Basic Concepts
In Spearman’s Correlation Hypothesis Testing we explore how to perform testing as to whether Spearman’s correlation is statistically different from zero. On this webpage, we show how to test whether Spearman’s correlation ρs is statistically different from ρ0 for any value of ρ0, even where ρ0 is not zero.
We again use the Fisher transformation, as described in Correlation Testing via Fisher Transformation, for hypothesis testing. In particular, to test the null hypothesis that ρs = ρ0, we can use the test
where rs is Spearman’s rank coefficient for your data. If the sample size n ≥ 10 and |ρs| ≤ .9, then the standard error in the denominator of the test can be replaced by
This version of the hypothesis test can also be used to estimate a confidence intervals, as described in Spearman’s Correlation Hypothesis Testing
Example
Example 1: Test whether Spearman’s rho is significantly different from -.5 based on the data in Example 1 of Spearman’s Correlation Hypothesis Testing (repeated in Figure 1).
We see from Figure 1 that there isn’t a significant difference between the population Spearman’s rho and -.5.
Figure 1 – Hypothesis testing using Fisher’s transformation
Worksheet Function
The Real Statistics Resource Pack provides the following worksheet function:
SCorrelTest(R1, R2, rho, lab, tails, alpha): an array function which outputs a column array consisting of the sample Spearman’s correlation coefficient, the z-stat (based on a Fisher transformation) and p-value which test the null hypothesis that ρ = rho and the lower/upper ends of a 1–alpha confidence interval (as described in Spearman’s Correlation Hypothesis Testing).
If lab = TRUE then a column of labels are added to the output (default is FALSE). tails = 1 or 2 (default) and alpha = significance level (default .05). R1 and R2 must have the same shape and size.
We can use the SCorrelTest function to obtain the results for Example 1 as shown in range G14:H18 of Figure 1.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
NCSS (2024) Correlation
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Correlation.pdf
Bonett, D. G. and Wright, T. A. (2000) Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika 65, no. 1: 23–28.
https://www.researchgate.net/publication/24063325_Sample_size_requirements_for_estimating_Pearson_Kendall_and_Spearman_correlations
Ruscio, J. (2008) Constructing confidence intervals for Spearman’s rank correlation and ordinal data: a simulation study comparing analytic and bootstrap methods
https://jmasm.com/index.php/jmasm/article/view/376/376
Bishara, A. J., Hittner, J. B. (2007) Confidence intervals for correlations when data are not normal. Behavior Research Methods 49, no. 1: 294–309. https://doi.org/10.3758/s13428-016-0702-8