18.6 Correlation Tests

This section covers three widely used statistical tests for assessing the significance of the relationship between two variables: Pearson's correlation, Spearman's rank correlation, and Kendall's tau.

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Setup: Shared Data Generation

The following modules generate three types of data to illustrate different correlation scenarios.

global_name_space.py

import argparse
import numpy as np

parser = argparse.ArgumentParser(description='Correlation Test Examples')
parser.add_argument('--seed', type=int, default=1, metavar='S',
                    help='random seed (default: 1)')
ARGS = parser.parse_args()

np.random.seed(ARGS.seed)
ARGS.size = 1000

load_data.py

import numpy as np
from global_name_space import ARGS

def load_data(data_type=0):
    data_dict = {}

    x = np.random.rand(ARGS.size) * 20
    eps = np.random.rand(ARGS.size) * 10

    # Dataset 0: No relationship (random scatter)
    y = np.random.rand(ARGS.size) * 20
    data_dict[0] = (x, y)

    # Dataset 1: Monotonic nonlinear relationship (cubic)
    y = (x + eps) ** 3
    data_dict[1] = (x, y)

    # Dataset 2: Non-monotonic relationship (sine)
    y = np.sin(x + eps)
    data_dict[2] = (x, y)

    return data_dict

The three datasets represent:

• Dataset 0: No relationship — random scatter, both Pearson and rank-based correlations should be near zero.
• Dataset 1: Monotonic nonlinear — Pearson may understate the relationship since it measures linearity, but Spearman and Kendall (which measure monotonicity) should detect it.
• Dataset 2: Non-monotonic (sinusoidal) — all correlation measures should be weak since the relationship is periodic, not monotonic or linear.

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Pearson's Correlation Test

Documentation: scipy.stats.pearsonr

Pearson's \(r\) measures the linear relationship between two variables. The test statistic under the null hypothesis \(H_0: \rho = 0\) follows a \(t\)-distribution with \(n-2\) degrees of freedom.

import matplotlib.pyplot as plt
import scipy.stats as stats
from load_data import load_data

def main():
    data_dict = load_data()
    _, axes = plt.subplots(1, len(data_dict), figsize=(12, 3))

    for ax, (x, y) in zip(axes, data_dict.values()):
        ax.plot(x, y, ".k")
        coef, p_val = stats.pearsonr(x, y)
        ax.set_title(f"Pearson's r: {coef:.4f}\np-value: {p_val:.4f}")
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    main()

When to use: When both variables are continuous and you expect a linear relationship. Pearson's \(r\) is sensitive to outliers and assumes bivariate normality for the test's p-value to be exact.

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Spearman's Rank Correlation Test

Video: Spearman's Rank Correlation | Documentation: scipy.stats.spearmanr

Spearman's \(\rho_s\) measures the monotonic relationship between two variables. It is computed as Pearson's \(r\) applied to the ranks of the data rather than the raw values. This makes it robust to outliers and applicable to non-linear but monotonic relationships.

import matplotlib.pyplot as plt
import scipy.stats as stats
from load_data import load_data

def main():
    data_dict = load_data()
    _, axes = plt.subplots(1, len(data_dict), figsize=(12, 3))

    for ax, (x, y) in zip(axes, data_dict.values()):
        ax.plot(x, y, ".k")
        coef, p_val = stats.spearmanr(x, y)
        ax.set_title(f"Spearman's ρ: {coef:.4f}\np-value: {p_val:.4f}")
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    main()

When to use: When the relationship may be monotonic but not necessarily linear, or when the data contains outliers or is ordinal.

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Kendall's Tau

Video 1: Kendall's Tau Explained | Video 2: Kendall's Tau Calculation | Documentation: scipy.stats.kendalltau

Kendall's \(\tau\) also measures the strength of a monotonic relationship, but it is based on the number of concordant and discordant pairs rather than ranks. It tends to be more robust for small sample sizes and has better statistical properties for hypothesis testing.

\[ \tau = \frac{(\text{number of concordant pairs}) - (\text{number of discordant pairs})}{\binom{n}{2}} \]

import matplotlib.pyplot as plt
import scipy.stats as stats
from load_data import load_data

def main():
    data_dict = load_data()
    _, axes = plt.subplots(1, len(data_dict), figsize=(12, 3))

    for ax, (x, y) in zip(axes, data_dict.values()):
        ax.plot(x, y, ".k")
        coef, p_val = stats.kendalltau(x, y)
        ax.set_title(f"Kendall's τ: {coef:.4f}\np-value: {p_val:.4f}")
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    main()

When to use: Similar situations to Spearman's \(\rho_s\), but preferred when sample sizes are small or when you want a more interpretable measure based on pairwise concordance.

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Comparison of the Three Tests

Feature	Pearson's \(r\)	Spearman's \(\rho_s\)	Kendall's \(\tau\)
Measures	Linear association	Monotonic association	Monotonic association
Data type	Continuous	Continuous or ordinal	Continuous or ordinal
Sensitivity to outliers	High	Low	Low
Assumption	Bivariate normality (for exact p-value)	None (rank-based)	None (rank-based)
Range	\([-1, 1]\)	\([-1, 1]\)	\([-1, 1]\)
Small samples	Less reliable	Moderate	More reliable

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Worked Example: Age and Income

Problem: Test whether age and income are related using \(\alpha = 0.05\).

age    = [18, 25, 57, 45, 26, 64, 37, 40, 24, 33]
income = [15000, 29000, 68000, 52000, 32000, 80000, 41000, 45000, 26000, 33000]

Solution

All three tests yield \(p \approx 0.0000\), indicating a strong, statistically significant relationship between age and income.

import matplotlib.pyplot as plt
import scipy.stats as stats

def main():
    x = [18, 25, 57, 45, 26, 64, 37, 40, 24, 33]
    y = [15_000, 29_000, 68_000, 52_000, 32_000, 80_000, 41_000, 45_000, 26_000, 33_000]

    coef, p_val = stats.pearsonr(x, y)
    print(f"Pearson's r:   coef = {coef:.4f},  p-value = {p_val:.4f}")

    coef, p_val = stats.spearmanr(x, y)
    print(f"Spearman's ρ:  coef = {coef:.4f},  p-value = {p_val:.4f}")

    coef, p_val = stats.kendalltau(x, y)
    print(f"Kendall's τ:   coef = {coef:.4f},  p-value = {p_val:.4f}")

    fig, ax = plt.subplots(figsize=(12, 3))
    ax.plot(x, y, "ok")
    ax.set_xlabel("Age")
    ax.set_ylabel("Income")
    plt.show()

if __name__ == "__main__":
    main()

Interpretation: Since all p-values are well below \(\alpha = 0.05\), we reject \(H_0: \rho = 0\) and conclude that there is a statistically significant positive relationship between age and income in this sample. Note that this does not establish a causal relationship—confounders such as experience, education, and industry could influence both variables.