Gamma and Kendall's Tau

Gamma is calculated by counting the number of concordant pairs of cases (i.e., those that display a positive relationship between the two variables) in a contingency table, subtracting from this the number of discordant pairs and then dividing the result by the total number of pairs. Kendall's tau is calculated similarly, but the denominator is more complex, resulting in a statistic that is more "conservative" (typically lower in value) than gamma. This has to do with tied pairs (i.e., those where the pair of cases have the same score on one variable); gamma does not consider tied pairs, while tau counts them negatively. So, tau will only reach 1.0 when all of the cases in a table are on the major diagonal of the table, while gamma can reach 1.0 with cases off the major diagonal.

Kendall's tau is often reported in two variations — tau-b and tau-c. Tau-b is used for square tables (tables in which the rows and columns are equal), while tau-c is used for rectangular tables (which don't have major diagonals). When a table is square, tau-b is virtually the same as tau-c.

If we constructed a table of vote for the two major presidential candidates in the 2016 election by party identification, we would have a seven by two table since party identification has seven categories and the two-party vote has two categories. Gamma calculated for this table is 0.91 while tau-c is 0.88. These are some of the highest gamma and tau values that you might observe in survey data. They indicate a very strong relationship between party identification and the vote.