Spearman correlation

Description

The Spearman Rank Correlation Coefficient is a form of the Pearson coefficient with the data converted to rankings (ie. when variables are ordinal). It can be used when there is non-parametric data and hence Pearson cannot be used.

The raw scores are converted to ranks and the differences (d_i) between the ranks of each observation on the two variables are calculated. The Spearman coefficient is denoted with the Greek letter rho (ρ).

ρ = 1 - (6 * SUM(d_i²)) / (n * (n² - 1))

Example

Two groups, x and y, are asked to rank ten items. The correlation between their rankings are then compared as below

Item	x	y	x-y	(x-y)^2
1	1	2	-1	1
2	3	5	-2	4
3	5	6	-1	1
4	6	6	0	0
5	8	7	1	1
6	9	7	2	4
7	6	5	1	1
8	4	3	1	1
9	3	1	2	4
10	2	1	1	1

n:	10	sum((x-y)^2):		18
n * (n^2 -1):	990		6 * sum((x-y)^2):	108

Spearman, ρ = 6 * sum((x-y)^2) / n*(n^2 -1)				0.89

The correlation of 0.89 is quite low, showing that they correlate well.

Discussion

The Spearman Coefficient can be used to measure ordinal data (ie. in rank order), not interval (as Pearson). It effectively works by first ranking the data then applying Pearson's calculation to the rank numbers.

This coefficient is also called Spearman's rho (after the Greek letter used).

SPSS: Analyze, Correlate, Bivariate (check Spearman's rho)

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Spearman correlation

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