1. Copyright © 2012 by Nelson Education Limited. Chapter 12 Association Between Variables Measured at the Ordinal Level 12-1

2. Copyright © 2012 by Nelson Education Limited. Explain the logic of pairs as it relates to measuring association. Use gamma, Kendall’s tau-b and tau-c, Somers’ d, and Spearman’s rho to analyze and describe a bivariate relationship. Testing Gamma and Spearman’s Rho for Statistical Significance In this presentation you will learn about: 12-2

3. Copyright © 2012 by Nelson Education Limited. Gamma is measure of association for two “collapsed" ordinal variables (variables with just a few values). –Gamma measures both strength and direction of relationship. –Gamma is a symmetrical; that is, the value of gamma will be the same regardless of which variable is taken as independent. Gamma 12-3

4. Copyright © 2012 by Nelson Education Limited. Gamma is a measure of association based on the logic of "pairs" of cases. To compute Gamma, two quantities must be found: n s and n d Gamma (continued) 12-4

5. Copyright © 2012 by Nelson Education Limited. n s is the total number of pairs of cases ranked in the same order on both variables. o For example, Joseph and Steven are among 50 respondents to a survey investigating the relationship between education (independent variable) and income (the dependent variable). o For this “pair” of cases, Joseph reports a higher level of education than Steven and Joseph also reports a higher level of income than Steven: o Thus this “pair” of cases is said to be similar (same). Gamma (continued) EducationIncome Joseph (High) (High) Steven (Low) (Low) 12-5

6. Copyright © 2012 by Nelson Education Limited. n d is the total number of pairs of cases ranked in different order on the variables. –For example, Caimile and Manjit are also among the 50 respondents. –For this “pair” of cases: Caimile reports a higher level of education than Manjit but Caimile has a lower level of income than Manjit: This “pair” of cases is said to be dissimilar (different). Gamma (continued) EducationIncome Caimile (High) (Low) Manjit (Low) (High) 12-6

7. Copyright © 2012 by Nelson Education Limited. If n = 50, the overall number of pairs of cases will be 1,225. –We can calculate the overall number of pairs of cases with this simple formula: (n *(n-1))/2 = (50*49)/2 = 1225 –The pairs “Joseph and Steven” and “Caimile and Manjit” are just 2 out of 1225 possible pairs of cases when n=50. Gamma (continued) 12-7

8. Let’s now consider the survey on education and income for all 50 respondents. The results are summarized in the bivariate table below: Gamma: An Example Copyright © 2012 by Nelson Education Limited.12-8 Income Education Low HighTotals Low151025 High 52025 Totals203050

9. Rather than looking at each pair individually to determine if it is similar or dissimilar (as we did above for the pairs Joseph-Steven and Caimile- Manjit), there is a short-cut for computing the total number of pairs of cases ranked in the same order on both variables (n s ) and the total number of pairs of cases ranked in different order on both variables (n d ). Note, to use the short-cut, the data must be summarized in a bivariate table. Gamma: An Example (continued) Copyright © 2012 by Nelson Education Limited.12-9

10. To compute n s in a bivariate table, multiply each cell frequency by all cell frequencies below and to the right: For this 2x2 table: n s :15 x 20 = 300 Gamma: An Example (continued) Income Education Low HighTotals Low151025 High 52025 Totals203050 Copyright © 2012 by Nelson Education Limited.12-10

11. Copyright © 2012 by Nelson Education Limited. To compute n d in a bivariate table, multiply each cell frequency by all cell frequencies below and to the left: n d : 5 x 10 = 50 Income Education Low HighTotals Low151025 High 52025 Totals203050 Gamma: An Example (continued) 12-11

12. Copyright © 2012 by Nelson Education Limited. Gamma is computed with Formula 12.2: Looking closely at this formula, we can see that Gamma is simply a measure of the preponderance of the total number of similar pairs relative to the total number of dissimilar pairs, and ranges from 0 (no association) to +1.00 (perfect association). Gamma: An Example (continued) 12-12

13. Copyright © 2012 by Nelson Education Limited. –As the preponderance of similar pairs increases (n s > n d ), gamma increases. It will also be positive in value, indicating a positive relationship. –As the preponderance of dissimilar pairs increases (n d > n s ), gamma increases. It will also be negative in value, indicating a negative relationship. –When n s = n d gamma will be zero (no association) Gamma: An Example (continued) 12-13

14. Copyright © 2012 by Nelson Education Limited. Recall, in a positive relationship, the variables change in the same direction (Fig. 12.4). In a negative relationship, the variables change in the opposite direction (Fig. 12.5). Gamma: An Example (continued) 12-14

15. Copyright © 2012 by Nelson Education Limited. Table 12.3 provides a guide to interpret the strength of gamma. –As before, the relationship between the values and the descriptive terms is arbitrary, so the scale in Table 12.3 is intended as a general guideline only: Gamma: An Example (continued) 12-15

16. Copyright © 2012 by Nelson Education Limited. Since there is a substantial preponderance of similar pairs (300) relative to dissimilar pairs (50), we know the value of Gamma will large (closer to 1) and positive. Using Formula 12.2: G = (300-50)/(300+50) = +250/350 = +.71 Gamma: An Example (continued) 12-16

17. Copyright © 2012 by Nelson Education Limited. A gamma of +.71 indicates is a strong and positive relationship: as education increases, income increases. –It is always useful to also compute column percentages for bivariate tables. Gamma: An Example (continued) 12-17

18. Copyright © 2012 by Nelson Education Limited. In addition to interpretation of strength and direction of association, gamma has a PRE (proportional reduction in error) interpretation. –Gamma, like lambda (Chapter 11), measures the proportional reduction in error gained by predicting one variable while taking the other into account. –In the case of gamma, we predict the order of pairs of respondents (i.e., we predict whether one respondent will have a higher or lower score than the other) rather than a score on the dependent variable. Gamma as a PRE Measure 12-18

19. Copyright © 2012 by Nelson Education Limited. So, the computed value of gamma of +.71 means that, when predicting the order of pairs of cases on the dependent variable (income), we would make 71% fewer errors by taking the independent variable (education) into account. Gamma as a PRE Measure (continued) 12-19

20. Copyright © 2012 by Nelson Education Limited. When variables are not coded from low to high (e.g., high education=1; low education =2), we must exercise caution in using the sign (+ or -) of gamma to determine actual direction of the relationship. Gamma ignores all tied pairs of cases, and tends to “exaggerate” the real strength of association. –In the example above, only 350 (n s + n d ) of the 1,225 number of possible pairs of cases were used in computing gamma; the rest (tied pairs of cases) were ignored. Limitations of Gamma 12-20

21. Copyright © 2012 by Nelson Education Limited. o For example, had Joseph and Steven both reported a high level of education, this pair would be tied on a variable, and excluded in the calculation of Gamma: Gamma ignores all types of tied pairs: Pairs that are tied on the independent variable (as above); pairs tied on the dependent variable; and pairs tied on both the independent variable and the dependent variable. Limitations of Gamma (continued) EducationIncome Joseph (High) (High) Steve(High) (Low) 12-21

22. Copyright © 2012 by Nelson Education Limited. Because gamma can “exaggerate” the actual strength of relationship, other ordinal measures that take ties into account, such as Kendall’s tau-b, Kendall’s tau-c, or Somers' d, should instead be used. Limitations of Gamma (continued) 12-22

23. Copyright © 2012 by Nelson Education Limited. Kendall’s tau-b includes pairs tied on the independent variable, t x, and pairs tied on the dependent variable t y. –Kendall’s tau-b can only reach a maximum of 1.00 when the independent and dependent variables have the same number of categories. –Kendall’s tau-c adjusts for number of categories, and is used as an alternative to Kendall’s tau-b when the variables have an unequal number of categories. Somers’ d includes only pairs tied on the dependent variable, t y. Kendall’s tau-b and tau-c, Somers' d 12-23

24. Copyright © 2012 by Nelson Education Limited. Kendall’s tau-b and tau-c are symmetrical measures of association. Somers’ d is as an asymmetric measure. –it assumes that you can identify one of the variables as the dependent. thus, Somers’ d corrects only for pairs tied on the dependent variable. Kendall’s tau-b and tau-c and Somers’ d range from -1.00 (perfect negative relationship) to +1.00 (perfect positive relationships). Kendall’s tau-b and tau-c, Somers' d (continued) 12-24

25. Copyright © 2012 by Nelson Education Limited. Kendall’s tau-b is computed with Formula 12.3: Kendall’s tau-c is computed with Formula 12.4: Kendall’s tau-b and tau-c, Somers' d (continued) 12-25

26. Copyright © 2012 by Nelson Education Limited. Somers’ d is computed with Formula 12.5: Kendall’s tau-b and tau-c, Somers' d (continued) 12-26

27. Looking again at the relationship between education and income, to compute t y in a bivariate table, multiply each cell frequency by all cell frequencies immediately to the right: For this 2x2 table: t y : (15 x 10)+ (5 x 20)= 150+100= 250 Kendall’s tau-b, tau-c, Somers' d: An Example Income Education Low HighTotals Low151025 High 52025 Totals203050 Copyright © 2012 by Nelson Education Limited.12-27

28. To compute t x in a bivariate table, multiply each cell frequency by all cell frequencies immediately below: For this 2x2 table: t x : (15 x 5)+ (10x 20)= 75+200= 275 Kendall’s tau-b, tau-c, Somers' d: An Example (continued) Income Education Low HighTotals Low151025 High 52025 Totals203050 Copyright © 2012 by Nelson Education Limited.12-28

29. Copyright © 2012 by Nelson Education Limited. The value of Kendall's tau-b, using Formula 12.3, is Kendall’s tau-b, tau-c, Somers' d: An Example (continued) 12-29

30. Copyright © 2012 by Nelson Education Limited. The value of Kendall's tau-c, using Formula 12.4, is Kendall’s tau-b, tau-c, Somers' d: An Example (continued) 12-30

31. Copyright © 2012 by Nelson Education Limited. The value of Somers' d, using Formula 12.5, is Kendall’s tau-b, tau-c, Somers' d: An Example (continued) 12-31

32. Copyright © 2012 by Nelson Education Limited. Each measure indicates a strong and positive association between education and income: G= 0.71 tau-b=0.41 tau-c=0.40 Somers' d=0.42 So, which measure should be used with “collapsed" ordinal variables? –No hard and fast rule. –When there are relatively few ties, use gamma because it has a PRE interpretation and the others do not. Otherwise use: Kendall’s tau-b (or Kendall’s tau-c when the variables have an unequal number of categories) or Somers’ d when one of the variables is specifically identified as the dependent. Which Measure to Use? 12-32

33. Copyright © 2012 by Nelson Education Limited. Spearman’s rho, r s, is measure of association for two ordinal-level variables that have a broad range of scores and many distinct values (i.e., few ties between cases on either variable). Spearman’s rho is based on the logic of the “differences in rank” between the two variables. Spearman’s rho measures both strength and direction of relationship, ranges from 0 (no association) to +1.00 (perfect association), and is a symmetrical. Spearman’s rho 12-33

34. Copyright © 2012 by Nelson Education Limited. The value of Spearman’s rho squared provides a PRE interpretation –The proportional reduction in errors of prediction when predicting rank on one variable from rank on the other variable, as compared to predicting rank while ignoring the other variable. Spearman’s rho (continued) 12-34

35. Copyright © 2012 by Nelson Education Limited. The formula for Spearman’s rho is: Spearman’s rho (continued) 12-35

36. The number of friends a person is hypothesized to have a positive effect on his/her quality of life. o Given the data below, how strong is the relationship between these two variables? Spearman’s rho: An Example No. of Quality Respondent Friendsof Life 1 5 7 2 9 10 3 7 6 4 3 5 5 1 3 6 10 10 7 4 5 8 15 8 9 6 9 10 3 6 11 5 7 Copyright © 2012 by Nelson Education Limited.12-36

37. To compute  D 2, the rank of each case on Y is subtracted from its rank on X,D, then squared,D 2. The sum of each difference squared, or  D 2, is entered directly into the Formula 14.2. Spearman’s rho: An Example (continued) No. of Qual. Friends Rank Life Rank D D 2 5 6.5 7 5.5 1 1 9 3 10 1.5 1.5 2.25 7 4 6 7.5 -3.5 12.25 3 9.5 5 9.5 0 0 1 11 3 11 0 0 10 2 10 1.5 0.5 0.25 4 8 5 9.5 -1.5 2.25 15 1 8 4 - 3 9 6 5 9 3 2 4 3 9.5 6 7.5 2 4 5 6.5 7 5.5 1 1  D=0  D 2 =36 Copyright © 2012 by Nelson Education Limited. 12-37

38. Copyright © 2012 by Nelson Education Limited. The computed value of +.84 indicates that there is a very strong and positive relationship between these two variables. When we square this number, we find that we make about 71% fewer errors in predicting quality of life with knowledge of number of friends. Spearman’s rho: An Example (continued) 12-38

39. Copyright © 2012 by Nelson Education Limited. In testing gamma and Spearman’s rho for statistical significance, the null hypothesis states that there is no association between the variables in the population. To test the significance of gamma and Spearman’s rho, the familiar five-step model should be used to organize the hypothesis testing procedures. Testing Statistical Significance of Gamma and Spearman’s rho 12-39

40. Copyright © 2012 by Nelson Education Limited. Z is used to test of the significance of gamma, and t is used for Spearman’s rho. Section 12.7 provides details on testing the null hypothesis of “no association” with gamma and Spearman’s rho. Testing Statistical Significance of Gamma and Spearman’s rho (continued) 12-40