Somers' D
Somers' D (Asymmetric Ordinal Association Measure) · Also known as: Somers D, Somers' delta, d_YX, asymmetric Kendall tau, ordinal association coefficient
Somers' D is an asymmetric ordinal association coefficient, introduced by Robert H. Somers in 1962, that quantifies how well one ordinal variable predicts another by measuring the excess of concordant over discordant pairs relative to all pairs that are not tied on the designated independent variable. It is the standard companion to Kendall's tau in ordinal regression and is central to ROC curve analysis and the c-statistic in logistic regression.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use Somers' D when both the predictor and the outcome are measured on ordinal scales (or when a binary/continuous predictor is to be assessed against an ordinal outcome) and the research question is directional — that is, X is conceptually the predictor and Y is the response. Key assumptions: (1) observations are independent; (2) both variables are at least ordinal; (3) no assumption of normality or linearity is required. It is especially appropriate when paired with ordinal regression (proportional-odds model) to report an effect size, and it is algebraically equivalent to the c-statistic (area under the ROC curve) when Y is binary, making it the natural measure in logistic regression evaluation. Somers' D is not suited when the relationship is expected to be symmetric or when interval-level measurement is available and linearity holds.
Strengths & limitations
- Explicitly asymmetric: the analyst designates X as the predictor and Y as the response, which matches most causal and predictive research questions.
- Directly interpretable as a proportional reduction in prediction error for ordinal outcomes.
- Algebraically equivalent to 2(AUC − 0.5) when Y is binary, unifying ordinal association and ROC analysis.
- Distribution-free: no normality or linearity assumption; robust to outliers and non-linear monotone relationships.
- Part of the Kendall tau family, enabling straightforward comparison with tau-b and tau-c.
- Less widely implemented in software than Kendall's tau-b; some packages compute only the symmetric version by default.
- Sensitive to the number of tied pairs: heavy ties on either variable compress the coefficient toward zero and reduce discriminating power.
- Does not capture non-monotone associations; a U-shaped relationship will yield a value near zero despite a strong pattern.
- Large-sample normal approximation for inference can be inaccurate with many ties or very small samples.
Frequently asked
How does Somers' D differ from Kendall's tau-b?
Kendall's tau-b is symmetric: it divides (C − D) by the geometric mean of the pairs untied on X and the pairs untied on Y, treating both variables equivalently. Somers' D is asymmetric: d_{Y|X} divides only by pairs untied on X, acknowledging that X is the predictor. When the number of X-tied and Y-tied pairs are equal, the two statistics coincide. In all other cases Somers' D is appropriate when one variable has a designated predictive role.
Is Somers' D the same as the c-statistic (AUC)?
When the outcome Y is binary, d_{Y|X} is algebraically identical to 2(AUC − 0.5), so AUC = 0.5(1 + d_{Y|X}). The c-statistic reported after logistic regression is therefore a rescaled version of Somers' D. For multi-category ordinal outcomes the equivalence generalises naturally, making Somers' D the standard effect-size complement to ordinal regression.
Which direction should I choose — d_{Y|X} or d_{X|Y}?
Choose the direction that matches your research question. If X is the independent variable and Y is the outcome you wish to predict, use d_{Y|X}. The denominator includes all pairs that differ on X, capturing how much information X provides about Y. Using the wrong direction is a conceptual error, not just a reporting preference.
How should I handle many tied values?
Ties on the predictor X reduce the effective sample size in the denominator of d_{Y|X} and shrink the coefficient toward zero. If ties are extensive (e.g., a predictor with only three categories), the coefficient may understate the actual association. In such cases report alongside the number of tied pairs, consider Goodman-Kruskal's gamma (which excludes all tied pairs from the denominator), and note the limitation explicitly.
Sources
- Somers, R. H. (1962). A new asymmetric measure of association for ordinal variables. American Sociological Review, 27(6), 799–811. DOI: 10.2307/2090408 ↗
- Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley. ISBN: 978-0470082898
- Harrell, F. E. (2001). Regression Modeling Strategies. Springer. ISBN: 978-0387952321
How to cite this page
ScholarGate. (2026, June 3). Somers' D (Asymmetric Ordinal Association Measure). ScholarGate. https://scholargate.app/en/statistics/somers-d
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Mann-Whitney U testStatistics↔ compare