Chapter 5
Association between
Categorical Variables
Copyright © 2011 Pearson Education, Inc.
5.1 Contingency Tables
Which hosts send more buyers to
Amazon.com?

To answer this question we must gather data on
two categorical variables: Host and Purchase

Host identifies the originating site: MSN,
RecipeSource, or Yahoo; Purchase indicates
whether or not the visit results in a sale
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5.1 Contingency Tables
Consider Two Categorical Variables
Simultaneously

A table that shows counts of cases on one
categorical variable contingent on the value of
another (for every combination of both variables)

Cells in a contingency table are mutually
exclusive
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5.1 Contingency Tables
Contingency Table for Web Shopping
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5.1 Contingency Tables
Marginal and Conditional Distributions
•
•
Marginal distributions appear in the “margins” of a
contingency table and represent the totals
(frequencies) for each categorical variable
separately
Conditional distributions refer to counts within a
row or column of a contingency table (restricted
to cases satisfying a condition)
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5.1 Contingency Tables
Conditional Distribution of Purchase for each
Host (Column Counts and Percentages)
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5.1 Contingency Tables
Conditional Distribution
•
Reveals the percentage of purchases
among visitors from RecipeSource to be
much less than for MSN and Yahoo
•
Host and Purchase are associated
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5.1 Contingency Tables
Segmented Bar Charts
•
Used to display conditional distributions
•
Divides the bars in a bar chart into
segments that are proportional to the
percentage in each category of a second
variable
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5.1 Contingency Tables
Contingency Table of Purchase by Region
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5.1 Contingency Tables
Segmented Bar Chart Shows Association
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5.1 Contingency Tables
Mosaic Plots

Alternative to segmented bar chart

A plot in which the size of each “tile” is
proportional to the count in a cell of a
contingency table
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5.1 Contingency Tables
Contingency Table of Shirt Size by Style
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5.1 Contingency Tables
Mosaic Plot Shows Association
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4M Example 5.1: CAR THEFT
Motivation
Should insurance companies vary the
premiums for different car models (are
some cars more likely to be stolen than
others)?
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4M Example 5.1: CAR THEFT
Method
Data obtained from the National Highway Traffic
Safety Administration (NHTSA) on car theft for
seven popular models (two categorical variables:
type of car and whether the car was stolen).
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4M Example 5.1: CAR THEFT
Mechanics
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4M Example 5.1: CAR THEFT
Mechanics
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4M Example 5.1: CAR THEFT
Message
The Dodge Intrepid is more likely to be stolen than
other popular models. The data suggest that
higher premiums for theft insurance should be
charged for models that are more likely to be
stolen.
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5.2 Lurking Variables
and Simpson’s Paradox
Association Not Necessarily Causation

Lurking Variable: a concealed variable that
affects the apparent relationship between two
other variables

Simpson’s Paradox: a change in the association
between two variables when data are separated
into groups defined by a third variable
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4M Example 5.2: AIRLINE ARRIVALS
Motivation
Does it matter which of two airlines a
corporate CEO chooses when flying to
meetings if he wants to avoid delays?
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4M Example 5.2: AIRLINE ARRIVALS
Method
Data obtained from US Bureau of
Transportation Statistics on flight delays for
two airlines (two categorical variables:
airline and whether the flight arrived on
time).
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4M Example 5.2: AIRLINE ARRIVALS
Mechanics
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4M Example 5.2: AIRLINE ARRIVALS
Mechanics –
Is destination a lurking variable?
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4M Example 5.2: AIRLINE ARRIVALS
Mechanics –
This is Simpson’s Paradox
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4M Example 5.2: AIRLINE ARRIVALS
Message
The CEO should book on US Airways as it is
more likely to arrive on time regardless of
destination.
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5.3 Strength of Association
Chi-Squared Statistic

A measure of association in a contingency
table

Calculated based on a comparison of the
observed contingency table to an artificial
table with the same marginal totals but no
association
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5.3 Strength of Association
Contingency Table
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5.3 Strength of Association
Calculating the Chi-Squared Statistic
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5.3 Strength of Association
Calculating the Chi-Squared Statistic
2
x 

 30  40 
40
 10 
40
2

2

 70  60 
10 
60
2
60
2

10 

2
40

 50  40 
40
 10 
2
(50  60)2

60
2
60
 2.5  1.67  2.5  1.67
 8.33
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5.3 Strength of Association
Cramer’s V

Derived from the Chi-Squared Statistic

Ranges in value from 0 (variables are not
associated) to 1(variables are perfectly
associated)
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5.3 Strength of Association
Calculating Cramer’s V
V
x2
nmin  r  1, c  1
V = 0.20 for our example
There is a weak association between group
(students or staff) and attitude toward sharing
copyrighted music
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5.3 Strength of Association
Checklist: Chi-Squared and Cramer’s V

Verify that variables are categorical

Verify that there are no obvious lurking
variables
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4M Example 5.3: REAL ESTATE
Motivation
Do people who heat their homes with gas
prefer to cook with gas as well? What
heating systems and appliances should a
developer select for newly built homes?
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4M Example 5.3: REAL ESTATE
Method
The developer contacts homeowners to
obtain the data. Two categorical variables:
type of fuel used for home heating (gas or
electric) and type of fuel used for cooking
(gas or electric).
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4M Example 5.3: REAL ESTATE
Mechanics
Chi-Squared = 98.62; Cramer’s V = 0.47
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4M Example 5.3: REAL ESTATE
Message
Homeowners prefer gas to electric heat by
about 2 to 1. The developer should build
about two-thirds of new homes with gas
heat. Put electric appliances in all homes
with electric heat and in half of the homes
with gas heat (assuming that buyers for
new homes have the same preferences).
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Best Practices

Use contingency tables to find and summarize
association between two categorical variables.

Be on the lookout for lurking variables.

Use plots to show association.

Exploit the absence of association.
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Pitfalls

Don’t interpret association as causation.

Don’t display too many numbers in a table.
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