Spearman's rank correlation coefficient: Definition from Answers.com

In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter $ρ$ (rho) or as $r s$ , is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any other assumptions about the particular nature of the relationship between the variables. Certain other measures of correlation are parametric in the sense of being based on possible relationships of a parameterised form, such as a linear relationship.

1 Calculation
2 Example
3 Determining significance
4 Correspondence analysis based on Spearman's rho
5 See also
6 References
7 External links

Calculation

In practice, however, a simpler procedure is normally used to calculate ρ. The raw scores are converted to ranks, and the differences $d i$ between the ranks of each observation on the two variables are calculated.

If there are no tied ranks, then ρ is given by:^[1]

$\rho = 1- {\frac {6 \sum d_i^2}{n(n^2 - 1)}}$

where:

d i = x i - y i

= the difference between the ranks of corresponding values

X i

and

Y i

, and

n = the number of values in each data set (same for both sets).

If tied ranks exist, classic Pearson's correlation coefficient between ranks has to be used instead of the above formula:^[1]

$\rho=\frac{n(\sum x_iy_i)-(\sum x_i)(\sum y_i)} {\sqrt{n(\sum x_i^2)-(\sum x_i)^2}~\sqrt{n(\sum y_i^2)-(\sum y_i)^2}}.$

One has to assign the same rank to each of the equal values. It is an average of their positions in the ascending order of the values:

An example of averaging ranks

In the table below, notice how the rank of values that are the same is the mean of what their ranks would otherwise be.

Variable $X i$	Position in the descending order	Rank $x i$
0.8	5	5
1.2	4	$\frac{4+3}{2}=3.5\$
1.2	3	$\frac{4+3}{2}=3.5\$
2.3	2	2
18	1	1

In this case the shortcut formula cannot be used (because of the tied ranks in the data) - the second, product-moment form, must be used instead.

Example

The raw data used in this example is shown below. The goal here is to calculate the correlation between the IQ of a person with the number of hours spent in front of TV per week.

IQ, $X i$	Hours of TV per week, $Y i$
106	7
86	0
100	27
101	50
99	28
103	29
97	20
113	12
112	6
110	17

The first step is to sort this data by the second column. Next, two more columns are created ( $x i$ and $y i$ ). The last of these columns ( $y i$ ) is assigned 1,2,3,...n, and then the data is sorted by the first original column ( $X i$ ). The first of the newly created columns ( $x i$ ) is assigned 1,2,3,...n. Then a column $d i$ is created to hold the differences between the two rank columns ( $x i$ and $y i$ ). Finally another column $d^2_i$ should be created. This is just column $d i$ squared.

After performing this process with the example data, the end result is something like the following:

IQ, $X i$	Hours of TV per week, $Y i$	rank $x i$	rank $y i$	$d i$	$d^2_i$
86	0	1	1	0	0
97	20	2	6	−4	16
99	28	3	8	−5	25
100	27	4	7	−3	9
101	50	5	10	−5	25
103	29	6	9	−3	9
106	7	7	3	4	16
110	17	8	5	3	9
112	6	9	2	7	49
113	12	10	4	6	36

The values in the $d^2_i$ column can now be added to find $\sum d_i^2 = 194$ . The value of n is 10. So these values can now be substituted back into the equation,

$\rho = 1- {\frac {6\times194}{10(10^2 - 1)}}$

which evaluates to ρ = −0.175757575..., which shows that the correlation between IQ and hours spent watching TV is very low (barely any correlation). In the case of ties in the original values, this formula should not be used. Instead, the Pearson correlation coefficient should be calculated on the ranks (where ties are given ranks, as described above).

Determining significance

One approach to testing whether an observed value of ρ is significantly different from zero (r will always maintain 1 ≥ r ≥ −1) is to calculate the probability that it would be greater than or equal to the observed r, given the null hypothesis, by using a permutation test. An advantage of this approach is that it automatically takes into account the number of tied data values there are in the sample, and the way they are treated in computing the rank correlation.

Another approach parallels the use of the Fisher transformation in the case of the Pearson product-moment correlation coefficient. That is, confidence intervals and hypothesis tests relating to the population value ρ can be carried out using the Fisher transformation:

$F(r) = {1 \over 2}\log{1+r \over 1-r} = \operatorname{arctanh}(r).$

If F(r) is the Fisher transformation of r, the sample Spearman rank correlation coefficient, and n is the sample size, then

$z = \sqrt{\frac{n-3}{1.06}}F(r)$

is a z-score for r which approximately follows a standard normal distribution under the null hypothesis of statistical independence (ρ = 0).^[2]^[3]

A generalization of the Spearman coefficient is useful in the situation where there are three or more conditions, a number of subjects are all observed in each of them, and it is predicted that the observations will have a particular order. For example, a number of subjects might each be given three trials at the same task, and it is predicted that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page^[4] and is usually referred to as Page's trend test for ordered alternatives.

Correspondence analysis based on Spearman's rho

Classic correspondence analysis is a statistical method which gives a score to every value of two nominal variables, in this way that Pearson's correlation coefficient between them is maximized.

There exists an equivalent of this method, called grade correspondence analysis, which maximizes Spearman's rho or Kendall's tau^[5].

References

This article includes a list of references or external links, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations where appropriate. (November 2009)

^ ^a ^b Myers, Jerome L.; Arnold D. Well (2003). Research Design and Statistical Analysis (second edition ed.). Lawrence Erlbaum. pp. 508. ISBN 0805840370.
^ Choi, S.C. (1977) Test of equality of dependent correlations. Biometrika, 64 (3), 645–647
^ Fieller, E.C. et al (1957) Tests for rank correlation coefficients :I. Biometrika 44, 470–481
^ Page, E. B. (1963). "Ordered hypotheses for multiple treatments: A significance test for linear ranks". Journal of the American Statistical Association 58: 216–230. doi:10.2307/2282965.
^ Kowalczyk, T.; Pleszczyńska E. , Ruland F. (eds.) (2004). Grade Models and Methods for Data Analysis with Applications for the Analysis of Data Populations. Studies in Fuzziness and Soft Computing vol. 151. Berlin Heidelberg New York: Springer Verlag. ISBN 9783540211204.

G.W. Corder, D.I. Foreman, "Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach", Wiley (2009)
C. Spearman, "The proof and measurement of association between two things" Amer. J. Psychol. , 15 (1904) pp. 72–101
M.G. Kendall, "Rank correlation methods" , Griffin (1962)
M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973)
J. C. Caruso, N. Cliff, "Empirical Size, Coverage, and Power of Confidence Intervals for Spearman's Rho", Ed. and Psy. Meas. , 57 (1997) pp. 637–654

External links

Table of critical values of ρ for significance with small samples
Online calculator
Chapter 3 part 1 shows the formula to be used when there are ties
Spearman's rank correlation: Simple notes for students with an example of usage by biologists and a spreadsheet for Microsoft Excel for calculating it (a part of materials for a Research Methods in Biology course).
Tests of Association Calculator

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Spearman's rank correlation coefficient

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