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Spearman's rho

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Spearman's rho

In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter $\rho$ (rho) or as $r_s$, is a nonparametric measure of statistical dependence between two variables.  It assesses how well the relationship between two variables can be described using a monotonic function.  If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.


Spearman's coefficient, like any correlation calculation, is appropriate for both continuous and discrete variables, including ordinal variables.[1][2]

Definition and calculation

The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the ranked variables.[3] For a sample of size n, the n raw scores $X_i, Y_i$ are converted to ranks $x_i, y_i$, and ρ is computed from these:

$\rho = \frac\left\{\sum_i\left(x_i-\bar\left\{x\right\}\right)\left(y_i-\bar\left\{y\right\}\right)\right\}\left\{\sqrt\left\{\sum_i \left(x_i-\bar\left\{x\right\}\right)^2 \sum_i\left(y_i-\bar\left\{y\right\}\right)^2\right\}\right\}.$

Identical values (rank ties or value duplicates) are assigned a rank equal to the average of their positions in the ascending order of the values. 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 ascending order Rank $x_i$
0.8 1 1
1.2 2 $\frac\left\{2+3\right\}\left\{2\right\}=2.5\$
1.2 3 $\frac\left\{2+3\right\}\left\{2\right\}=2.5\$
2.3 4 4
18 5 5

In applications where duplicate values (ties) are known to be absent, a simpler procedure can be used to calculate ρ.[3][4] Differences $d_i = x_i - y_i$ between the ranks of each observation on the two variables are calculated, and ρ is given by:

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

Note that this latter method should not be used in cases where the data set is truncated; that is, when the Spearman correlation coefficient is desired for the top X records (whether by pre-change rank or post-change rank, or both), the user should use the Pearson correlation coefficient formula given above.

The standard error of the coefficient (σ) was determined by Pearson in 1907 and Gosset in 1920. It is

$\sigma = \frac\left\{ 0.6325 \right\}\left\{ \left( n - 1 \right)^\left\{ \frac\left\{ 1 \right\}\left\{ n \right\} \right\} \right\}$

Related quantities

There are several other numerical measures that quantify the extent of statistical dependence between pairs of observations. The most common of these is the Pearson product-moment correlation coefficient, which is a similar correlation method to Spearman's rank, that measures the “linear” relationships between the raw numbers rather than between their ranks.

An alternative name for the Spearman rank correlation is the “grade correlation”;[5] in this, the “rank” of an observation is replaced by the “grade”. In continuous distributions, the grade of an observation is, by convention, always one half less than the rank, and hence the grade and rank correlations are the same in this case. More generally, the “grade” of an observation is proportional to an estimate of the fraction of a population less than a given value, with the half-observation adjustment at observed values. Thus this corresponds to one possible treatment of tied ranks. While unusual, the term “grade correlation” is still in use.[6]

Interpretation

The sign of the Spearman correlation indicates the direction of association between X (the independent variable) and Y (the dependent variable). If Y tends to increase when X increases, the Spearman correlation coefficient is positive. If Y tends to decrease when X increases, the Spearman correlation coefficient is negative. A Spearman correlation of zero indicates that there is no tendency for Y to either increase or decrease when X increases. The Spearman correlation increases in magnitude as X and Y become closer to being perfect monotone functions of each other. When X and Y are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfect monotone increasing relationship implies that for any two pairs of data values XiYi and XjYj, that Xi − Xj and Yi − Yj always have the same sign. A perfect monotone decreasing relationship implies that these differences always have opposite signs.

The Spearman correlation coefficient is often described as being "nonparametric". This can have two meanings. First, the fact that a perfect Spearman correlation results when X and Y are related by any monotonic function can be contrasted with the Pearson correlation, which only gives a perfect value when X and Y are related by a linear function. The other sense in which the Spearman correlation is nonparametric in that its exact sampling distribution can be obtained without requiring knowledge (i.e., knowing the parameters) of the joint probability distribution of X and Y.

Example

In this example, we will use the raw data in the table below 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

First, we must find the value of the term $d^2_i$. To do so we use the following steps, reflected in the table below.

1. Sort the data by the first column ($X_i$). Create a new column $x_i$ and assign it the ranked values 1,2,3,...n.
2. Next, sort the data by the second column ($Y_i$). Create a fourth column $y_i$ and similarly assign it the ranked values 1,2,3,...n.
3. Create a fifth column $d_i$ to hold the differences between the two rank columns ($x_i$ and $y_i$).
4. Create one final column $d^2_i$ to hold the value of column $d_i$ squared.
 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

With $d^2_i$ found, we can add them to find $\sum d_i^2 = 194$. The value of n is 10. So these values can now be substituted back into the equation,

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

which evaluates to ρ = -29/165 = −0.175757575... with a P-value = 0.6864058 (using the t distribution)

This low value shows that the correlation between IQ and hours spent watching TV is very low. 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:

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

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

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

is a z-score for r which approximately follows a standard normal distribution under the null hypothesis of statistical independence (ρ = 0).[7][8]

One can also test for significance using

$t = r \sqrt\left\{\frac\left\{n-2\right\}\left\{1-r^2\right\}\right\}$

which is distributed approximately as Student's t distribution with n − 2 degrees of freedom under the null hypothesis.[9] A justification for this result relies on a permutation argument.[10]

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[11] 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 that gives a score to every value of two nominal variables. In this way the Pearson 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.[12]