on the
next page.
Agresti/Franklin Statistics, 11 of 62
Example: How to Get A Better
Tan
Agresti/Franklin Statistics, 12 of 62
Example: How to Get A Better
Tan

Graph of the Sampling Distribution:
Agresti/Franklin Statistics, 13 of 62
Example: How to Get A Better
Tan

The student who planned the experiment
hypothesized that the tanning studio
would give a better tan than the tanning
lotion.
Agresti/Franklin Statistics, 14 of 62
Example: How to Get A Better
Tan

She wanted to test the null hypothesis:
• H0: The treatments are identical in tanning
quality.

Against
• Ha: Better tanning quality results with the
tanning studio.
Agresti/Franklin Statistics, 15 of 62
Example: How to Get A Better
Tan

This alternative hypothesis is one-sided.

If Ha were true, we would expect the ranks
to be smaller (better) for the tanning studio.

Thus, if Ha were true, we would expect the
differences between the sample mean rank
for the tanning lotion and the sample mean
rank for the tanning studio to be positive.
Agresti/Franklin Statistics, 16 of 62
Wilcoxon Test

The test comparing two groups based
on the sampling distribution of the
difference between the sample mean
ranks is called the Wilcoxon test.
Agresti/Franklin Statistics, 17 of 62
Wilcoxon Nonparametric Test
for Comparing Two Groups
1.
Assumptions: Independent random
samples from two groups.
Agresti/Franklin Statistics, 18 of 62
Wilcoxon Nonparametric Test
for Comparing Two Groups
2.
Hypotheses:
•
•
H0: Identical population distributions for
the two groups (this implies equal
expected values for the sample mean
ranks).
Ha: Higher expected value for the sample
mean rank for a specified group (onesided).
Agresti/Franklin Statistics, 19 of 62
Wilcoxon Nonparametric Test
for Comparing Two Groups
3.
Test Statistic:
•
Difference between sample mean ranks
for the two groups (Equivalently, can use
sum of ranks for one sample).
Agresti/Franklin Statistics, 20 of 62
Wilcoxon Nonparametric Test
for Comparing Two Groups
4.
P-value: One-tail or two-tail probability,
depending on Ha, that the difference
between the sample mean ranks is as
extreme or more extreme than observed.
5.
Conclusion: Report the P-value and
interpret it. If a decision is needed,
reject H0 if the P-value ≤ significance
level such as 0.05.
Agresti/Franklin Statistics, 21 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

For the actual experiment:
• the ranks were (2,4,5) for the girls using
the tanning lotion
• the ranks were (1,3) for the girls using the
tanning studio.
Agresti/Franklin Statistics, 22 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

The mean rank for the tanning lotion
is:
(2+4+5)/3 = 3.7

The mean rank for the tanning studio
is:
(1+3)/2=2
Agresti/Franklin Statistics, 23 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

The test statistic is the difference
between the sample mean ranks:
• 3.7 – 2 = 1.7
Agresti/Franklin Statistics, 24 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

The one-sided alternative hypothesis
states that the tanning studio gives a
better tan.

This means that the expected mean rank
would be larger for the tanning lotion than
for the tanning studio, if Ha is true.

And, the difference between the mean
ranks would be positive.
Agresti/Franklin Statistics, 25 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

The test statistic we obtained from the
data was:
• Difference between the sample mean
ranks = 1.7.

P-value = P(difference between sample
mean ranks at least as large as 1.7)
Agresti/Franklin Statistics, 26 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

The P-value can be obtained from the graph of the
sampling distribution (as seen on a previous page
and displayed again here):

Agresti/Franklin Statistics, 27 of 62
Example: Does the UV Tanning
Studio Give a Better Tan than the
Tanning Lotion?

P-value = 0.20.

This is not a very small P-value.

The evidence does not strongly
support the claim that the tanning
studio gives a better tan.
Agresti/Franklin Statistics, 28 of 62
The Wilcoxon Rank Sum

The Wilcoxon test can, equivalently, use
as the test statistic the sum of the ranks
in just one of the samples.

This statistic will have the same
probabilities as the differences between
the sample mean ranks.

Some software reports the sum of ranks
as the Wilcoxon rank sum statistic.
Agresti/Franklin Statistics, 29 of 62
Example: Is there a treatment
difference between the UV Tanning
Studio and the Tanning Lotion?

Suppose the experiment was
designed with a two-sided alternative
hypothesis:
• H0: The treatments are identical in tanning
•
quality.
Ha: The treatments are different in tanning
quality.
Agresti/Franklin Statistics, 30 of 62
Example: Is there a treatment
difference between the UV Tanning
Studio and the Tanning Lotion?
Agresti/Franklin Statistics, 31 of 62
Using the Wilcoxon Test with a
Quantitative Response

When the response variable is
quantitative, the Wilcoxon test is applied
by converting the observations to ranks.

For the combined sample, the
observations are ordered from smallest to
largest.

The test compares the mean ranks for the
two samples.
Agresti/Franklin Statistics, 32 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

Experiment:
• A sample of 64 college students were
•
randomly assigned to a cell phone group
or a control group, 32 to each.
On a machine that simulated driving
situations, participants were instructed to
press a “brake button” when they detected
a red light.
Agresti/Franklin Statistics, 33 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

Experiment:
• The control group listened to the radio
•
•
while they performed the simulated
driving.
The cell phone group carried out a
conversation on a cell phone.
Each subject’s response time to the red
lights is recorded and averaged over all of
his/her trials.
Agresti/Franklin Statistics, 34 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

Boxplots of the data:
Agresti/Franklin Statistics, 35 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?



The box plots do not show any substantial
skew, but there is an extreme outlier for the
cell phone group.
The t inferences that we have used
previously assume normal population
distributions.
The Wilcoxon Test does not assume
normality. This test can be used in place of
the t test if the normality assumption is
questioned.
Agresti/Franklin Statistics, 36 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?


To use the Wilcoxon test, we need to
rank the data (response times) from 1
(smallest reaction time) to 64 (largest
reaction time).
The test statistic is then calculated
from the ranks.
Agresti/Franklin Statistics, 37 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

The next page shows the output for
the hypothesis test:
• H0: The distribution of reaction times is
•
identical for the two groups.
Ha: The distribution of reaction times
differs for the two groups.
Agresti/Franklin Statistics, 38 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?
Agresti/Franklin Statistics, 39 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

The small P-value (.019) shows strong
evidence against the null hypothesis.

The sample mean ranks suggest that
reaction times tend to be slower for
those using cell phones.
Agresti/Franklin Statistics, 40 of 62
Example: Do Drivers Using Cell
Phones Have Slower Reaction Times?

Insight:
• The Wilcoxon test is not affected by
outliers.
• No matter how far the largest observation
falls from the next largest, it still gets the
same rank.
Agresti/Franklin Statistics, 41 of 62
Nonparametric Estimation
Comparing Groups

When the response variable is
quantitative, we can compare a
measure of center for the two groups.
• One way to do this is by comparing
•
means.
This method requires the assumption of
normal population distributions.
Agresti/Franklin Statistics, 42 of 62
Nonparametric Estimation
Comparing Groups

When the response distribution is
highly skewed, nonparametric
methods are preferred.
• For highly skewed distributions, a better
•
measure of the center is the median.
We can then estimate the difference
between the population medians for the
two groups.
Agresti/Franklin Statistics, 43 of 62
Nonparametric Estimation
Comparing Groups

Most software for the Wilcoxon test
reports point and interval estimates
comparing medians.
• Some software refers to the equivalent
Mann-Whitney test.
Agresti/Franklin Statistics, 44 of 62
Nonparametric Estimation
Comparing Groups


The Wilcoxon test (and the MannWhitney test) does not require a
normal population assumption.
It does require an extra assumption:
the population distributions for the
two groups are symmetric and have
the same shape.
Agresti/Franklin Statistics, 45 of 62
For a study on the effects of hypnosis, subjects
were divided into a control group and treatment
group and a measure of respiratory ventilations
was taken on each subject.
Controls: 3.99 4.19 4.21 4.54 4.64 4.69 4.84 5.48
Treated: 4.36 4.67 4.78 5.08 5.16 5.20 5.52 5.74
What is the mean rank for the Control Group?
a. 8
b. 49
c. 6.125
d. 7.5
Agresti/Franklin Statistics, 46 of 62
For a study on the effects of hypnosis, subjects
were divided into a control group and treatment
group and a measure of respiratory ventilations
was taken on each subject.
Controls: 3.99 4.19 4.21 4.54 4.64 4.69 4.84 5.48
Treated: 4.36 4.67 4.78 5.08 5.16 5.20 5.52 5.74
What is the mean rank for the Treatment
Group?
a. 8
b. 10.875
c. 13.25
d. 16
Agresti/Franklin Statistics, 47 of 62
For a study on the effects of hypnosis, subjects
were divided into a control group and treatment
group and a measure of respiratory ventilations
was taken on each subject.
Controls: 3.99 4.19 4.21 4.54 4.64 4.69 4.84 5.48
Treated: 4.36 4.67 4.78 5.08 5.16 5.20 5.52 5.74
A test of the hypothesis that subjects in the
treatment group tended to ventilate more
resulted in a P-value of 0.025. Does this Pvalue support the claim that subjects in the
treatment group ventilated more?
a. yes
b. no
Agresti/Franklin Statistics, 48 of 62
Section 14.2
Nonparametric Methods for Several
Groups and for Matched Pairs
Agresti/Franklin Statistics, 49 of 62
Comparing Mean Ranks of
Several Groups

The Wilcoxon test for comparing
mean ranks of two groups extends to
a comparison of mean ranks for
several groups.
• This test is called the KruskalWallis test.
Agresti/Franklin Statistics, 50 of 62
ANOVA test vs. KruskalWallis test

Both tests are used to compare many
groups.
• The ANOVA F test assumes normal
•
•
population distributions.
The Kruskal-Wallis test does not make this
assumption.
The Kruskal-Wallis test is a “safer” method
to use with small samples when not much
information is available about the shape of
the distributions.
Agresti/Franklin Statistics, 51 of 62
ANOVA test vs. KruskalWallis test

The Kruskal-Wallis test is also useful
when the data are merely ranks and
we don’t have a quantitative
measurement of the response
variable.
Agresti/Franklin Statistics, 52 of 62
Summary: Kruskal-Wallis Test
1.
Assumptions: Independent random
samples from several (g) groups.
2.
Hypotheses:
H0: Identical population distributions for
the g groups
Ha: Population distributions not all
identical.
Agresti/Franklin Statistics, 53 of 62
Summary: Kruskal-Wallis Test
3.
Test statistic: Uses between-groups
variability of sample mean ranks.
Software easily calculates this.
4.
P-value: Right-tail probability above
observed test statistic value from chisquared distribution.
5.
Conclusion: Report the P-value and
interpret in context.
Agresti/Franklin Statistics, 54 of 62
Example: Does Heavy Dating
Affect College GPA?

Experiment: A student in a statistics
class (Tim) decided to study whether
dating was associated with college
GPA.

He wondered whether students who
data a lot tend to have poorer GPAs.
Agresti/Franklin Statistics, 55 of 62
Example: Does Heavy Dating
Affect College GPA?

Experiment: He asked 17 students in
the class to anonymously fill out a
short questionnaire in which they
were asked to give their college GPA
and to indicate whether, during their
college careers, they had dated
regularly, occasionally, or rarely.
Agresti/Franklin Statistics, 56 of 62
Example: Does Heavy Dating
Affect College GPA?

Dot plots of the GPA data for the 3 dating groups:
Agresti/Franklin Statistics, 57 of 62
Example: Does Heavy Dating
Affect College GPA?

Since the dot plots showed evidence
of severe skew to the left and since
the sample size was small in each
group, Tim felt safer analyzing the
data with the Kruskal-Wallis test than
with the ordinary ANOVA F test.
Agresti/Franklin Statistics, 58 of 62
Example: Does Heavy Dating
Affect College GPA?

The hypotheses for the Kruskal-Wallis
test:
H0: Identical population distributions for the
three dating groups
Ha: Population distributions for the three
dating groups are not all identical.
Agresti/Franklin Statistics, 59 of 62
Example: Does Heavy Dating
Affect College GPA?

This table shows the data with the GPA values ordered
from smallest to largest for each dating group.
Agresti/Franklin Statistics, 60 of 62
Example: Does Heavy Dating
Affect College GPA?

MINITAB output for the Kruskal-Wallis test:
Agresti/Franklin Statistics, 61 of 62
Example: Does Heavy Dating
Affect College GPA?

The test statistic reported in the output is
H = 0.72.

The corresponding P-value reported in
the output is 0.696.

This large P-value does not give any
evidence against H0.

It is plausible that GPA is independent of
dating group.
Agresti/Franklin Statistics, 62 of 62