1. Chapter Eighteen MEASURES OF ASSOCIATION

2. Parametirc vs. Nonparametric Measures of Association
Parametirc measure of association is for continuous variables measured on an interval or ratio scale.
Bivariate correlation (Pearson correlation) is a typical parametric measure of association.
The coefficient does not distinguish between independent and dependent variables.
Nonparametric measure of association is for nominal or ordinal data.

3. Bivariate Correlation Analysis
Pearson correlation coefficient
r symbolized the coefficient's estimate of linear association based on sampling data
Correlation coefficients reveal the magnitude and direction of relationships
Coefficient’s sign (+ or -) signifies the direction of the relationship
Assumptions of r
Linearity
Bivariate normal distribution

4. Bivariate Correlation Analysis
Scatterplots
Provide a means for visual inspection of data
the direction of a relationship
the shape of a relationship
the magnitude of a relationship
(with practice)

5. Interpretation of Coefficients
Relationship does not imply causation
Y could cause X
X could cause Y
X and Y could influence each other.
X and Y could be affected by a third variable.
Statistical significance is measured by t-value.
Statistical significance does not imply a relationship is practically meaningful

6. Interpretation of Coefficients
Be careful about artifact correlations
Coefficient of determination (r2)
The amount of common variance in X and Y
F-test is used for goodness of fit.
Correlation matrix
used to display coefficients for more than two variables

7. Bivariate Linear Regression
Establish a linear relationship between a independent variable (X) and a dependent variable (Y).
Use the observed value of X to estimate or predict corresponding Y value.
Regression coefficients
Slope: β1 = Δ Y / Δ X
Intercept: βo = Y bar – β1 X bar

8. Bivariate Linear Regression
Error term: deviation of the ith observation from the regression line represented by βo + β1 Xi , i.e., εi = Yi - βo - β1 Xi
Method of Least Squares
Regression line is line of best fit for the data.
To find the best fit, the method of least squares is used.
Method of least squares is to minimize Σ εi2 (the total squared errors of estimate).
Technically, calculus (differentiation) is used to solve for β1 and βo..

9. Interpreting Linear Regression
Goodness of fit
T test for individual coefficients
Zero slope (β1 = 0) means
Y completely unrelated to X and no systematic pattern is evident
constant values of Y for every value of X
data are related, but represented by a nonlinear function
F test for the model
F value is related to r2 (coefficient of determination)

10. Interpreting Linear Regression
Residuals
What remain after the line is fitted
Estimated error terms
Standardized residuals are comparable to Z scores with a mean of 0 and a standard deviation of 1.
Confidence band vs. prediction band

11. Measures for Nominal Data
When there is no relationship at all, coefficient is 0
When there is complete dependency, the coefficient displays unity or 1
The following measures are used for nominal data (next slide).

12. Measures for Nominal Data
Chi-square based measure
Phi
Cramer’s V
Contingency coefficient of C
Proportional reduction in error (PRE)
Lambda
Tau

13. Characteristics of Ordinal Data
Concordant- subject who ranks higher on one variable also ranks higher on the other variable
Discordant- subject who ranks higher on one variable ranks lower on the other variable

14. Measures for Ordinal Data
No assumption of bivariate normal distribution
Most based on concordant/discordant pairs
Values range from +1.0 to -1.0

15. Measures for Ordinal Data
The following test statistics are used.
Gamma
Somer’s d
Spearman’s rho
Kendall’s tau b
Kendall’s tau c