{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T18:34:06Z","timestamp":1774290846746,"version":"3.50.1"},"reference-count":55,"publisher":"Public Library of Science (PLoS)","issue":"3","license":[{"start":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T00:00:00Z","timestamp":1774224000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["CHRC (UIDP\/04923\/2020)"],"award-info":[{"award-number":["CHRC (UIDP\/04923\/2020)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04152"],"award-info":[{"award-number":["UIDB\/04152"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>\n                    A new inner weighting scheme for Partial Least Squares \u2013 Path Modelling (PLS-PM) is proposed to detect and approximate nonlinear structural relationships in Structural Equation Models (SEM). PLS-PM is an iterative method used for the estimation of Structural Equation Models (SEM), a widely used analytical tool for assessing causal relationships between latent variables. However, PLS-PM struggles to address the structural nonlinear relationships. To address this limitation, a new PLS-PM inner weighting scheme,\n                    <jats:italic>smooth weighting<\/jats:italic>\n                    , is proposed as an additional option to the traditional\n                    <jats:italic>centroid<\/jats:italic>\n                    ,\n                    <jats:italic>factor<\/jats:italic>\n                    , and\n                    <jats:italic>path weighting<\/jats:italic>\n                    schemes. A real marketing dataset is used to demonstrate the usefulness of the method for finding evidence of nonlinearity, and a simulated dataset is used to assess its ability to approximate underlying (unknown) nonlinear structural relationships. The results show that the proposed scheme can recover several nonlinear functional forms, outperforming existing inner weighting schemes for commonly used sample sizes (larger than 75 units), regardless of the level of error contamination in the observed manifest variables.\n                  <\/jats:p>","DOI":"10.1371\/journal.pone.0345111","type":"journal-article","created":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T17:46:46Z","timestamp":1774288006000},"page":"e0345111","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["Beyond linearity - a new Partial Least Squares - Path Modelling (PLS-PM) inner weighting scheme for detecting and approximating nonlinear structural relationships in Structural Equation Models"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2251-3803","authenticated-orcid":true,"given":"Jorge M.","family":"Mendes","sequence":"first","affiliation":[]},{"given":"Pedro S.","family":"Coelho","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2026,3,23]]},"reference":[{"key":"pone.0345111.ref001","doi-asserted-by":"crossref","first-page":"139","DOI":"10.2753\/MTP1069-6679190202","article-title":"PLS-SEM: Indeed a silver bullet","volume":"19","author":"JF Hair","year":"2011","journal-title":"Journal of Marketing Theory and Practice"},{"key":"pone.0345111.ref002","doi-asserted-by":"crossref","DOI":"10.1002\/9781118619179","volume-title":"Structural equations with latent variables","author":"KA Bollen","year":"1989"},{"key":"pone.0345111.ref003","volume-title":"Structural equation modeling with LISREL: essentials and advances","author":"LA Hayduk","year":"1987"},{"issue":"1","key":"pone.0345111.ref004","article-title":"Power Analysis for Parameter Estimation in Structural Equation Modeling: A Discussion and Tutorial","volume":"4","author":"YA Wang","year":"2021","journal-title":"Advances in Methods and Practices in Psychological Science"},{"issue":"2","key":"pone.0345111.ref005","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1134\/S0006350918020100","article-title":"SEM: Structural Equation Modeling in Molecular Biology","volume":"63","author":"AA Igolkina","year":"2018","journal-title":"BIOPHYSICS"},{"issue":"1","key":"pone.0345111.ref006","doi-asserted-by":"crossref","first-page":"543","DOI":"10.5465\/19416520903065683","article-title":"12\u2003Structural Equation Modeling in Management Research: A Guide for Improved Analysis","volume":"3","author":"LJ Williams","year":"2009","journal-title":"ANNALS"},{"key":"pone.0345111.ref007","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.1016\/S1573-4412(07)06064-3","article-title":"Structural econometric modeling: rationales and examples from industrial organization.","volume-title":"Handbook of econometrics","author":"PC Reiss","year":"2007"},{"key":"pone.0345111.ref008","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1108\/03090561311285484","article-title":"Structural equation modelling in marketing and business research: Critical issues and practical recommendations","volume":"47","author":"FJ Mart\u00ednez-L\u00f3pez","year":"2013","journal-title":"European Journal of Marketing"},{"key":"pone.0345111.ref009","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1186\/1756-0500-3-267","article-title":"Structural equation modeling in medical research: a primer","volume":"3","author":"TN Beran","year":"2010","journal-title":"BMC Res Notes"},{"key":"pone.0345111.ref010","unstructured":"Elangovan N, Rajendran R. 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