MLstatkit

MLstatkit is a comprehensive Python library designed to seamlessly integrate established statistical methods into machine learning projects. It encompasses a variety of tools, including Delong's test for comparing areas under two correlated Receiver Operating Characteristic (ROC) curves, Bootstrapping for calculating confidence intervals, AUC2OR for converting the Area Under the Receiver Operating Characteristic Curve (AUC) into several related statistics such as Cohen’s d, Pearson’s rpb, odds-ratio, and natural log odds-ratio, and Permutation_test for assessing the statistical significance of the difference between two models' metrics by randomly shuffling the data and recalculating the metrics to create a distribution of differences. With its modular design, MLstatkit offers researchers and data scientists a flexible and powerful toolkit to augment their analyses and model evaluations, catering to a broad spectrum of statistical testing needs within the domain of machine learning.

Installation

Install MLstatkit directly from PyPI using pip:

pip install MLstatkit

Usage

Delong's Test

Delong_test function enables a statistical evaluation of the differences between the areas under two correlated Receiver Operating Characteristic (ROC) curves derived from distinct models. This facilitates a deeper understanding of comparative model performance.

Parameters:

• true : array-like of shape (n_samples,)
  True binary labels in range {0, 1}.

• prob_A : array-like of shape (n_samples,)
  Predicted probabilities by the first model.

• prob_B : array-like of shape (n_samples,)
  Predicted probabilities by the second model.

Returns:

• z_score : float
  The z score from comparing the AUCs of two models.

• p_value : float
  The p value from comparing the AUCs of two models.

Example:

from MLstatkit.stats import Delong_test

# Example data
true = np.array([0, 1, 0, 1])
prob_A = np.array([0.1, 0.4, 0.35, 0.8])
prob_B = np.array([0.2, 0.3, 0.4, 0.7])

# Perform DeLong's test
z_score, p_value = Delong_test(true, prob_A, prob_B)

print(f"Z-Score: {z_score}, P-Value: {p_value}")

This demonstrates the usage of Delong_test to statistically compare the AUCs of two models based on their probabilities and the true labels. The returned z-score and p-value help in understanding if the difference in model performances is statistically significant.

Bootstrapping for Confidence Intervals

The Bootstrapping function calculates confidence intervals for specified performance metrics using bootstrapping, providing a measure of the estimation's reliability. It supports calculation for AUROC (area under the ROC curve), AUPRC (area under the precision-recall curve), and F1 score metrics.

Parameters:

• true : array-like of shape (n_samples,)
  True binary labels, where the labels are either {0, 1}.
• prob : array-like of shape (n_samples,)
  Predicted probabilities, as returned by a classifier's predict_proba method, or binary predictions based on the specified scoring function and threshold.
• metric_str : str, default='f1'
  Identifier for the scoring function to use. Supported values include 'f1', 'accuracy', 'recall', 'precision', 'roc_auc', 'pr_auc', and 'average_precision'.
• n_bootstraps : int, default=1000
  The number of bootstrap iterations to perform. Increasing this number improves the reliability of the confidence interval estimation but also increases computational time.
• confidence_level : float, default=0.95
  The confidence level for the interval estimation. For instance, 0.95 represents a 95% confidence interval.
• threshold : float, default=0.5
  A threshold value used for converting probabilities to binary labels for metrics like 'f1', where applicable.
• average : str, default='macro'
  Specifies the method of averaging to apply to multi-class/multi-label targets. Other options include 'micro', 'samples', 'weighted', and 'binary'.
• random_state : int, default=0
  Seed for the random number generator. This parameter ensures reproducibility of results.

Returns:

• original_score : float
  The score calculated from the original dataset without bootstrapping.
• confidence_lower : float
  The lower bound of the confidence interval.
• confidence_upper : float
  The upper bound of the confidence interval.

Examples:

from MLstatkit.stats import Bootstrapping

# Example data
y_true = np.array([0, 1, 0, 0, 1, 1, 0, 1, 0])
y_prob = np.array([0.1, 0.4, 0.35, 0.8, 0.2, 0.3, 0.4, 0.7, 0.05])

# Calculate confidence intervals for AUROC
original_score, confidence_lower, confidence_upper = Bootstrapping(y_true, y_prob, 'roc_auc')
print(f"AUROC: {original_score:.3f}, Confidence interval: [{confidence_lower:.3f} - {confidence_upper:.3f}]")

# Calculate confidence intervals for AUPRC
original_score, confidence_lower, confidence_upper = Bootstrapping(y_true, y_prob, 'pr_auc')
print(f"AUPRC: {original_score:.3f}, Confidence interval: [{confidence_lower:.3f} - {confidence_upper:.3f}]")

# Calculate confidence intervals for F1 score with a custom threshold
original_score, confidence_lower, confidence_upper = Bootstrapping(y_true, y_prob, 'f1', threshold=0.5)
print(f"F1 Score: {original_score:.3f}, Confidence interval: [{confidence_lower:.3f} - {confidence_upper:.3f}]")

# Calculate confidence intervals for AUROC, AUPRC, F1 score
for score in ['roc_auc', 'pr_auc', 'f1']:
    original_score, conf_lower, conf_upper = Bootstrapping(y_true, y_prob, score, threshold=0.5)
    print(f"{score.upper()} original score: {original_score:.3f}, confidence interval: [{conf_lower:.3f} - {conf_upper:.3f}]")

Permutation Test for Statistical Significance

The Permutation_test function assesses the statistical significance of the difference between two models' metrics by randomly shuffling the data and recalculating the metrics to create a distribution of differences. This method does not assume a specific distribution of the data, making it a robust choice for comparing model performance.

Parameters:

• y_true : array-like of shape (n_samples,)
  True binary labels, where the labels are either {0, 1}.
• prob_model_A : array-like of shape (n_samples,)
  Predicted probabilities from the first model.
• prob_model_B : array-like of shape (n_samples,)
  Predicted probabilities from the second model.
• metric_str : str, default='f1'
  The metric for comparison. Supported metrics include 'f1', 'accuracy', 'recall', 'precision', 'roc_auc', 'pr_auc', and 'average_precision'.
• n_bootstraps : int, default=1000
  The number of permutation samples to generate.
• threshold : float, default=0.5
  A threshold value used for converting probabilities to binary labels for metrics like 'f1', where applicable.
• average : str, default='macro'
  Specifies the method of averaging to apply to multi-class/multi-label targets. Other options include 'micro', 'samples', 'weighted', and 'binary'.
• random_state : int, default=0
  Seed for the random number generator. This parameter ensures reproducibility of results.

Returns:

• metric_a : float
  The calculated metric for model A using the original data.
• metric_b : float
  The calculated metric for model B using the original data.
• p_value : float
  The p-value from the permutation test, indicating the probability of observing a difference as extreme as, or more extreme than, the observed difference under the null hypothesis.
• benchmark : float
  The observed difference between the metrics of model A and model B.
• samples_mean : float
  The mean of the permuted differences.
• samples_std : float
  The standard deviation of the permuted differences.

Examples:

from MLstatkit.stats import Permutation_test

y_true = np.array([0, 1, 0, 0, 1, 1, 0, 1, 0])
prob_model_A = np.array([0.1, 0.4, 0.35, 0.8, 0.2, 0.3, 0.4, 0.7, 0.05])
prob_model_B = np.array([0.2, 0.3, 0.25, 0.85, 0.15, 0.35, 0.45, 0.65, 0.01])

# Conduct a permutation test to compare F1 scores
metric_a, metric_b, p_value, benchmark, samples_mean, samples_std = Permutation_test(
    y_true, prob_model_A, prob_model_B, 'f1'
)

print(f"F1 Score Model A: {metric_a:.5f}, Model B: {metric_b:.5f}")
print(f"Observed Difference: {benchmark:.5f}, p-value: {p_value:.5f}")
print(f"Permuted Differences Mean: {samples_mean:.5f}, Std: {samples_std:.5f}")

Conversion of AUC to Odds Ratio (OR)

The AUC2OR function converts an Area Under the Curve (AUC) value to an Odds Ratio (OR) and optionally returns intermediate values such as t, z, d, and ln_OR. This conversion is useful for understanding the relationship between AUC, a common metric in binary classification, and OR, which is often used in statistical analyses.

Parameters:

• AUC : float
  The Area Under the Curve (AUC) value to be converted.
• return_all : bool, default=False
  If True, returns intermediate values (t, z, d, ln_OR) in addition to OR.

Returns:

• OR : float
  The calculated Odds Ratio (OR) from the given AUC value.
• t : float, optional
  Intermediate value calculated from AUC.
• z : float, optional
  Intermediate value calculated from t.
• d : float, optional
  Intermediate value calculated from z.
• ln_OR : float, optional
  The natural logarithm of the Odds Ratio.

Examples:

from MLstatkit.stats import AUC2OR

AUC = 0.7  # Example AUC value

# Convert AUC to OR and retrieve all intermediate values
t, z, d, ln_OR, OR = AUC2OR(AUC, return_all=True)

print(f"t: {t:.5f}, z: {z:.5f}, d: {d:.5f}, ln_OR: {ln_OR:.5f}, OR: {OR:.5f}")

# Convert AUC to OR without intermediate values
OR = AUC2OR(AUC)
print(f"OR: {OR:.5f}")

References

Delong's Test

The implementation of Delong_test in MLstatkit is based on the following publication:

• Xu Sun and Weichao Xu, "Fast implementation of DeLong’s algorithm for comparing the areas under correlated receiver operating characteristic curves," in IEEE Signal Processing Letters, vol. 21, no. 11, pp. 1389-1393, 2014, IEEE.

Bootstrapping

The Bootstrapping method for calculating confidence intervals does not directly reference a single publication but is a widely accepted statistical technique for estimating the distribution of a metric by resampling with replacement. For a comprehensive overview of bootstrapping methods, see:

• B. Efron and R. Tibshirani, "An Introduction to the Bootstrap," Chapman & Hall/CRC Monographs on Statistics & Applied Probability, 1994.

Permutation Test

The Permutation_tests are utilized to assess the significance of the difference in performance metrics between two models by randomly reallocating observations to groups and computing the metric. This approach does not make specific distributional assumptions, making it versatile for various data types. For a foundational discussion on permutation tests, refer to:

• P. Good, "Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses," Springer Series in Statistics, 2000.

These references lay the groundwork for the statistical tests and methodologies implemented in MLstatkit, providing users with a deep understanding of their scientific basis and applicability.

AUC2OR

The AUR2OR function converts the Area Under the Receiver Operating Characteristic Curve (AUC) into several related statistics including Cohen’s d, Pearson’s rpb, odds-ratio, and natural log odds-ratio. This conversion is particularly useful in interpreting the performance of classification models. For a detailed explanation of the mathematical formulas used in this conversion, refer to:

• Salgado, J. F. (2018). "Transforming the area under the normal curve (AUC) into Cohen’s d, Pearson’s rpb, odds-ratio, and natural log odds-ratio: Two conversion tables." European Journal of Psychology Applied to Legal Context, 10(1), 35-47.

These references provide the mathematical foundation for the AUR2OR function, ensuring that users can accurately interpret the statistical significance and practical implications of their model performance metrics.

Contributing

We welcome contributions to MLstatkit! Please see our contribution guidelines for more details.

License

MLstatkit is distributed under the MIT License. For more information, see the LICENSE file in the GitHub repository.

Update log

• 0.1.7 Update README.md
• 0.1.6 Debug.
• 0.1.5 Update README.md, Add AUC2OR function.
• 0.1.4 Update README.md, Add Permutation_tests function, Re-do Bootstrapping Parameters.
• 0.1.3 Update README.md.
• 0.1.2 Add Bootstrapping operation process progress display.
• 0.1.1 Update README.md, setup.py. Add CONTRIBUTING.md.
• 0.1.0 First edition