1. Contingency Tables – Part II – Getting Past Chi-Square?

2. Measures of Association – A Review
What is the difference between a significance test statistic and a measure of association?
How are they related?
The basic questions about associations between variables?
Does an association exist (vs. independence)?
What is form (& direction) of the association?
What is the strength (“size”) of the association?

3. “Strength of Association
What does “association” mean?
Shared or common elements
Degree of agreement
Predictability (reduction in errors/ignorance)
Characteristics of association measures?
Coefficient should range between -1 & +1
Coefficient should not be directly affected by N
Coefficient should be independent of a variable’s scale of measurement (its “metric”)
Coefficient values should be interpretable (intuitively or methematically)

4. “Strength of Association (cont.)
A number of different measures of association (coefficients) are available:
Based on different levels of measurement
Based on different analytical models
How to choose among them?
Identify levels-of-measurement of both variables
Identify if you have a clear independent variable  use a directional or a nondirectional coefficient
Identify which coefficients are most commonly used

5. Measurement Level Situations:
Association between 2 numerical variables?
Coefficient = Pearson’s r
r2 = proportion of variance “in common”
May use Spearman’s r if data are ranks
Association between 1 categoric and 1 numeric variable? (as in ANOVA)
Coefficient of Association = eta (ή)
eta-squared = proportion of variance “between groups”
In SPSS, use Descriptives  Cross-tabs or Compare Means  Means procedures

6. Association between 2 categoric variables
Different approaches to nonparametric measures of association
Chi-square-based
 Correct for degrees of freedom and sample size
Uncertainty/Errors of Prediction
 Predictability of Y given knowledge of X
Concordance/agreement
 Proportion of shared or correspondent values
Note: coefficients for Ordinal and Nominal variables are different
 Coeff. limited to the lower measurement level

7. Strength of Association (continued)
Association between 2 Nominal variables (or 1 nominal + 1 ordinal variable)
Chi-square-derived:
Contingency coefficient, C
Cramer’s V coefficient  use this for 3x3 or larger tables
Phi coefficient, Φ  use this for 2x2 tables (or 2x3 tables)
PRE-derived :
Lambda (asymmetric) (λyx <> λxy)

8. Strength of Association (continued)
Association between 2 Ordinal variables
Concordance-based (PRE) statistics:
Gamma, γ  most commonly used (note: in cases of 2x2 tables, gamma = Yules Q)
Others? Kendall’s tau; Somer’s d (less used)
Rank-order statistics:
Spearman’s Rho , 
Use if many categories & few ties
Must convert scores to ranks
Can also use Chi-square-based measures
Computing Phi as ordered coefficient

9. Nonparametric Measures of Association: Summary Recap
Nominal variables
Phi, Φ for 2x2 tables (or 2x3)
Kramer’s V for 3x3 tables or larger
Ordinal variables
Gamma, γ  most commonly used
Yules Q  same statistic in a 2x2 table
Spearman’s r if many values & few ties
Can also use Phi and Kramer’s V

10. Nonparametric Measures of Association: Summary (continued)
Different kinds of coefficients will not yield the same values on the same crosstab
Gamma (& Yules Q) will almost always compute higher values than Kramer’s V (& Phi) on the same tables
Note that 2x2 tables (with binary variables) are somewhat of a special case

11. Non-Parametric measures of association
How to Compute them?
By Hand: see formulas in the textbook
Chi-square-based = easiest to compute
Gamma = more laborious by hand
Note: X & Y variables in crosstab must be formatted in the same direction for ordinal statistics (e.g., Gamma)
In SPSS: Click Statistics box in Crosstabs pop-up menu, then select appropriate coefficients (Note: do not select them all)

14. II. Multivariate analysis of associations
Going beyond bivariate analysis to multivariate analyses
We often wish to consider more than two variables at a time because other variables may be involved in more complex patterns
Termed “Partialling” or “Elaborating”  statistically consider:
confounding effects of additional variables  “spurious relationships”
Complicating effects of additional variables  “contingent relationships”

15. Multivariate Analysis (continued)
In cross-tabulations, crosstabs are “nested within levels of other variables
Compute separate sub-crosstabs within each category or level of the 3rd variable
See the example on the handout
Partialing is only useful when the extra variable is associated with both X and Y
Then we wish to remove the extra covariation
Otherwise, it’s a waste of time