1. Week 12
Chapter 13 – Association between variables measured at the ordinal level
&
Chapter 14: Association Between Variables Measured at the Interval-Ratio Level

2. Association Between Variables Measured at the Ordinal Level
Chapter 13
Association Between Variables Measured at the Ordinal Level

3. This Presentation Two Types of Ordinal Variables Gamma Spearman’s Rho
Hypothesis Tests for Gamma and Rho

4. Two Types of Ordinal Variables
Continuous ordinal variables:
Have many possible scores
Resemble interval-ratio level variables
Use Spearman’s Rho: rs
Example: a scale measuring attitudes toward handgun control with scores ranging from 0 to 20
Collapsed ordinal variables:
Have just a few values or scores
Use Gamma: G
Can also use Somer’s d and Kendall’s tau-b (see text website)
Example: social class measured as lower, middle, upper

5. Gamma
Gamma is used to measure the strength and direction of the relationship between two ordinal level variables that have been arrayed in a bivariate table
Before computing and interpreting Gamma, it will always be useful to find and interpret the column percentages

6. Gamma
Interpretation:
Use the table below as a guide to interpret the strength of gamma in overall terms

7. Gamma
In addition to strength, gamma also identifies the direction of the relationship
In a negative relationship, the variables change in different directions
Example: As age increases, income decreases (or, as age decreases, income increases)
In a positive relationship, the variables change in the same direction
Example: As education increases, income increases (or, as education decreases, income decreases)

8. Gamma

9. Gamma

10. Gamma
In addition to strength and direction, a hypothesis test of Gamma can also indicate if the two variables share a relationship in the population, or if the two variables are significantly related
Hypothesis Test of Gamma:
Step 1: Make Assumptions and Meet Test Requirements
Random sampling
Ordinal level of measurement
Normal sampling distribution

11. Gamma Hypothesis Test of Gamma: Step 2: State the Null Hypothesis
Ho: γ = 0
No relationship exists between the variables in the population
H1: γ ≠ 0
A relationship exists between the variables in the population

12. Gamma Hypothesis Test of Gamma:
Step 3: Select the Sampling Distribution and Establish the Critical Region
Sampling distribution = Z distribution
Set alpha (two-tailed)
Look up Z(critical) in Appendix A

13. Gamma
Hypothesis Test of Gamma:
Step 4: Compute the Test Statistic

14. Gamma Hypothesis Test of Gamma:
Step 5: Make a Decision and Interpret the Results
Compare Z(obtained) to Z(critical)
If Z(obtained) falls in the critical region, reject Ho
If Z(obtained) does not fall in the critical region, fail to reject Ho
Interpret results

15. Spearman’s Rho (rs)
Measure of association for ordinal-level variables with a broad range of different scores and few ties between cases on either variable
Computing Spearman’s Rho
Rank cases from high to low on each variable
Use ranks, not the scores, to calculate Rho

16. Spearman’s Rho (rs)

17. Spearman’s Rho (rs)

18. Spearman’s Rho (rs)

19. Spearman’s Rho (rs)
Rho is positive, therefore jogging and self-image share a positive relationship: as jogging rank increases, self-image rank also increases
On its own, Rho does not have a good strength interpretation
But Rho2 is a PRE measure
For this example, Rho2 = (0.86)2 = 0.74
Therefore, we would make 74% fewer errors if we used the rank of jogging to predict the rank on self-image compared to if we ignored the rank on jogging

20. Spearman’s Rho (rs) Hypothesis Test of Spearman’s Rho:
In addition to strength and direction, a hypothesis test of Rho can also indicate if the two variables share a relationship in the population, or if the two variables are significantly related
Hypothesis Test of Spearman’s Rho:
Step 1: Make Assumptions and Meet Test Requirements
Random sampling
Ordinal level of measurement
Normal sampling distribution

21. Spearman’s Rho (rs) Hypothesis Test of Spearman’s Rho:
Step 2: State the Null Hypothesis
Ho: ρs = 0
No relationship exists between the variables in the population
H1: ρs ≠ 0
A relationship exists between the variables in the population

22. Spearman’s Rho (rs) Hypothesis Test of Spearman’s Rho:
Step 3: Select the Sampling Distribution and Establish the Critical Region
Sampling distribution = Student’s t
Alpha = 0.05 (two-tailed)
Degrees of freedom = N-2 = 8
t(critical) = ±2.306

23. Spearman’s Rho (rs) Hypothesis Test of Gamma:
Step 4: Compute the Test Statistic

24. Spearman’s Rho (rs) Hypothesis Test of Gamma:
Step 5: Make a Decision and Interpret the Results
t(obtained) = 4.77
t(critical) = ±2.306
t(obtained) falls in the critical region, so reject Ho
Jogging and self-image are related in the population from which the sample was drawn

25. Association Between Variables Measured at the Interval-Ratio Level
Chapter 14
Association Between Variables Measured at the Interval-Ratio Level

26. This Presentation Scattergrams
Graphs that display relationships between two interval-ratio variables
Regression Coefficients and the Regression Line
Regression line summarizes the linear relationship between X and Y
Regression coefficients predict scores on Y from scores on X
Pearson’s r
Preferred measure of association for two interval-ratio variables
Coefficient of determination: r2
Correlation matrix

27. Scattergrams Scattergrams have two dimensions:
The X (independent) variable is arrayed along the horizontal axis
The Y (dependent) variable is arrayed along the vertical axis
Each dot on a scattergram is a case
The dot is placed at the intersection of the case’s scores on X and Y

28. Scattergrams
A regression line, which summarizes the linear relationship between X and Y, is added to the graph
“Eyeball” a straight line that connects all of the dots or comes as close as possible to connecting all of the dots
To be more precise: calculate the conditional mean of Y for each value of X, plot those values, and connect the dots
Inspection of a scattergram should always be the first step in assessing the relationship between two interval-ratio level variables

29. Scattergrams Linearity
A key assumption of scattergrams and regression analysis is that X and Y share a linear relationship
In a linear relationship the dots of a scattergram form a straight line pattern

30. Linear Relationship: Example

31. Scattergrams Linearity
In a nonlinear relationship the dots do not form a straight line pattern

32. Scattergrams Three Questions Does a relationship exist?
A relationship exists if the conditional means of Y change across values of X
As long as the regression line lies at an angle to the X axis (and is not parallel to the X axis), we can conclude that a relationship exists between the two variables

33. Scattergrams Three Questions How strong is the relationship?
Strength of the relationship is determined by the spread of the dots around the regression line
In a perfect association, all dots fall on the regression line
In a stronger association, the dots fall close (are clustered tightly around) the regression line
In a weaker association, the dots are spread out relatively far from the regression line

34. Scattergrams Three Questions
What is the direction of the relationship? (Direction of association is determined by the angle of the regression line)

35. What is the Direction of the Relationship?

36. Scattergrams
Based on this scattergram for percent college educated (X) and voter turnout (Y) on election day for 50 states:
Does a relationship exist? How strong is the relationship?
What is the direction of the relationship? Is the relationship linear?

37. Scattergrams Does a relationship exist?
The regression line falls at an angle to the X axis (it is not parallel), therefore we can conclude that an association exists between voter turnout and college education

38. Scattergrams How strong is the relationship?
The greater the extent to which dots are clustered around the regression line, the stronger the relationship
This relationship is weak to moderate in strength

39. Scattergrams What is the direction of the relationship?
Positive: Regression line rises from lower-left to upper-right
Negative: Regression line falls from upper-left to lower-right
This is a positive relationship: As percent college educated increases, voter turnout increases

40. Scattergrams Is the relationship linear?
The conditional means on Y form a straight line, as demonstrated by the regression line
Therefore, the relationship is linear

41. Pearson’s r
Pearson’s r is a measure of association for interval-ratio level variables
Pearson’s r can indicate the direction of association, but it does not have an acceptable strength interpretation
But, by squaring r, we obtain a PRE measure called the coefficient of determination
The coefficient of determination indicates the percentage of the variation in Y that is explained by X

42. Pearson’s r
Calculate r

43. Pearson’s r
r = 0.50
r is positive, therefore the relationship between X and Y is positive
As the number of children in dual-career families increases, husbands’ hours of housework per week also increases
r2 = (0.50)2 = 0.25
r2 is 0.25, therefore the number of children in dual-career families explains 25% of the variation in husbands’ hours of housework per week

44. Pearson’s r Hypothesis Test of Pearson’s r
Step 1: Make Assumptions and Meet Test Requirements
Random sample
Interval-ratio level measurement
Bivariate normal distributions
Linear relationship
Homoscedasticity
Normal sampling distribution

45. Pearson’s r Hypothesis Test of Pearson’s r
Step 2: State the Null Hypothesis
Ho: ρ = 0
H1: ρ ≠ 0
Step 3: Select the Sampling Distribution and Establish the Critical Region
Sampling distribution = Student’s t
Alpha = 0.05 (two-tailed)
Degrees of freedom = N-2 = 10
t(critical) = ±2.228

46. Pearson’s r Hypothesis Test of Pearson’s r
Step 4: Compute Test Statistic

47. Pearson’s r Hypothesis Test of Pearson’s r
Step 5: Make a Decision and Interpret the Results
t(critical) = ±2.228
t(obtained) = 1.83
t(obtained) does not fall in the critical region, so we fail to reject Ho
The two variables are not related in the population

48. Correlation Matrix
A correlation matrix is a table that shows the relationships between all possible pairs of variables

49. Correlation Matrix Using the matrix below:
What is the correlation between GDP and inequality?
Of all the variables correlated with Inequality, which has the strongest relationship? The weakest?