Chapter 6Chapter 6Two-Way Tables

BPS - 5th Ed. Chapter 6 1

Categorical VariablesCategorical Variables

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.

BPS - 5th Ed. Chapter 6 2

Two-Way TableTwo-Way Table

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 variableThe number of observations falling into each

combination of categories is entered into each cell of the table

BPS - 5th Ed. Chapter 6 3

Two-Way TableTwo-Way TableData from Wissahickon High School

and their AP course involvement

What are the two categorical data pieces?◦# of AP courses◦Year in school

BPS - 5th Ed. Chapter 6 4

Marginal DistributionsMarginal Distributions

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)

BPS - 5th Ed. Chapter 6 5

Marginal DistributionsMarginal Distributions

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 percentsA bar graph could be used to

graphically display marginal distributions for categorical variables

BPS - 5th Ed. Chapter 6 6

Case StudyCase Study

BPS - 5th Ed. Chapter 6 7

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

Case StudyCase Study

BPS - 5th Ed. Chapter 6 8

Age and Education

Variables

Marginal distributions

Case StudyCase Study

BPS - 5th Ed. Chapter 6 9

Age and Education

Variables

Marginal distributions

21.6% 46.5% 32.0%

15.9%33.1%25.4%25.6%

Case StudyCase Study

BPS - 5th Ed. Chapter 6 10

Age 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%

Conditional DistributionsConditional Distributions

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

BPS - 5th Ed. Chapter 6 11

Case StudyCase Study

BPS - 5th Ed. Chapter 6 12

Age and EducationCompare 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.

Case StudyCase Study

BPS - 5th Ed. Chapter 6 13

Age and EducationCompare 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

Case StudyCase Study

BPS - 5th Ed. Chapter 6 14

Age 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%

Conditional DistributionsConditional DistributionsThe 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.

BPS - 5th Ed. Chapter 6 15

Case StudyCase Study

BPS - 5th Ed. Chapter 6 16

Age and Education

Conditional Distributions of Age for each level of Education:

Simpson’s ParadoxSimpson’s ParadoxWhen 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.

BPS - 5th Ed. Chapter 6 17

Discrimination?Discrimination?(Simpson’s Paradox)(Simpson’s Paradox)

Consider the acceptance rates for the following group of men and women who applied to college.

BPS - 5th Ed. Chapter 6 18

counts AcceptedNot

acceptedTotal

Men 198 162 360

Women 88 112 200

Total 286 274 560

percentsAcceptedNot

accepted

Men 55% 45%

Women 44% 56%

A higher percentage of men were accepted: Discrimination?

Discrimination?Discrimination?(Simpson’s Paradox)(Simpson’s Paradox)Lurking variable: Applications were split between the Business School (240) and the Art School (320).

BPS - 5th Ed. Chapter 6 19

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

BUSINESS SCHOOL

Discrimination?Discrimination?(Simpson’s Paradox)(Simpson’s Paradox)Lurking variable: Applications were split between the Business School (240) and the Art School (320).

BPS - 5th Ed. Chapter 6 20

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

Discrimination?Discrimination?(Simpson’s Paradox)(Simpson’s Paradox) 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.

BPS - 5th Ed. Chapter 6 21