1. Measures of Association February 25, 2011

2. Objectives By the end of this meeting, participants should be able to: a)Calculate ordinal measures of association and discuss how appropriate each is for various types of variables. b)Calculate nominal measures of association. c)Perform a chi-square significance test of a cross tabulation.

3. What is Association? Association can be thought of in two general different ways: a)Covariation- the extent to which two variables change together Positive covariation- as one variable increases another variable increases as well (be aware of coding!!) Positive covariation- as one variable increases another variable increases as well (be aware of coding!!) Negative covariation- as one variable increases another variable decreases (or vice versa) Negative covariation- as one variable increases another variable decreases (or vice versa) No relationship- changes in one variable have no systematic effect on another variable No relationship- changes in one variable have no systematic effect on another variable

4. What is Association? b)Probability- how well can one variable predict changes in another variable c)Most measures of association are bounded between -1 and 1

5. Ordinal Measures a)Kendall ’ s tau ( τ ) Most commonly used ordinal measure Most commonly used ordinal measure Measures covariation, meaning that it will have the same value regardless of which variable is the independent and the dependent variable (it is a symmetric measure) Measures covariation, meaning that it will have the same value regardless of which variable is the independent and the dependent variable (it is a symmetric measure) The value of the measure ranges from -1 to 1 The value of the measure ranges from -1 to 1 Generally speaking a value over.7 is considered strong, one between.3 and.7 moderate and less than.3 weak. This is a loose standard and will differ depending on the state of the literature. Generally speaking a value over.7 is considered strong, one between.3 and.7 moderate and less than.3 weak. This is a loose standard and will differ depending on the state of the literature.

6. Ordinal Measures b)More Specific Taus Tau-b- Will only reach a value of 1 if both variables have the same number of categories Tau-b- Will only reach a value of 1 if both variables have the same number of categories Tau-c- Corrects for unequal number of categories between the independent and dependent variables (although remember the measure is symmetric). Tau-c- Corrects for unequal number of categories between the independent and dependent variables (although remember the measure is symmetric). c)Somer ’ s d Is a measure of association based on the differences in percentages Is a measure of association based on the differences in percentages d is a measure of how much the dependent variable changes as a result of the independent variable (this is an asymmetric measure). You need to be careful that you are analyzing the expected relationship rather than the opposite d is a measure of how much the dependent variable changes as a result of the independent variable (this is an asymmetric measure). You need to be careful that you are analyzing the expected relationship rather than the opposite

7. Ordinal Measures d)Goodman and Kruskal ’ s Gamma ( γ ) Computes whether two variables measure the same underlying dimension Computes whether two variables measure the same underlying dimension γ is based on the logic of concordant (similar) and discordant (dissimilar) pairs γ is based on the logic of concordant (similar) and discordant (dissimilar) pairs γ ranges from -1 to +1 γ ranges from -1 to +1 e)Spearman ’ s Rho ( ρ ) also known as the Spearman ’ s rank order coefficient This measure computes the association between two variables that are rank orders (like income) This measure computes the association between two variables that are rank orders (like income) This measure should not be used with variables that are ordered categories such as ideology This measure should not be used with variables that are ordered categories such as ideology

8. Caveats a) a)Ordinal level measures of association assume a monotonic relationship. b) b)That is, as one variable increases, the other variable will consistently increase or consistently decrease (but not both). c) c)The relationship between age and voter turnout is not likely to be monotonic, therefore ordinal level measures of association are problematic. d) d)Make sure that the categories are ordered in a logical way. This may mean excluding missing or rare cases (such as did not vote, third party, etc.).

9. How to choose among the measures? a)When examining rank ordered variables, the choice is clearly Spearman ’ s Rho ( ρ ). b)What about other cases? Tau is the most commonly used ordinal measure of association Tau is the most commonly used ordinal measure of association Gamma should generally only be used when examining a scale Gamma should generally only be used when examining a scale Presenting more than one measure is an acceptable option provided that it aids the readers ’ understanding. Presenting more than one measure is an acceptable option provided that it aids the readers ’ understanding.

10. Nominal Measure of Association a)This measure should be used for nominal data or data where the categories cannot be ordered. b)Lambda ( λ ) The most commonly used measure, lambda, is asymmetric, having different values depending on which variable is the independent and dependent variable. The most commonly used measure, lambda, is asymmetric, having different values depending on which variable is the independent and dependent variable. Is more of predictive measure, it computes the proportional reduction in error (PRE). In other words lambda estimates how much better your prediction about the value of dependent variable is when you know the independent variable. Is more of predictive measure, it computes the proportional reduction in error (PRE). In other words lambda estimates how much better your prediction about the value of dependent variable is when you know the independent variable.

11. Significance Tests a) a)We want to know whether the association we observe is due to chance—whether we observe an association because of the sample we happened to draw—or whether the relationship between variables exists in the underlying population of interest. b) b)Chi–Square Test ( χ 2 ) Tests the null hypothesis that there is no relationship between the two variables in the population. If we reject the null hypothesis, we conclude that the relationship between variables is statistically significant.

12. Chi–Square Test ( χ 2 ) a)Once a Chi-Square is completed, using the degrees of freedom a p value can be found. b)Researchers almost want the lowest possible p value or, in other words, the lowest probability that the relationship between the two variables occurred by chance. c)The standard value in the social science is p values of less than.05 but values of less than.1 or.01 are not uncommon. Whatever standard is chosen it should be clearly marked on the table.

13. Substantive Criteria a)Significance tests, while important, are not the sole determinant of good research. b)Small samples are more likely to be found not significant. c)It is important to determine whether the predicted relationship is substantively important. d)Recall that association is part of causation but not the same thing as causation.

14. Think About What is the difference between statistical significance and substantive importance?

15. For March 7 a)Two separate homeworks for two separate grades. b)HW 1: Select two ordinal variables with a possible relationship from the PS-ARE data. (You may use the same two as last time.) Recode the variables (if necessary) so that they are both ordinal scales. Recode the variables (if necessary) so that they are both ordinal scales. Compute the cross tabulation of the two variables with both frequencies and percentages. Compute the cross tabulation of the two variables with both frequencies and percentages. Does a chi-square test show a statistically significant relationship between the two variables? Does a chi-square test show a statistically significant relationship between the two variables? Find the most appropriate measure of association and interpret it. Find the most appropriate measure of association and interpret it. c)HW 2: Read WKB chapter 13 and turn-in answers to questions 2 & 3 on pp. 296-297.