Nonparametrics Statistics Notes - Correlations (Spearman, Kendall tau, Gamma)

You can compute three different alternatives to the parametric Pearson product-moment correlation coefficient: Spearman rank R, Kendall Tau, and Gamma. Select Correlations (Spearman, Kendall tau, gamma) from the Nonparametric Statistics Startup Panel - Quick tab to display the Nonparametric Correlations dialog box, from which you select variables and the specific type of correlation to be computed (see Nonparametric Correlations). You can choose to compute single nonparametric correlations or matrices of nonparametric correlations. A useful Graph of Input Data is the scatterplot with which you can easily graph the correlations. A scatterplot matrix for all variables can also be produced. Note that alternative nonparametric measures of contingency between two variables are also available in Basic Statistics.

Spearman rank R
Spearman rank R can be thought of as the regular Pearson product-moment correlation coefficient (Pearson r); that is, in terms of the proportion of variability accounted for, except that Spearman R is computed from ranks. Spearman R assumes that the variables under consideration were measured on at least an ordinal (rank order) scale; that is, the individual observations (cases) can be ranked into two ordered series. Detailed discussions of the Spearman R statistic and its power and efficiency can be found in Gibbons (1985), Hays (1981), McNemar (1969), Siegel and Castellan (1988), Kendall (1948), Olds (1949), or Hotelling and Pabst (1936).
Kendall Tau
Kendall Tau is equivalent to Spearman R with regard to the underlying assumptions. It is also comparable in terms of its Statistical power. However, Spearman R and Kendall Tau are usually not identical in magnitude because their underlying logic as well as their computational formulas are very different. Siegel and Castellan (1988) express the relationship of the two measures in terms of the inequality:

-1 <= 3 * Kendall tau - 2 * Spearman R <= 1

More importantly, Kendall Tau and Spearman R imply different interpretations: Spearman R can be thought of as the regular Pearson product-moment correlation coefficient; that is, in terms of proportion of variability accounted for, except that Spearman R is computed from ranks. Kendall Tau, on the other hand, represents a probability; that is, it is the difference between the probability that the two variables are in the same order in the observed data versus the probability that the two variables are in different orders.

There are two p-values that are reported for testing the significance of the Kendall’s Tau statistic. The first is based on a large sample normal approximation. The test statistic, Z, is computed as:

where T is Kendall’s Tau statistic and N is the total number of observations. The p-value is then computed using the standard normal distribution.

The second p-value reported is referred to as the exact 1-sided p-value and is only computed for n > 4 and n <= 10. The p-value is computed based on the exact frequency distribution of all possible values of Kendall’s Tau under the null hypothesis with the additional assumption that there are no ties present in the data.

Gamma
The Gamma statistic is preferable to Spearman R or Kendall Tau when the data contain many tied observations. In terms of the underlying assumptions, Gamma is equivalent to Spearman R or Kendall Tau; in terms of its interpretation and computation, it is more similar to Kendall Tau than Spearman R. In short, Gamma is also a probability; specifically, it is computed as the difference between the probability that the rank ordering of the two variables agree minus the probability that they disagree, divided by 1 minus the probability of ties. Thus, Gamma is basically equivalent to Kendall Tau, except that ties are explicitly taken into account.

For more information, see Correlation and Nonparametric Correlations - Advanced tab.