1. Association Between Variables Measured at the Ordinal Level
Chapter 14
Association Between Variables Measured at the Ordinal Level

2. Chapter Outline Introduction Proportional Reduction in Error (PRE)
The Computation of Gamma
Determining the Direction of Relationships

3. Chapter Outline
Interpreting Association with Bivariate Tables: What Are the Sources of Civic Engagement in U.S. Society?
Spearman’s Rho (rs )
Testing the Null Hypothesis of “No Association” with Gamma and Spearman’s Rho

4. Gamma
Gamma is used to measure the strength and direction of 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.

5. An Ordinal Measure: Gamma
To compute Gamma, two quantities must be found:
Ns is the number of pairs of cases ranked in the same order on both variables.
Nd is the number of pairs of cases ranked in different order on the variables.

6. An Ordinal Measure: Gamma
To compute Ns, multiply each cell frequency by all cell frequencies below and to the right.
For this table, Ns is 10 x 5 = 50.
Low
High 10 12 17 5

7. An Ordinal Measure: Gamma
To compute Nd, multiply each cell frequency by all cell frequencies below and to the left.
For this table, Nd is 12 x 17 = 204.
Low auth
High author
Low effic 10 12
High
effic 17 5

8. An Ordinal Measure: Gamma
Gamma is computed with Formula 14.1

9. Calculate and interpret Gamma
Ns = 10(5)=50 Nd=12(17) = 204
G = (Ns+Nd)/(Ns-Nd) =
(50-204)/(50+204) = -.61
PRE interpretation: We reduce our errors in predicting the efficiency of a workplace by 61% if we know the management style

10. An Ordinal Measure: Gamma
In addition to strength, gamma also identifies the direction of the relationship.
This is a negative relationship: as authoritarianism increases, efficiency decreases.
In a positive relationship, the variables would change in the same direction.

11. Let’s look at a more complicated problem requiring Gamma

12. Let’s look at a more complicated calculation of gamma
High job security
Med. Job security
Low job security
High job satisf a. 16 B 8 C 14
Medium job satisf D 19 E 17 F 60
Low job satisf G 9 H 11 I 56

13. Calculating Gamma Ns = 2304+1273+928+952 = 5,457
Nd = 2361
G = ( )/( )=.396
How do we express the PRE interpretation?
What is the direction of the relationship and what does that mean?

14. Spearman’s rho
Beginning with problem on p. 386, your text gives several examples in which not gamma but Spearman’s rho is the appropriate measure.
What do those problems have in common?
How do you convert Spearman’s rho into a PRE measure?

15. Spearman’s rho and SPSS
Which variables in our GSS2002 data set might be suitable for rho?
How do we get SPSS to calculate rho?
Just ask for Analyze/cross tabs/ gamma
and they’ll throw in what they call the Spearman’s coefficient (I think that’s the square of rho)
Example with polyview and attend