Ordinal Association: Gamma, Kendall's Tau-B and Tau-C, Somers'd
Ordinal association refers to that type of association in which the categorical variables follow a pattern, like a scale from best to worst or from most likely to least likely, etc. Ordinal association refers to the extensive study of the relationship between the different positions of the same set of items.
Gamma is a type of Ordinal association that is a symmetrical kind of measure. Gamma, in Ordinal association, varies from -1 to +1. Gamma can also be called Kruskall’s gamma. This measure of the ordinal association is based on the subtraction between the discordant pairs and the concordant pairs. This measure of Ordinal association is computed as the division between the difference of the concordant and the discordant pairs and the sum of the concordant and the discordant pairs.
This measure of the Ordinal association explains a perfect relationship under the given condition of weak monotonicity. However, under the given condition of statistical independence, the value of gamma would remain zero. The researcher should note that the value of gamma can be zero in those situations where the difference between the concordant and the discordant pairs are zero.
There is also another measure of ordinal association called Kendall’s tau- b. This measure of Ordinal association called Kendall’s tau-b is calculated as the excess of the concordant over the discordant pairs, and it is divided by the geometric mean between the number of tied pairs and untied pairs.
This measure of Ordinal association called Kendall’s tau-b explains the perfect association under the given condition of strict monotonicity. This measure of Ordinal association called the Kendall’s tau-b explains the null relationship as the statistical independence. The level of data in the measure of the Ordinal association called the Kendall’s tau-b needs the binary or ordinal level of data. This measure of Ordinal association is a non directional coefficient.
Another Ordinal association called Kendall’s tau-c is a variant of Kendall’s tau-b for larger tables. This measure of the Ordinal association is calculated as the excess of the difference between the concordant and discordant pairs times the term, which represents an adjustment for the size of the table.
This measure of the Ordinal association called Kendall's tau-c assumes different values that depend upon the size of the table. This measure of Ordinal association called Kendall's tau-c shows perfect association under the given condition of strict monotonicity. Under the given condition of statistical independence, Kendall's tau-c defines the null relationship.
This measure of the Ordinal association is a symmetrical measure. The level of data variables in this type of Ordinal association is generally of an ordinal type of level. This type of Ordinal association is often inefficient in cases of higher level data.
The value of this type of Ordinal association generally varies between the ranges of -1 to +1. Kendall's tau-c has been exclusively designed for the purpose of attaining these variations from non square tables.
There is also another type of Ordinal association called Somer’s d. Somer’s d reaches the maximum value of 1 in cases of positive relationships, and the minimum value of -1 in cases of negative relationships under the condition of strict monotonicity. The null relationship in Somer’s d is defined as statistical independence. Somer’s d is used only with the ordinal type of data level. In the case of two by two tables, Somer’s d is equivalent to the percent differences measured in dichotomous association. Somer’s d is an asymmetric kind of measure of ordinal association.
