1. Chapter 12 (Ch. 11 in 2/3 Can. Ed.) Bivariate Association for Tabular Data: Basic Concepts

2. Introduction Two variables are said to be associated when they vary together - when one variable changes as the other changes. Association can be important evidence for causal relationships, particularly if the association is strong. Criteria for causality:  1. Association or correlation between variables  2. Establishment of time order.  3. Elimination of alternatives.

3. Introduction If variables are associated, then the score on one variable can be predicted from the score of the other variable. The stronger the association, the more accurate the predictions.

4. Association and Bivariate Tables Bivariate association can be investigated by finding answers to three questions: 1. Is there an association? 2. How strong is the association? 3. What is the pattern or direction of the association?

5. Association and Bivariate Tables: Healey, #12.1 (1e) or #11.1 in 2/3e) The table below shows the relationship between authoritarianism of bosses (X) and the efficiency of workers (Y) for 44 workplaces. Efficiency (y)LowHigh Low101222 High17522 Total271744 Authoritarianism (x)

6. 1. Is There an Association? An association exists if the conditional distributions of one variable change across the values of the other variable. Conditional means that the values of one variable are dependent on those of another variable. With bivariate tables, column percentages are the conditional distributions of Y for each value of X. If the column % changes, the variables are associated.

7. 1. Is There an Association? (cont.) Remember: The column % is (cell frequency / column total) * 100. Calculate %:(10/27)*100 = 37.04% (12/17)* 100 = 70.59% (17/27)*100 = 62.96% (5/17)*100 = 29.41% Efficiency (y)LowHigh Low10 (37.04%)12 (70.59%)22 High17 (62.96%)5 (29.41%)22 Total27 (100.00%)17 (100.00%)44 Authoritarianism (x)

8. 1. Is There an Association? (cont.) The column % show efficiency of workers (Y) by authoritarianism of supervisor (X). The column % change, so these variables are associated. Efficiency (y)LowHigh Low37.04%70.59% High62.96%29.41% 100% Authoritarianism (x)

9. 2. How Strong is the Association? The stronger the relationship, the greater the change in column % (or the conditional distributions). In weak relationships, there is little or no change in column %. In strong relationships, there is large change in column %.

10. 2. How Strong is the Association? (cont.) One way to measure strength is to find the “maximum difference”. This is the largest difference in column % for any row of the table. If…… DifferenceStrength Between 0 and 10%Weak Between 10 and 30%Moderate Greater than 30%Strong

11. 2. How Strong is the Association? (cont.) The Maximum Difference in Problem 12.1 is 70.59 – 37.04 = 33.55. This is a strong relationship. Efficiency (y)LowHigh Low37.04%70.59% High62.96%29.41% 100% Authoritarianism (x)

12. 3. What is the Pattern of the Relationship? “Pattern” = which scores of the variables go together? To detect, find the cell in each column which has the highest column %.

13. 3. What is the Pattern of the Relationship? (cont.) Low on Authoritarianism goes with High on efficiency. High on Authoritarianism goes with Low in efficiency. Efficiency (y)LowHigh Low37.04 %70.59 % High62.96 %29.41 % 100% Authoritarianism (x)

14. Alternatively… 3. What is the Direction of the Relationship? If both variables are ordinal, we can discuss direction as well as pattern. In positive relationships, the variables vary in the same direction.  As one increases, the other increases. In negative relationships, the variables vary in opposite directions.  As one increases, the other decreases.

15. Direction of the Relationship? (cont.) As authoritarianism increases, efficiency decreases. Workplaces high in authoritarianism are low on efficiency. Relationship in Problem 12.1 (or 11.1) is negative. Efficiency (y)LowHigh Low37.04 %70.59 % High62.96 %29.41 % 100% Authoritarianism (x)

16. Summary: Problem 12.1 (11.1 in 2 nd ) There is a strong, negative relationship between authoritarianism and efficiency. These results would be consistent with the idea that authoritarian bosses cause inefficient workers. Efficiency (y)LowHigh Low37.04 %70.59 % High62.96 %29.41 % 100% Authoritarianism (x)

17. What is the Direction of this Relationship? Low on X is associated with low on Y. High on X is associated with high on Y. As X increase, Y increases. This relationship is positive. (y)LowHigh Low60%30% High40%70% 100% (x)

18. Correlation vs. Causation The results above are also consistent with the idea that inefficient workers cause bosses to become authoritarian. Correlation (or association) and causation are not the same things. Strong associations may be used as evidence of causal relationships but they do not prove variables are causally related. What else would we need to know to be sure there is a causal relationship between authoritarianism and efficiency?