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  1. 1. Cross-Tabulation <ul><li>Andrew Martin </li></ul><ul><li>PS 372 </li></ul><ul><li>University of Kentucky </li></ul>
  2. 2. Cross-Tabs Continued <ul><li>10. Gamma coefficients sometimes overstate the strength of the relationship because they do not count tied pairs. </li></ul>
  3. 3. What about nominal data? <ul><li>Two statistical measures can be used to measure association for nominal data -- the proportional-reduction-in-error statistic and odds ratio. </li></ul>
  4. 4. Interpreting ROE <ul><li>Essentially reduction-of-error statistics work by calculating the number of percentage of errors reduced by using the information of the independent variable (x) to calculate the dependent variable (y). </li></ul><ul><li>One common measure of ROE is the Goodman-Kruskal’s lambda (λ). </li></ul>
  5. 5. Interpreting ROE <ul><li>The Goodman-Kruskal lambda (λ) specifies the percentage of errors reduced by using X to predict Y. </li></ul><ul><li>For example, if λ = .24, X reduced the number of false predictions about Y by 24 percent. </li></ul>
  6. 6. Odds Ratio <ul><li>If you have a table with two dichotomous variables (meaning the variables take on values of 0 or 1) you can use the odds ratio statistic to describe the association. </li></ul>
  7. 7. Do you favor the death penalty?
  8. 8. Calculating Odds Ratios The odds ratio statistic is simply two fractions. If the gender variable can be split according to support for the death penalty, we go about as such: A = Men supporting death penalty B = Men opposing death penalty C = Women supporting death penalty D = Women opposing death penalty
  9. 9. A/B C/D which you calculate by multiplying A * D and dividing by B * C so A * D B * C
  10. 10. So Men supporting death penalty Men opposing death penalty over Women supporting death penalty Women opposing death penalty
  11. 12. <ul><li>The odds of men favoring the death penalty are about one and a half times greater than the odds of women favoring it. </li></ul><ul><li>-- 1.54, with men as numerator </li></ul><ul><li>The odds of a female favoring the death penalty are only about two-thirds of a male doing so. </li></ul><ul><li>-- .65, with women as numerator </li></ul>
  12. 13. Interpretation <ul><li>Doesn’t matter which variable is the numerator and which is the denominator as long as you interpret the odds ratio correctly. </li></ul><ul><li>Remember, the odds ratio compares chances or likelihoods of something being chosen or happening. </li></ul><ul><li>In practice it is applied to discrete or categorical variables. </li></ul>
  13. 14. Interpretation <ul><li>Unlike most measures, the odds ratio has a null value of 1.0, not 0. If an odds ratio equals 1.0, the odds are the same, and the groups do not differ in their response propensities </li></ul><ul><li>The odds ratio's boundaries are 0 and (plus) infinity. In other words, the odds ratio will always be a positive number. </li></ul><ul><li>The farther from 1.0, in either direction, the stronger the association. </li></ul>
  14. 15. Odds Ratio Properties <ul><li>The inverse, or reciprocal of an odds ratio shows the strength when the order of categories is switched. </li></ul><ul><li>The odds ratio is standardized. It is always expressed in the units of odds, not the measurement scale of a variable. </li></ul><ul><li>The odds ratio can be applied to investigate patterns of association in tables larger than 2 X 2 and in multi-way cross tabs having more than three variables. </li></ul>