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Overview

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!

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Overview

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  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. • E1 = 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… • Now, choose the mode of each independent variable category. • 78 + 85 + 98 = 261 correct predictions E2 = 320 BTW: χ2 = 31.7

  4. A simple PRE example (cont.) • Proportional reduction in error = (E1 – E2)/E1 = (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 predictingBiblical 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

  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.

  13. 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.

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