Model-free variable importance testing with machine learning methods

Model-free variable importance testing with machine learning methods | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Model-free variable importance testing with machine learning methods Xinyu Zhang, Xu Guo, Niwen Zhou, Xuejun Jiang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7110795/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this paper, we investigate variable importance testing problems in a model-free framework. Some remarkable procedures have been developed recently. Despite their success, existing procedures suffer from a significant limitation, that is, they generally require a larger training sample and do not have the fastest possible convergence rate under alternative hypothesis. In this paper, we propose a new procedure to test variable importance. Flexible machine learning methods are adopted to estimate unknown functions. Under the null hypothesis, our proposed test statistic converges to the standard chi-squared distribution. While under local alternative hypotheses, it converges to the non-central chi-square distribution. It has non-trivial power against the local alternative hypothesis which converges to the null at the fastest possible rate. We also extend our procedure to test conditional independence. Asymptotic properties are also developed. Numerical studies and two real data examples are conducted to illustrate the performance of our proposed test statistic. Variable importance machine learning methods conditional independence Full Text Additional Declarations No competing interests reported. Supplementary Files suppModelfreevariableimportancetestingwithmachinelearningmethods1.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7110795","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":489323374,"identity":"f456b5f7-6eb2-48f9-b716-bbd367df5224","order_by":0,"name":"Xinyu Zhang","email":"","orcid":"","institution":"Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xinyu","middleName":"","lastName":"Zhang","suffix":""},{"id":489323377,"identity":"0002bd1b-6bd3-4b5b-aa9f-cf94e4f226d5","order_by":1,"name":"Xu Guo","email":"","orcid":"","institution":"Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xu","middleName":"","lastName":"Guo","suffix":""},{"id":489323380,"identity":"ed68fda0-7ac8-4e21-81f9-eda29df4124c","order_by":2,"name":"Niwen Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVRIiWNgGAWjYHACNiC2SQAzeUjQkka6lsMkaDGXSH/24GPb+TxziQTGB2/bGOTNCWmxnJGQbjiz7XYxkMFsOLeNwXBnAwEtBrcTjknztt1O3HAjgQ3IYEgwOEBQS2IbUOU5kBb230RqSQYZfgBsCzNxWu4/Y5OccS652ODMw2bJOeckDDcQ1HLm+DOJD2V2eQbHkw9+eFNmI0/QFiTA2AAkJIhXPwpGwSgYBaMANwAAVQVANxM5lZcAAAAASUVORK5CYII=","orcid":"","institution":"Beijing Normal University","correspondingAuthor":true,"prefix":"","firstName":"Niwen","middleName":"","lastName":"Zhou","suffix":""},{"id":489323383,"identity":"6ab1a68b-2ff8-4293-86b8-9dfb184a10ed","order_by":3,"name":"Xuejun Jiang","email":"","orcid":"","institution":"Southern University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xuejun","middleName":"","lastName":"Jiang","suffix":""}],"badges":[],"createdAt":"2025-07-13 02:53:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7110795/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7110795/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89991131,"identity":"e261c476-fb60-4463-b013-26cba7cdcdfb","added_by":"auto","created_at":"2025-08-27 07:18:22","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":440695,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7110795/v1_covered_2ab30990-6731-4bf2-992c-868656b56b2a.pdf"},{"id":87455273,"identity":"0c03b299-965b-43af-b8a5-76cbb7210dbd","added_by":"auto","created_at":"2025-07-24 04:13:04","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":89892,"visible":true,"origin":"","legend":"","description":"","filename":"suppModelfreevariableimportancetestingwithmachinelearningmethods1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7110795/v1/2f8c5715fac7ee7763a65229.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Model-free variable importance testing with machine learning methods","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Variable importance, machine learning methods, conditional independence","lastPublishedDoi":"10.21203/rs.3.rs-7110795/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7110795/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, we investigate variable importance testing problems in a model-free framework. 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