What is Spearman Correlation?
Spearman Correlation Coefficient is also referred to as Spearman Rank Correlation or Spearman’s rho. It is typically denoted either with the Greek letter rho (ρ), or rs. It is one of the few cases where a Greek letter denotes a value of a sample and not the characteristic of the general population. Like all correlation coefficients, Spearman’s rho measures the strength of association of two variables. As such, the Spearman Correlation Coefficient is a close sibling to Pearson’s Bivariate Correlation Coefficient, Point-Biserial Correlation, and the Canonical Correlation.
All correlation analyses express the strength of linkage or co-occurrence between to variables in a single value between -1 and +1. This value is called the correlation coefficient. A positive correlation coefficient indicates a positive relationship between the two variables (the larger A, the larger B) while a negative correlation coefficients expresses a negative relationship (the larger A, the smaller B). A correlation coefficient of 0 indicates that no relationship between the variables exists at all. However correlations are limited to linear relationships between variables. Even if the correlation coefficient is zero a non-linear relationship might exist.
Compared to Pearson’s bivariate correlation coefficient the Spearman Correlation does not require continuous-level data (interval or ratio), because it uses ranks instead of assumptions about the distributions of the two variables. This allows us to analyze the association between variables of ordinal measurement levels. Moreover the Spearman Correlation is a non-paracontinuous-level test, which does not assume that the variables approximate multivariate normal distribution. Spearman Correlation Analysis can therefore be used in many cases where the assumptions of Pearson’s Bivariate Correlation (continuous-level variables, linearity, heteroscedasticity, and multivariate normal distribution of the variables to test for significance) are not met.
Typical questions the Spearman Correlation Analysis answers are as follows:
- Sociology: Do people with a higher level of education have a stronger opinion of whether or not tax reforms are needed?
- Medicine: Does the number of symptoms a patient has indicate a higher severity of illness?
- Biology: Is mating choice influenced by body size in bird species A?
- Business: Are consumers more satisfied with products that are higher ranked in quality?
Theoretically, the Spearman correlation calculates the Pearson correlation for variables that are converted to ranks. Similar to Pearson’s bivariate correlation, the Spearman correlation also tests the null hypothesis of independence between two variables. However this can lead to difficult interpretations. Kendall’s Tau-b rank correlation improves this by reflecting the strength of the dependence between the variables in comparison.
Since both variables need to be of ordinal scale or ranked data, Spearman’s correlation requires converting interval or ratio scales into ranks before it can be calculated. Mathematically, Spearman correlation and Pearson correlation are very similar in the way that they use difference measurements to calculate the strength of association. Pearson correlation uses standard deviations while Spearman correlation difference in ranks. However, this leads to an issue with the Spearman correlation when tied ranks exist in the sample. An example of this is when a sample of marathon results awards two silver medals but no bronze medal. A statistician is even crueler to these runners because a rank is defined as average position in the ascending order of values. For a statistician, the marathon result would have one first place, two places with a rank of 2.5, and the next runner ranks 4. If tied ranks occur, a more complicated formula has to be used to calculate rho, but SPSS automatically and correctly calculates tied ranks.
Spearman Correlation in SPSS
We have shown in the Pearson’s Bivariate Correlation Analysis that the Reading Test Scores and the Writing test scores are positively correlated. Let us assume that we never did this analysis; the research question posed is then, “Are the grades of the reading and writing test correlated?” We assume that all we have to test this hypothesis are the grades achieved (A-F). We could also include interval or ratio data in this analysis because SPSS converts scale data automatically into ranks.
The Spearman Correlation requires ordinal or ranked data, therefore it is very important that measurement levels are correctly defined in SPSS. Grade 2 and Grade 3 are ranked data and therefore measured on an ordinal scale. If the measurement levels are specified correctly, SPSS will automatically convert continuous-level data into ordinal data. Should we have raw data that already represents rankings but without specification that this is an ordinal scale, nothing bad will happen.
Spearman Correlation can be found in SPSS in Analyze/Correlate/Bivariate…
This opens the dialog for all Bivariate Correlations, which also includes Pearson’s Bivariate Correlation. Using the arrow, we add Grade 2 and Grade 3 to the list of variables for analysis. Then we need to tick the correlation coefficients we want to calculate. In this case the ones we want are Spearman and Kendall’s Tau-b.
The Output of Spearman’s Correlation Analysis
The output of Spearman’s Correlation Analysis is fairly straightforward and consists of only one table.
The first coefficient in the output table is Kendall’s Tau-b. Kendall’s Tau is a simpler correlation coefficient that calculates how many concordant pairs (same rank on both variables) exist in a sample. Tau-b measures the strength of association when both variables are measured at the ordinal level. It adjusts the sample for tied ranks.
The second coefficient is Spearman’s rho. SPSS shows that for example the Bivariate Correlation Coefficient between Grade 2 and Grade 3 is τ = 0.507 and ρ = 0.634. In both cases the significance is p < 0.001. For small samples, SPSS automatically calculates a permutation test of significance instead of the classical t-test, which violates the assumption of multivariate normality when sample size is small.
One possible interpretation and write-up of Spearman’s Correlation Coefficient rho and the test of significance is as follows:
We analyzed the question of whether the grade achieved in the reading test and the grade achieved in the writing test are somewhat linked. Spearman’s Correlation Coefficient indicates a strong association between these two variables (ρ = 0.634). The test of significance indicates that with p < 0.001 we can reject the null hypothesis that both variables are independent in the general population. Thus we can say that with a confidence of more than 95% the observed positive correlation between grade writing and grade reading is not caused by random effects and both variables are interdependent.
Please always bear in mind that correlation alone does not make for causality.
Syntax
NONPAR CORR
/VARIABLES=Grade2 Grade3
/PRINT=BOTH TWOTAIL NOSIG
/MISSING=PAIRWISE.
