---

Somers’ D, short for Somers’ Delta, is a measure of the strength and direction of the association between an ordinal dependent variable and an ordinal independent variable.

An ordinal variable is one in which the values have a natural order (e.g. “bad”, “neutral”, “good”).

The value for Somers’ D ranges between -1 and 1 where:

• -1: Indicates that all pairs of the variables disagree
• 1: Indicates that all pairs of the variables agree

Somers’ D is used in practice for many nonparametric statistical methods.

Somers’ D: Definition

Given two variables, X and Y, we can say :

• Two pairs (x_i, y_i) and (x_j, y_j) are concordant if the ranks of both elements agree.
• Two pairs (x_i, y_i) and (x_j, y_j) are discordantif the ranks of both elements agree.

We can then calculate Somers’ D using the following formula:

Somers’ D = (N_C – N_D) / (N_C + N_D + N_T)

where:

• N_C: The number of concordant pairs
• N_D: The number of discordant pairs
• N_T: The number of tied pairs

The resulting value will always be between -1 and 1.

Somers’ D: Example in R

Suppose a grocery store would like to assess the relationship between the following two ordinal variables:

• The overall niceness of the cashier (ranked from 1 to 3)
• The overall satisfaction of the customer’s experience (also ranked from 1 to 3)

They collect the following ratings from a of 10 customers:

To quantify the relationship between the two variables, we can calculate Somers’ D using the following code in R:

#enter data
nice <- c(1, 1, 1, 2, 2, 2, 3, 3, 3, 3)
satisfaction <- c(2, 2, 1, 2, 3, 2, 2, 3, 3, 3)

#load DescTools package
library(DescTools)

#calculate Somers' D
SomersDelta(nice, satisfaction)

[1] 0.6896552

Somers’ D turns out to be 0.6896552.

Since this value is fairly close to 1, this indicates that there is a fairly strong positive relationship between the two variables.

This makes sense intuitively: Customers who rate the cashiers as nicer also tend to rate their overall satisfaction higher.

Additional Resources