Chapter 10
Analyzing the Association
Between Categorical Variables
 Learn
….
How to detect and describe
associations between
categorical variables
Agresti/Franklin Statistics, 1 of 90
 Section 10.1
What Is Independence and What
is Association?
Agresti/Franklin Statistics, 2 of 90
Example: Is There an Association
Between Happiness and Family Income?
Agresti/Franklin Statistics, 3 of 90
Example: Is There an Association
Between Happiness and Family Income?
Agresti/Franklin Statistics, 4 of 90
Example: Is There an Association
Between Happiness and Family Income?

The percentages in a particular row of
a table are called conditional
percentages

They form the conditional distribution
for happiness, given a particular
income level
Agresti/Franklin Statistics, 5 of 90
Example: Is There an Association
Between Happiness and Family Income?
Agresti/Franklin Statistics, 6 of 90
Example: Is There an Association
Between Happiness and Family Income?

Guidelines when constructing tables
with conditional distributions:
• Make the response variable the column
variable
• Compute conditional proportions for the
response variable within each row
• Include the total sample sizes
Agresti/Franklin Statistics, 7 of 90
Independence vs Dependence


For two variables to be independent,
the population percentage in any
category of one variable is the same
for all categories of the other variable
For two variables to be dependent (or
associated), the population
percentages in the categories are not
all the same
Agresti/Franklin Statistics, 8 of 90
Example: Happiness and
Gender
Agresti/Franklin Statistics, 9 of 90
Example: Happiness and
Gender
Agresti/Franklin Statistics, 10 of 90
Example: Belief in Life After
Death
Agresti/Franklin Statistics, 11 of 90
Example: Belief in Life After
Death

Are race and belief in life after death
independent or dependent?
• The conditional distributions in the table
are similar but not exactly identical
• It is tempting to conclude that the
variables are dependent
Agresti/Franklin Statistics, 12 of 90
Example: Belief in Life After
Death

Are race and belief in life after death
independent or dependent?
• The definition of independence between
variables refers to a population
• The table is a sample, not a population
Agresti/Franklin Statistics, 13 of 90
Independence vs Dependence


Even if variables are independent, we
would not expect the sample
conditional distributions to be
identical
Because of sampling variability, each
sample percentage typically differs
somewhat from the true population
percentage
Agresti/Franklin Statistics, 14 of 90
 Section 10.2
How Can We Test whether
Categorical Variables are
Independent?
Agresti/Franklin Statistics, 15 of 90
A Significance Test for
Categorical Variables

The hypotheses for the test are:
H0: The two variables are independent
Ha: The two variables are dependent
(associated)
• The test assumes random sampling and a
large sample size
Agresti/Franklin Statistics, 16 of 90
What Do We Expect for Cell Counts if
the Variables Are Independent?

The count in any particular cell is a
random variable
• Different samples have different values for
the count

The mean of its distribution is called
an expected cell count
• This is found under the presumption that
H0 is true
Agresti/Franklin Statistics, 17 of 90
How Do We Find the Expected
Cell Counts?

Expected Cell Count:
•
For a particular cell, the expected cell count
equals:
(Row total) (Column total)
Expected cell count 
Total sample size
Agresti/Franklin Statistics, 18 of 90
Example: Happiness by Family
Income
Agresti/Franklin Statistics, 19 of 90
The Chi-Squared Test Statistic
The chi-squared statistic summarizes how
far the observed cell counts in a
contingency table fall from the expected
cell counts for a null hypothesis

2
(observed count - expected count)
 
expected count
2
Agresti/Franklin Statistics, 20 of 90
Example: Happiness and Family
Income
Agresti/Franklin Statistics, 21 of 90
Example: Happiness and Family
Income

State the null and alternative
hypotheses for this test

H0: Happiness and family income
are independent

Ha: Happiness and family income
are dependent (associated)
Agresti/Franklin Statistics, 22 of 90
Example: Happiness and Family
Income

Report the  statistic and explain how it

was calculated:
To calculate the  statistic, for each cell,
calculate:
2
2
2
(observed count - expected count)
expected count


Sum the values for all the cells
The  value is 73.4
2
Agresti/Franklin Statistics, 23 of 90
Example: Happiness and Family
Income

The larger the  value, the greater the
evidence against the null hypothesis of
independence and in support of the
alternative hypothesis that happiness and
income are associated
2
Agresti/Franklin Statistics, 24 of 90
The Chi-Squared Distribution

To convert the
test statistic to a Pvalue, we use the sampling distribution
2
of the  statistic

For large sample sizes, this sampling
distribution is well approximated by the
chi-squared probability distribution
Agresti/Franklin Statistics, 25 of 90
The Chi-Squared Distribution
Agresti/Franklin Statistics, 26 of 90
The Chi-Squared Distribution

Main properties of the chi-squared
distribution:
• It falls on the positive part of the real
number line
• The precise shape of the distribution
depends on the degrees of freedom:
df = (r-1)(c-1)
Agresti/Franklin Statistics, 27 of 90
The Chi-Squared Distribution

Main properties of the chi-squared
distribution:
• The mean of the distribution equals the
df value
• It is skewed to the right
• The larger the value, the greater the
evidence against H0: independence
Agresti/Franklin Statistics, 28 of 90
The Chi-Squared Distribution
Agresti/Franklin Statistics, 29 of 90
The Five Steps of the ChiSquared Test of Independence
1. Assumptions:
• Two categorical variables
• Randomization
• Expected counts ≥ 5 in all cells
Agresti/Franklin Statistics, 30 of 90
The Five Steps of the ChiSquared Test of Independence
2. Hypotheses:

H0: The two variables are independent

Ha: The two variables are dependent
(associated)
Agresti/Franklin Statistics, 31 of 90
The Five Steps of the ChiSquared Test of Independence
3. Test Statistic:
2
(observed count - expected count)
 
expected count
2
Agresti/Franklin Statistics, 32 of 90
The Five Steps of the ChiSquared Test of Independence
4. P-value: Right-tail probability above the
observed
value, for the chi-squared
distribution with df = (r-1)(c-1)
5. Conclusion: Report P-value and interpret
in context
•
If a decision is needed, reject H0 when P-value ≤
significance level
Agresti/Franklin Statistics, 33 of 90
Chi-Squared is Also Used as a
“Test of Homogeneity”

The chi-squared test does not depend on
which is the response variable and which
is the explanatory variable

When a response variable is identified
and the population conditional
distributions are identical, they are said to
be homogeneous
• The test is then referred to as a test of
homogeneity
Agresti/Franklin Statistics, 34 of 90
Example: Aspirin and Heart
Attacks Revisited
Agresti/Franklin Statistics, 35 of 90
Example: Aspirin and Heart
Attacks Revisited

What are the hypotheses for the chisquared test for these data?

The null hypothesis is that whether a
doctor has a heart attack is independent
of whether he takes placebo or aspirin

The alternative hypothesis is that there’s
an association
Agresti/Franklin Statistics, 36 of 90
Example: Aspirin and Heart
Attacks Revisited

Report the test statistic and P-value for
the chi-squared test:
•

The test statistic is 25.01 with a P-value of 0.000
This is very strong evidence that the
population proportion of heart attacks
differed for those taking aspirin and for
those taking placebo
Agresti/Franklin Statistics, 37 of 90
Example: Aspirin and Heart
Attacks Revisited

The sample proportions indicate that
the aspirin group had a lower rate of
heart attacks than the placebo group
Agresti/Franklin Statistics, 38 of 90
Limitations of the
Chi-Squared Test
If the P-value is very small, strong
evidence exists against the null
hypothesis of independence
But…
 The chi-squared statistic and the Pvalue tell us nothing about the nature
of the strength of the association

Agresti/Franklin Statistics, 39 of 90
Limitations of the
Chi-Squared Test

We know that there is statistical
significance, but the test alone does
not indicate whether there is practical
significance as well
Agresti/Franklin Statistics, 40 of 90
 Section 10.3
How Strong is the Association?
Agresti/Franklin Statistics, 41 of 90
The following is a table on Gender and Happiness:
Gender:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
In a study of the two variables (Gender
and Happiness), which one is the
response variable?
a. Gender
b. Happiness
Agresti/Franklin Statistics, 42 of 90
The following is a table on Gender and Happiness:
Gender:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
What is the Expected Cell Count for
‘Females’ who are ‘Pretty Happy’?
a. 898
b. 801.5
c. 902
d. 521
Agresti/Franklin Statistics, 43 of 90
The following is a table on Gender and Happiness:
Gender:
Not
Pretty
Very
Females
163
898
502
Males
130
Calculate the
705
 value
2
a. 1.75
b. 0.27
c. 0.98
d. 10.34
Agresti/Franklin Statistics, 44 of 90
379
The following is a table on Gender and Happiness:
Gender:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
At a significance level of 0.05, what is the
correct decision?
a. ‘Gender’ and ‘Happiness’ are
independent
b. There is an association between
‘Gender’ and ‘Happiness’
Agresti/Franklin Statistics, 45 of 90
Analyzing Contingency Tables

Is there an association?
• The chi-squared test of
independence addresses this
• When the P-value is small, we infer
that the variables are associated
Agresti/Franklin Statistics, 46 of 90
Analyzing Contingency Tables

How do the cell counts differ from
what independence predicts?

To answer this question, we
compare each observed cell count to
the corresponding expected cell
count
Agresti/Franklin Statistics, 47 of 90
Analyzing Contingency Tables

How strong is the association?

Analyzing the strength of the
association reveals whether the
association is an important one, or if
it is statistically significant but weak
and unimportant in practical terms
Agresti/Franklin Statistics, 48 of 90
Measures of Association

A measure of association is a statistic
or a parameter that summarizes the
strength of the dependence between
two variables
Agresti/Franklin Statistics, 49 of 90
Difference of Proportions

An easily interpretable measure of
association is the difference between
the proportions making a particular
response
Agresti/Franklin Statistics, 50 of 90
Difference of Proportions
Agresti/Franklin Statistics, 51 of 90
Difference of Proportions

Case (a) exhibits the weakest possible
association – no association
Accept Credit Card
Income
High
Low

No
60%
60%
Yes
40%
40%
The difference of proportions is 0
Agresti/Franklin Statistics, 52 of 90
Difference of Proportions

Case (b) exhibits the strongest possible
association:
Income
High
Low

Accept Credit Card
No
Yes
0%
100%
100%
0%
The difference of proportions is 100%
Agresti/Franklin Statistics, 53 of 90
Difference of Proportions

In practice, we don’t expect data to
follow either extreme (0% difference
or 100% difference), but the stronger
the association, the large the absolute
value of the difference of proportions
Agresti/Franklin Statistics, 54 of 90
Example: Do Student Stress and
Depression Depend on Gender?
Agresti/Franklin Statistics, 55 of 90
Example: Do Student Stress and
Depression Depend on Gender?

Which response variable, stress or
depression, has the stronger sample
association with gender?
Agresti/Franklin Statistics, 56 of 90
Example:
Example: Do
Do Student
Student Stress
Stress and
and
Depression
DepressionDepend
Dependon
onGender?
Gender?
Stress:
Gender
Yes
Female
Male

35%
16%
No
65%
84%
The difference of proportions between females
and males was 0.35 – 0.16 = 0.19
Agresti/Franklin Statistics, 57 of 90
Example: Do Student Stress and
Depression Depend on Gender?
Depression:

Gender
Yes
No
Female
8%
92%
Male
6%
94%
The difference of proportions between females
and males was 0.08 – 0.06 = 0.02
Agresti/Franklin Statistics, 58 of 90
Example: Do Student Stress and
Depression Depend on Gender?

In the sample, stress (with a
difference of proportions = 0.19) has a
stronger association with gender than
depression has (with a difference of
proportions = 0.02)
Agresti/Franklin Statistics, 59 of 90
The Ratio of Proportions:
Relative Risk

Another measure of association, is
the ratio of two proportions: p1/p2

In medical applications in which the
proportion refers to an adverse
outcome, it is called the relative risk
Agresti/Franklin Statistics, 60 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents
Agresti/Franklin Statistics, 61 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

Treating the auto accident outcome
as the response variable, find and
interpret the relative risk
Agresti/Franklin Statistics, 62 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

The adverse outcome is death

The relative risk is formed for that outcome

For those who wore a seat belt, the
proportion who died equaled 510/412,878 =
0.00124

For those who did not wear a seat belt, the
proportion who died equaled 1601/164,128
= 0.00975
Agresti/Franklin Statistics, 63 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

The relative risk is the ratio:
• 0.00124/0.00975 = 0.127
• The proportion of subjects wearing a
seat belt who died was 0.127 times the
proportion of subjects not wearing a
seat belt who died
Agresti/Franklin Statistics, 64 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

Many find it easier to interpret the
relative risk but reordering the rows
of data so that the relative risk has
value above 1.0
Agresti/Franklin Statistics, 65 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

Reversing the order of the rows, we
calculate the ratio:
• 0.00975/0.00124 = 7.9
• The proportion of subjects not wearing
a seat belt who died was 7.9 times the
proportion of subjects wearing a seat
belt who died
Agresti/Franklin Statistics, 66 of 90
Example: Relative Risk for Seat Belt
Use and Outcome of Auto Accidents

A relative risk of 7.9 represents a
strong association
• This is far from the value of 1.0 that would
occur if the proportion of deaths were the
same for each group
• Wearing a set belt has a practically
significant effect in enhancing the chance
of surviving an auto accident
Agresti/Franklin Statistics, 67 of 90
Properties of the Relative Risk



The relative risk can equal any
nonnegative number
When p1= p2, the variables are
independent and relative risk = 1.0
Values farther from 1.0 (in either
direction) represent stronger
associations
Agresti/Franklin Statistics, 68 of 90
Large Does Not Mean There’s
a Strong Association



A large chi-squared value provides
strong evidence that the variables are
associated
It does not imply that the variables have a
strong association
This statistic merely indicates (through
its P-value) how certain we can be that
the variables are associated, not how
strong that association is
Agresti/Franklin Statistics, 69 of 90
 Section 10.4
How Can Residuals Reveal the
Pattern of Association?
Agresti/Franklin Statistics, 70 of 90
Association Between
Categorical Variables

The chi-squared test and measures of
association such as (p1 – p2) and p1/p2 are
fundamental methods for analyzing
contingency tables

The P-value for
summarized the
strength of evidence against H0:
independence
Agresti/Franklin Statistics, 71 of 90
Association Between
Categorical Variables


If the P-value is small, then we conclude
that somewhere in the contingency table
the population cell proportions differ from
independence
The chi-squared test does not indicate
whether all cells deviate greatly from
independence or perhaps only some of
them do so
Agresti/Franklin Statistics, 72 of 90
Residual Analysis


A cell-by-cell comparison of the
observed counts with the counts that
are expected when H0 is true reveals
the nature of the evidence against H0
The difference between an observed
and expected count in a particular cell
is called a residual
Agresti/Franklin Statistics, 73 of 90
Residual Analysis


The residual is negative when fewer
subjects are in the cell than expected
under H0
The residual is positive when more
subjects are in the cell than expected
under H0
Agresti/Franklin Statistics, 74 of 90
Residual Analysis

To determine whether a residual is
large enough to indicate strong
evidence of a deviation from
independence in that cell we use a
adjusted form of the residual: the
standardized residual
Agresti/Franklin Statistics, 75 of 90
Residual Analysis

The standardized residual for a cell:
(observed count – expected count)/se
• A standardized residual reports the number of
standard errors that an observed count falls from
its expected count
• Its formula is complex
• Software can be used to find its value
• A large value provides evidence against
independence in that cell
Agresti/Franklin Statistics, 76 of 90
Example: Standardized Residuals for
Religiosity and Gender

“To what extent do you consider
yourself a religious person?”
Agresti/Franklin Statistics, 77 of 90
Example: Standardized Residuals for
Religiosity and Gender
Agresti/Franklin Statistics, 78 of 90
Example: Standardized Residuals for
Religiosity and Gender

Interpret the standardized residuals in
the table
Agresti/Franklin Statistics, 79 of 90
Example: Standardized Residuals for
Religiosity and Gender



The table exhibits large positive residuals
for the cells for females who are very
religious and for males who are not at all
religious.
In these cells, the observed count is much
larger than the expected count
There is strong evidence that the
population has more subjects in these cells
than if the variables were independent
Agresti/Franklin Statistics, 80 of 90
Example: Standardized Residuals for
Religiosity and Gender



The table exhibits large negative residuals
for the cells for females who are not at all
religious and for males who are very
religious
In these cells, the observed count is much
smaller than the expected count
There is strong evidence that the
population has fewer subjects in these cells
than if the variables were independent
Agresti/Franklin Statistics, 81 of 90
 Section 10.5
What if the Sample Size is Small?
Fisher’s Exact Test
Agresti/Franklin Statistics, 82 of 90
Fisher’s Exact Test



The chi-squared test of independence
is a large-sample test
When the expected frequencies are
small, any of them being less than
about 5, small-sample tests are more
appropriate
Fisher’s exact test is a small-sample
test of independence
Agresti/Franklin Statistics, 83 of 90
Fisher’s Exact Test



The calculations for Fisher’s exact
test are complex
Statistical software can be used to
obtain the P-value for the test that the
two variables are independent
The smaller the P-value, the stronger
is the evidence that the variables are
associated
Agresti/Franklin Statistics, 84 of 90
Example: Tea Tastes Better
with Milk Poured First?


This is an experiment conducted by
Sir Ronald Fisher
His colleague, Dr. Muriel Bristol,
claimed that when drinking tea she
could tell whether the milk or the tea
had been added to the cup first
Agresti/Franklin Statistics, 85 of 90
Example: Tea Tastes Better
with Milk Poured First?

Experiment:
• Fisher asked her to taste eight cups of tea:
• Four had the milk added first
• Four had the tea added first
• She was asked to indicate which four
had the milk added first
• The order of presenting the cups was
randomized
Agresti/Franklin Statistics, 86 of 90
Example: Tea Tastes Better
with Milk Poured First?
Results:
Agresti/Franklin Statistics, 87 of 90
Example: Tea Tastes Better
with Milk Poured First?
Analysis:
Agresti/Franklin Statistics, 88 of 90
Example: Tea Tastes Better
with Milk Poured First?

The one-sided version of the test
pertains to the alternative that her
predictions are better than random
guessing

Does the P-value suggest that she
had the ability to predict better than
random guessing?
Agresti/Franklin Statistics, 89 of 90
Example: Tea Tastes Better
with Milk Poured First?


The P-value of 0.243 does not give
much evidence against the null
hypothesis
The data did not support Dr. Bristol’s
claim that she could tell whether the
milk or the tea had been added to the
cup first
Agresti/Franklin Statistics, 90 of 90