1. Contingency Tables

2. Homework problems
Correlation coefficient
Two-sample t test

3. Contingency tables
An method of analyzing data measured at the nominal or ordinal level
Commonly used in
Survey research
Epidemiology
Cross-sectional designs

4. Contingency tables The table itself is descriptive
What percentage of women in a sample plan to vote for the Democratic candidate for president?
What percentage of adults over 50 in a sample have been diagnosed with high blood pressure?

5. Contingency tables
In conjunction with contingency tables, measures of association can also be used to provide evidence regarding the strength of the relationship between the variables.

6. Contingency tables
A contingency table shows the frequency of each value of the dependent variable for each value of the independent variable.
It also shows the relative frequency of each value of the dependent variable.

7. Example
A county and its largest city are considering adoption of a consolidated government.
Population of city residents is much larger than of the county
Local university conducts randomized telephone survey of residents to assess their opinion on consolidation

8. Survey of 650 county residents
The analysis question is “Are the opinions on consolidation of residents inside the city different from those outside the city?”
Survey of 650 county residents
505 city residents
145 outside city residents
Difficult to assess differences because sample size is different for the two groups

9. INSIDE CITY
OUTSIDE CITY
CONSOLIDATION
FOR
273
54% 54
37%
NO OPINION
134
27% 34
23%
AGAINST 98
19% 57
39%
TOTAL
505
100%
145

10. Construction of Contingency Tables
Independent variable is across the top of the table - labeling the columns
Dependent variable is down the side – labeling the rows
Arrange values of both dependent and independent variable in logical order (especially if ordinal data)

11. Contingency Tables Are referred to by their size and dimensions
Size – the number of rows and columns
Example is a 3 by 2 table
Dimension – the number of variables whose joint distribution is being displayed
Example is a 2-dimensional table

12. Construction of Contingency Tables
Number of cases contained in each cell
Number of cases totaled for each value of the independent variable
Thus, percentages are computed in the direction of the independent variable.

13. Hypotheses
The percentage distribution may provide some support for a hypothesis
City residents are more likely to support city/county consolidation than those who do not live inside the city.
It also suggests an association and the strength of that association.
In our example, a higher percentage of city residents was in favor of consolidation.

14. Measure of statistical significance
However, we may also want to know
how strong is the association
is the difference between city and non-city residents statistically significant?
To assess this, we need a measure of association.
For interval level data, we used the Pearson’s r.

15. Measure of statistical significance
For contingency tables using nominal data, we use a chi square (2) measure of statistical significance to determine the existence of a relationship.
It does not assess the strength or direction of the relationship.
Chi square is partially a function of sample size.
2 tends to increase as sample size increases.

16. Measure of statistical significance
Use the same hypothesis testing steps we have used for t-test and one sample z test.
The null hypothesis is “no relationship between the DV and IV.”
To do so, we compare the observed frequency in each cell to the expected frequency for the corresponding cell.

17. Example
Overhead
IV: three different job training programs (vocational education, on-the-job training, work skills training)
DV: outcome (working, in school, unemployed)
Why comparing percentages isn’t enough

18. Overhead - job training programs

19. Cramer’s v
The chi square only measures the existence of an association
If we have nominal-level data, we can determine the strength of the association by calculating Cramer’s v.
Ranges from 0 (no relationship) to 1.0 (perfect relationship)

20. Cramer’s v M whichever is smaller: # rows or # columns
subtract 1 from this # N
total number of cases in table