Properties of Balanced Permutations

When permutation methods are used in practice, often a limited number of random permutations are used to decrease the computational burden. However, most theoretical literature assumes that the whole permutation group is used, and methods based on random permutations tend to be seen as approximate. There exists a very limited amount of literature on exact testing with random permutations, and only recently a thorough proof of exactness was given. In this paper, we provide an alternative proof, viewing the test as a "conditional Monte Carlo test" as it has been called in the literature. We also provide extensions of the result. Importantly, our results can be used to prove properties of various multiple testing procedures based on random permutations.

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“…The role of the group structure has recently been emphasized (Southworth et al. 2009 ; Goeman and Solari 2010 ). Southworth et al.…”

Section: Introductionmentioning

confidence: 99%

“…Southworth et al. ( 2009 ) note that in particular the set of ‘balanced permutations’ cannot be used, since it is not a group.…”

Section: Introductionmentioning

confidence: 99%

Exact testing with random permutations

Hemerik

Goeman

2017

TEST

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“…It is worth noting that the group structure of

\overset{𝕆}{true}

is essential for the validity of the result, as remarked in Southworth, Kim and Owen (). Furthermore, the proof extends that given in Lehmann and Romano (), which is based on a group with finite elements (such as the permutation group), to a group with uncountably infinite elements (such as the orthogonal group).…”

Section: Rotation‐based K‐fwer Controlmentioning

confidence: 62%

Rotation‐based multiple testing in the multivariate linear model

2014

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Abstract:Permutation methods are very useful in several scientic elds. They have the advantage of making fewer assumptions about the data and of providing more reliable inferential results. They are also particularly useful in case of high-dimensional problems since they easily account for dependence between tests, thereby allowing for more powerful multiplicity control procedures. Indeed, Westfall and Young's min-p procedure often improves on the Holm procedure by providing more rejections. The advantage of being able to make fewer assumptions about the process generating the data unfortunately involves an inherent limitation in the way a process can be modeled (e.g. through multiple linear models). In this work, we propose a permutation (and rotation) method which allows the inference in the multivariate linear model even in the presence of covariates (i.e. nuisance parameters, i.e. confounders). Also, the method allows for the immediate application of the min-p procedure. We make clear how permutations are a particular case of rotations of the data. Permutation tests are exact, while rotation tests retain exactness under multiple-multivariate linear model with normal errors. When errors are not normal, the rotation tests are weakly exchangeable (i.e. approximated and asymptotically exact). A real application to genetic data is presented and discussed.

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“…In contrast to identification (see footnote 15) randomness of the chosen permutations is important here: non-random permutation sets do not generally guarantee valid inference (e.g. Southworth et al 2009).…”

Section: A2 Consistency Of Recentered Ivsmentioning

confidence: 99%

Non-random exposure to exogenous shocks: theory and applications

2020

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We develop new tools for causal inference in settings where exogenous shocks affect the treatment status of multiple observations jointly, to different extents. In these settings researchers may construct treatments or instruments that combine the shocks with predetermined measures of shock exposure. Examples include measures of spillovers in social and transportation networks, simulated eligibility instruments, and shift-share instruments. We show that leveraging the exogeneity of shocks for identification generally requires a simple but nonstandard recentering, derived from the specification of counterfactual shocks that might as well have been realized. We further show how specification of counterfactual shocks can be used for finite-sample inference and specification tests, and we characterize the recentered instruments that are asymptotically efficient. We use this framework to estimate the employment effects of Chinese market access growth due to high-speed rail construction and the insurance coverage effects of expanded Medicaid eligibility.

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Properties of Balanced Permutations

Cited by 23 publications

References 16 publications

Exact testing with random permutations

Exact testing with random permutations

Rotation‐based multiple testing in the multivariate linear model

Non-random exposure to exogenous shocks: theory and applications

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