Concordance and disagreement

Concordance and discordance are labels that we apply to sets of two elements when we want to see the association relationship between both elements. The association evaluates the behavior that follows a variable given another variable.

In other words, determining the degree of association between the two variables would be to see how B behaves when it increases A. If, when it increases, A also increases B, variable A and variable B are a matching AB pair. On the contrary, when A increases and there is a decrease in B, we say that the pair AB is discordant.

The matching pairs are the pairs that are arranged in the same direction in each variable.

Discordant pairs are pairs that are arranged in the opposite direction in each variable.

Schematically:

  • Increase of A => Increase of B => Par AB is concordant.
  • Increase of B => Decrease of B => Par AB is jarring.

Applications

In economics and finance it is very important to establish the degree of association between two variables. For example, when we are evaluating the price of a financial asset and want to diversify our portfolio by decreasing Pearson’s correlation coefficient between assets.

The classic assumptions about financial assets define that their returns must be identical and independently distributed following a normal distribution. When these assumptions are not met, we cannot use Pearson’s correlation coefficient as a measure of dependency.

When we cannot apply the Pearson correlation coefficient, we can go to the classified correlations, from English, rank correlations. These classified correlations are non-parametric dependency measures based on ordered observations. The concordant and discordant pairs participate in some well-known measures such as the Spearman Rho, the Kendall Tau and the Goodman and Kruskal Gamma.

Practical example

We assume that we want to see if the skiers classify their preferences in the same order for alpine skiing or Nordic skiing at station i. Your ratings can range from 1 (very preferable) to 5 (very little preferable).

We define:

X = assessment of skiers for alpine skiing at station i.

Z = assessment of skiers for Nordic skiing at station i.

The observations obtained are:

Ski resort (i) X Z
TO one 5
B two 3
C 3 4
D 4 one
AND 5 two

We note that we have sorted the elements of column X in order to compare them with the elements of column Z. In this way, we can answer our question.

Some concordant pairs would be:

  • BC – CB: the two types of skiers have classified station B as worse for both activities compared to station C.
  • DE – ED: the two types of skiers have classified station E as best for both activities compared to station D.

Some discordant pairs would be:

  • CD – DC, AB – BA: the two types of skiers have classified the stations in opposite directions.

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