# Spearman correlation coefficient¶

Spearman’s correlation coefficient aims to measure the strength of a monotonic relationship between two random variables and . It is in fact equivalent to the Pearson’s correlation coefficient after having transformed and to linearize any monotonic relationship (remember that Pearson’s correlation coefficient may only be used to measure the strength of linear relationships, see Pearson’s correlation coefficient):

where and denote the cumulative distribution functions of and .

If we arrange a sample made up of pairs , the estimation of Spearman’s correlation coefficient first of all requires a ranking to produce two samples and . The ranking of the observation is defined as the position of in the sample reordered in ascending order: if is the smallest value in the sample , its ranking would equal 1; if is the second smallest value in the sample, its ranking would equal 2, and so forth. The ranking transformation is a procedure that takes the sample ) as input data and produces the sample as an output result.

For example, let us consider the sample . We therefore have . is in fact the second smallest value in the original, the smallest, etc.

The estimation of Spearman’s correlation coefficient is therefore equal to Pearson’s coefficient estimated with the aid of the pairs , , …, :

where and represent the empirical means of the samples and .

The Spearman’s correlation coefficient takes values between -1 and 1. The closer its absolute value is to 1, the stronger the indication is that a monotonic relationship exists between variables and . The sign of Spearman’s coefficient indicates if the two variables increase or decrease in the same direction (positive coefficient) or in opposite directions (negative coefficient). We note that a correlation coefficient equal to 0 does not necessarily imply the independence of variables and . There are two possible situations in the event of a zero Spearman’s correlation coefficient:

the variables and are in fact independent,

or a non-monotonic relationship exists between and .

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Spearman’s coefficient is often referred to as the rank correlation coefficient.