Spearman's Rank Correlation Statistic

A Spearman's Rank Correlation Statistic is a non-parametric ordinal rank correlation statistic that can measure monotonic relationships between ranked variables.

• AKA: Spearman's Rho, Spearman Correlation Coefficient, Spearman Rank Correlation Coefficient, Spearman R.
• Context:
  • It can typically compute Rank Correlation Values between ranked observations using rank differences.
  • It can typically assess Monotonic Association Strength between ordinal variables without requiring normal distribution assumptions.
  • It can typically utilize Rank-Based Calculations through the formula ρ = 1 - (6 Σd²ᵢ)/(n(n²-1)) where dᵢ represents rank differences.
  • It can typically detect Non-Linear Monotonic Relationships that parametric correlation measures might miss.
  • It can typically provide Correlation Coefficient Values ranging from -1 to +1 for inverse relationships to direct relationships.
  • ...
  • It can often serve as Distribution-Free Alternatives to Pearson correlation coefficients for non-normal data.
  • It can often enable Ordinal Data Analysis in behavioral research, social sciences, and educational assessments.
  • It can often support Outlier-Robust Analysis through rank transformations of continuous data.
  • It can often facilitate Statistical Hypothesis Testing through Spearman correlation tests with significance levels.
  • ...
  • It can range from being a Simple Rank Correlation Statistic to being a Complex Rank Correlation Statistic, depending on its rank correlation computational complexity.
  • It can range from being a Small-Sample Rank Correlation Statistic to being a Large-Sample Rank Correlation Statistic, depending on its rank correlation sample size.
  • ...
  • It can integrate with Statistical Software Packages for automated rank correlation calculations.
  • It can support Fisher's Z Transformation for confidence interval estimations.
  • It can enable Rank-Based Inferences through asymptotic normal distributions.
  • It can complement Kendall's Tau Statistics in non-parametric correlation analysis.
  • ...
• Example(s):
  • Educational Rank Correlation Applications, such as:
    • Student Performance Rank Correlation measuring grade rankings versus test score rankings.
    • Academic Achievement Rank Correlation assessing study time ranks against achievement ranks.
  • Social Science Rank Correlation Applications, such as:
    • Survey Response Rank Correlation analyzing ordinal survey responses.
    • Preference Ranking Correlation evaluating consumer preference rankings.
  • Biomedical Rank Correlation Applications, such as:
    • Clinical Rating Scale Correlation comparing physician rankings with patient outcome rankings.
    • Treatment Effectiveness Rank Correlation assessing treatment ranks versus recovery time ranks.
  • ...
• Counter-Example(s):
  • Kendall's Tau Rank Correlation, which uses concordant pairs and discordant pairs rather than rank differences.
  • Pearson Product-Moment Correlation Coefficient, which requires interval data and linear relationship assumptions.
  • Point-Biserial Correlation, which measures associations between dichotomous variables and continuous variables.
• See: Spearman Correlation Test, Rank Correlation Statistic, Non-Parametric Statistics, Correlation and Dependence, Ordinal Scale Measurement.

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References

2017a

• (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).

2017b

• (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.

2017c

• (Stat 509, 2017) ⇒ Design and Analysis of Clinical Trials, The Pennsylvania State University 18.2 - Spearman Correlation Coefficient https://onlinecourses.science.psu.edu/stat509/node/157
  • The Spearman rank correlation coefficient, [math]\displaystyle{ r_s }[/math] , is a nonparametric measure of correlation based on data ranks. It is obtained by ranking the values of the two variables ([math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]) and calculating the Pearson [math]\displaystyle{ r_p }[/math] on the resulting ranks, not the data itself. Again, PROC CORR will do all of these actual calculations for you.

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

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Spearman's_rank_correlation_coefficient Retrieved:2015-8-20.
  • 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 statistical dependence between two variables. It assesses how well the relationship between two variables can be described using a monotonic function. 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.

    Spearman's coefficient, like any correlation calculation, is appropriate for both continuous and discrete variables, including ordinal variables. 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.

2011

• (Sammut & Webb, 2011) ⇒ Claude Sammut, and Geoffrey I. Webb. (2011). “Rank Correlation.” In: (Sammut & Webb, 2011) p.828
  • QUOTE: Spearman’s Rank correlation calculates the sum of squared rank distances and is normalized such that it evaluates to − 1 for reversed and to + 1 for identical rankings. Formally, it is defined as follows: [math]\displaystyle{ (τ,τ′)↦ 1 − \frac{6∑^m_{i=1}(τ(i)−τ′(i))}{2m(m^2−1)} \ (1) }[/math]

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