Measures of association for Ordinal Logistic Regression

Find definitions and interpretation guidance for every statistic in the Measures of association table.

In This Topic

Pairs
Somers' D
Goodman-Kruskal Gamma
Kendall's Tau-a

Pairs

For ordinal logistic regression, Minitab calculates the cumulative probabilities of each observation and compares these values for each pair of observations. These categories describe the pairs for a response with values 1, 2, and 3:

Concordant: For pairs that include the response value of 1, a pair is concordant if the cumulative probability for the response value of 1 is greater for the observation with the response value of 1 than for the observation with the response values of 2 or 3. For pairs with response values 2 and 3, a pair is concordant if the cumulative probability for response up to 2 is greater for the observation with the response value 2 than for the observation with the response value 3.
Discordant: For pairs that include the response value 1, a pair is discordant if the cumulative probability for the response value 1 is greater for the observation with response value 2 or 3. For pairs with response values 2 and 3, a pair is discordant if the cumulative probability for responses up to 2 is greater for the observation with the response value 3 than for the observation with the response value 2.
Tie: Please see my comment in BLRA pair is tied if the observations have equal cumulative probabilities.

Interpretation

Use the numbers of pairs to compare the predictive performance of models. The higher the percentage of concordant pairs, the better the model performs.

Somers' D

Somers' D is the proportion difference between concordant and discordant pairs, including ties.

Interpretation

Use Somers' D to compare the predictive performance of models. Higher values indicate better predictive performance. For example, if 75% of the pairs are concordant and 25% are discordant, then Somers' D is 0.5.

Somers' D and the Goodman-Kruskal Gamma statistic are identical when the model predicts 0 tied pairs. The more tied pairs, the more the Goodman-Kruskal Gamma statistic exceeds Somers' D.

Goodman-Kruskal Gamma

Goodman-Kruskal Gamma is the proportion difference between concordant and discordant pairs, excluding ties.

Interpretation

Use the Goodman-Kruskal Gamma to compare the predictive performance of models. Higher values indicate better predictive performance. For example, if 75% of the non-tied pairs are concordant and 25% are discordant, the Goodman-Kruskal Gamma is 0.5.

Somers' D and the Goodman-Kruskal Gamma statistic are identical when the model predicts 0 tied pairs. The more tied pairs, the more the Goodman-Kruskal Gamma statistic exceeds Somers' D.

Kendall's Tau-a

Kendall's Tau-a is the proportion difference of concordant and discordant pairs out of all possible pairs, including pairs with the same response value.

Interpretation

Use Kendall's Tau-a to compare the predictive performance of models. Higher values indicate better predictive performance. Kendall's Tau-a is always lower than Somers' D and the Goodman-Kruskal Gamma statistic because those two statistics do not include pairs with the same response value.