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

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?

“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)

“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

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

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

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)

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

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

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

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)

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”

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