1. POLS 7000X STATISTICS IN POLITICAL SCIENCE CLASS 9 BROOKLYN COLLEGE-CUNY SHANG E. HA Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

2. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chapter 8: Relationships Between Two Variables  Independent and Dependent Variables  How to Construct and Percentage a Bivariate Table  Properties of a Bivariate Relationship  The Concept of Chi-Square as a Statistical Test  The Concept of Statistical Independence  The Structure of Hypothesis Testing with Chi-Square

3. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Introduction  Bivariate Analysis: A statistical method designed to detect and describe the relationship between two variables.  Cross-Tabulation: A technique for analyzing the relationship between two variables that have been organized in a table.

4. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Constructing a Bivariate Table  Bivariate table: A table that displays the distribution of one variable across the categories of another variable.  Column variable: A variable whose categories are the columns of a bivariate table.  Row variable: A variable whose categories are the rows of a bivariate table.  Cell: The intersection of a row and a column in a bivariate table.  Marginals: The row and column totals in a bivariate table.

5. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Percentaging a Bivariate Table

6. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Percentaging a Bivariate Table

7. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Percentages Can Be Computed in Different Ways: 1. Column Percentages: column totals as base 2. Row Percentages: row totals as base

8. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Properties of a Bivariate Relationship 1. Does there appear to be a relationship? 2. How strong is it? 3. What is the direction of the relationship?

9. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Existence of the Relationship

10. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Determining the Strength of the Relationship  A quick method is to examine the percentage difference across the different categories of the independent variable.  The larger the percentage difference across the categories, the stronger the association.  We rarely see a situation with either a 0% or a 100% difference.

11. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Direction of the Relationship  Positive relationship: A bivariate relationship between two variables measured at the ordinal level or higher in which the variables vary in the same direction.  Negative relationship: A bivariate relationship between two variables measured at the ordinal level or higher in which the variables vary in opposite directions.

12. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications A Positive Relationship

13. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications A Negative Relationship

14. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Two Discrete/Nominal Variables  We would like to see whether the two discrete/nominal variables are statistically “independent” or “dependent”  Independent: NO relationships between two discrete/nominal variables  Dependent: STATISTICALLY SIGNIFICANT relationships between two discrete/nominal variables

15. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Cross-Tabulation Confidence in Big Business Confidence in Labor Union Very confident Somewhat confident Not confident at all Total Very confident 117 (18.84%) 300 (6.00%) 152 (5.97%) 569 (6.97%) Somewhat confident 266 (42.83%) 2,834 (56.67%) 1,189 (46.72%) 4,289 (52.52%) Not confident at all 238 (38.33%) 1,867 (37.33%) 1,204 (47.31%) 3,309 (40.52%) Total 621 (100.00%) 5,001 (100.00%) 2,545 (100.00%) 8,167 (100.00%)

16. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chi-Square Test  We would like to know whether any differences in each cell of the previous table stem from sampling variability or from some statistically-detectable fundamental differences

17. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications If there is NO relationship… Confidence in Big Business Confidence in Labor Union Very confident Somewhat confident Not confident at all Total Very confident ? (6.97%) ? (6.97%) ? (6.97%) 569 (6.97%) Somewhat confident ? (52.52%) ? (52.52%) ? (52.52%) 4,289 (52.52%) Not confident at all ? (40.52%) ? (40.52%) ? (40.52%) 3,309 (40.52%)

18. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Test Procedure  Ho: Two variables are statistically independent (no relationship)  Ha: Two variables are statistically dependent (some relationship)  fo: Observed frequencies in each cell  fe: Expected frequencies in each cell (assuming no relationship)

19. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Observed Frequencies (black), Expected Frequencies (red) Confidence in Big Business Confidence in Labor Union Very confident Somewhat confident Not confident at all Total Very confident 117 43.3 300 348.4 152 177.3 569 Somewhat confident 266 326.1 2,834 2,626.3 1,189 1,336.5 4,289 Not confident at all 238 251.6 1,867 2,026.2 1,204 1,031.2 3,309

20. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Interpretation  If Ho is correct, χ 2 will be small, as fo and fe are close to each other in each cell  Therefore, as χ 2 becomes larger, we are more likely to reject the null hypothesis

21. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Interpretation  Use χ 2 Table Compare critical value (under a certain confidence level; 95%) with the calculated χ 2 Compare significance level ( α ) and p-value: if p-value is small than α (usually, 0.05), then reject the null hypothesis  Degrees of freedom: df = (j – 1)*(k – 1) j: number of columns k: number of rows

22. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Table

23. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Summary  In the above example, χ 2(d.f.=4) = 222.02 Critical value 9.488 Reject the null hypothesis There is statistically significant relationship between two variables  We are able to see whether two discrete/nominal variables are statistically related, but we are not able to know its direction or its strength  In Chi-square test, there is no difference between the independent variable and the dependent variable