Measures for Ordinal Data

Gamma

Gamma is an ordinal statistic which is computed by using the ordinal statistical operations of "greater than" ("GT" or ">"), "less than" ("LT" or "<"), and "equal to" ("EQ" or "<>" or "="). Using these ordinal statistics each pair of data can be classified as either tied (T), concordant (P), or discordant (Q). The formula for gamma is

g = (P - Q) / (P + Q)

where P is the total number of concordant comparisons, and Q is the total number of discordant comparisons.

Gamma is the proportion of concordant-discordant (P-Q) pairs over the total number of pairs (P+Q). It takes on a positive value if the number of concordant pairs (P) is larger than the number of discordant pairs (Q), a negative value if the number of discordant pairs is greater than the number of concordant pairs, and zero is the number of concordant pairs equals the number of discordant pairs.

Note that gamma ignores tied cases. There are three other statistics available in CROSSTABS which are based on concordant (P) and discordant (Q) data pairs. They are: Kendall's tau-b; Kendall's tau-c; and Somers' d. Each of them deal with tied data pairs (T) in different ways.

Tau c

Kendall's tau-c is conceptually similar to gamma, but it makes an adjustment for the number of rows and columns and uses the total number of cases rather than just the total number of concordant and discordant pairs as in gamma.

tau c = 2m(P-Q) / ((N*N)(m-1))

where P and Q are the number of concordant and discordant pairs, m is the number of rows or column, whichever is smallest, and N is the total number of cases.

Tau b

Kendall's Tau b is also conceptually similar to gamma, but it makes a correction for tied pairs on both the dependent variable, Ty, and the independent variable, Tx. Its formula is

tau b = P-Q/SQRT((P+Q+Tx)(P+Q+Ty))

where Tx is the number of pairs tied on X but not Y, and Ty is the number of pairs tied on Y but not X.

In our movie example the movie rating would normally be considered the dependent variable and the violence ratings the independent variable. Then tau b would be --

tau b = (360-87)/SQRT((360+87+524)(360+87+182))

      = 273/SQRT((971)(629)) 

      = 273/SQRT(610759) 

      = 273/781.5107 

      = .349

Spearman Correlation, r_s

The Spearman rank order correlation is a correlational measure that is used when both variables are ordinal. The traditional formula for calculating the Spearman rank-order correlation is

where d² is the difference between paired ranks, and n is the number of paired ranks. To use this formula you would first rank the data for each variable and then find the differences in the ranks, d, for each case.

CAUTION: Please note that the formula given above is inappropriate when there are tied ranks. So how should the Spearman rank-order correlation be computed if there are ties? The Spearman correlation is a special case of the Pearson product-moment correlation. If you compute a Pearson product-moment correlation on the ranked data the result will be the correct value of the Spearman rank order correlation.

Directional Measures for Ordinal Data

Somers' d 

Gamma, tau-b, tau-c, and Spearman's rank order correlation make no distinction between the independent and the dependent variable. Somers' d assumes that you can identify one of the variables as the dependent variable. The formula only includes ties on the dependent variable (Ty).

Somers' d = P-Q/(P+Q+Ty)

For the sake of discussion lets assume that the movie rating was the dependent variable in the example from the gamma discussion. Ty would be the number of ties on movie ratings (column ties), but not violence ties (row ties).

Somers' d = (360-87)/(360+87+182) 

          = 273/629 

          = .434