Concordant pairs and discordant pairs

Used to describe the relationship between sets of paired observations. A pair is concordant if the pair of observations are in the same direction. A pair is discordant if the pair of observations are in opposite directions.

So, if you have two sets of paired observations - (Xi, Yi) and (Xj, Yj):

·    The pairs are concordant if Xi > Xj and Yi > Yj or Xi < Xj and Yi < Yj   

·    The pairs are discordant if Xi > Xj and Yi < Yj or Xi < Xj and Yi > Yj   

For example suppose you and a friend are rating pizza from various local restaurants to decide which to recommend to other friends. You rate 5 different pizzas on a scale from best to worse (1 to 5).

Arranging your rankings in order helps to determine which pairs are concordant and which are discordant. .

Looking at the your first choice, Pizza B - your friend gave it a 2 while you rated it a 1. Compare your friend's 2 rating with your other ratings in order. If your rating are higher, you say the pair is concordant. If your ratings are not higher, you say the pair is discordant.

(X1, Y1) and (X2, Y2) = (1, 2) and (2, 3);  1 < 2 and 2 < 3 so the pair is concordant.

(X1, Y1) and (X3, Y3) = (1, 2) and (3, 1);  1 < 3 and 2 > 1 so the pair is discordant.

(X1, Y1) and (X4, Y4) = (1, 2) and (4, 5);  1 < 4 and 2 < 5 so the pair is concordant.

(X1, Y1) and (X5, Y5) = (1, 2) and (5, 4);  1 < 5 and 2 < 4 so the pair is concordant.

Now moving on to compare the next sets:

(X2, Y2) and (X3, Y3) = (2, 3) and (3, 1);  2 < 3 and 3 > 1 so the pair is discordant.

(X2, Y2) and (X4, Y4) = (2, 3) and (4, 5);  2 < 4 and 3 < 5 so the pair is concordant.

(X2, Y2) and (X5, Y5) = (2, 3) and (5, 4);  2 < 5 and 3 < 4 so the pair is concordant.

(X3, Y3) and (X4, Y4) = (3, 1) and (4, 5);  3 < 4 and 1 > 5 so the pair is discordant.

(X3, Y3) and (X5, Y5) = (3, 1) and (5, 4);  3 < 5 and 1 > 4 so the pair is discordant.

(X4, Y4) and (X5, Y5) = (4, 5) and (5, 4);  4 < 5 and 5 > 4 so the pair is discordant.

There are 5 concordant pairs and 5 discordant pairs.

The number of concordant and discordant pairs are used in calculations for Kendall's tau.