Association Between Variables Measured at the Ordinal Level

1. Association Between Variables Measured at the Ordinal Level

2. Introduction • Data of two different forms • Continuous ordinal data • Similar to interval data • Will use Spearman’s rho for this • Collapsed ordinal data • Has many fewer values or categories • Will use gamma for this

3. Proportional Reduction in Error (PRE) • For ordinal data • Predict the order of a pair of cases on the dependent variable (Y) while ignoring the order on the independent variable (X) • Predict the order of a pair of cases on the Y while taking their order on the X into account • Gamma measures the PRE when predicting the order of pairs of cases on both variables

4. PRE for Ordinal Variables: Example • A researcher hypothesizes that “burnout” among elementary school teachers might be related to the number of years a teacher has been teaching • In a positive association, if Mark ranked above Julie on one variable, he would tend to rank above her on the other • In a negative association, if Mark ranked above Julie on one variable, he would tend to rank below her on the other

5. Gamma • If the two variables are associated, we will reduce our errors when making predictions about the dependent variable when first taking the independent variable into account • With no association between the variables, gamma will be 0 • Gamma ranges from -1 to +1

6. The Computation of Gamma • First, two sums are needed • Need the number of pairs of cases that are ranked the same on both variables, which is N sub s • Then, need the number of pairs of cases ranked differently on the variables, which is N sub d • We do this by working with the cell frequencies cell by cell

7. Interpreting Gamma • A gamma of .57 indicates that we would make 57% fewer errors if we predicted the order of pairs of cases on one variable (the dependent variable) from the order of pairs of cases on the other (the independent variable) • Length of service is associated with degree of burnout

8. Interpreting Gamma, cont. Gamma is a symmetrical measure of association The value of gamma will be the same regardless of which variable is taken as independent If gamma is positive, the variables are moving both away from the top left corner of the table If gamma is negative, as the independent variable moves to the right, the dependent variable moves up (rather than down)

9. Spearman’s Rho (r sub s) • Used for ordinal-level variables that have a broad range of many different scores and few ties between cases on either variable • Adding several ordinal variables together to make a new composite measure of a concept results in continuous scores that is well suited to Spearman’s Rho

10. Interpretation of Spearman’s Rho • It is an index of the strength of association between the variables • It ranges from 0 (no association) to plus or minus 1 (perfect association) • A perfect positive association would exist if there were no disagreements in ranks between the two variables

11. PRE for Spearman’s Rho • If the value of rho is first squared, a PRE interpretation is possible • Rho squared represents the proportional reduction in error when predicting the rank on one variable from rank on the other variable compared to predicting rank on one variable while ignoring the other variable • So, a Spearman’s rho of .30 indicates that our PRE for these data is 9 percent

12. Testing the Null Hypothesis for Gamma and Spearman’s Rho • If you are working with random samples, you will need to find if sample findings can be generalized to the population • The measures of association can be tested for their statistical significance • For ordinal-level variables (for gamma and rho), the null hypothesis is that there is no association between the variables in the population