CHAPTER 17
BIVARIATE STATISTICS:
NONPARAMETRIC TESTS
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-1
What The Experts Say
A parametric procedure is an exact solution to an
approximate problem, whereas a nonparametric
procedure is an approximate solution to an exact
problem.
--Amir D. Aczel, Complete
Business Statistics (Homewood,
Ill.: Irwin/McGraw-Hill,
1999).
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-2
Learning Objectives
• Explain the use of various nonparametric
tests
• Run various nonparametric tests
• Interpret the results obtained from
nonparametric tests
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-3
Get This! Does Color Matter?
• Officials in the U.S. government responsible for
several million questionnaires every year have
reported that yellow paper produces the highest
percentage of returns, with pink next in
effectiveness
• Francis Buttle, a professor of marketing at
Cranfield University in the United Kingdom, and
Gavin Thomas, an independent marketing
research consultant, were not convinced that color
makes a difference
• They conducted an experiment as part of a large
survey into the perceived cost and benefits of ISO
9000 in certified organizations.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-4
Get This! Does Color Matter? – cont’d
• The experiment was designed to evaluate
whether yellow-colored paper stock produced a
significantly different response rate than white.
• More white questionnaires (29.32%) were
returned than yellow (28.24%). The
researchers used a statistical technique called
“chi-square test of significance” to analyze the
data. They concluded that the difference in
response rates between yellow and white
returns was not significant.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-5
Now Ask Yourself
• Do you think paper color matters when it
comes to deciding whether or not a
questionnaire will be completed and mailed
back to the researcher?
• Do you think the chi-square test of
significance was really necessary to generate
the findings? Why or why not?
• Could the Internet have been used to
conduct a similar study? Why or why not?
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-6
Nonparametric Tests
• Nonparametric methods of testing hypotheses are
•
•
•
•
not concerned with particular population
parameters.
We do not have to know the shape of the
population or make assumptions for the purpose of
testing hypotheses.
They have relatively easy computations and are
relatively easy to understand.
They can be used with ranked information and for
tests involving small samples sizes.
Their applications are limited to certain types of
information.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-7
Nonparametric Tests – cont’d
• They tend to be less precise and efficient than
parametric methods.
• They can be used in any of the following
situations:
– When the population distribution from which
the sample is drawn cannot be assumed to be
normal.
– When the sampling distribution is known not
to be normal.
– When nonmetric (nominal- or ordinal-scaled)
data are used.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-8
Contingency Tables –
Chi-Square Test of Independence
• A contingency table or cross-tabulation table is a
statistical table in which row entries classify data
according to one variable and column entries classify
data according to another variable. When there are r
rows and c columns in theΧtable, it is called an r x c
contingency table.
– The frequencies in the cells are called cell
frequencies.
– The total of the frequencies in each row or each
column is called the marginal frequency.
– The same formula used to explain the goodness-offit test can be used to compute the value of 2.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-9
Contingency Tables –
Chi-Square Test of Independence – cont’d
(Oi  Ei ) 2
 
Ei
i 1
k
2
where
Oi = observed frequency in cell i
E i = expected or theoretical frequency in cell i
k
= number of mutually exclusive categories
The number of degrees of freedom of an r x c contingency
table is (r - 1)(c - 1).
Contingency tables may be constructed in various ways.
Sometimes they simply show the numbers of elements in
each cell. Other times, raw percentages are provided in each
cell.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-10
Rank-Sum Tests
Use some sort of ranking totals in their calculations
• Wilcoxon Test
• Mann-Whitney (U) Test
• Kruskal-Wallis (H) Test
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-11
Wilcoxon Test for Matched Pairs
 a.k.a., the signed-rank test
 Analyzes ordinal data for differences between two related
samples by using plus and minus signs and considers both
the magnitudes of the differences and ranks of the
differences between the paired values.
– Test is used when there is a natural way to pair data.
– Test works well for before and after data to test for a
median difference of zero.
– Test can take the place of a one-sample t-test when we
cannot assume a normal distribution.
– Purpose of the test is to decide whether the difference
between a computed signed-rank sum and the
expected rank sum of the same sign is large enough to
be significant.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-12
Wilcoxon Test for Matched Pairs – cont’d
The sum of all ranks: 1, 2, 3, … n can be obtained by
(n/2)(1+n)
The expected value of T is ½ of the sum of all ranks, or
E(T) = n(1+n)/4
When the sample size is large, preferably 10 or more,
the sampling distribution of T is approximately normal.
The standard error of the statistic T is
T 
n(n  1)( 2n  1)
24
and the standard normal z is
z
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
T  E (T )
T
PPT-13
Mann-Whitney (U) Test
• a.k.a., the U test
• Is another type of rank-sum test. This test
can be used to determine whether two
independent samples are drawn from
identical populations or from two
populations with the same median.
• This test can take the place of the twosample t-test when the researcher is unsure
about the normality assumption.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-14
Mann-Whitney (U) Test – cont’d
Let n1 = the number of items in sample A
n2 = the number of items in sample B
The sum of cumulative items of first sample A, denoted by
the letter U, can be obtained by two methods.
Method 1.
Count the number of A items that precede each
B item. The sum of the numbers counted is the
value of U.
Method 2.
Use the following formula:
U  n1  n2 
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
n1 (n1  1)
 R1
2
where R1 = total of ranks of A
items
PPT-15
Mann-Whitney (U) Test – cont’d
When the sample sizes are large, preferably both n1 and
n2 larger than 10, the sampling distribution of the U is
approximately normal.
The expected value of U is
E (U ) 
n1n2
2
the standard error of the statistic U is
n1n2 (n1  n2  1)
U 
12
and the standard normal z is
z
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
U  E (U )
U
PPT-16
Kruskal-Wallis (H) Test
• A type of rank-sum test. This test can be used to
determine whether k independent samples are
drawn from identical populations or from k
populations with the same median.
• This test may be used to substitute for the method
of one-way analysis of variance.
• a.k.a., the H test
• If the null hypothesis that k samples are drawn
from identical populations is true and each sample
size is 5 or more, the sampling distribution of the
statistic H can be approximated by the 2
distribution with D (degrees of freedom) = k - 1.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-17
Groupings of Two Sample
Nonparametric Tests
NONPARAMETRIC TEST
TWO SAMPLES
Independent
Paired
Chi-Square
Chi-Square
Mann-Whitney
Wilcoxin
Kruskal-Wallis
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-18
Decision Time!
We ran some of the nonparametric tests using
Microsoft Excel and SPSS. As a marketing manager,
you must be careful not to squander your company’s
financial resources.
In some of the computer runs, SPSS provided much
more detailed results, but the SPSS software tends to
be an added expense for companies since SPSS does
not come with most computers like Excel does.
Based on your brief experiences with the two types of
software, would you spend a little more money to
obtain more detailed results? Why or why not?
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-19
Net Impact
The Internet:
• Does not aid in the analysis of
nonparametric data, but it can be used to
help explain why certain events occur.
• Can contribute information that helps
researchers better understand the
relationships that the tests reveal.
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-20
Chapter 17
End of Presentation
Marketing Research, 2nd Edition
Alan T. Shao
Copyright © 2002 by South-Western
PPT-21