Somers' D Calculator

What is Somers' D?

Somers' d is a measure of ordinal association between two variables that takes a directional (asymmetric) approach. Unlike Goodman-Kruskal gamma, Somers' d accounts for tied pairs on the dependent variable, making it a more conservative measure. It ranges from −1 to +1, where 0 indicates no ordinal association.

Three Forms of Somers' D

CrossTabs.com calculates all three forms:

• d(Y|X): Treats X (row variable) as independent — includes ties on Y in the denominator
• d(X|Y): Treats Y (column variable) as independent — includes ties on X in the denominator
• Symmetric d: The harmonic mean of d(Y|X) and d(X|Y)

Somers' D Formula

Based on concordant (C) and discordant (D) pairs:

Where Ty = number of pairs tied on Y but not X

Relationship to Other Measures

Somers' d is closely related to Kendall's tau-b and gamma. Gamma ignores all ties, tau-b corrects for both row and column ties, and Somers' d corrects for ties on the dependent variable only. For prediction problems, Somers' d is often preferred because it penalizes ties on the outcome variable.

Frequently Asked Questions

When should I use Somers' d vs. Kendall's tau-b?

Use Somers' d when you have a clear dependent variable (e.g., predicting satisfaction from income level). Use tau-b when both variables are symmetric and neither is clearly dependent. Somers' d is more appropriate for predictive contexts.

How do I choose the direction (d(Y|X) vs d(X|Y))?

The row variable is X and the column variable is Y. Use d(Y|X) when predicting Y from X (the most common case). CrossTabs.com reports all three forms so you can choose the appropriate one for your research question.

What is the confidence interval based on?

CrossTabs.com uses a jackknife variance estimator to compute confidence intervals for Somers' d, which provides robust standard errors even for moderate sample sizes.