Permutation Tests

Description

Permutation tests assess the statistical significance of observed results (such as model accuracy, feature importance, or group differences) by comparing them to what would occur purely by chance. The technique randomly shuffles labels or data thousands of times, recalculating the metric of interest each time to build an empirical null distribution. If the actual observed result falls in the extreme tail of this distribution (typically beyond the 95th or 99th percentile), it provides strong evidence that the relationship is genuine rather than due to random chance, without requiring parametric assumptions about data distributions.

Example Use Cases

Reliability

Validating feature importance in medical diagnosis models by permuting each feature 10,000 times to ensure that identified risk factors (e.g., blood pressure, cholesterol) have statistically significant predictive power beyond random chance.

Verifying that a model's claimed 95% accuracy on test data is genuinely better than random guessing by permuting labels 5,000 times and confirming the actual accuracy falls beyond the 99th percentile of the null distribution.

Explainability

Testing whether observed differences in loan approval rates between demographic groups are statistically significant by permuting group labels and calculating the approval rate difference distribution under the null hypothesis of no discrimination.

Limitations

Computationally expensive as it requires thousands of model evaluations or metric calculations, scaling poorly with dataset size and model complexity.
Requires many permutations (typically 5,000-10,000) to achieve reliable p-values for strict significance thresholds like p < 0.01.
Assumes exchangeability of observations under the null hypothesis, which may be violated in time series or hierarchical data structures.
Cannot be easily parallelised for some metrics that require global model retraining, limiting scalability for complex machine learning pipelines.

Resources

Research Papers

Permutation Tests for Classification

Golland, Polina, Mukherjee, Sayan, and Panchenko, Dmitry•Jan 1, 2003

The Exchangeability Assumption for Permutation Tests of Multiple Regression Models: Implications for Statistics and Data Science Educators

Johanna Hardin et al.•Jun 11, 2024

Permutation tests are a powerful and flexible approach to inference via resampling. As computational methods become more ubiquitous in the statistics curriculum, use of permutation tests has become more tractable. At the heart of the permutation approach is the exchangeability assumption, which determines the appropriate null sampling distribution. We explore the exchangeability assumption in the context of permutation tests for multiple linear regression models, including settings where the assumption is not tenable. Various permutation schemes for the multiple linear regression setting have been proposed and assessed in the literature. As has been demonstrated previously, in most settings, the choice of how to permute a multiple linear regression model does not materially change inferential conclusions with respect to Type I errors. However, some violations (e.g., when clustering is not appropriately accounted for) lead to issues with Type I error rates. Regardless, we believe that understanding (1) exchangeability in the multiple linear regression setting and also (2) how it relates to the null hypothesis of interest is valuable. We close with pedagogical recommendations for instructors who want to bring multiple linear regression permutation inference into their classroom as a way to deepen student understanding of resampling-based inference.

Tutorials

How to use Permutation Tests | Towards Data Science

Michael Berk•Sep 21, 2021

Permutation test in R | Towards Data Science

Serafim Petrov•Mar 15, 2021

Documentations

scikit-learn permutation_importance

Scikit-learn Developers•Jan 1, 2007

Related Techniques

Name	Description	Assurance Goals
SHapley Additive exPlanations	SHAP explains model predictions by quantifying how much each input feature contributes to the outcome. It assigns an importance score to every feature, indicating whether it pushes the prediction towards or away from the average. The method systematically evaluates how predictions change as features are included or excluded, drawing on game theory concepts to ensure a fair distribution of contributions.	Explainability Fairness Reliability
Permutation Importance	Permutation Importance quantifies a feature's contribution to a model's performance by randomly shuffling its values and measuring the resulting drop in predictive accuracy. If shuffling a feature significantly degrades the model's performance, that feature is considered important. This model-agnostic technique helps identify which inputs are genuinely driving predictions, rather than just being correlated with the outcome.	Explainability Reliability
Sobol Indices	Sobol Indices quantify how much each input feature contributes to the total variance in a model's predictions through global sensitivity analysis. The technique calculates first-order indices (individual feature contributions) and total-order indices (including all interaction effects involving that feature). By systematically sampling the input space and decomposing output variance, Sobol Indices reveal which features drive model uncertainty and which interactions between features are most important for predictions.	Explainability Fairness
Bootstrapping	Bootstrapping estimates uncertainty by repeatedly resampling the original dataset with replacement to create many new training sets, training a model on each sample, and analysing the variation in predictions. This approach provides confidence intervals and stability measures without making strong statistical assumptions. By showing how predictions change with different random samples of the data, it reveals how sensitive the model is to the specific training examples and provides robust uncertainty estimates.	Reliability Transparency Fairness