1. Describing Association for Discrete Variables

2. Discrete variables can have one of two different qualities: 1. ordered categories 2. non-ordered categories

3. 1. Ordered categories e.g., “High,” “Medium,” and “Low” [both variables must be ordered] 2. Non-ordered categories e.g., “Yes” and “No”

4. Relationships between two variables may be either 1. symmetrical or 2. asymmetrical

5. Symmetrical means that we are only interested in describing the extent to which two variables “hang around together” [non-directional] Symbolically, X  Y

6. Asymmetrical means that we want a measure of association that yields a different description of X’s influence on Y from Y’s influence on X [directional] Symbolically, X  Y Y  X

7. Ordered Categories Asymmetrical Relationship No Yes Yule’s Q Cramer’s V Gamma (G) Lambda ( ) Somers’ d yx No Yes

8. For symmetrical relationships between two non-ordered variables, there are two choices: 1. Yule’s Q (for 2x2 tables) 2. Cramer’s V (for larger tables)

9. Respondents in the 1997 General Social Survey (GSS 1997) were asked: Were they strong supporters of any political party (yes or no)?; and, Did they vote in the 1996 presidential election (yes or no)? Party Identification Not Strong Strong Total Voting Voted a b a + b Turnout Not Voted c d c + d Total a + c b + d a+b+c+d

11. Party Identification Not Strong Strong Total Voting Voted615 339 954 Turnout Not Voted 318 59 377 Total933 398 1,331

12. Q = [(339)(318) - (615)(59)] / [(339)(318) + (615)(59)] = [(107,801) - (36,285)] / [(107,801) + (36,285)] = (71,516) / (144,086) = 0.496

13. What does this mean? Yule’s Q varies from 0.00 (statistical independence; no association) to + 1.00 (perfect direct association) and – 1.00 (perfect inverse association)

14. Use the following rule of thumb (for now): 0.00 to 0.24"No relationship" 0.25 to 0.49"Weak relationship" 0.50 to 0.74"Moderate relationship" 0.75 to 1.00"Strong relationship" Yule’s Q = + 0.496 "... represents a moderate positive association between party identification strength and voting turnout."

15. Party Identification Not Strong Strong Total Voting Voted 0 954 954 Turnout Not Voted 377 0 377 Total377 954 1,331

16. What would be the value of Yule's Q? Q = [(954)(377) - (0)(0)] / [(954)(377) + (0)(0)] = [(359,658) - (0)] / [(359,658) + (0)] = (359,658) / (359,658) = 1.000

17. Party Identification Not Strong Strong Total Voting Voted477 477 954 Turnout Not Voted 189 188 377 Total666 665 1,331

18. In this case, Yule's Q would be: Q = [(477)(189) - (477)(188)] / [(477)(189) + (477)(188)] = [(90,153) - (89,676)] / [(90,153) + (89,676)] = (477) / (179,829) = 0.003

19. Obviously Yule's Q can only be calculated for 2 x 2 tables. For larger tables (e.g., 3 x 4 tables having three rows and four columns), most statistical programs such as SAS report the Cramer's V statistic. Cramer's V has properties similar to Yule's Q, but since it is computed from  2 it cannot take negative values: Where min(R – 1) or (C – 1) means either number of rows less one or number of columns less one, whichever is smaller, and N is sample size.

20. In the example above,  2 = 50.968 and Cramer's V is = 0.196

21. For asymmetrical relationships between two non-ordered variables, the statistic of choice is: Lambda ( )

22. Lambda is calculated as follows: = [(Non-modal responses on Y) - (Sum of non-modal responses for each category of X)] / (Non-modal responses on Y)

23. Party Identification Not Strong Strong Total Voting Voted615 339 954 Turnout Not Voted 318 59 377 Total933 398 1,331

24. In this example, = [(377) - (318 + 59)] / (377) = [(377) - (377)] / (377) = (0) / (377) = 0.00

25. For symmetrical relationships between two variables having ordered categories, the statistic of choice is: Gamma (G)

26. where n s are concordant pairs and n d are discordant pairs

27. The concepts of concordant and discordant pairs are simple and are based on a generalization of the diagonal and off-diagonal in the Yule’s Q statistic.

29. To construct concordant pairs: "Starting with the upper right cell (i.e., the first row, last column in the table), add together all frequencies in cells below AND to the left of this cell, then multiply that sum by the cell frequency. Move to the next cell (i.e., still row one, but now one column to the left) and do the same thing. Repeat until there are NO cells to the left AND below the target cell. Then sum up all these products to form the value for the concordant pairs."

30. To illustrate, take the crosstabulation below which shows the relationship between a measure of social class and respondents' satisfaction with their current financial situation: Social Class Financially SatisfiedLowerWorking Middle UpperTotal Very well 10 131 251 36 428 More or less 19 309 343 19 690 Not at all 43 190 84 7 324 Total 72 630 678 62 1,442

31. Social Class Financially SatisfiedLowerWorking Middle UpperTotal Very well 10 131 251 36 428 More or less 19 309 343 19 690 Not at all 43 190 84 7 324 Total 72 630 678 62 1,442

35. For this table, the calculations are: 36 x (343 + 309 + 19 + 84 + 190 + 43) = 35,568 251 x (309 + 19 + 190 + 43) = 140,811 131 x (19 + 43) = 8,122 19 x (84 + 190 + 43) = 6,023 343 x (190 + 43) = 79,919 309 x (43) = 13,287 These are NOT the value of the concordant pairs; they are the values that must be added together to determine the value of concordant pairs. n s = (35,568 + 140,811 + 8,122 + 6,023 + 79,919 + 13,287) n s = 283,730

36. To construct discordant pairs: "Starting with the upper left cell (i.e., the first row, first column in the table), add together all frequencies in cells below AND to the right of this cell, then multiply that sum by the cell frequency. Move to the next cell (i.e., still row one, but now one column to the right) and do the same thing. Repeat until there are NO cells to the left AND below the target cell. Then sum up all these products to form the value for the discordant pairs."

37. Social Class Financially SatisfiedLowerWorking Middle UpperTotal Very well 10 131 251 36 428 More or less 19 309 343 19 690 Not at all 43 190 84 7 324 Total 72 630 678 62 1,442

41. For the discordant pairs in this table, the calculations are: 10 x (309 + 343 + 19 + 190 + 84 + 7) = 9,520 131 x (343 + 19 + 84 + 7) = 59,343 251 x (19 + 7) = 6,526 19 x (190 + 84 + 7) = 5,339 309 x (84 + 7) = 28,119 343 x (7) = 2,401 Again, these are NOT the value of the disconcordant pairs; they are the values that must be added together to determine the value of disconcordant pairs. n d = (9,520 + 59,343 + 6,526 + 5,339 + 28,119 + 2,401) n d = 111,248

42. G = [(283,730) - (111,248)] / [(283,730) + (111,248)] = (172,482) / (394,978) = 0.437

43. For asymmetrical relationships between two variables having ordered categories, the statistic of choice is: Somers’ d yx

44. For this crosstabulation, we specify Social Class (the column variable) as the independent variable (X) and Financial Satisfaction (the row variable) as the dependent variable (Y). Social Class (X) Financially Satisfied (Y)LowerWorking Middle UpperTotal Very well 10 131 251 36 428 More or less 19 309 343 19 690 Not at all 43 190 84 7 324 Total 72 630 678 62 1,442

45. Somers' d yx statistic is created by adjusting concordant and discordant pairs for tied pairs on the dependent variable (Y). In the example we have been using example, the only asymmetrical relationship that makes sense is one with the dependent variable (Y) as the row variable. Therefore Somers' d yx will be shown only for this situation, that is, for tied pairs on the row variable. (Tied pairs for the column variable follow the identical logic.) A tied pair is all respondents who are identical with respect to categories of the dependent variable but who differ on the category of the independent variable to which they belong. In the case of financial satisfaction, it is all respondents who express the same satisfaction level but who identify themselves with different social classes. In other words, for ties for a dependent row variable it is all the observations in the other cells in the same row.

46. The computational rule is: Target the upper left hand cell (in the first row, first column); multiply its value by the sum of the cell frequencies to right in the same row; move to the cell to the right and multiply its value by the sum of the cell frequencies to right in the same row; repeat until there are no more cells to the right in the same row; then move to the first cell in the next row (first column) and repeat until there are no more cells in the table having cells to the right. Add up these products.

47. Social Class Financially SatisfiedLowerWorking Middle UpperTotal Very well 10 131 251 36 428 More or less 19 309 343 19 690 Not at all 43 190 84 7 324 Total 72 630 678 62 1,442

51. Here, the products are: 10 x (131 + 251 + 36) = 4,180 131 x (251 + 36) = 37,597 251 x (36) = 9,036 19 x (309 + 343 + 19) = 12,749 309 x (343 + 19) = 111,858 343 x (19) = 6,517 43 x (190 + 84 + 7) = 12,083 190 x (84 + 7) = 17,290 84 x (7) = 588 Thus, tied pairs (T r ) for rows equals T r = (4,180 + 37,597 + 9,036 + 12,749 + 111,858 + 6,517 + 12,083 + 17,290 + 588) = 211,898

52. In this example, Somers' d yx = [(283,730) - (111,248)] / [(283,730) + (111,248) + (211,898)] = (172,482) / (606,976) = 0.284

53. Ordered Categories Asymmetrical Relationship No Yes Yule’s Q Cramer’s V Gamma (G) Lambda ( ) Somers’ d yx No Yes

54. Using SAS to Produce Two-Way Frequency Distributions and Statistics Using SAS to Produce Two-Way Frequency Distributions and Statistics libname mystuff 'a:\'; libname library 'a:\'; options formchar='|----|+|---+=|-/\ *' ps=66 nodate nonumber; proc freq data=mystuff.marriage; tables church*married / expected all; title1 ‘Crosstabulation for Discrete Variables'; run;

55. Crosstabulation for Discrete Variables TABLE OF CHURCH BY MARRIED CHURCH MARRIED Frequency| Expected | Percent | Row Pct | Col Pct |Divorced|Married |Never |Separate|Widowed | Total ---------+--------+--------+--------+--------+--------+ Annually | 74 | 269 | 129 | 18 | 43 | 533 | 62.318 | 290.33 | 101.17 | 18.695 | 60.485 | | 5.09 | 18.50 | 8.87 | 1.24 | 2.96 | 36.66 | 13.88 | 50.47 | 24.20 | 3.38 | 8.07 | | 43.53 | 33.96 | 46.74 | 35.29 | 26.06 | ---------+--------+--------+--------+--------+--------+ Monthly | 30 | 149 | 50 | 10 | 26 | 265 | 30.983 | 144.35 | 50.303 | 9.295 | 30.072 | | 2.06 | 10.25 | 3.44 | 0.69 | 1.79 | 18.23 | 11.32 | 56.23 | 18.87 | 3.77 | 9.81 | | 17.65 | 18.81 | 18.12 | 19.61 | 15.76 | ---------+--------+--------+--------+--------+--------+ Never | 32 | 85 | 34 | 6 | 16 | 173 | 20.227 | 94.234 | 32.839 | 6.0681 | 19.632 | | 2.20 | 5.85 | 2.34 | 0.41 | 1.10 | 11.90 | 18.50 | 49.13 | 19.65 | 3.47 | 9.25 | | 18.82 | 10.73 | 12.32 | 11.76 | 9.70 | ---------+--------+--------+--------+--------+--------+ Weekly | 34 | 289 | 63 | 17 | 80 | 483 | 56.472 | 263.09 | 91.684 | 16.942 | 54.811 | | 2.34 | 19.88 | 4.33 | 1.17 | 5.50 | 33.22 | 7.04 | 59.83 | 13.04 | 3.52 | 16.56 | | 20.00 | 36.49 | 22.83 | 33.33 | 48.48 | ---------+--------+--------+--------+--------+--------+ Total 170 792 276 51 165 1454 11.69 54.47 18.98 3.51 11.35 100.00

56. Crosstabulation for Discrete Variables STATISTICS FOR TABLE OF CHURCH BY MARRIED Statistic DF Value Prob ------------------------------------------------------ Chi-Square 12 57.792 0.000 Likelihood Ratio Chi-Square 12 57.806 0.000 Mantel-Haenszel Chi-Square 1 8.152 0.004 Phi Coefficient 0.199 Contingency Coefficient 0.196 Cramer's V 0.115 Statistic Value ASE ------------------------------------------------------ Gamma 0.052 0.033 Kendall's Tau-b 0.035 0.022 Stuart's Tau-c 0.031 0.020 Somers' D C|R 0.033 0.021 Somers' D R|C 0.037 0.024 Pearson Correlation 0.075 0.026 Spearman Correlation 0.041 0.026 Lambda Asymmetric C|R 0.000 0.000 Lambda Asymmetric R|C 0.062 0.027 Lambda Symmetric 0.036 0.016 Uncertainty Coefficient C|R 0.016 0.004 Uncertainty Coefficient R|C 0.015 0.004 Uncertainty Coefficient Symmetric 0.016 0.004 Sample Size = 1454

57. Exercise Compute values for Lambda ( ), Gamma (G) and Somers' d yx for the following two-way frequency distribution. Assume that the row variable, self-described health, is the dependent (Y) variable. Education Degree Level Self-Described Health Less than H.S. H.S. Jr.Co. Col. Grad.Sch. Total Excellent 69 227 20 82 37 435 Good 156 403 26 77 34 696 Fair 122 111 8 12 5 258 Poor 50 16 0 3 0 69 Total 397 757 54 174 76 1,458

58. Answers 1. The modal responses on Y (self-described health) are 696. Therefore, the non-modal responses are 435 + 258 + 69 = 762. For each category of self-described health, the non-modal responses total 754. Therefore, Lambda = (762 - 754) / 762 = 0.010 2. Concordant pairs (n s ) = 320,060 and discordant pairs (n d ) = 130,272 Gamma = (320060 - 130272) / (320060 + 130272) = 189788 / 450332 = 0.421 3. Tied pairs (T r ) = 227,737 Therefore, Somers' d yx = (320060 - 130272) / (320060 + 130272 + 227737) = 189788 / 678069 = 0.280