Permutation Tests for Linear Models
Abstract
Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.
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Anderson and Robinson (2001) find that in samples of the size of Perry, the Freedman-Lane procedure generates test statistics distributed most like those generated by the exact method and in close agreement with p-values from the true distribution when regression coefficients are known.
“They find that in samples of the size of Perry, the Freedman–Lane procedure generates test statistics that are distributed most like those generated by the exact method, and are in close agreement with the p-values from the true distribution when regression coefficients are known.”
From Analyzing social experiments as implemented: A reexamination of the evidence from the HighScope Perry Preschool Program (2010) · cites this paper as Anderson and Robinson (2001)
Cites
Statistical Tests Based on Permutations of the Observations
Wald, A., & Wolfowitz, J. (1944) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177731207 · Cited by 260 · see citing works
Tests for no treatment effect in randomized clinical trials
GAIL, M. H., TAN, W. Y., & PIANTADOSI, S. (1988) · Biometrika · doi:10.1093/biomet/75.1.57 · Cited by 143 · see citing works
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Permutation inference for the general linear model
Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., & Nichols, T. E. (2014) · NeuroImage · doi:10.1016/j.neuroimage.2014.01.060 · Cited by 3,563 · see citing works
Distance‐Based Tests for Homogeneity of Multivariate Dispersions
Anderson, M. J. (2006) · Biometrics · doi:10.1111/j.1541-0420.2005.00440.x · Cited by 3,304 · see citing works
Self-citationMRtrix3: A fast, flexible and open software framework for medical image processing and visualisation
Tournier, J.-D., Smith, R., Raffelt, D., Tabbara, R., Dhollander, T., Pietsch, M., Christiaens, D., Jeurissen, B., Yeh, C.-H., & Connelly, A. (2019) · NeuroImage · doi:10.1016/j.neuroimage.2019.116137 · Cited by 2,974 · see citing works
(2010) · Quantitative Economics · doi:10.3982/qe8 · Cited by 304 · see citing works
“In another paper, Anderson and Robinson (2001) compare an exact permutation method (where δ is known) with a variety of permutation-based methods.”
Young, J. M., Vandewouw, M. M., Mossad, S. I., Morgan, B. R., Lee, W., Smith, M. L., Sled, J. G., & Taylor, M. J. (2019) · NeuroImage: Clinical · doi:10.1016/j.nicl.2019.101855 · Cited by 57 · see citing works
Unique white matter microstructural patterns in ADHD presentations—a diffusion tensor imaging study
Svatkova, A., Nestrasil, I., Rudser, K., Goldenring Fine, J., Bledsoe, J., & Semrud‐Clikeman, M. (2016) · Human Brain Mapping · doi:10.1002/hbm.23243 · Cited by 44 · see citing works
“The confidence limits for the TFCE analysis with number of permutation set on 5,000 was 0.05 6 0.006 (for P-value 5 0.05) [Anderson and Robinson, 2001].”
MethodInfluentialA cross‐comparative analysis of in vivo versus ex vivo MRI indices in a mouse model of concussion
Vinh To, X., Kurniawan, N. D., Cumming, P., & Nasrallah, F. A. (2023) · Brain Research · doi:10.1016/j.brainres.2023.148562 · Cited by 5 · see citing works
- Authors
- Anderson, M. J., & Robinson, J.
- Affiliations
- The University of Sydney (AU)
- Corresponding author
- Robinson, John
- Venue
- Australian & New Zealand Journal of Statistics
- Publisher
- Wiley
- Published
- 2001-03
- Volume
- 43
- Issue
- 1
- Pages
- 75-88
- ISSN
- 1369-1473 (ISSN-L)
- 1467-842X
- OpenAlex
- W1990510254
- Cited by
- 555
- References
- 2
- Citations by year
- 2026: 22 · 2025: 23 · 2024: 20 · 2023: 23 · 2022: 24 · 2021: 46 · 2020: 37 · 2019: 35 · 2018: 34 · 2017: 28 · 2016: 40 · 2015: 38 · 2014: 58 · 2013: 28 · 2012: 31
- FWCI
- 4.43
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Australian & New Zealand Journal of Statistics (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- permutation tests, linear models, partial regression coefficients, exact tests, asymptotic normality, asymptotic correlation, local alternatives, asymptotic power, critical values
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