MTest: 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.

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Installation

# From CRAN (if available):
# install.packages("MTest")

# From source (package root):
# R CMD build .
# R CMD INSTALL MTest_x.y.z.tar.gz

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Quick start

library(MTest)

data(simDataMTest)
fit <- lm(y ~ ., data = simDataMTest)

set.seed(123)
out <- MTest(fit, nboot = 200, valor_vif = 0.90, trace = FALSE)

# ASL (bootstrap p-values)
out$pval_vif      # ASL_VIF(j)   = Pr(R^2_j > valor_vif)
out$pval_klein    # ASL_Klein(j) = Pr(R^2_g < R^2_j)

# Inspect distributions / summary
head(out$Bvals)   # nboot x (p+1): columns = "global" and one per predictor
print(out)        # compact console summary (print.MTest)

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Returned components

MTest() returns an object of class "MTest" with:

• Bvals — matrix nboot x (p+1) with bootstrap $R^2$: first column "global" for $R^2_g$, then one column per predictor for $R^2_j$.
• VIFvals — matrix nboot x p with bootstrap VIF per predictor.
• pval_vif — named vector with $\mathbb{P}(R^2_j >$ valor_vif$)$.
• pval_klein — named vector with $\mathbb{P}(R^2_g < R^2_j)$.
• vif.tot, R.tot — observed (non-bootstrap) VIF and $R^2$.
• nsam, nboot — bootstrap sample size and iterations used.

Note (factors): VIF is returned per design column. If you need group-wise GVIFs (as in car::vif), compute them separately.

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Interpreting ASL

• High pval_klein[j] → $R^2_j$ often exceeds $R^2_g$ (Klein's rule frequently triggered).
• High pval_vif[j] → $R^2_j$ often exceeds valor_vif. Equivalently, $\mathrm{VIF}_j$ exceeds the implied cutoff $1/(1-c)$.

Do not rely on a single rule: inspect both and consider the modeling context.

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Pairwise KS helper

pairwiseKStest() runs pairwise Kolmogorov–Smirnov tests on matrix columns (e.g., the $R^2_j$ columns of Bvals), returning a p-value matrix and a simple Suggestion ranking that mirrors the original behavior.

# Compare only predictors, exclude "global":
ks_res <- pairwiseKStest(out$Bvals[, -1],
                         alternative = "greater",  # "less" or "two.sided"
                         use = "asis",             # same behavior as original
                         exact = NULL)             # let ks.test decide

# Full p-value matrix (rows = x, cols = y)
print(ks_res)             # shows the matrix and a compact summary
ks_res$KSpwMatrix

# Suggestion field:
#   alternative = "greater" -> sort row sums (descending)
#   alternative = "less"    -> sort column sums (descending)
ks_res$Suggestion

# Object class (S3):
class(ks_res)
# [1] "pairwiseKStest" "list"

The package also provides a print.pairwiseKStest() method that always prints the p-value matrix and, optionally, a row-wise minima summary.

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Reproducibility tips

• Set seed= in MTest() (and/or call set.seed() before running).
• Use trace = TRUE for a progress bar on long runs.

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Minimal API

MTest(object, nboot = 100, nsam = NULL, trace = FALSE, seed = NULL,
      valor_vif = 0.9)

pairwiseKStest(X,
               alternative = c("greater","less","two.sided"),
               use = c("asis","pairwise.complete.obs"),
               exact = NULL)

• valor_vif is a threshold on $R^2_j$. The implied VIF cutoff is $1/(1-$ valor_vif $)$ (e.g., 0.90 ↔ 10).
• pairwiseKStest() expects a numeric matrix/data frame; for Bvals you typically pass Bvals[, -1] to exclude "global".

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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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Acknowledgements

Thanks to the R community and the maintainers of stats, car, and related tooling.

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Changelog (from earlier README)

• Clarified that ASL are bootstrap proportions for Klein and VIF events.
• Documented the pairs bootstrap scheme and robustness to transformed terms (log(), I(), interactions, factors, poly()).
• Added the pairwiseKStest helper with class c("pairwiseKStest","list") and a print() method that prints the full p-value matrix.