1. Overview Chi-square showed us how to determine whether two (nominal or ordinal) variables are statistically significantly related to each other. But statistical significance ≠ substantive significance. The p value does not measure strength of relationship! So, how do we tell how strong a relationship is? Is the curiosity killing you?

2. Measures of Association (for nominal and ordinal variables) The Proportional Reduction in Error (PRE) Approach –How much better can we predict the dependent variable by knowing the independent variable? An example –How do you feel about school vouchers? (NES 2000) –Knowing only a d.v., how do you predict an outcome? –Mode = 238 –How many errors? # mis- predictions. –E 1 = 84 + 65 + 194 = 343

3. A simple PRE example Now suppose you have knowledge about how a nominal independent variable relates to the dependent variable. Say, religion… School vouchersProtestantJewishCatholic Total Favor strongly55 (29.9%)85 (39.5%)98 (53.8%)238 (41.0%) Favor not strongly22 (12.0%)32 (14.9%)30 (16.5)84 (14.5) Oppose not strongly29 (15.8%)22 (10.2%)14 (7.7%)65 (11.2%) Oppose strongly78 (42.2%)76 (35.3%)40 (22.0%)194 (33.4%) Total184 (99.9%)215 (99.9%)182 (100.0%)581 (100.1%) Now, choose the mode of each independent variable category. 78 + 85 + 98 = 261 correct predictions E 2 = 320 BTW: χ 2 = 31.7

4. A simple PRE example (cont.) Proportional reduction in error = (E 1 – E 2 )/E 1 = (343 – 320)/343 =.067 We call this a 6.7% reduction in error… This calculation is aka “Lambda.” Note – Lambda can be used when one or both variables are nominal. It bombs when one dv category has a preponderance of the observations (Cramér’s V is useful then).

5. PRE with two ordinal variables When both variables are ordinal, you have many options for measuring the strength of a relationship Gamma, Kendall’s tau-b, Kendall’s tau-c, etc. Choices, choices, choices…

6. Biblical Literalism and Education Is the Bible the word of God or of men? (NES 2000) Chi-sq = 105.4 at 4 df  p =.000  reject the null hypothesis

7. Gamma, Tau-b, Tau-c… So our independent variable, education, reduces our error in predicting Biblical literalism by either 22.2% (tau-b), 18.8% (tau-c) or 38.3 whopping % (gamma) And, SPSS reports sign. level, but let me come back to that later.

8. Either of these might be considered a perfect relationship, depending on one’s reasoning about what relationships between variables look like. Why are there multiple measures of association? Statisticians over the years have thought of varying ways of characterizing what a perfect relationship is: tau-b = 1, gamma = 1 tau-b <1, gamma = 1 55 1025 3 730 55 35 40

9. I’m so confused!!

10. Rule of Thumb Gamma tends to overestimate strength but gives an idea of upper boundary. If table is square use tau-b; if rectangular, use tau-c. Pollock (and we agree): τ <.1 is weak;.1<τ<.2 is moderate;.2<τ<.3 moderately strong;.3< τ<1 strong.

11. A last example Theory: People’s partisanship leads them to develop distinct ideas about public policies. A case in point: Dem’s, Ind’s, and Rep’s develop different ideas about whether immigration should be increased, kept the same, or decreased

12. Last example (cont.) Specifically, Dem’s have tended to favor minorities and those with less power. Therefore, I anticipate that Dem’s will be most in favor of increasing immigration, Rep’s will be most in favor of decreasing it. Let’s test this out using NES data.

15. Last example (cont.) Conclusion There is a relationship between partisan- ship and feelings about immigration— i.e., what we saw in the table is not a result of chance. The relationship is weak (tau-c =.04). Dem’s are only a little more likely to favor immigration, Reps only a little more likely to oppose it.