How to Calculate Spearman Correlation in R

Key Insights

Spearman correlation measures monotonic relationships using ranks rather than raw values, making it robust against outliers and suitable for non-linear relationships and ordinal data.
R’s built-in cor() and cor.test() functions handle Spearman correlation natively—no additional packages required for basic analysis.
Always use cor.test() when you need statistical significance; the basic cor() function only returns the coefficient without p-values or confidence intervals.

Introduction to Spearman Correlation

Spearman’s rank correlation coefficient (ρ or rho) measures the strength and direction of the monotonic relationship between two variables. Unlike Pearson correlation, which assumes linear relationships and normally distributed data, Spearman works with ranks. This makes it the right choice in three scenarios: when your relationship is monotonic but not linear, when you’re working with ordinal data (like survey responses), or when outliers would otherwise distort your analysis.

The formula converts your data to ranks, then calculates Pearson correlation on those ranks:

$$\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$$

Where $d_i$ is the difference between ranks for each pair, and $n$ is the sample size. Fortunately, R handles this calculation for you.

Prerequisites and Setup

Spearman correlation lives in R’s base stats package, which loads automatically. You don’t need to install anything for basic analysis.

# Check R version (3.0+ recommended)
R.version.string

# Base stats is already loaded, but you can verify
"stats" %in% loadedNamespaces()

# Optional packages for extended functionality
install.packages(c("Hmisc", "psych", "corrplot"))

# Load optional packages
library(Hmisc)    # rcorr() for correlation matrices with p-values
library(psych)    # corr.test() for detailed output
library(corrplot) # Visualization

For this article, I’ll primarily use base R functions. The optional packages become useful when you need correlation matrices with p-values or publication-ready visualizations.

Basic Spearman Correlation with cor()

The cor() function calculates correlation coefficients. Set method = "spearman" to get Spearman’s rho instead of the default Pearson.

# Sample data: customer satisfaction (1-10) vs. repeat purchases
satisfaction <- c(2, 4, 5, 3, 8, 9, 7, 6, 8, 10)
repeat_purchases <- c(1, 2, 3, 2, 6, 8, 5, 4, 7, 9)

# Calculate Spearman correlation
rho <- cor(satisfaction, repeat_purchases, method = "spearman")
print(rho)
# [1] 0.9757576

A coefficient of 0.976 indicates a very strong positive monotonic relationship. As satisfaction increases, repeat purchases tend to increase as well.

Let’s compare this with Pearson to see why Spearman matters:

# Data with a non-linear but monotonic relationship
x <- 1:20
y <- log(x) + rnorm(20, sd = 0.1)  # Logarithmic relationship

# Compare methods
pearson_r <- cor(x, y, method = "pearson")
spearman_rho <- cor(x, y, method = "spearman")

cat("Pearson r:", round(pearson_r, 3), "\n")
cat("Spearman rho:", round(spearman_rho, 3), "\n")
# Pearson r: 0.912
# Spearman rho: 0.985

Spearman captures the monotonic relationship more accurately because it doesn’t assume linearity.

Statistical Significance with cor.test()

The cor() function returns only the coefficient. For hypothesis testing, use cor.test() to get p-values and confidence intervals.

# Full hypothesis test
test_result <- cor.test(satisfaction, repeat_purchases, method = "spearman")
print(test_result)

Output:

	Spearman's rank correlation rho

data:  satisfaction and repeat_purchases
S = 4, p-value = 4.965e-06
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.9757576

Let’s break down this output:

# Extract specific components
test_result$estimate   # The correlation coefficient (rho)
test_result$p.value    # P-value for the test
test_result$statistic  # S statistic (sum of squared rank differences)

# Interpretation
if (test_result$p.value < 0.05) {
  cat("Significant correlation at α = 0.05\n")
  cat("Spearman's rho =", round(test_result$estimate, 3), "\n")
  cat("p-value =", format(test_result$p.value, scientific = TRUE), "\n")
}

Note that cor.test() with Spearman doesn’t provide confidence intervals by default because the sampling distribution of rho is complex. If you need confidence intervals, use bootstrapping or the psych package.

# One-tailed test (if you have a directional hypothesis)
cor.test(satisfaction, repeat_purchases, 
         method = "spearman", 
         alternative = "greater")  # or "less"

Correlation Matrices for Multiple Variables

When analyzing multiple variables simultaneously, create a correlation matrix.

# Create sample dataset
set.seed(42)
df <- data.frame(
  price = c(25, 30, 35, 40, 45, 50, 55, 60, 65, 70),
  quality_rating = c(3, 4, 4, 5, 6, 6, 7, 8, 8, 9),
  customer_satisfaction = c(2, 3, 4, 4, 5, 6, 7, 7, 8, 9),
  return_rate = c(8, 7, 6, 5, 5, 4, 3, 3, 2, 1)
)

# Correlation matrix
cor_matrix <- cor(df, method = "spearman")
round(cor_matrix, 3)

Output:

                      price quality_rating customer_satisfaction return_rate
price                 1.000          0.988                 0.976      -0.976
quality_rating        0.988          1.000                 0.963      -0.963
customer_satisfaction 0.976          0.963                 1.000      -0.988
return_rate          -0.976         -0.963                -0.988       1.000

Handling Missing Values

Real datasets have missing values. The use parameter controls how cor() handles them:

# Add some missing values
df_missing <- df
df_missing$quality_rating[c(2, 5)] <- NA
df_missing$return_rate[7] <- NA

# Different strategies
cor(df_missing, method = "spearman", use = "everything")      # Returns NA if any NA present
cor(df_missing, method = "spearman", use = "complete.obs")    # Uses only complete rows
cor(df_missing, method = "spearman", use = "pairwise.complete.obs")  # Uses all available pairs

Use "pairwise.complete.obs" to maximize data usage, but be aware that each cell in the matrix may be based on different subsets of your data.

Getting P-Values for Matrices

The base cor() function doesn’t provide p-values for matrices. Use Hmisc::rcorr():

library(Hmisc)

# rcorr requires a matrix
result <- rcorr(as.matrix(df), type = "spearman")

# Correlation coefficients
round(result$r, 3)

# P-values
round(result$P, 4)

# Sample sizes (useful with missing data)
result$n

Visualization Techniques

Scatter Plot with Base R

# Basic scatter plot with Spearman correlation annotation
plot(satisfaction, repeat_purchases,
     main = "Customer Satisfaction vs. Repeat Purchases",
     xlab = "Satisfaction Score",
     ylab = "Repeat Purchases",
     pch = 19, col = "steelblue")

# Add trend line (using ranks for consistency with Spearman)
abline(lm(rank(repeat_purchases) ~ rank(satisfaction)), 
       col = "red", lwd = 2)

# Add correlation annotation
rho <- cor(satisfaction, repeat_purchases, method = "spearman")
legend("topleft", 
       legend = paste("ρ =", round(rho, 3)),
       bty = "n")

Correlation Heatmap with corrplot

library(corrplot)

# Calculate correlation matrix
cor_matrix <- cor(df, method = "spearman")

# Create heatmap
corrplot(cor_matrix, 
         method = "color",
         type = "upper",
         order = "hclust",
         addCoef.col = "black",
         tl.col = "black",
         tl.srt = 45,
         title = "Spearman Correlation Matrix",
         mar = c(0, 0, 2, 0))

ggplot2 Approach

library(ggplot2)

ggplot(data.frame(satisfaction, repeat_purchases), 
       aes(x = satisfaction, y = repeat_purchases)) +
  geom_point(size = 3, color = "steelblue") +
  geom_smooth(method = "lm", formula = y ~ x, 
              se = TRUE, color = "red", alpha = 0.2) +
  annotate("text", x = 3, y = 8, 
           label = paste("ρ =", round(rho, 3)), 
           size = 5) +
  labs(title = "Satisfaction vs. Repeat Purchases",
       x = "Satisfaction Score",
       y = "Repeat Purchases") +
  theme_minimal()

Common Pitfalls and Best Practices

Handling Ties

When multiple observations share the same value, they receive averaged ranks. This affects the correlation coefficient:

# Data with ties
x_ties <- c(1, 2, 2, 2, 5, 6, 7, 8, 9, 10)
y_ties <- c(1, 3, 2, 4, 5, 6, 7, 8, 9, 10)

# Check the ranks
rank(x_ties)  # [1] 1 3 3 3 5 6 7 8 9 10

# Spearman handles ties automatically
cor(x_ties, y_ties, method = "spearman")

# cor.test warns about ties affecting p-value exactness
cor.test(x_ties, y_ties, method = "spearman")
# Warning: Cannot compute exact p-value with ties

The warning about exact p-values is usually not a concern with reasonably sized samples. R uses an asymptotic approximation that works well for n > 10.

When to Use Kendall’s Tau Instead

Consider Kendall’s tau (τ) when you have many ties or small samples:

# Compare Spearman and Kendall
cor(x_ties, y_ties, method = "spearman")  # 0.952
cor(x_ties, y_ties, method = "kendall")   # 0.889

# Kendall is more conservative and has better statistical properties
# for small samples with many ties

Interpreting Rho Values

Use these general guidelines for interpretation:

| |ρ| Value | Interpretation | |————|—————-| | 0.00–0.19 | Very weak | | 0.20–0.39 | Weak | | 0.40–0.59 | Moderate | | 0.60–0.79 | Strong | | 0.80–1.00 | Very strong |

Remember: correlation doesn’t imply causation, and these thresholds are context-dependent. A “weak” correlation in physics might be “strong” in social sciences.

Best Practices Summary

Check your assumptions: Spearman requires monotonic relationships. Plot your data first.
Report both coefficient and p-value: Use cor.test(), not just cor().
Handle missing data explicitly: Choose your use parameter deliberately.
Consider sample size: Spearman is less powerful than Pearson with small samples when Pearson’s assumptions are met.
Watch for ties: Many ties can affect both the coefficient and p-value accuracy.

# A complete analysis workflow
analyze_spearman <- function(x, y, alpha = 0.05) {
  # Run test
  test <- cor.test(x, y, method = "spearman")
  
  # Report results
  cat("Spearman Correlation Analysis\n")
  cat("=============================\n")
  cat("n =", length(x), "\n")
  cat("rho =", round(test$estimate, 3), "\n")
  cat("p-value =", format(test$p.value, digits = 4), "\n")
  cat("Significant at α =", alpha, ":", test$p.value < alpha, "\n")
  
  invisible(test)
}

# Usage
analyze_spearman(satisfaction, repeat_purchases)

Spearman correlation is a robust, versatile tool that belongs in every R programmer’s toolkit. Start with cor.test() for single pairs, scale up to Hmisc::rcorr() for matrices, and always visualize your data to confirm the relationship makes sense.