Applying spearman's correlation

What does the Speaman Correlation measure?

  • The Spearman correlation (denoted as p (rho) or rs) 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

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
  • rs : Spearman correlation
  • di : rg(Xi) - rg(Yi): 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:

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

  • x : value of x
  • x̄ : mean of x
  • y : value of y
  • ȳ : mean of y
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Example: how to calculate Spearman Correlation?

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.

Answer

  1. Step 1: rank each student
Student #History ScoreHistory RankAlgebra ScoreAlgebra Rank
1353305
2235333
3471452
4176236
510788
6432491
798127
86949
9284314
  1. step 2: calculate difference between the ranks (d) and square d
Student #History ScoreHistory RankAlgebra ScoreAlgebra Rankdd2
135330524
223533324
347145211
417623600
51078811
643249111
79812711
8694900
928431400
  1. step 3: sum (add up) all the d2 scores
  • Σ d2 = 4 + 4 + 1 + 0 + 1 + 1 + 1 + 0 + 0 = 12
  1. step 4: insert the values in the formula. These ranks are not tied, so use the first formula:

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

= 1 – (6*12)/(9(81-1))
= 1 – 72/720
= 1-0.1
= 0.9

> The Spearman Rank Correlation for this set of data is 0.9.

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Last updated
05-01-2022
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