TwoWay Tables

1
Chapter 6

• Two-Way Tables

2
Association

• To study associations between quantitative
  variables ? correlation regression (Ch 4 Ch
  5)
• To study associations between categorical
  variables ? cross-tabulate frequencies
  calculate conditional percents (this Chapter)

3
Example Age and Education
Age groups is the categorical explanatory
variable Education level is the categorical
response variable
Marginal distributions
4
Example Marginal Totals
5
Marginal Distributions
Marginal distributions are used as background
information only. They do not address association
6
Marginal Distribution, Row Variable
7
Marginal Distribution, Column Variable
BPS
Chapter 6
7
8
Association

• To determine associations, calculate conditional
  distributions (conditional percents)
• Two types of conditional distributions
• Conditioned on row variable
• Conditioned on column variable

BPS
Chapter 6
8
9
Association

• If explanatory variable is in rows
• calculate row percents
• analyze row conditional distributions

10
Association

• If explanatory variable is in columns
• calculate column percents
• analyze column conditional distribution

BPS
Chapter 6
10
11
Example Column Percents
Is AGE associated with EDUCATION? AGE is
explanatory var. ? use column percents
12
Example Association
As age goes up, completing college goes
down NEGATIVE association between age and
education
13
Association

• No association conditional percents nearly equal
  at all levels of explanatory variable
• Positive association as explanatory variable
  rises ? conditional percentages increase
• Negative associations as explanatory variable
  rises ? conditional percentages go down

14
Example 2 Row Percent

• Statement of problem Is ACCEPTANCE into a
  graduate program (response variable) predicted by
  GENDER (explanatory variable)?

Explanatory variable (gender) is in rows ? use
row percents
BPS
Chapter 6
14
15
Example 2
Statement of problem Is ACCEPTANCE associated
with GENDER?
Explanatory variable in rows ? use row percents
Therefore positive association with maleness
BPS
Chapter 6
15
16
Simpsons Paradox
Lurking variables can change or even reverse the
direction of an association

• In example 2, consider the lurking variable
  "major
• Business School (240 applicants)
• Art School (320 applicants)
• Does this lurking variable explain the
  association?
• To address this potential problem, subdivide the
  data according to the lurking variable

17
Simpsons Paradox Illustration
18
Simpsons Paradox Illustration

• Overall higher proportion of men accepted than
  women
• Within majors ? higher proportion of women
  accepted than men
• Reason ? Men applied to easier majors ? the
  initial association was an artifact of the
  lurking variable MAJOR applied to