Spearman's rank correlation coefficient

spearman fig1

spearman fig2

spearman fig3

Spearman's Rank chart

Spearman's rank correlation coefficient, often denoted by the symbol ρ (the Greek letter rho) or as r_s, is a non-parametric measure of correlation that assesses how well the relationship between two variables can be described using a monotonic function. It is named after Charles Spearman and is often used in the fields of psychology, education, and other social sciences to measure the strength and direction of association between two ranked variables.

Overview

Spearman's rank correlation coefficient is based on the ranked values for each variable rather than the raw data. The coefficient can take values from +1 to -1. A coefficient of +1 indicates a perfect positive monotonic relationship, -1 indicates a perfect negative monotonic relationship, and 0 indicates no monotonic relationship. This measure is a form of a Pearson correlation coefficient that is applied to rank data, making it a useful measure of correlation when the assumptions of Pearson's correlation are not met.

Calculation

The formula to calculate Spearman's rank correlation coefficient for a sample is:

ρ = 1 - {{(6 Σd²)/(n(n² - 1))}}

where d is the difference between the ranks of corresponding variables, and n is the number of observations.

For a population, Spearman's rank correlation coefficient is often denoted as ρ, and its calculation is similar but considers the entire population.

Interpretation

The value of Spearman's rank correlation coefficient indicates the strength and direction of the monotonic relationship between the two variables. A positive value indicates that as one variable increases, the other variable tends to increase, in a monotonic fashion. Conversely, a negative value indicates that as one variable increases, the other variable tends to decrease. The closer the coefficient is to either -1 or +1, the stronger the monotonic relationship.

Applications

Spearman's rank correlation coefficient is widely used in various fields to measure the association between variables when the data do not meet the normality assumption required for Pearson's correlation coefficient. It is particularly useful in studies involving ordinal variables or when the data are not linearly related but are monotonically related.

Advantages and Limitations

One of the main advantages of Spearman's rank correlation coefficient is its non-parametric nature, which means it does not assume a normal distribution of the data. However, it is limited to identifying monotonic relationships and may not accurately reflect the strength of non-monotonic relationships.

See Also

• Pearson correlation coefficient
• Kendall rank correlation coefficient
• Correlation and dependence
• Non-parametric statistics