1. Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions
Copyright © 2005 by Evan Schofer
Do not copy or distribute without permission

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3. Review: Chi-square Test
Chi-Square test is a test of independence
Null hypothesis: the two categorical variables are statistically independent
There is no relationship between them
H0: Gender and political party are independent
Alternate hypothesis: the variables are related, not independent of each other
H1: Gender and political party are not independent
Test is based on comparing the observed cell values with the values you’d expect if there were no relationship between variables.

4. Review: Expected Cell Values
If two variables are independent, cell values will depend only on row & column marginals
Marginals reflect frequencies… And, if frequency is high, all cells in that row (or column) should be high
The formula for the expected value in a cell is:
fi and fj are the row and column marginals
N is the total sample size

5. Review: Chi-square Test
The Chi-square formula:
Where:
R = total number of rows in the table
C = total number of columns in the table
Eij = the expected frequency in row i, column j
Oij = the observed frequency in row i, column j
Assumption for test: Large N (>100)
Critical value DofF: (R-1)(C-1).

6. Chi-square Test of Independence
Example: Gender and Political Views
Let’s pretend that N of 68 is sufficient
Women
Men
Democrat
O11: 27
E11: 23.4
O12 : 10
E12 : 13.6
Republican
O21 : 16
E21 : 19.6
O22 : 15
E22 : 11.4

7. Chi-square Test of Independence
Compute (E – O)2 /E for each cell
Women
Men
Democrat
(23.4 – 27)2/23.4
= .55
(13.6 – 10)2/13.6 = .95
Republican
(19.6 – 16)2/19.6
= .66
(11.4 – 15)2/15
= .86

8. Chi-Square Test of Independence
Finally, sum up to compute the Chi-square
c2 = = 3.02
What is the critical value for a=.05?
Degrees of freedom: (R-1)(C-1) = (2-1)(2-1) = 1
According to Knoke, p. 509: Critical value is 3.84
Question: Can we reject H0?
No. c2 of 3.02 is less than the critical value
We cannot conclude that there is a relationship between gender and political party affiliation.

9. Chi-square Test of Independence
Weaknesses of chi-square tests:
1. If the sample is very large, we almost always reject H0.
Even tiny covariations are statistically significant
But, they may not be socially meaningful differences
2. It doesn’t tell us how strong the relationship is
It doesn’t tell us if it is a large, meaningful difference or a very small one
It is only a test of “independence” vs. “dependence”
Measures of Association address this shortcoming.

10. Measures of Association
Separate from the issue of independence, statisticians have created measures of association
They are measures that tell us how strong the relationship is between two variables
Weak Association Strong Association
Women
Men
Dem. 51 49
Rep.
Women
Men
Dem.
100
Rep.

11. Crosstab Association:Yule’s Q
Appropriate only for 2x2 tables (2 rows, 2 columns)
Label cell frequencies a through d: a b c d
Recall that extreme values along the “diagonal” (cells a & d) or the “off-diagonal” (b & c) indicate a strong relationship.
Yule’s Q captures that in a measure
0 = no association. -1, +1 = strong association

12. Crosstab Association:Yule’s Q
Rule of Thumb for interpreting Yule’s Q:
Bohrnstedt & Knoke, p. 150
Absolute value of Q
Strength of Association
0 to .24
“virtually no relationship”
.25 to .49
“weak relationship”
.50 to .74
“moderate relationship”
.75 to 1.0
“strong relationship”

13. Crosstab Association:Yule’s Q
Example: Gender and Political Party Affiliation
Women
Men
Dem 27 10
Rep 16 15
Calculate “bc”
bc = (10)(16) = 160
a b
c d
Calculate “ad”
ad = (27)(15) = 405
-.48 = “weak association”, almost “moderate”

14. Association: Other Measures
Phi ()
Very similar to Yule’s Q
Only for 2x2 tables, ranges from –1 to 1, 0 = no assoc.
Gamma (G)
Based on a very different method of calculation
Not limited to 2x2 tables
Requires ordered variables
Tau c (tc) and Somer’s d (dyx)
Same basic principle as Gamma
Several Others discussed in Knoke, Norusis.

15. Crosstab Association: Gamma
Gamma, like Q, is based on comparing “diagonal” to “off-diagonal” cases.
But, it does so differently
Jargon:
Concordant pairs: Pairs of cases where one case is higher on both variables than another case
Discordant pairs: Pairs of cases for which the first case (when compared to a second) is higher on one variable but lower on another

16. Crosstab Association: Gamma
Example: Approval of candidates
Cases in “Love Trees/Love Guns” cell make concordant pairs with cases lower on both
Hate Trees
Trees OK
Love Trees
Love Guns
1205
603 71
Guns = OK
659
1498
452
Hate Guns
431
467
1120
All 71 individuals can be a pair with everyone in the lower cells. Just Multiply!
(71)( ) = 216,905 conc. pairs

17. Crosstab Association: Gamma
More possible concordant pairs
The “Love Guns/Trees are OK” cell and the “Trees = OK/Love Guns” cells also can have concordant pairs
These 603 can pair with all those that score lower on approval for Guns & Trees
(603)( ) = 657,270 conc. pairs
Hate Trees
Trees = OK
Love Trees
Love Guns
1205
603 71
Guns = OK
659
1498
452
Hate Guns
431
467
1120
These can pair lower too!
(452)( ) = 405,896 conc. pairs

18. Crosstab Association: Gamma
Discordant pairs: Pairs where a first person ranks higher on one dimension (e.g. approval of Trees) but lower on the other (e.g., app. of Guns)
Hate Trees
Trees = OK
Love Trees
Love Guns
1205
603 71
Guns = OK
659
1498
452
Hate Guns
431
467
1120
The top-left cell is higher on Guns but lower on Trees than those in the lower right. They make pairs:
(1205)( ) = 4,262,085 discordant pairs

19. Crosstab Associaton: Gamma
If all pairs are concordant or all pairs are discordant, the variables are strongly related
If there are an equal number of discordant and concordant pairs, the variables are weakly associated.
Formula for Gamma:
ns = number of concordant pairs
nd = number of discordant pairs

20. Crosstab Association: Gamma
Calculation of Gamma is typically done by computer
Zero indicates no association
+1 = strong positive association
-1 = strong negative association
It is possible to do hypothesis tests on Gamma
To determine if population gamma differs from zero
Requirements: random sample, N > 50
See Knoke, p

21. Crosstab Association Final remarks:
You have a variety of possible measures to assess association among variables. Which one should you use?
Yule’s Q and Phi require a 2x2 table
Larger ordered tables: use Gamma, Tau-c, Somer’s d
Ideally, report more than one to show that your findings are robust.

22. Odds Ratios
Odds ratios are a powerful way of analyzing relationships in crosstabs
Many advanced categorical data analysis techniques are based on odds ratios
Review: What is a probability?
p(A) = # of outcomes that are “A” divided by total number of outcomes
To convert a frequency distribution to a probability distribution, simply divide frequency by N
The same can be done with crosstabs: Cell frequency over N is probability.

23. Odds Ratios If total N = 68, probability of drawing cases is: Women
Dem
27 / 68
10 / 68
Rep
16 / 68
15 / 68
Women
Men
Dem
.397
.147
Rep
.235
.220

24. Odds Ratios Odds are similar to probability… but not quite
Odds of A = Number of outcomes that are A, divided by number of outcomes that are not A
Note: Denominator is different that probability
Ex: Probability of rolling 1 on a 6-sided die = 1/6
Odds of rolling a 1 on a six-sided die = 1/5
Odds can also be calculated from probabilities:

25. Odds Ratios
Conditional odds = odds of being in one category of a variable within a specific category of another variable
Example: For women, what are the odds of being democrat?
Instead of overall odds of being democrat, conditional odds are about a particular subgroup in a table
Women
Men
Dem 27 10
Rep 16 15
Conditional odds of being democrat are: 27 / 16 = 1.69
Note: Odds for women are different than men

26. Odds Ratios
If variables in a crosstab are independent, their conditional odds are equal
Odds of falling into one category or another are same for all values of other variable
If variables in a crosstab are associated, conditional odds differ
Odds can be compared by making a ratio
Ratio is equal to 1 if odds are the same for two groups
Ratios much greater or less than 1 indicate very different odds.

27. Odds Ratios c d Formula for Odds Ratio in 2x2 table:
Women
Men
Dem 27 10
Rep 16 15
a b
c d
Ex: OR = (10)(16)/(27)(15) = 160 / 405 = .395
Interpretation: men have .395 times the odds of being a democrat compared to women
Inverted value (1/.395=2.5) indicates odds of women being democrat = 2.5 is times men’s odds

28. Odds Ratios: Final Remarks
1. Cells with zeros cause problems for odds ratios
Ratios with zero in denominator are undefined.
Thus, you need to have full cells
2. Odds ratios can be used to measure assocation
Indeed, Yule’s Q is based on them
3. Odds ratios form the basis for most advanced categorical data analysis techniques
For now it may be easier to use Yule’s Q, etc. But, if you need to do advanced techniques, you will use odds ratios.

31. Tests for Difference in Proportions
Another approach to small (2x2) tables:
Instead of making a crosstab, you can just think about the proportion of people in a given category
More similar to T-test than a Chi-square test
Ex: Do you approve of Pres. Bush? (Yes/No)
Sample: N = 86 women, 80 men
Proportion of women that approve: PW = .70
Proportion of men that approve: PM = .78
Issue: Do the populations of men/women differ?
Or are the differences just due to sampling variability

32. Tests for Difference in Proportions
Hypotheses:
Again, the typical null hypothesis is that there are no differences between groups
Which is equivalent to statistical independence
H0: Proportion women = proportion men
H1: Proportion women not = proportion men
Note: One-tailed directional hypotheses can also be used.

33. Tests for Difference in Proportions
Strategy: Figure out the sampling distribution for differences in proportions
Statisticians have determined relevant info:
1. If samples are “large”, the sampling distribution of difference in proportions is normal
The Z-distribution can be used for hypothesis tests
2. A Z-value can be calculated using the formula:

34. Tests for Difference in Proportions
Standard error can be estimated as:
Where:

35. Difference in Proportions: Example
Q: Do you approve of Pres. Bush? (Yes/No)
Sample: N = 86 women, 80 men
Women: N = 86, PW = .70
Men: N = 80, PW = .78
Total N is “Large”: 166 people
So, we can use a Z-test
Use a = .05, two-tailed Z = 1.96

36. Difference in Proportions: Example
Use formula to calculate Z-value
And, estimate the Standard Error as:

37. Difference in Proportions: Example
First: Calculate Pboth:

38. Difference in Proportions: Example
Plug in Pboth=.739:

39. Difference in Proportions: Example
Finally, plug in S.E. and calculate Z:

40. Difference in Proportions: Example
Results:
Critical Z = 1.96
Observed Z = .739
Conclusion: We can’t reject null hypothesis
Women and Men do not clearly differ in approval of Bush