1. 1. Nominal Measures of Association 2. Ordinal Measure s of Association

2. ASSOCIATION Association
The strength of relationship between 2 variables
Knowing how much variables are related may enable you to predict the value of 1 variable when you know the value of another
As with test statistics, the proper measure of association depends on how variables are measured

3. Significance vs. Association
Association = strength of relationship
Test statistics = how different findings are from null
They do capture the strength of a relationship
t = number of standard errors that separate means
Chi-Square = how different our findings are from what is expected under null
If null is no relationship, then higher Chi-square values indicate stronger relationships.
HOWEVER --- test statistics are also influenced by other stuff (e.g., sample size)

4. MEASURES OF ASSOCIATION FOR NOMINAL-LEVEL VARIABLES
“Chi-Square Based” Measures
2 indicates how different our findings are from what is expected under null
2 also gets larger with higher sample size (more confidence in larger samples)
To get a “pure” measure of strength, you have to remove influence of N
Phi
Cramer's V

5. PHI Phi (Φ) = 2 √ N Formula standardizes 2 value by sample size
Measure ranges from 0 (no relationship) to values considerably >1
(Exception: for a 2x2 bivariate table, upper limit of Φ= 1)

6. PHI Example: 2 x 2 table LIMITATION OF Φ: 2=5.28
Lack of clear upper limit makes Φ an undesirable measure of association

7. CRAMER’S V
Cramer’s V = 2
√ (N)(Minimum of r-1, c-1)
Unlike Φ, Cramer’s V will always have an upper limit of 1, regardless of # of cells in table
For 2x2 table, Φ & Cramer’s V will have the same value
Cramer’s V ranges from 0 (no relationship) to +1 (perfect relationship)

8. 2-BASED MEASURES OF ASSOCIATION
Sample problem 1:
The chi square for a 5 x 3 bivariate table examining the relationship between area of Duluth one lives in & type of movie preference is 8.42, significant at .05 (N=100). Calculate & interpret Cramer’s V.
ANSWER:
(Minimum of r-1, c-1) = 3-1 = 2
Cramer’s V = .21
Interpretation: There is a relatively weak association between area of the city lived in and movie preference.
Cogdon
Lincoln Park
Observation Hill
Lakeside
Chester Park
Comedy
Drama
Action

9. 2-BASED MEASURES OF ASSOCIATION
Sample problem 2:
The chi square for a 4 x 4 bivariate table examining the relationship between type of vehicle driven & political affiliation is 12.32, sig. at .05 (N=300). Calculate & interpret Cramer’s V.
ANSWER:
(Minimum of r-1, c-1) = 4 -1 = 3
Cramer’s V = .12
Interpretation: There is a very weak association between type of vehicle driven & political affiliation.
Compact
Truck
SUV
Luxury

10. SUMMARY: 2 -BASED MEASURES OF ASSOCIATION
Limitation of Φ & Cramer’s V:
No direct or meaningful interpretation for values between 0-1
Both measure relative strength (e.g., .80 is stronger association than .40), but have no substantive meaning; hard to interpret
“Rules of Thumb” for what is a weak, moderate, or strong relationship vary across disciplines
AGAIN, THESE ARE FOR NOMINAL-LEVEL VARIABLES ONLY.

11. LAMBDA (λ)
PRE (Proportional Reduction in Error) is the logic that underlies the definition & computation of lambda
Tells us the reduction in error we gain by using the IV to predict the DV
Range 0-1 (i.e., “proportional” reduction)
E1 – Attempt to predict the category into which each case will fall on DV or “Y” while ignoring IV or “X”
E2 – Predict the category of each case on Y while taking X into account
The stronger the association between the variables the greater the reduction in errors

12. LAMBDA: EXAMPLE 1 Risk Classification Re-arrested Low Medium High
Does risk classification in prison affect the likelihood of being rearrested after release? (2=43.7)
Risk Classification
Re-arrested
Low
Medium
High
Total
Yes 25 20 75
120 No 50 15 85 40 90
205

13. LAMBDA: EXAMPLE Find E1 (# of errors made when ignoring X)
E1 = N – (largest row total)
= = 85
Risk Classification
Re-arrested
Low
Medium
High
Total
Yes 25 20 75
120 No 50 15 85 40 90
205

14. LAMBDA: EXAMPLE Find E2 (# of errors made when accounting for X)
E2 = (each column’s total – largest N in column)
= (75-50) + (40-20) + (90-75) = = 60
Risk Classification
Re-arrested
Low
Medium
High
Total
Yes 25 20 75
120 No 50 15 85 40 90
205

15. CALCULATING LAMBDA: EXAMPLE
Calculate Lambda
λ = E1 – E2 = = 25 = 0.294 E
Interpretation – when multiplied by 100, λ indicates the % reduction in error achieved by using X to predict Y, rather than predicting Y “blind” (without X)
0.294 x 100 = 29.4% - “Knowledge of risk classification in prison improves our ability to predict rearrest by 29%.”

16. LAMBDA: EXAMPLE 2
What is the strength of the relationship between citizens’ race and attitude toward police?
(obtained chi square is > (2[critical])
Calculate & interpret lambda to answer this question
Attitude
towards police
Race
Totals
Black
White
Other
Positive 40
150 35
225
Negative 80 95 55
230
120
245 90
455

17. LAMBDA: EXAMPLE 2 E1 = N – (largest row total) 455 – 230 = 225
E2 = (each column’s total – largest N in column)
(120 – 80) + (245 – 150) + (90 – 55) =
= 170
λ = E1 – E2 = = = 0.244 E
INTERPRETATION:
x 100 = 24.4% - “Knowledge of an individual’s race improves our ability to predict attitude towards police by 24%”
Attitude
towards police
Race
Totals
Black
White
Other
Positive 40
150 35
225
Negative 80 95 55
230
120
245 90
455

18. SPSS EXAMPLE
IS THERE A SIGNIFICANT RELATIONSHIP B/T GENDER & VOTING BEHAVIOR?
If so, what is the strength of association between these variables?
ANSWER TO Q1: “YES”

19. SPSS EXAMPLE ANSWER TO QUESTION 2:
By either measure, the association between these variables appears to be weak

20. 2 LIMITATIONS OF LAMBDA 1. Asymmetric
Value of the statistic will vary depending on which variable is taken as independent
2. Misleading when one of the row totals is much larger than the other(s)
For this reason, when row totals are extremely uneven, use a chi square-based measure instead

21. ORDINAL MEASURE OF ASSOCIATION
GAMMA
For examining STRENGTH & DIRECTION of “collapsed” ordinal variables (<6 categories)
Like Lambda, a PRE-based measure
Range is -1.0 to +1.0

22. GAMMA Logic: Applying PRE to PAIRS of individuals Prejudice
Lower Class
Middle Class
Upper Class
Low
Kenny
Tim
Kim
Middle
Joey
Deb
Ross
High
Randy
Eric
Barb

23. GAMMA CONSIDER KENNY-DEB PAIR
In the language of Gamma, this is a “same” pair
direction of difference on 1 variable is the same as direction on the other
If you focused on the Kenny-Eric pair, you would come to the same conclusion
Prejudice
Lower Class
Middle Class
Upper Class
Low
Kenny
Tim
Kim
Middle
Joey
Deb
Ross
High
Randy
Eric
Barb

24. GAMMA NOW LOOK AT THE TIM-JOEY PAIR
In the language of Gamma, this is a “different” pair
direction of difference on one variable is opposite of the difference on the other
Prejudice
Lower Class
Middle Class
Upper Class
Low
Kenny
Tim
Kim
Middle
Joey
Deb
Ross
High
Randy
Eric
Barb

25. GAMMA Logic: Applying PRE to PAIRS of individuals Formula:
same – different
same + different

26. GAMMA Prejudice Lower Class Middle Class Upper Class Low Kenny Tim Kim
If you were to account for all the pairs in this table, you would find that there were 9 “same” & 9 “different” pairs
Applying the Gamma formula, we would get:
9 – 9 = = 0.0
Prejudice
Lower Class
Middle Class
Upper Class
Low
Kenny
Tim
Kim
Middle
Joey
Deb
Ross
High
Randy
Eric
Barb
Meaning: knowing how 2 people differ on social class would not improve your guesses as to how they differ on prejudice.
In other words, knowing social class doesn’t provide you with any info on a person’s level of prejudice.

27. GAMMA 3-case example Applying the Gamma formula, we would get:
3 – 0 = = 1.00
Prejudice
Lower Class
Middle Class
Upper Class
Low
Kenny
Middle
Deb
High
Barb

28. Gamma: Example 1 Examining the relationship between:
FEHELP (“Wife should help husband’s career first”) &
FEFAM (“Better for man to work, women to tend home”)
Both variables are ordinal, coded 1 (strongly agree) to 4 (strongly disagree)

29. Gamma: Example 1
Based on the info in this table, does there seem to be a relationship between these factors?
Does there seem to be a positive or negative relationship between them?
Does this appear to be a strong or weak relationship?

30. GAMMA
Do we reject the null hypothesis of independence between these 2 variables?
Yes, the Pearson chi square p value (.000) is < alpha (.05)
It’s worthwhile to look at gamma.
Interpretation:
There is a strong positive relationship between these factors.
Knowing someone’s view on a wife’s “first priority” improves our ability to predict whether they agree that women should tend home by 75.5%.

31. USING GSS DATA
Construct a contingency table using two ordinal level variables
Are the two variables significantly related?
How strong is the relationship?
What direction is the relationship?