Asymmetric Measures

Available Coefficients

Most statistical software packages, such as SPSS, offer asymmetric coefficients only for special combinations of measurement levels. These coefficients are described first.

Measures for Nominal-Nominal Tables

Measures for nominal variables are lambda (λ), Goodman and Kruskal's tau (τ), and the uncertainty coefficient U. All measures are based on the definition of PROPORTIONAL REDUCTION OF ERROR (PRE):

where E0 is the error in the dependent variable Y without using the independent variable X to predict or explain Y, and E1 is the error in Y if X is used to predict or explain Y.

Both τ and U use definitions of variance to compute E0 and E1. Similar to the well-known eta (η) from the ANALYSIS OF VARIANCE (see below), E0 is the total variance of Y, and E1 is the residual variance (variance within the categories of X). For τ, the so-called Gini concentration is used to measure the variation (Agresti, 1990, pp. 24–25), and U applies the concept of entropy. For λ, a different definition is used: It analyzes to what extent the dependent variable can be predicted both without knowing X and with knowledge of X.

All three coefficients vary between 0 (no reduction in error) and 1 (perfect reduction in error). Even if a clear association exists, λ may be zero or near zero. This is the case if one category of the dependent variable is dominant, so that the same category of Y has the highest frequencies and becomes the best predictor within each condition X.

Measure for Ordinal-Ordinal Tables

The most prominent measure for this pattern is Somers' d. Somers' d is defined as the least squares regression slope (dy/x = sxy/s2x) between the dependent variable Y and the independent variable X, if both variables are treated as ordinal (Agresti, 1984, pp. 161–163), and Daniels' formula for generalized correlation coefficient (see Association) is used to compute the variance s2x of X and the covariance sxy between X and Y. Somers' d is not a PRE coefficient. The corresponding PRE coefficient is the symmetric Kendall's τ2b, which can be interpreted as explained variance and is equal to the geometric mean if Somers' d is computed for X and Y as dependent variables (= dy/x and dx/y). If both variables are dichotomous, Somers' d is equal to the difference of proportions (Agresti, 1984, p. 161).

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