mtest-py 0.1.4

A Procedure for Multicollinearity Testing using Bootstrap

Functions to detect and quantify multicollinearity via a nonparametric pairs bootstrap.

MTest reports achieved significance levels (ASL; bootstrap proportions) for two widely used rules:

• Klein's rule: flag multicollinearity if $R^2_j > R^2_g$
• VIF rule: flag multicollinearity if $\mathrm{VIF}_j$ is large, with $\mathrm{VIF}_j = \dfrac{1}{1 - R^2_j}$

Reference: Morales-Oñate & Morales-Oñate (2023). MTest: a Bootstrap Test for Multicollinearity. Revista Politécnica, 51(2), 53–62.
DOI: https://doi.org/10.33333/rp.vol51n2.05

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What MTest does

Given a fitted linear model, MTest:

1. Resamples rows of the model frame (pairs bootstrap) nboot times.
2. At each bootstrap replicate, recomputes the global $R^2_g$ and the auxiliary $R^2_j$ (regressing each predictor on the rest), using the same expanded design matrix as the original fit. This is robust to log(), I(), interactions, factors, poly(), etc.
3. Returns bootstrap distributions and ASL (bootstrap proportions) for:
   • VIF rule (threshold on $R^2_j$):

$$ \mathrm{ASL}_{\mathrm{VIF}}(j) = \mathbb{P}\big(R^2_j > c\big) $$

Example: `valor_vif = 0.90` implies a VIF cutoff of $1 / (1 - 0.90) = 10$.

• Klein's rule:

$$ \mathrm{ASL}_{\mathrm{Klein}}(j) = \mathbb{P}\big(R^2_g < R^2_j\big). $$

These ASLs are simple bootstrap proportions of the corresponding events (no additional parametric assumptions).

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Model context

Linear regression model:

$$ Y_i = \beta_0 + \beta_1 X_{1i} + \cdots + \beta_p X_{pi} + u_i, \quad i=1,\ldots,n. $$

Auxiliary regressions (one per predictor):

$$ X_{ji} = \gamma_0 + \sum_{k \ne j} \gamma_k X_{ki} + e_{ji}, \quad j=1,\ldots,p. $$

Let $R^2_g$ be the global $R^2$ and $R^2_j$ the $R^2$ of the $j$-th auxiliary regression.

Installation

pip install mtest_py

Quickstart

Example 1: Multicollinearity Test (MTest)

import pandas as pd
from mtest import mtest, mtest_summary

# Load dataset (mtcars equivalent in R)
url = "https://raw.githubusercontent.com/selva86/datasets/master/mtcars.csv"
mtcars = pd.read_csv(url)

X = mtcars[["disp", "hp", "wt", "qsec"]]   # predictors
y = mtcars["mpg"].to_numpy()               # response

# Run MTest
res = mtest(X, y, n_boot=500, r2_threshold=0.9, seed=123, add_intercept=True)

# Print results
print("R² global:", res["R2_global"])
print("VIF:", res["VIF_named"])
print("p-values VIF rule:", res["p_vif"])
print("p-values Klein rule:", res["p_klein"])

# Tabular summary
df_sum = mtest_summary(res, sort_by="VIF")
print(df_sum)

Example 2: Pairwise Kolmogorov–Smirnov Test

from mtest import pairwise_ks_test, ks_summary

X = mtcars[["disp", "hp", "wt", "qsec"]]

ks_res = pairwise_ks_test(X, alternative="greater")
summary = ks_summary(ks_res, digits=6)

print(summary["summary_text"])

API

mtest(X, y, n_boot=1000, nsam=None, r2_threshold=0.9, seed=None, return_distributions=True)

• X: array-like (n, p) predictors. Intercept is not added automatically.
• y: array-like (n,) response.
• n_boot: bootstrap replicates.
• nsam: bootstrap sample size (default: n).
• r2_threshold: threshold on auxiliary R² used for VIF rule.
• seed: RNG seed.
• return_distributions: if True, returns bootstrap arrays.

Return: dict with keys

• R2_global, R2_aux (original sample),
• VIF (original sample),
• B_R2_global (n_boot,),
• B_R2_aux (n_boot, p), columns aligned with predictors,
• p_vif (dict), p_klein (dict).

Notes

• For the VIF rule we use Pr(R²_j > r2_threshold) — pass r2_threshold accordingly.
• Klein's rule p-value is Pr(R²_global < R²_j) across bootstrap replicates.
• Numerical stability: we use least squares and guard divisions-by-zero.

Citation

Morales-Oñate, V., & Morales-Oñate, B. (2023).
MTest: a Bootstrap Test for Multicollinearity. Revista Politécnica, 51(2), 53–62.
https://doi.org/10.33333/rp.vol51n2.05

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License

MIT (or your package license). Include the corresponding LICENSE file in the repo.

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