Simulating the spurious correlation of ratios. Run this script multiple times and see the correlations between variables.

pearson_1986_spurious_correlations.py

"""Simulate the spurious correlation of ratios.

Run this script multiple times and see the correlations between variables.

Reference
---------
- Pearson, K. (1896). Mathematical Contributions to the Theory of Evolution.
On a Form of Spurious Correlation Which May Arise When Indices Are Used in the
Measurement of Organs. Proceedings of the Royal Society of London, 60, 489–498.
<https://doi.org/10.1098/rspl.1896.0076>

"""

import numpy as np

# Sample size
N = 1000

# Select three measurements at random
x = np.random.random(N)
y = np.random.random(N)
z = np.random.random(N)

# Compute their x, y, z pair correlations
corr_xy = np.corrcoef(x, y)[0, 1]
corr_yz = np.corrcoef(y, z)[0, 1]
corr_xz = np.corrcoef(x, z)[0, 1]

# Ratio pairs
u = x / y
v = z / y

# Compute ratio correlations
corr_uv = np.corrcoef(u, v)[0, 1]

print(f"Correlation xy = {corr_xy:.2f}")
print(f"Correlation yz = {corr_yz:.2f}")
print(f"Correlation xz = {corr_xz:.2f}")
print(f"Correlation uv = {corr_uv:.3f}")