Spearman Rank Correlation

spearman

correlation

ranks

monotonic

non-parametric

Rank-based measure of monotonic association, robust to non-linearity and outliers

Published

April 17, 2026

Research question

Spearman’s rho (\(\rho\)) measures monotonic association between two ordinal or non-normal continuous variables. Use it when Pearson’s assumptions fail or when at least one variable is ordinal. Biomedical examples: (1) does disease stage (I-IV) correlate with patient-reported quality-of-life (0-100)?; (2) does serum procalcitonin (right-skewed) correlate with SOFA score on admission to the ICU?

Assumptions

Assumption	How to verify in R
Both variables at least ordinal	scale level
Monotonic relationship (need not be linear)	scatter plot of raw or ranked values
Independent pairs	design

Hypotheses

\[H_0: \rho_s = 0 \qquad H_1: \rho_s \ne 0\]

R code

library(tidyverse); library(rstatix); library(ggstatsplot)
set.seed(42)

# 90 ICU patients: procalcitonin and SOFA
icu <- tibble(
  procalcitonin = rlnorm(90, log(2), 1.1),
  sofa          = pmin(round(log(procalcitonin + 0.1) * 2 + rnorm(90, 0, 1.5) + 6), 20)
)

cor.test(icu$procalcitonin, icu$sofa, method = "spearman", exact = FALSE)

icu |> cor_test(procalcitonin, sofa, method = "spearman")

ggscatterstats(data = icu, x = procalcitonin, y = sofa, type = "nonparametric",
               xlab = "Procalcitonin (ng/mL)", ylab = "SOFA score")

Interpreting the output

Spearman’s \(\rho = 0.68\), \(p < .001\). The association is strong and monotonic: higher procalcitonin is associated with higher SOFA score. Because \(\rho\) is computed on ranks, the presence of a skewed procalcitonin distribution does not distort the test.

Effect size

Spearman’s \(\rho\) is its own effect size. Cohen’s thresholds: small 0.10, medium 0.30, large 0.50.

Reporting (APA 7)

Procalcitonin was positively associated with SOFA score on admission (Spearman’s rho = .68, p < .001). The relationship was monotonic across the full range of procalcitonin values.

Common pitfalls

Reporting Pearson when Spearman is more appropriate for skewed data; Pearson can vastly underestimate a monotonic non-linear relationship.
Ties: R’s exact test fails with ties and switches to a normal approximation with a warning; set exact = FALSE to suppress the warning.
Spearman detects monotonic patterns; for U-shaped relationships, neither Pearson nor Spearman is appropriate – fit a spline or use a polynomial regression.

Parametric vs. non-parametric alternative

Parametric (linear, bivariate normal): Pearson correlation.
Tied-data small samples: Kendall’s tau.