jonckheere-test

A Python implementation of the Jonckheere-Terpstra test for ordered alternatives in k independent samples.

Overview

The Jonckheere-Terpstra test is a nonparametric statistical test used to determine whether there is a statistically significant trend across ordered groups. Unlike the Kruskal-Wallis test which only detects any difference between groups, the Jonckheere-Terpstra test specifically tests for an ordered alternative hypothesis—that the populations are either increasing or decreasing across groups.

When to Use This Test

• You have k ≥ 2 independent samples (groups)
• The groups have a natural ordering (e.g., dosage levels, time points, severity grades)
• You want to test if there's an increasing or decreasing trend across groups
• Your data is at least ordinal (ranked)
• You prefer a nonparametric approach (no normality assumption)

Installation

pip install jonckheere-test

Or with uv:

uv add jonckheere-test

Quick Start

import numpy as np
from jonckheere_test import jonckheere_test

# Sample data: scores across three treatment groups
control = [40, 35, 38, 43, 44, 41]
low_dose = [38, 40, 47, 44, 40, 42]
high_dose = [48, 40, 45, 43, 46, 44]

# Combine data and create group labels
data = np.concatenate([control, low_dose, high_dose])
groups = np.repeat([1, 2, 3], 6)

# Test for increasing trend
result = jonckheere_test(data, groups, alternative='increasing')

print(f"JT statistic: {result.statistic}")
print(f"p-value: {result.p_value:.4f}")
print(f"Method: {result.method}")

Output:

JT statistic: 79.0
p-value: 0.0207
Method: asymptotic

Features

Feature	Description
Exact p-values	Computed via convolution for small samples (n ≤ 100) without ties
Asymptotic approximation	Normal approximation with tie-corrected variance for large samples or data with ties
Permutation test	Monte Carlo permutation test for any sample size
Automatic method selection	Intelligently chooses the best method based on your data
Tie handling	Proper variance correction when ties are present

API Reference

jonckheere_test(x, groups, alternative='two-sided', n_perm=None, random_state=None, method=None)

Perform the Jonckheere-Terpstra test for ordered alternatives.

Parameters

Parameter	Type	Description
x	array-like	Sample data values
groups	array-like	Group labels (must be orderable). Values are sorted to determine group order
alternative	'two-sided', 'increasing', or 'decreasing'	The alternative hypothesis to test (default: 'two-sided')
n_perm	int, optional	Number of permutations for permutation test. If provided, uses permutation method
random_state	int, optional	Random seed for reproducibility (permutation test only)
method	'exact', 'asymptotic', or None	Method for p-value computation. If None, automatically selected

Returns

JonckheereResult dataclass with the following attributes:

Attribute	Type	Description
statistic	float	The JT test statistic (J)
p_value	float	The p-value for the test
alternative	str	The alternative hypothesis used
method	str	The method used: 'exact', 'asymptotic', or 'permutation'
mean	float or None	Expected value of J under the null (asymptotic only)
variance	float or None	Variance of J under the null (asymptotic only)
z_score	float or None	Standardized test statistic (asymptotic only)

Method Selection

When method=None (default), the function automatically selects:

1. Permutation: If n_perm is provided
2. Asymptotic: If n > 100 or ties are present
3. Exact: Otherwise (small samples without ties)

Examples

Testing for a Decreasing Trend

# Data showing decreasing performance over time
time_1 = [95, 92, 88, 91, 94]
time_2 = [85, 82, 79, 83, 80]
time_3 = [72, 68, 75, 70, 71]

data = np.concatenate([time_1, time_2, time_3])
groups = np.repeat(['T1', 'T2', 'T3'], 5)

result = jonckheere_test(data, groups, alternative='decreasing')
print(f"p-value: {result.p_value:.4f}")  # Significant decreasing trend

Using Permutation Test

# For more robust p-values or when exact computation is too slow
result = jonckheere_test(
    data, 
    groups, 
    alternative='increasing',
    n_perm=10000,
    random_state=42  # For reproducibility
)
print(f"Permutation p-value: {result.p_value:.4f}")

Forcing a Specific Method

# Force asymptotic method even for small samples
result = jonckheere_test(data, groups, method='asymptotic')

# Access additional statistics
print(f"Expected value (μ): {result.mean}")
print(f"Variance (σ²): {result.variance:.2f}")
print(f"Z-score: {result.z_score:.2f}")

Statistical Background

The Test Statistic

The Jonckheere-Terpstra statistic J is computed as the sum of Mann-Whitney U statistics for all pairs of groups:

$$J = \sum_{i < j} U_{ij}$$

where $U_{ij}$ counts the number of times an observation from group $i$ is less than an observation from group $j$.

Null Distribution

• Exact method: The null distribution is computed via convolution of individual Mann-Whitney distributions
• Asymptotic method: Uses normal approximation with:
  • Mean: $E(J) = \frac{N^2 - \sum n_i^2}{4}$
  • Variance: Includes corrections for ties

Interpretation

• Increasing alternative: Tests if group medians increase with group order
• Decreasing alternative: Tests if group medians decrease with group order
• Two-sided alternative: Tests for any monotonic trend

Dependencies

• Python ≥ 3.9
• NumPy ≥ 1.21
• SciPy ≥ 1.7

References

• Jonckheere, A. R. (1954). "A distribution-free k-sample test against ordered alternatives". Biometrika, 41(1/2), 133-145.
• Terpstra, T. J. (1952). "The asymptotic normality and consistency of Kendall's test against trend". Indagationes Mathematicae, 14, 327-333.
• Hollander, M., Wolfe, D. A., & Chicken, E. (2014). Nonparametric Statistical Methods (3rd ed.). John Wiley & Sons. https://doi.org/10.1002/9781119196037

License

MIT License - see LICENSE for details.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

Author

Ariel Bereslavsky