Association

The analysis of the relation (association) between variables is a fundamental task of social research. The association between two (categorical) variables can be expressed in a two-way CONTINGENCY TABLE or in cross-classification by calculating row percentages, column percentages, or total percentages. This technique can also be implemented if variables are continuous. The variables must be grouped into categories. A summary measure of the association between two variables is given by association coefficients, which measure the strength of the relation between two variables X and Y and, for ordinal, interval, and ratio variables, the direction of the association. The direction can be positive (concordant; higher values in X correspond to higher values in Y) or negative (discordant; higher values in X correspond to lower values in Y).

Two broad classes of association coefficients can be distinguished:

• Asymmetric (or directional) Measures or association coefficients assume that one of the two variables (e.g., X) can be identified as the independent variable and the other variable (e.g., Y) as the dependent variable. For example, the variable X might be gender, and variable Y might be occupational status.
• Symmetric measures or association coefficients do not require differentiation between dependent and independent variables. They can, therefore, be used if such a distinction is impossible; an example is a researcher who is interested in the relations between different leisure-time activities (doing sports and visiting the theater). However, symmetric coefficients can also be computed if identification as independent and dependent variables is possible.

Some examples of symmetric coefficients (e.g., Healey, 1995; SPSS, 2002) are the following: phi (φ); Cramér's V; the contingency coefficient C for nominal variables; Kendall's taua(τa), taub (τb), and tauc (τc); Goodman and Kruskal's gamma (γ) for ordinal variables; Spearman's rho (ρ) for rank (ordinal) data; and Pearson's r for interval or ratio variables. Both the size of the table and the number of both variables' categories determine which measure is used within one measurement level. Directed symmetric coefficients are often synonymously labeled as correlation coefficients; sometimes, the term correlation is reserved for directed symmetric associations between two continuous variables.

Examples of asymmetric coefficients (e.g., Healey, 1995; SPSS, 2002) are lambda (λ), Goodman and Kruskal's τ, the uncertainty coefficient for nominal variables, Somers's d for ordinal variables, and eta (η), if the dependent variable is interval or ratio scaled and the independent variable is nominal. Further symmetric coefficients, such as symmetric lambda, symmetric Somers, and so on, can be derived from asymmetric coefficients by means of averaging.

The quoted coefficients (except η) assume that both variables have the same measurement level. If the analyzed variables have different measurement levels, it is recommended to reduce the measurement level and then compute an appropriate association measure for the lower measurement level. Unfortunately, this is often necessary as coefficients for mixed measurement levels are not available in standard statistical software packages. Coefficients for mixed measurement levels can be derived from Daniels' generalized coefficient of correlation, for example (Daniels, 1944; Kendall, 1962, pp. 19–33).

The concept of association can also be generally applied to the multivariate case. Examples of multivariate association coefficients include the canonical correlation analysis between two sets of variables, X ={X1, X2,…, Xp} and Y ={Y1, Y2,…, Yp}; the multiple correlation coefficient R (from multiple regression) for a set of variables, X = {X1, X2,…Xp} and Y ={Y1}; and the inertia between two nominal variables, X and Y, from BIVARIATE ANALYSIS.

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