Spearman's Rank Correlation Statistic

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A Spearman's Rank Correlation Statistic is a non-parametric rank correlation between the ranking of two variables.



References

2017a

The raw scores are converted to ranks and the differences ([math]\displaystyle{ d_i }[/math]) between the ranks of each observation on the two variables are calculated. The Spearman coefficient is denoted with the Greek letter rho ([math]\displaystyle{ \rho }[/math]).
[math]\displaystyle{ \rho = 1 - (6 * SUM(d_i^2)) / (n * (n^2 - 1)) }[/math]
(...) The Spearman Coefficient can be used to measure ordinal data (ie. in rank order), not interval (as Pearson). It effectively works by first ranking the data then applying Pearson's calculation to the rank numbers.
This coefficient is also called Spearman's rho (after the Greek letter used).

2017b

2017c

The Spearman rank correlation coefficient has properties similar to those of the Pearson correlation coefficient, although the Spearman rank correlation coefficient quantifies the degree of linear association between the ranks of [math]\displaystyle{ X }[/math] and the ranks of [math]\displaystyle{ Y }[/math]. Also, [math]\displaystyle{ r_s }[/math] does not estimate a natural population parameter (unlike Pearson's [math]\displaystyle{ r_p }[/math] which estimates [math]\displaystyle{ \rho_p }[/math] ).
An advantage of the Spearman rank correlation coefficient is that the [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] values can be continuous or ordinal, and approximate normal distributions for [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are not required. Similar to the Pearson [math]\displaystyle{ r_p }[/math] , Fisher's Z transformation can be applied to the Spearman [math]\displaystyle{ r_s }[/math] to get a statistic, [math]\displaystyle{ z_s }[/math], that has an asymptotic normal distribution for calculating an asymptotic confidence interval.

2015

2011