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 (xi, yi) and (xj, yj) are concordant if the ranks of both elements agree.
- Two pairs (xi, yi) and (xj, yj) are discordant if the ranks of both elements agree.
We can then calculate Somers’ D using the following formula:
Somers’ D = (NC – ND) / (NC + ND + NT)
- NC: The number of concordant pairs
- ND: The number of discordant pairs
- NT: 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 sample 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)  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.