1. Cross-Tabs Continued Andrew Martin PS 372 University of Kentucky

2. Statistical Independence Statistical independence is a property of two variables in which the probability that an observation is in a particular category of one variable and a particular category of the other variable equals the simple or marginal probability of being in those categories. Contrary to other statistical measures discussed in class, statistical independence indicators test for a lack of a relationship between two variables.

3. Statistical Independence Let us assume two nominal variables, X and Y. The values for these variables are as follows: X: a, b, c, ... Y: r, s, t , ...

4. Statistical Independence P(X= a ) stands for the probability a randomly selected case has property or value a on variable X. P(Y=r) stands for the probability a randomly selected case has property or value r on variable Y P(X=a, Y=r) stands for the joint probability that a randomly selected observation has both property a and property r simultaneously.

5. Statistical Independence If X and Y are statistically independent: P(X= a , Y= r ) = [P(X= a )][P(Y= r )] for all a and r .

6. Statistical Independence

7. If gender and turnout are independent: Total obs in column m * Total obs in row v N = mv

8. Statistical Independence Total obs in column m * Total obs in row v N = mv 210 * 100 300 = 70 70 is the expected frequency. Because the observed and expected frequencies are the same, the variables are independent.

9. 150 * 150 300 = 75

10. Here, the relationship is not independent (or dependent) because 75 (expected frequency) is less than 100 (observed frequency).

11. Testing for Independence How do we test for independence for an entire cross-tabulation table? A statistic used to test the statistical significance of a relationship in a cross-tabulation table is a chi-square test (χ 2 ).

12. Chi-Square Statistic The chi-square statistic essentially compares an observed result—the table produced by the data—with a hypothetical table that would occur if, in the population, the variables were statistically independent.

13. How is the chi-square statistic calculated? The chi-square test is set up just like a hypothesis test. The observed chi-square value is compared to the critical value for a certain critical region. A statistic is calculated for each cell of the cross-tabulation and is similar to the independence statistic.

14. How is the chi-square statistic calculated? (Observed frequency – expected frequency) 2