1. Association Between Two Variables Measured at the Nominal Level
Chapter 13
Association Between Two Variables Measured at the Nominal Level

2. Nominal Level Measures of Association
It is always useful to compute column percentages for bivariate tables.
But, it is also useful to have a summary measure – a single number – to indicate the strength of the relationship. That’s what we’ll learn about in this chapter.

3. Nominal Level Measures of Association
For nominal level variables, there are two commonly used measures of association:
Phi or Cramer’s V
Lambda

4. Nominal Measures: Phi
Phi is used for 2x2 tables.
The formula for Phi:

5. Nominal Measures: Cramer’s V
Cramer’s V is used for tables larger than 2x2.
Formula for Cramer’s V:

6. SPSS: Phi and Kramer’s V
SPSS has both instructions combined into one
You need to know which one applies
Phi if it’s a 2 x 2 table
Kramer’s V for any other cross tabulation

7. Strength of Phi or Kramer’s V
Value
Strength
Between 0.0 and 0.10
Weak
Between 0.10 and 0.30
Moderate
Greater than 0.30
Strong

8. Let’s ask SPSS to calculate a few chi square based measures
Class and happiness
Ager3 and grass
Ager3 and attend
Attend and grass
Attend and happy

9. Nominal Measures: Lambda
Like Phi and Kramer’s V, Lambda is used to measure the strength of the relationship between nominal variables in bivariate tables.
Unlike Phi, Lambda is a PRE(proportional reduction of error) measure and its value has a more direct interpretation.
While Phi is only an index of strength, the value of Lambda tells us the improvement in predicting Y while taking X into account.

10. Nominal Measures: Lambda
Formula for Lambda:

11. Lambda as PRE measure
E1 = errors made in predicting the dependent variable without knowing the independent variable = N – largest row total
E2 = For each column, subtract the largest cell frequency from the col. total and add those values
This will become more clear when we look at an example

12. Association and Bivariate Tables
To compute λ, we must first find E1 and E2:
E1 = N – largest row total = 44 – 22 = 22
E2 = For each column, subtract the largest cell frequency from the col. total = (27 – 17) + (17 – 12) = = 15
Lambda = (E1-E2)/E1 = (22-15)/22 = 7/22 = .32
Low Author.
High
Author.
Totals
Low Efficiency 10 12 22
High Efficiency 17 5 27 44

13. Nominal Measures: Lambda
Lambda is a PRE measure.
A Lambda of .32 means that knowing authoritarianism (X) increases our ability to predict efficiency (Y) by 32%.

14. The Limitations of Lambda
Lambda gives an indication of the strength of the relationship only.
It does not give information about pattern.
To analyze the pattern of the relationship, use the column %s in the bivariate table.
When the mode is the same in each column of the independent variable, lambda will be zero even if a relationship exists. Thus we request both lambda and Kramer’s V/Phi.

15. Calculate lambda for this example.
Low Author.
High
Author.
Totals
Low Efficiency
2 (7%)
8 (47%)
10(23%)
High Efficiency
25 (93%)
9 (53%)
34(77%)
27(100%)
17(100%)
44(100%)