Spearman's Correlation

Last updated

24-05-2019

When to use the Spearman Correlation test

What does the Speaman Correlation measure?

The Spearman correlation (denoted as p (rho) or r_s) measures the strength and direction of association between two ranked variables.
It is most commonly used to measure the degree and direction of a linear relation between two variables that are of the ordinal type.

What are the assumptions of the Spearman Correlation test?

two variables that are either ordinal, interval or ratio (note: if the data are interval or ratio, normally you would use a Pearson correlation test)
the relationship between the variables is monotonic, or (curvi)linear

Spearman Correlation formula

What is the definition of the Spearman Correlation test?

\[\rho = r_s = 1 - \frac{6\sum{d^2_i}}{n(n^2-1)}\]

p : Spearman correlation
r_s: Spearman correlation
d_i : rg(X_i) - rg(Y_i): difference between the two ranks of each observation (for example, one can have the second best score on variable X, but the ninth on variable Y.)
n : number of scores

Note: if there are tied ranks, the following method can be used to calculate the Spearman Correlation:

\[\rho = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt {\sum_i (x_i - \bar{x})^2 \sum_i (y_i - \bar{y})^2}}\]

Example: how to calculate Spearman Correlation?

Sample question

The scores for nine students in history and algebra are as follows:

History: 35, 23, 47, 17, 10, 43, 9, 6, 28
Algebra: 30, 33, 45, 23, 8, 49, 12, 4, 31

Compute the Spearman rank correlation.

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