BPS - 3rd Ed. Chapter 6 1
Chapter 6
Two-Way Tables
BPS - 3rd Ed. Chapter 6 2
In this chapter we will study the relationship between two categorical variables (variables whose values fall in groups or categories).
To analyze categorical data, use the counts or percents of individuals that fall into various categories.
Categorical Variables
BPS - 3rd Ed. Chapter 6 3
When there are two categorical variables, the data are summarized in a two-way table– each row in the table represents a value of the row
variable– each column of the table represents a value of the
column variable
The number of observations falling into each combination of categories is entered into each cell of the table
Two-Way Table
BPS - 3rd Ed. Chapter 6 4
A distribution for a categorical variable tells how often each outcome occurred – totaling the values in each row of the table
gives the marginal distribution of the row variable (totals are written in the right margin)
– totaling the values in each column of the table gives the marginal distribution of the column variable (totals are written in the bottom margin)
Marginal Distributions
BPS - 3rd Ed. Chapter 6 5
It is usually more informative to display each marginal distribution in terms of percents rather than counts– each marginal total is divided by the table
total to give the percents A bar graph could be used to graphically
display marginal distributions for categorical variables
Marginal Distributions
BPS - 3rd Ed. Chapter 6 6
Case Study
Data from the U.S. Census Bureau for the year 2000 on the level of education reached by Americans
of different ages.
(Statistical Abstract of the United States, 2001)
Age and Education
BPS - 3rd Ed. Chapter 6 7
Case StudyAge and Education
Variables
Marginal distributions
BPS - 3rd Ed. Chapter 6 8
Case StudyAge and Education
Variables
Marginal distributions
21.6% 46.5% 32.0%
15.9%33.1%25.4%25.6%
BPS - 3rd Ed. Chapter 6 9
Case StudyAge and Education
Marginal Distributionfor Education Level
Not HS grad 15.9%
HS grad 33.1%
College 1-3 yrs 25.4%
College ≥4 yrs 25.6%
BPS - 3rd Ed. Chapter 6 10
Relationships between categorical variables are described by calculating appropriate percents from the counts given in the table– prevents misleading comparisons due to
unequal sample sizes for different groups
Conditional Distributions
BPS - 3rd Ed. Chapter 6 11
Case StudyAge and Education
Compare the 25-34 age group to the 35-54 age group in terms of success in completing at least 4 years of college:
Data are in thousands, so we have that 11,071,000 persons in the 25-34 age group have completed at least 4 years of college, compared to 23,160,000 persons in the 35-54 age group.
The groups appear greatly different, but look at the group totals.
BPS - 3rd Ed. Chapter 6 12
Case StudyAge and Education
Compare the 25-34 age group to the 35-54 age group in terms of success in completing at least 4 years of college:
Change the counts to percents: Now, with a fairer comparison using percents, the groups appear very similar.group age 54-35 for (28.4%) .284
81,435
23,160
group age 34-25 for (29.3%) .29337,786
11,071
BPS - 3rd Ed. Chapter 6 13
Case StudyAge and Education
If we compute the percent completing at least four years of college for all of the age groups, this would give us the conditional distribution of age, given that the education level is “completed at least 4 years of college”:
Age: 25-34 35-54 55 and over
Percent with≥ 4 yrs college: 29.3% 28.4% 18.9%
BPS - 3rd Ed. Chapter 6 14
The conditional distribution of one variable can be calculated for each category of the other variable.
These can be displayed using bar graphs. If the conditional distributions of the second variable
are nearly the same for each category of the first variable, then we say that there is not an association between the two variables.
If there are significant differences in the conditional distributions for each category, then we say that there is an association between the two variables.
Conditional Distributions
BPS - 3rd Ed. Chapter 6 15
Case StudyAge and Education
Conditional Distributions of Age for each level of Education:
BPS - 3rd Ed. Chapter 6 16
When studying the relationship between two variables, there may exist a lurking variable that creates a reversal in the direction of the relationship when the lurking variable is ignored as opposed to the direction of the relationship when the lurking variable is considered.
The lurking variable creates subgroups, and failure to take these subgroups into consideration can lead to misleading conclusions regarding the association between the two variables.
Simpson’s Paradox
BPS - 3rd Ed. Chapter 6 17
Consider the acceptance rates for the following group of men and women who applied to college.
Discrimination?(Simpson’s Paradox)
counts AcceptedNot
acceptedTotal
Men 198 162 360
Women 88 112 200
Total 286 274 560
percents AcceptedNot
accepted
Men 55% 45%
Women 44% 56%
A higher percentage of men were accepted: Discrimination?
BPS - 3rd Ed. Chapter 6 18
Discrimination?(Simpson’s Paradox)
counts AcceptedNot
acceptedTotal
Men 18 102 120
Women 24 96 120
Total 42 198 240
percents AcceptedNot
accepted
Men 15% 85%
Women 20% 80%
A higher percentage of women were accepted in Business
Lurking variable: Applications were split between the Business School (240) and the Art School (320).
BUSINESS SCHOOL
BPS - 3rd Ed. Chapter 6 19
Discrimination?(Simpson’s Paradox)
counts AcceptedNot
acceptedTotal
Men 180 60 240
Women 64 16 80
Total 244 76 320
percents AcceptedNot
accepted
Men 75% 25%
Women 80% 20%
ART SCHOOL
A higher percentage of women were also accepted in Art
Lurking variable: Applications were split between the Business School (240) and the Art School (320).
BPS - 3rd Ed. Chapter 6 20
So within each school a higher percentage of women were accepted than men.There is not any discrimination against women!!!
This is an example of Simpson’s Paradox. When the lurking variable (School applied to: Business or Art) is ignored the data seem to suggest discrimination against women. However, when the School is considered the association is reversed and suggests discrimination against men.
Discrimination?(Simpson’s Paradox)