1. Association Between Variables Measured at the Ordinal Level
Chapter 12
Association Between Variables Measured at the Ordinal Level
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2. In this presentation you will learn about:
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
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Gamma
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.
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Gamma (continued)
Gamma is a measure of association based on the logic of "pairs" of cases.
To compute Gamma, two quantities must be found: ns and nd
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Gamma (continued)
ns is the total number of pairs of cases ranked in the same order on both variables.
For example, Joseph and Steven are among 50 respondents to a survey investigating the relationship between education (independent variable) and income (the dependent variable).
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:
Thus this “pair” of cases is said to be similar (same).
Education Income
Joseph (High) (High) Steven (Low) (Low)
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Gamma (continued)
nd 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).
Education Income
Caimile (High) (Low) Manjit (Low) (High)
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Gamma (continued)
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.
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Gamma: An Example
Let’s now consider the survey on education and income for all 50 respondents. The results are summarized in the bivariate table below:
Income
Education
Low High
Totals
Low 15 10 25
High 5 20 30 50
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9. Gamma: An Example (continued)
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 (ns) and the total number of pairs of cases ranked in different order on both variables (nd).
Note, to use the short-cut, the data must be summarized in a bivariate table.
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10. Gamma: An Example (continued)
To compute ns in a bivariate table, multiply each cell frequency by all cell frequencies below and to the right:
For this 2x2 table: ns:15 x 20 = 300
Income
Education
Low High
Totals
Low 15 10 25
High 5 20 30 50
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11. Gamma: An Example (continued)
To compute nd in a bivariate table, multiply each cell frequency by all cell frequencies below and to the left:
nd: 5 x 10 = 50
Income
Education
Low High
Totals
Low 15 10 25
High 5 20 30 50
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12. Gamma: An Example (continued)
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 (perfect association).
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13. Gamma: An Example (continued)
As the preponderance of similar pairs increases (ns > nd ), gamma increases. It will also be positive in value, indicating a positive relationship.
As the preponderance of dissimilar pairs increases (nd > ns), gamma increases. It will also be negative in value, indicating a negative relationship.
When ns = nd gamma will be zero (no association)
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14. Gamma: An Example (continued)
Recall, in a positive relationship, the variables change in the same direction (Table 12.4). In a negative relationship, the variables change in the opposite direction (Table 12.5).
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15. Gamma: An Example (continued)
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:
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16. Gamma: An Example (continued)
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
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17. Gamma: An Example (continued)
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.
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Gamma as a PRE Measure
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.
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19. Gamma as a PRE Measure (continued)
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.
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Limitations of Gamma
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 (ns + nd ) of the 1,225 number of possible pairs of cases were used in computing gamma; the rest (tied pairs of cases) were ignored.
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21. Limitations of Gamma (continued)
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.
Education Income
Joseph (High) (High)
Steve (High) (Low)
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22. Limitations of Gamma (continued)
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.
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Kendall’s tau-b and tau-c, Somers' d
Kendall’s tau-b includes pairs tied on the independent variable, tx, and pairs tied on the dependent variable ty.
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, ty.
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24. Kendall’s tau-b and tau-c, Somers' d (continued)
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 (perfect negative relationship) to (perfect positive relationships).
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25. Kendall’s tau-b and tau-c, Somers' d (continued)
Kendall’s tau-b is computed with Formula 12.3:
Kendall’s tau-c is computed with Formula 12.4:
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26. Kendall’s tau-b and tau-c, Somers' d (continued)
Somers’ d is computed with Formula 12.5:
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27. Kendall’s tau-b, tau-c, Somers' d: An Example
Looking again at the relationship between education and income, to compute ty in a bivariate table, multiply each cell frequency by all cell frequencies immediately to the right:
For this 2x2 table: ty: (15 x 10)+ (5 x 20)= = 250
Income
Education
Low High
Totals
Low 15 10 25
High 5 20 30 50
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28. Kendall’s tau-b, tau-c, Somers' d: An Example (continued)
To compute tx in a bivariate table, multiply each cell frequency by all cell frequencies immediately below:
For this 2x2 table: tx: (15 x 5)+ (10x 20)= = 275
Income
Education
Low High
Totals
Low 15 10 25
High 5 20 30 50
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29. Kendall’s tau-b, tau-c, Somers' d: An Example (continued)
The value of Kendall's tau-b, using Formula 12.3, is
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30. Kendall’s tau-b, tau-c, Somers' d: An Example (continued)
The value of Kendall's tau-c, using Formula 12.4, is
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31. Kendall’s tau-b, tau-c, Somers' d: An Example (continued)
The value of Somers' d, using Formula 12.5, is
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Which Measure to Use?
Each measure indicates a strong and positive association between education and income:
G= tau-b= tau-c= 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.
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Spearman’s rho
Spearman’s rho, rs , 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 (perfect association), and is a symmetrical.
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34. Spearman’s rho (continued)
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.
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35. Spearman’s rho (continued)
The formula for Spearman’s rho is:
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36. Spearman’s rho: An Example
The number of friends a person is hypothesized to have a positive effect on his/her quality of life.
Given the data below, how strong is the relationship
between these two variables?
No. of Quality
Respondent Friends of Life
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37. Spearman’s rho: An Example (continued)
To compute D2, the rank of each case on Y is subtracted from its rank on X ,D, then squared,D2.
The sum of each difference squared, or D2, is entered directly into the Formula 12.6.
No. of Qual.
Friends Rank Life Rank D D2
D= D2=36
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38. Spearman’s rho: An Example (continued)
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.
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39. Testing Statistical Significance of Gamma and Spearman’s rho
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.
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Testing Statistical Significance of Gamma and Spearman’s rho (continued)
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.
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