### Categorical Data Analysis

```Categorical Data Analysis
CDA
Outline
• Contingency Table
• Graphical display of Categorical Data
Bar Chart, Pie Chart, Mosaic Plot
• Measures of Association
Pearson Correlation Coefficient, Cramer’s V
• Test of Independence
• Test of Symmetry
Contingency Table
• A contingency table is a rectangular table
having I rows for categories of X and J columns
for categories of Y.
• The cells of the table represent the I×J
possible outcomes.
Contingency Table: Example 1_Heart
attack vs. Aspirin use
• The table below is from a report on the
relationship between aspirin use and heart
attacks by the Physicians’ Health Study
Research Group at Harvard Medical School.
• The 2×3 contingency table is
Myocardial Infarction
Treatment
Fatal Attack
Nonfatal Attack
No Attack
Placebo
18
171
10,845
Aspirin
5
99
10,933
Generating Contingency Table in R
• Input the 2×3 table in R as a 2×3 matrix
• Change the matrix to table using the function
as.table(), because some functions are happier
with tables than matrices
Graphical Display of Categorical Data
• One Categorical Variable
Bar Chart: a chart with rectangular bars with
lengths proportional to the values that they
represent
Pie Chart: a circular chart divided into sectors,
illustrating proportion.
Graphical Display of Categorical Data
• Two Categorical Variables
Mosaic Plot: a graphical display that examine
the relationship among two or more
categorical variables.
Mosaic Plot Construction
• A mosaic plot starts with a square with length
one. The square is divided firstly into
horizontal bars whose widths are proportional
to the probabilities associated with the first
categorical variable. Then each bar is split
vertically into bars that are proportional to the
conditional probabilities of the second
categorical variables. Additional splits can be
made if wanted using a third, fourth variable,
etc.
Mosaic Plot: Example 2_HairEyeColor
• The HiarEyeColor data comes from a survey of
students at the University of Delaware (1974).
It has 592 observations on 3 variables (Hair,
Eye, Sex). Here we omit Sex.
Mosaic Plot in R
• Option 1: install package vcd, use function
mosaic()
• Option 2: use function mosaicplot()
Measures of Association
• Continuous Variables-Pearson Correlation
Coefficient
• Ordinal Variables-Pearson Correlation
Coefficient
• Nominal Variables-Cramer’s V
Cramer’s V
• Cramer’s V measures the association between
two nominal variables. It varies from 0 (no
association) to 1 (complete association) and
can reach 1 only when the two variables are
equal to each other.
Cramer’s V (cont’d)
1, When the two variables are binary, Cramer’s
V is the same as Phi Coefficient (which
measures the association between two binary
variables)
2, In R, under library(vcd), use function
assocstats()
Contingency Table Analysis
• Large Sample Size
Chi-square Test
• Small Sample Size
Fisher’s Exact Test
Test of Independence (Chi-square Test)
Column 1
Column 2
Total
Row 1
π11
π12
π1+
Row 2
π21
π22
π2+
Total
π+1
π+2
1
H0: Row and Column are independent
πij=πi+π+j for all i,j
Ha: Row and Column are not independent
πij≠πi+π+j for some i and j
Test of Independence (Chi-square Test)
Under H0: πij=πi+π+j for all i,j
Expected Counts in each cell is
Test of Independence (Fisher’s Exact Test)
• When any of the expected counts fall below 5,
Chi-square test is not appropriate. Instead, we
use Fisher’s Exact Test.
Example 3: The following data are from a
Stanford University study of the effectiveness
of the antidepressant Celexa in the treatment
of compulsive shopping.
Outcome
Treatment
Worse
Same
Better
Celexa
2
3
7
Placebo
2
8
2
Test of Independence in R
• Chi-Square Test
Use R function chisq.test()
• Fisher’s Exact Test
Use R function fisher.test()
Test of Symmetry: Matched Pairs
• Example 4: Suppose two surveys on
President’s job approval were conducted onemonth apart on 1600 Americans and the
result is summarized in the following table.
(Source: Agresti, 1990) Is there a significant
difference in job approval rating?
2nd Survey
1st Survey
Approve
Disapprove
Approve
794
150
Disapprove
86
570
Test of Symmetry: Matched Pairs
Useful Resource
• Quick R
http://www.statmethods.net/index.html
```