In statistics, Spearman's rank correlation coefficient or Spearman's ρ is a number ranging from −1 to 1 that indicates how strongly two sets of ranks are correlated. It is used in situations where the ranking of a dataset is important, for instance in sports. If a statistician wanted to know whether people who are high ranking in sprinting are also high ranking in long-distance running, they would use a Spearman rank correlation coefficient.
The coefficient is named after Charles Spearman and often denoted by the Greek letter (rho) or as . It is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.
The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). 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.
Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e., relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a correlation of −1) rank between the two variables.
Spearman's coefficient is appropriate for both continuous and discrete ordinal variables.Scale types. Both Spearman's and Kendall's can be formulated as special cases of a more general correlation coefficient.
Applications
The coefficient can be used to determine how well data fits a model, like when determining the similarity of text documents.
Definition and calculation The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the rank variables.
For a sample of size the pairs of raw scores are converted to ranks and is computed as
where denotes the conventional Pearson correlation coefficient operator, but applied to the rank variables, is the covariance of the rank variables, and are the standard deviations of the rank variables.
Only when all ranks are distinct integers (no ties), it can be computed using the popular formula
where is the difference between the two ranks of each observation, is the number of observations.
Consider a bivariate sample with corresponding rank pairs Then the Spearman correlation coefficient of is
where, as usual,
and
We shall show that can be expressed purely in terms of provided we assume that there be no ties within each sample.
Under this assumption, we have that can be viewed as random variables distributed like a uniformly distributed discrete random variable on Hence
and
where
and thus
(These sums can be computed using the formulas for the triangular numbers and square pyramidal numbers, or basic summation results from umbral calculus.)
Observe now that
Putting this all together thus yields
Identical values are usually each assigned fractional ranks equal to the average of their positions in the ascending order of the values, which is equivalent to averaging over all possible permutations.
If ties are present in the data set, the simplified formula above yields incorrect results: Only if in both variables all ranks are distinct, then (calculated according to biased variance). The first equation — normalizing by the standard deviation — may be used even when ranks are normalized to [0, 1] ("relative ranks") because it is insensitive both to translation and linear scaling.
The simplified method should also not be used in cases where the data set is truncated; that is, when the Spearman's 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.
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"; 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.
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 perfectly monotonic functions of each other. When X and Y are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfectly monotonic increasing relationship implies that for any two pairs of data values and , that and always have the same sign. A perfectly monotonic 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, a perfect Spearman correlation results when X and Y are related by any monotonic function. Contrast this 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 is 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.
The strength of correlations also are an important consideration for interpretation of correlation analyses but descriptions of the strength of correlation are not universally accepted. A small correlation coefficient calculated for a very large sample may be statistically significant without providing a quantitative relation between variables. Similarly, a large correlation coefficient calculated with a small sample size may indicate that a quantitative equation between variables may be possible even though the correlation coefficient is not statistically significant. Akoglu (2018)Akoglu, Haldun, 2018, Review Article--User's guide to correlation coefficients: Turkish Journal of Emergency Medicine, Volume 18, Issue 3, Pages 91-93, https://doi.org/10.1016/j.tjem.2018.08.001 noted the need for consistent descriptions of strength and provides different correlation-strength descriptors for psychology, politics, and medicine that have different descriptors and thresholds. Because there is a general lack of consensus on correlation strength, Granato (2014)Granato, G.E., 2014, Statistics for stochastic modeling of volume reduction, hydrograph extension, and water-quality treatment by structural stormwater runoff best management practices (BMPs): U.S. Geological Survey Scientific Investigations Report 2014–5037, 37 p., https://dx.doi.org/10.3133/sir20145037. defined operational definitions for the absolute values of correlation coefficients as weak (less than 0.5), moderate (greater than or equal to 0.5 and less than 0.75), semi-strong (greater than or equal to 0.75 and less than 0.85), and strong (greater than or equal to 0.85) for use in hydrologic analyses of stormwater treatment statistics. Similarly, Schober and others (2018)Schober, Patrick, Boer, Christa, Schwarte, Lothar, 2018, Correlation coefficients: Appropriate use and interpretation: Anesthesia & Analgesia 126(5):p 1763-1768, 10.1213/ANE.0000000000002864 note that "...cutoff points are arbitrary and inconsistent and should be used judiciously..." Despite their warning Schober and others (2018) provide a table indicating that correlations less than or equal to 0.1 are negligible; correlations greater than 0.1 and less than or equal to 0.39 are weak; correlations greater than 0.39 and less than or equal to 0.69 are moderate; correlations greater than 0.69 and less than or equal to 0.89 are strong; and correlations greater than 0.89 are very strong. Descriptions of strength indicate the potential to develop quantitative relations among variables but these descriptors, as with the correlation coefficients themselves, do not indicate causation.
Example In this example, the arbitrary raw data in the table below is used to calculate the correlation between the IQ of a person with the number of hours spent in front of TV per week [fictitious values used]. IQ, Hours of TV per week, 10671002786210150992810329972011312112611017
Firstly, evaluate . To do so use the following steps, reflected in the table below. Sort the data by the first column (). Create a new column and assign it the ranked values 1, 2, 3, ..., n. Next, sort the augmented (with ) data by the second column (). Create a fourth column and similarly assign it the ranked values 1, 2, 3, ..., n. Create a fifth column to hold the differences between the two rank columns ( and ). Create one final column to hold the value of column squared.
IQ, Hours of TV per week, rank rank 8621100972026 −416992838 −5251002747 −3910150510 −5251032969 −3910677341611017853911269274911312104636
With found, add them to find . The value of n is 10. These values can now be substituted back into the equation
to give
which evaluates to with a p-value = 0.627188 (using the t-distribution).
That the value is close to zero shows that the correlation between IQ and hours spent watching TV is very low, although the negative value suggests that the longer the time spent watching television the lower the IQ. 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).
Confidence intervals Confidence intervals for Spearman's ρ can be easily obtained using the Jackknife Euclidean likelihood approach in de Carvalho and Marques (2012). The confidence interval with level is based on a Wilks' theorem given in the latter paper, and is given by
where is the quantile of a chi-square distribution with one degree of freedom, and the are jackknife pseudo-values. This approach is implemented in the R package spearmanCI.
Determining significance One approach to test whether an observed value of ρ is significantly different from zero (r will always maintain ) 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 in the sample and the way they are treated in computing the rank correlation.