1. Statistical Analysis of Categorical Variables
Cross Tabulation
Statistical Analysis of Categorical Variables

2. To date….
We have examined statistical tests for differences of means and proportions
The statistics are measured at the interval level.

3. New Test…
Now we wish to examine statistical tests for questions involving nominal and ordinal variables. To do so we introduce the Chi Square Test.

4. Cross Tabulation
We are interested in the counting the number of cases for the categories of one variable in terms of the categories of a second variable, and….
Implicitly, we are asking if there are differences in the patterns of the counts….

5. Cross Tabulation and Chi Square Test
A cross tabulation cross classifies one variable by another variable. Below is a cross classification of occupational groups and wards for the Simon data for 1905.

6. Cross Tabulation and Chi Square Test
We count the number of cases in each occupational category for each ward. At the edges of the table we total the rows and columns.

7. Graphic Illustration of the Counts of Occupational Groups by Ward

8. The Simplest Example….a 2 by 2 Table
Do the opinions of men and women differ on the War in Iraq?
Do the opinions of men and women differ on the importance of capturing Osama Bin Laden?
Data: September 2006 ABC News Poll on the War on Terror. A Sample of about 1000 respondents.

9. Basic Frequencies

10. Basic Frequencies Broken Down by Gender

11. Another Graphical Illustration: 4 Bins of Counts

12. Tabular Data

13. Some Terms and Assumptions
Cell frequency: number in the body of the table
Marginal total: total of the row or the column
Row percent: the proportion of cases in the cell for the particular row.
Column percent: the proportion of cases in the cell for the particular column
Expected frequency: the number of cases expected based upon the marginal proportions
Deviation: the difference between the expected frequency and the actual frequency

14. Tabular Data Cell Frequency Marginals
Q12 War worth fighting NET * Q921 GENDER Crosstabulation
Count
Q921 GENDER
Male
Female
Total
Q12 War worth
Worth fighting NET
213
217
430
fighting NET
Not worth fighting NET
245
307
552
Total
458
524
982
Marginals

15. 213
245
217
307
Table Counts and Graph

16. Row Percents

17. Column Percents

18. Frequencies, Row and Column Percents

19. New Concept: Expected Frequencies
What would the counts in the cells be if there was no impact of gender on attitudes towards the Iraq War?
The marginal proportions would define the cell counts.

20. Expected Frequencies Row Total * Column Total/ Grand Total Or…
Row Proportion * Column Total
Column Proportion * Row Total

21. Another Example: The Importance of Capturing Osama Bin Laden

23. Frequencies by Gender

24. Frequencies By Gender

25. Row Percents

26. Column Percents

28. Expected Frequencies

29. Actual Frequencies, Expected Frequencies, and Deviations (Residual)

30. Chi Square
Chi Square = Sum of [ (Expected – Observed)2 / Expected Frequency ]
Chi Square Table:

31. Examples of Chi Square Distribution
see also

32. Degrees of Freedom for Chi Square
Degrees of Freedom = (r-1)* (c-1)
So, 2 by 2 table has 1 degree of freedom
3 by 2 table has (3-1)(2-1)= 2 degrees of freedom

33. Calculations: Catching Osama bin Laden by Gender
640.09/198.3 =
640.09/220.7 =
640.09/251.7 =
640.09/280.3 =
Chi Square (SUM) =

34. Attitudes toward Iraq War by Gender

35. Chi Square Test
For a larger table, calculation is the same, but the number of terms increases. The number of terms is equal to the number of cells.

36. Concentration of Occupational Groups by Ward

37. Cross Tabulation
Are the occupational patterns different in the four wards?
Or….are the patterns a result of chance? (null hypothesis)
How would we decide?

38. Illustration: Frequencies and Marginals

39. Row and Column Percents

40. Expected and Actual Frequencies
OCC$ (rows) by WARD (columns)
Total
profcler | |
prop | |
skilled | |
skillpart | |
unskilled | |
Total
Expected values
OCC$ (rows) by WARD (columns)
profcler | |
prop | |
skilled | |
skillpart | |
unskilled | |

41. Deviates Deviates: (Observed-Expected) OCC$ (rows) by WARD (columns)
profcler | |
prop | |
skilled | |
skillpart | |
unskilled | |

42. Calculations Case number OCC$ WARD FREQUENCY EXPECTED RESIDUAL CHITERM
1 profcler
2 profcler
3 profcler
4 profcler
5 prop
6 prop
7 prop
8 prop
9 skilled
skilled
skilled
skilled
skillpart
skillpart
skillpart
skillpart
unskilled
unskilled
unskilled
unskilled

43. Review: Terms and Assumptions
Cell frequency: number in the body of the table
Marginal total: total of the row or the column
Row percent: the proportion of cases in the cell for the particular row.
Column percent: the proportion of cases in the cell for the particular column
Expected frequency: the number of cases expected based upon the marginal proportions
Deviation: the difference between the expected frequency and the actual frequency

44. Strength of Relationships
Phi: Square root of (Chi Square/N)
Cramer’s V: Square root of (Chi Square/n*min(r-1, c-1))
Contingency Coefficient: Square root of (Chi Square/(Chi Square+n))