The Permutation Spearman’s ρ Test tests whether there is a monotonic association between two variables (i.e., the values either increase, decrease together). Rather than relying on distributional assumptions, it builds a null distribution by repeatedly shuffling one variable and recalculating ρ, then asks how often a value as extreme as the observed ρ arises by chance.

Spearman’s ρ is appropriate when data may be non-normal, contain outliers, or where the relationship is monotonic but not necessarily linear. It is also robust when variables are measured on ordinal or bounded scales (such as proportional cover or index scores). The permutation approach means no sample-size assumptions are required, making it suitable for small datasets.

For background on permutation testing, see Cobb (2007), and Allen Downey’s blog articles there is only one test (2011) and there is still only one test (2016).

Hypotheses

Null Hypothesis (always the same) There is no monotonic association between Variable X and Variable Y (ρ = 0).

Alternate hypothesis (set this): Variable X and Variable Y are associated (ρ ≠ 0) Variable X and Variable Y are positively associated (ρ > 0) Variable X and Variable Y are negatively associated (ρ < 0)

Data

Paste your paired data from Excel into the boxes below with one value per row. Both columns must contain the same number of values (paired observations).

Variable X (e.g. Human Impact Index)

Variable Y (e.g. Tree Cover)

Permutations: 999 4,999 9,999
Run the Test!   Clear
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Results

Note that the p value may change slightly between multiple runs on the same dataset. Don’t worry, this is expected and is not an error — it reflects the random permutations sampled in each run. This does not affect the rigour of the method, and you can safely report a single p value.

Interpretation

For a relatively straightforward discussion on the interpretation of p-values (including a list of common misinterpretations), see Dahiru (2008). For a more detailed exploration of the challenges associated with the misinterpretation of statistical testing, see Ioannidis (2005). For a discussion on the importance of testing for effect size (not just significance), see Sullivan and Feinn (2012). A common heuristic for Spearman’s ρ treats |ρ| < 0.10 as negligible, 0.10–0.29 as small, 0.30–0.49 as medium, and ≥ 0.50 as large, though what constitutes a meaningful effect always depends on your study context.