MEASURES OF ASSOCIATION I. Introduction. A. Chi-square only tells you if a relationship exists. It tells you nothing about the strength or direction (for ordinal categorical variables only) of the relationship. B. However, there are several measures of association that are frequently calculated with chi-square that do provide this information. II. Measures of association for nominal variables. A. Chi-square based measures. 1. Phi is a chi-square based measure of association appropriate for 2x2 tables. 2. Calculating Phi. 3. Cramer's V is a chi-square based measure of association appropriate for all other tables. 4. Calculating Cramer's V. 5. Both phi and Cramer's V range from 0 to 1 with zero meaning no association and 1 meaning a perfect association. 6. Although easy to calculate, phi and Cramer's V are difficult to interpret and not terribly sensitive measures. B. Preportional Reduction in Error Measures. 1. Proportional Reduction in Error (PRE) measures are more meaningful and sensitive. 2. PRE measures indicate how much more accurately we can predict scores on the dependent variable when we know scores on the independent variable compared to when we do not know scores on the independent variable. 3. Lamda. a. Calculating lamda b. Interpreting lamda. i. Ranges from 0 to 1 with 0 meaning no association and 1 meaning perfect association. ii. It represents proportionally how much more accurately we can predict the dependent variable when we know the independent variable. iii. Lamda is asymmetrical meaning its value differs depending on which variable is taken as the dependent variable. 4. Tau. a. Calculating tau. b. Interpreting tau. III. Measures of association for ordinal variables. A. Ordinal measures of association not only indicate how strong an association exists, but also whether the association is negative or positive. B. Ordinal measures of association can only be used if both variables are at least ordinal. C. Gamma is the simplest PRE measure of association for ordinal variables. 1. Calculating gamma. 2. Interpreting gamma. a. Gamma ranges from -1 to +1 with -1 indicating a perfect negative association, +1 a perfect positive association, and 0 indicating no association. b. It represents the increase in our ability to predict whether a case will be higher or lower on one variable if we know their rank on the other variable. c. Gamma is symmetrical which means its value is the same regardless of which variable is the dependent variable. D. Other ordinal measures of association. 1. Somer's d is another PRE measure of association for ordinal variables. It is similar to gamma but differs in that it takes into account cases whose ranks tie one the dependent variable. 2. Like gamma it ranges from -1 to +1, but unlike gamma, it is asymmetrical. 3. Kendall's tau-b is also a PRE measure of association for ordinal variables. It is similar to gamma and Somer's d but differs in that it takes into account cases who are tied on both variables. 4. Like gamma and tau-b, it ranges from -1 to +1. Tau-b is symmetrical like gamma. IV. Testing hypotheses about association. A. Some measures of association (like gamma) can be used without chi-square to test hypotheses about the relationship between two variables. B. This is because the sampling distribution of these measures is basically normal allowing a z or t-test to be used.