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Spearman’s correlation
Introduction
Before learning about Spearman’s correllation it is important to understand Pearson’s correlation which is a statistical measure of the strength of a linear relationship between paired data. Its calculation and subsequent significance testing of it requires the following data assumptions to hold:
interval or ratio level; linearly related; bivariate normally distributed.
If your data does not meet the above assumptions then use Spearman’s rank correlation!
Monotonic function
To understand Spearman’s correlation it is necessary to know what a monotonic function is. A monotonic function is one that either never increases or never decreases as its independent variable increases. The following graphs illustrate monotonic functions:
Monotonically increasing Monotonically decreasing Not monotonic
Monotonically increasing - as the x variable increases the y variable never decreases; Monotonically decreasing - as the x variable increases the y variable never increases; Not monotonic - as the x variable increases the y variable sometimes decreases and sometimes increases.
Spearman’s correlation coefficient
Spearman’s correlation coefficient is a statistical measure of the strength of a monotonic relationship between paired data. In a sample it is denoted by and is by design constrained as follows
And its interpretation is similar to that of Pearsons, e.g. the closer is to the stronger the monotonic relationship. Correlation is an effect size and so we can verbally describe the strength of the correlation using the following guide for the absolute value of :
.00-.19 “very weak” .20-.39 “weak” .40-.59 “moderate” .60-.79 “strong” .80-1.0 “very strong”
The calculation of Spearman’s correlation coefficient and subsequent significance testing of it requires the following data assumptions to hold:
interval or ratio level or ordinal; monotonically related.
Note, unlike Pearson’s correlation, there is no requirement of normality and hence it is a nonparametric statistic.
Let us consider some examples to illustrate it. The following table gives x and y values for the relationship. From the graph we can see that this is a perfectly increasing monotonic relationship.
Invariably what we observe in a sample are values as follows:
very strong -ve weak +ve monotonic correlation monotonic correlation
Note: Spearman’s correlation coefficient is a measure of a monotonic relationship and thus a value of does not imply there is no relationship between the variables. For example in the following scatterplot which implies no (monotonic) correlation however there is a perfect quadratic relationship:
perfect quadratic relationship
Example
The following data comprises 23 groundwater samples that were collected recording the Uranium concentration (ppb) and the total dissolved solids (mg/L). It is of interest to know if the two variables are correlated?
We should initial consider if Pearson’s correlation is appropriate or whether we should resort to Spearman’s if there are assumption violations.
The scatterplot suggests a definite positive correlation between Uranium and TDS. However, there is possibly slight evidence of non-linearity for TDS values close to zero. However, this is debateable and so we shall move on and consider the other normality assumption.
We need to perform some normality checks for the two variables. One simple way of doing this is to examine boxplots of the data. These are given below.
correlation in the population against the alternative hypothesis, H1, that there is monotonic correlation; our data will indicate which of these opposing hypotheses is most likely to be true. Let s be the Spearman’s population correlation coefficient
then we can thus express this test as:
H 0 : s 0 H 1 : s 0
i.e. the null hypothesis of no monotonic correlation present in population against the alternative that there is monotonic correlation present.
Since SPSS reports the p-value for this test as being .000 we can say that we have very strong evidence to believe H 1 , i.e. we have some evidence to believe that groundwater uranium and TDS values are monotonically correlated in the population.
This could be formally reported as follows:
"A Spearman's correlation was run to determine the relationship between 23 groundwater uranium and TDS values. There was a strong, positive monotonic correlation between Uranium and TDS ( = .71, n = 23, p < .001)."
