1. Chapter 14 in 1e Ch. 12 in 2/3 Can. Ed. Association Between Variables Measured at the Ordinal Level Using the Statistic Gamma and Conducting a Z-test for Significance

2. Introduction to Gamma Gamma is the preferred measure to test strength and direction of two ordinal-level variables that have been arrayed in a bivariate table. Before computing and interpreting Gamma, it is always useful to find and interpret the column percentages. Gamma can answer the questions:  1. Is there an association?  2. How strong is the association?  3. What direction (because level is ordinal) is it?

3. Introduction to Gamma (cont.) Gamma can also be tested for significance using a Z or t-test to see if the association (relationship) between two ordinal level variables is significant. In this case, you would use the 5 step method, as for χ 2 and conduct a hypothesis test.

4. Introduction to Gamma (cont.) Like Lambda, Gamma is a PRE (Proportional Reduction in Error) measure: it tells us how much our error in predicting y is reduced when we take x into account. With Gamma, we try to predict the order of pairs of cases (predict whether one case will have a higher or lower score than another) For example, if case A scores High on Variable1 and High on Variable 2, will case B also score High-High on both variables?

5. Introduction to Gamma (cont.) To compute Gamma, two quantities must be found:  N s is the number of pairs of cases ranked in the same order on both variables.  N d is the number of pairs of cases ranked differently on the variables. Gamma is calculated by finding the ratio of cases that are ranked the same on both variables minus the cases that are not ranked the same (N s – N d ) to the total number of cases (N s + N d ).

6. Computing Gamma This ratio can vary from +1.00 for a perfect positive relationship to -1.00 for a perfect negative relationship. Gamma = 0.00 means no association or no relationship between two variables. Note that when N s is greater than N d, the ratio with be positive, and when N s is less than N d the ratio will be negative.

7. Formula for Gamma Formula for Gamma:

8. A Simple Example for Gamma using Healey #12.1 in 1e or #11.1 in 2/3 e As previously seen, this table shows the relationship between authoritarianism of bosses (X) and the efficiency of workers (Y) for 44 workplaces. Since the variables are at the ordinal level, we can measure the association using the statistic Gamma. Efficiency (y)LowHigh Low101222 High17522 Total271744 Authoritarianism (x)

9. Simple Example (cont.) For Ns, start with the Low-Low cell (upper left) and multiply the cell frequency by the cell frequency below and to the right. N s = 10(5) = 50 Efficiency (y)LowHigh Low101222 High17522 Total271744 Authoritarianism (x)

10. Simple Example (cont.) For N d, start with the High-Low cell (upper right) and multiply each cell frequency by the cell frequency below and to the left. N d = 12(17) = 204 Efficiency (y)LowHigh Low101222 High17522 Total271744 Authoritarianism (x)

11. Simple Example (cont.) Using the table, we can see that G =-0.61 is a strong negative association. ValueStrength Between 0.0 and 0.30 Weak Between 0.30 and 0.60 Moderate Greater than 0.60 Strong

12. Simple Example (cont.) In addition to strength, gamma also identifies the direction of the relationship. We can look at the sign of Gamma (+ or -). In this case, the sign is negative (G = - 0.61). This is a negative relationship: as Authoritarianism increases, Efficiency decreases. In a negative relationship, the variables change in opposite directions.

13. Example: Healey #14.7 (1e), #12.7 in 2/3e) This question involves a more complicated calculation for Gamma. The question asks if aptitude test scores are related to job performance rating for 75 city employees. Part a.  Are the two variables, Aptitude, (measured as Low, Medium and High) and Job Performance (Low, Medium, and High) associated?  How strong is this association?  What direction is the association? Part b.  Is the association significant?

14. Part A: Calculating Gamma For Ns, start with the Low-Low cell (upper left) and multiply each cell frequency by total of all cell frequencies below and to the right and add together. For this table, N s is 11(10+9+9+9) + 6(9+9) + 9(9+9) + 10 (9) = 767 Efficiency (y)LowModerateHighTotal Low116724 Moderate910928 High59923 Totals25 75 Test Scores (x)

15. Part A: Calculating Gamma (cont.) For N d, start with High-Low cell (upper right) and multiply each cell frequency by total of all cell frequencies below and to the left and add together. For this table, N d = 7 (10+9+9+5) + 6 (9+5) + 9(9+5) + 10 (5) = 491 Efficiency (y)LowModerateHighTotal Low116724 Moderate910928 High59923 Totals25 75 Test Scores (x)

16. Part A: Calculating Gamma (cont.) Using the table, we can see that G =+0.22 is a weak positive association. ValueStrength Between 0.0 and 0.30 Weak Between 0.30 and 0.60 Moderate Greater than 0.60 Strong

17. Part A: Calculating Gamma (cont.) As noted before, gamma also identifies the direction of the relationship. We can look at the sign of Gamma (+ or -). In this case, the sign is positive (G = + 0.22). This is a positive relationship: as Aptitude Test Scores increase, Job Performance increases. Next, we test the association for significance, using the 5 step method.

18. Part B: Testing Gamma for Significance The test for significance of Gamma is a hypothesis test, and the 5 step model should be used. Step 1: Assumptions  Random sample, ordinal, Sampling Dist. is normal Step 2: Null and Alternate hypotheses  H o : γ=0, H 1 : γ≠0 (Note: γ is the population value of G) Step 3: Sampling Distribution and Critical Region  Z-distribution, α =.05, z = +/-1.96

19. Part B: Testing Gamma for Significance (cont.) Part 4: Calculating Test Statistic: Formula : Calculate:

20. Part B: Testing Gamma for Significance (cont.) Step 5: Make Decision and Interpret Z obs =.92 < Z crit = +/-1.96 Fail to reject H o The association between aptitude tests and job performance is not significant. *Part C: No, the aptitude test should not be continued, because there is no significant association.

21. Practice Question: Healey #14.8 (1e) or #12.8 (2/3e) Try this question as a homework assignment. The solution to the question can be found in the Final Review powerpoint. Note: We will not cover Spearman’s rho (also shown in Chapter 14 (12 in 2nd). This statistic will not be included on the final exam.

22. Kendall’s Tau b* (not in 1 st Can. Ed.) *do not need to calculate – for SPSS only The statistic Tau b is the preferred measure of strength to report when a bivariate table has many “tied pairs” (when cases are scored the same on both variables in a table) In this case, gamma will tend to overestimate the strength of the association. Rule of thumb: when the value of gamma is double that of Tau b, report Tau b instead, because it will be a better measure of strength. *omit Tau c and Spearman’s rho

23. Using SPSS to Calculate Gamma Go to Analyze>Descriptives>Crosstabs (as with Chi-square). Click on Cells for column % and on Statistics, asking for both Gamma and Tau b. Note that SPSS uses a t-test rather than a Z- test to test Gamma for significance. Compare the significance of Gamma (this is the p- value) to your alpha value. If your p-value is less than your alpha, then the association is significant.