Rank Correlation Statistic

A Rank Correlation Statistic is a non-parametric test statistic that measures the statistical relationship between rankings of different ordinal variables or different rankings of the same variable.

• AKA: Rank Correlation Coefficient, Rank Correlation Measure, Ordinal Association Measure.
• Context:
  • It can typically measure Ordinal Association Strength between ranked observations through rank-based calculations.
  • It can typically assess Monotonic Relationships without requiring parametric assumptions about data distributions.
  • It can typically quantify Ranking Agreement between judge rankings, preference orderings, or ordinal measurements.
  • It can typically provide Correlation Values within [-1, 1] indicating perfect disagreement to perfect agreement.
  • It can typically support Non-Parametric Hypothesis Testing for ordinal data analysis.
  • ...
  • It can often enable Robust Statistical Analysis when continuous data violates normality assumptions.
  • It can often facilitate Inter-Rater Reliability Assessment in behavioral research and social sciences.
  • It can often detect Non-Linear Monotonic Patterns that linear correlation measures might miss.
  • It can often handle Tied Observations through tie correction procedures.
  • ...
  • It can range from being a Simple Rank Correlation Statistic to being a Complex Rank Correlation Statistic, depending on its rank correlation computational method.
  • It can range from being a Pairwise Rank Correlation Statistic to being a Multivariate Rank Correlation Statistic, depending on its rank correlation dimensionality.
  • It can range from being a Complete Rank Correlation Statistic to being a Partial Rank Correlation Statistic, depending on its rank correlation scope.
  • ...
  • It can utilize Rank Transformation Methods to convert ordinal data into numerical ranks.
  • It can support Statistical Significance Testing through permutation tests or asymptotic distributions.
  • It can integrate with Statistical Software Packages for automated rank analysis.
  • It can complement Parametric Correlation Tests in comprehensive data analysis.
  • ...
• Example(s):
  • Concordance-Based Rank Correlation Statistics, such as:
    • Kendall's Tau Rank Correlation Statistic (τ), which counts concordant pairs and discordant pairs.
    • Goodman and Kruskal's Gamma, which extends Kendall's tau for ordinal variables with many tied values.
  • Rank-Difference-Based Rank Correlation Statistics, such as:
    • Spearman's Rank Correlation Statistic (ρ), which applies Pearson correlation to rank values.
    • Quadrant Correlation, which uses rank quadrants for correlation assessment.
  • Specialized Rank Correlation Statistics, such as:
    • Somers' D, which measures asymmetric ordinal associations.
    • Hoeffding's D Statistic, which detects general dependence patterns beyond monotonic relationships.
  • ...
• Counter-Example(s):
  • Pearson Product-Moment Correlation Coefficient, which requires interval data and assumes linear relationships.
  • Chi-Square Test of Independence, which tests categorical associations without considering ordinal structure.
  • Cramer's V, which measures nominal variable associations without rank information.
• See: Wilcoxon Signed-Rank Test, Non-Parametric Statistics, Ordinal Data Analysis, Statistical Significance Test, Mann-Whitney U Test, Preference Learning, ROC Analysis, Inter-Rater Agreement.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/rank_correlation Retrieved:2015-8-20.
  • In statistics, a rank correlation is any of several statistics that measure the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/rank_correlation#Correlation_coefficients Retrieved:2015-8-20.
  • Some of the more popular rank correlation statistics include
    1. Spearman's ρ.
    2. Kendall's τ.
    3. Goodman and Kruskal's γ.
  • An increasing rank correlation coefficient implies increasing agreement between rankings. The coefficient is inside the interval [−1, 1] and assumes the value:
    • 1 if the agreement between the two rankings is perfect; the two rankings are the same.
    • 0 if the rankings are completely independent.
    • −1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other.
  • Following , a ranking can be seen as a permutation of a set of objects. Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group. We can then introduce a metric, making the symmetric group into a metric space. Different metrics will correspond to different rank correlations.

2011

• (Sammut & Webb, 2011) ⇒ Claude Sammut, and Geoffrey I. Webb. (2011). “Rank Correlation.” In: (Sammut & Webb, 2011) p.828
  • QUOTE: Rank correlation measures the correspondence between two rankings τ and τ′ of a set of m objects. Various proposals for such measures have been made, especially in the field of statistics. Two of the best-known measures are Spearman’s Rank Correlation and Kendall’s tau:

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