Spearman Correlation Test

A Spearman Correlation Test is a non-parametric correlational hypothesis test that is based on a Spearman's rank correlation statistic.

• AKA: Spearman's Rank Correlation Test.
• Context:
  • It is a non-parametric analog to the Pearson Correlation Test.
  • It assumes that the following conditions are met:
    • Each pair of data are interval or ratio level or ordinal; linearly and monotonically related.
  • It tests the following hypotheses:
    • Null hypothesis: There no correlation present in population , i.e [math]\displaystyle{ H_0:\;\;\rho_s= 0 }[/math]
    • Alternative hypothesis: There is correlation present in the population , i.e [math]\displaystyle{ H_A: \;\;\rho_s \ne 0 }[/math]

where [math]\displaystyle{ -1 \lt \rho_s \lt 1 }[/math] is the Spearman's rank correlation statistic calculated for each observation.

• • …
• Counter-Example(s):
  • Mantel Correlation Test.
  • Pearson Correlation Test.
  • Kendall Tau Correlation Test.
  • Gamma Correlation Test
• See: Correlational Hypothesis Test, Correlation, Autocorrelation, Cointegration.

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References

2017a

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Spearman's_rank_correlation_coefficient
  • In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter [math]\displaystyle{ \rho }[/math] (rho) or as [math]\displaystyle{ r_s }[/math], is a nonparametric measure of rank correlation (statistical dependence between the ranking of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.

The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.

Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a correlation of -1) rank between the two variables.

Spearman's coefficient is appropriate for both continuous and discrete variables, including ordinal variables. Both Spearman's [math]\displaystyle{ \rho }[/math] and Kendall's [math]\displaystyle{ \tau }[/math] can be formulated as special cases of a more general correlation coefficient.

2017b

• (CM, 2017) ⇒ http://changingminds.org/explanations/research/analysis/spearman.htm
  • The Spearman Rank Correlation Coefficient is a form of the Pearson coefficient with the data converted to rankings (ie. when variables are ordinal). It can be used when there is non-parametric data and hence Pearson cannot be used.

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).

2017c

• (Quest Software Inc., 2017) ⇒ Statistics – Textbook, Nonparametric Statistics https://documents.software.dell.com/statistics/textbook/nonparametric-statistics#correlations
  • Spearman R (Siegel & Castellan, 1988) assumes that the variables under consideration were measured on at least an ordinal (rank order) scale, that is, that the individual observations can be ranked into two ordered series. Spearman R can be thought of as the regular Pearson product moment correlation coefficient, that is, in terms of proportion of variability accounted for, except that Spearman R is computed from ranks.

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