Spearman's rank correlation coefficient

Spearman’s rank correlation coefficient

Spearman’s Rank Correlation Coefficient

  • Spearman’s rank correlation coefficient, also known as Spearman’s rho, is a non-parametric measure of rank correlation.
  • It assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables.
  • This statistical measure is a unit-free measure of association and ranges from -1 to +1.

Calculation of Spearman’s Rho

  • To calculate Spearman’s rank correlation coefficient, each of the variables should be ranked individually.
  • Once ranks have been assigned, the difference d between the ranks of each observation on the two variables is calculated.
  • Spearman’s rho is then calculated using the formula: ρ = 1 - [(6 Σd²) / (n³ - n)] Where:
    • ρ is Spearman’s rho
    • Σ refers to the sum of
    • d is the difference in the ranks
    • n is the count of observations

Interpretation of Spearman’s Rho

  • A Spearman correlation of 1 results when each of the variables is a perfect monotone function of the other.
  • A Spearman correlation of 0 means that there is no such monotonic function.
  • A Spearman correlation of -1 results when one variable is a perfect monotone decreasing function of the other.
  • The closer the coefficient is to either -1 or 1, the stronger the correlation between the ranks of the variables.

Assumptions for using Spearman’s Rho

  • Ordinal scale: Both variables are ordinal or continuous and can be ranked like exam results.
  • Mutually dependent pairs: Each observation in the data set consists of a pair of mutually dependent values.

Limitations

  • Spearman’s rank correlation coefficient is not a full-proof statistical tool. It is based on a rank-order relationship, so if the data doesn’t rank perfectly, you’ll get misleading results.
  • It doesn’t measure the linear relationship between variables. It can be used to identify monotonic relationships, but it doesn’t tell you how steep the relationship is.
  • This method can only detect a monotonic relationship between two variables, not more complex relationships.
  • Like all correlation methods, spearman’s rho does not imply causation. Correlation is simply a measure of association, not causality.