Somers' D Calculator

Somers' D for predicting Y from X is defined as:

$$d_{Y|X} = \frac{C - D}{C + D + T_Y}$$

Where:

• C — number of concordant pairs (both variables move in the same direction)
• D — number of discordant pairs (variables move in opposite directions)
• T_Y — number of pairs tied on the dependent variable Y but not on X

Compared to Gamma (γ = (C−D)/(C+D)), Somers' D adds T_Y to the denominator. This means D is always smaller in magnitude than Gamma (or equal when there are no ties on Y). The relationship is:

$$d_{Y|X} = \gamma \cdot \frac{C + D}{C + D + T_Y}$$

This formula shows that Somers' D equals Gamma multiplied by the proportion of untied pairs among all non-X-tied pairs. When ties on Y are rare, D approximates Gamma; when ties on Y are common, D can be substantially smaller.

For the reverse direction:

$$d_{X|Y} = \frac{C - D}{C + D + T_X}$$

where T_X are pairs tied on X but not Y. The symmetric version averages both: d_sym = (d_{Y|X} + d_{X|Y}) / 2, which equals Kendall's Tau-b when there are no ties on both variables simultaneously.

Somers' D ranges from −1 to +1 and is interpreted as a measure of predictive association:

• |D| < 0.10: Negligible association
• 0.10–0.30: Weak predictive association
• 0.30–0.50: Moderate predictive association
• 0.50–0.70: Strong predictive association
• |D| > 0.70: Very strong predictive association

Because Somers' D accounts for ties on the dependent variable, it provides a more honest assessment of predictive power than Gamma, which can overstate association by ignoring ties. Comparing D to Gamma for the same data reveals how much the ties inflate the apparent association.