Spearman vs Pearson Correlation

Spearman correlation is a rank-based measure of association, whereas Pearson correlation quantifies the strength of a linear relationship between two continuous variables.

Key Differences

1. Data Type and Distribution

   • Pearson assumes:

     • Continuous variables
     • Linear relationships
     • Normally distributed variables (or large enough samples for CLT)

   • Spearman can be applied to:

     • Discrete, ordinal, or continuous variables
     • Non-normal distributions
     • Nonlinear but monotonic relationships

2. Sensitivity to Outliers

   • Pearson is sensitive to outliers, as it directly uses raw values.
   • Spearman works on ranks, making it more robust to extreme values or skewed distributions.

3. Functional Form of Relationship

   • Pearson measures linear correlation (i.e., whether $y$ changes linearly with $x$).
   • Spearman measures monotonic correlation (i.e., whether $y$ tends to increase or decrease as $x$ increases, regardless of the exact form).

Summary

Spearman correlation is better suited for ordinal, skewed, or discrete data, especially when the relationship is monotonic but non-linear, or when outliers are present. Pearson is preferable when the relationship is expected to be linear and homoscedastic between two continuous variables.

Example: Visualising the Difference

We create a synthetic dataset with:

• A non-linear but monotonic relationship between $x$ and $y$
• Some outliers to test robustness

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import pearsonr, spearmanr
 
np.random.seed(0)
 
# Simulate x (discrete low-cardinality variable)
x = np.random.choice([0, 1, 2, 3, 4], size=100, p=[0.2, 0.3, 0.3, 0.15, 0.05])
 
# Simulate y: nonlinear monotonic trend with noise
y = x**2 + np.random.normal(0, 2, size=100)
 
# Inject outliers
y[::15] += np.random.randint(10, 20, size=7)
 
# Correlations
pearson_corr, _ = pearsonr(x, y)
spearman_corr, _ = spearmanr(x, y)
 
# Plot
plt.figure(figsize=(6, 4))
plt.scatter(x, y, alpha=0.7)
plt.xlabel("x")
plt.ylabel("y")
plt.title("Non-linear Relationship with Outliers")
plt.grid(True)
plt.tight_layout()
plt.show()
 
print(f"Pearson correlation: {pearson_corr:.3f}")
print(f"Spearman correlation: {spearman_corr:.3f}")

Interpretation of Results

Typical output (may vary):

Pearson correlation: 0.68
Spearman correlation: 0.83

• Pearson is reduced due to:

  • Nonlinearity (e.g., quadratic growth)
  • Outliers inflating variance

• Spearman remains strong:

  • Preserves rank order
  • Diminishes influence of outliers

Conclusion

This example highlights that Spearman correlation is more robust and flexible in many real-world settings where:

• Variables are discrete, ranked, or not normally distributed
• The relationship is monotonic but not linear
• Outliers may distort direct measurements of association

For rigorous statistical analysis, it’s important to examine data characteristics and the underlying assumptions before selecting a correlation metric.