{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T10:29:42Z","timestamp":1787653782613,"version":"build-2736575974"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T00:00:00Z","timestamp":1787356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The concept of ranked sparsity, originally introduced in the context of penalized regression, arises in modeling applications when an expected disparity exists in the quality of information between different feature sets. Its presence can cause traditional and modern model selection methods to fail because such procedures commonly presume \u201ccovariate equipoise\u201d\u2014that each potential parameter is equally worthy of entering into the final model. However, this presumption does not always hold, especially in the presence of derived variables or with highly disparate feature sets (i.e., multi-modal data). For instance, when all possible interactions are considered as candidate predictors, the sheer number of them grossly inflates the number of false discoveries, resulting in unnecessarily complex and difficult-to-interpret models with many (truly spurious) interactions. In this work, we motivate a ranked sparsity extension to the Bayesian Information Criterion (RBIC) that requires a stronger level of evidence in order to allow certain variables (e.g., interactions vs main effects and genetic vs clinical covariates) into a model. We compare the performance of RBIC relative to competing methods for selecting polynomials and interactions in a simulation study and in two applications, showing that stepwise selection guided by RBIC produces better-predicting, more transparent models (with fewer false interactions) compared to existing alternatives.<\/jats:p>","DOI":"10.3390\/e28090943","type":"journal-article","created":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:49:34Z","timestamp":1787582974000},"page":"943","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Ranked Sparsity Extension to the Bayesian Information Criterion: A Tool for Selecting Variables from Multiple Data Modalities"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4650-5798","authenticated-orcid":false,"given":"Ryan A.","family":"Peterson","sequence":"first","affiliation":[{"name":"Department of Internal Medicine, Carver College of Medicine, University of Iowa, 375 Newton Road, Iowa City, IA 52242, USA"},{"name":"Department of Biostatistics, College of Public Health, University of Iowa, 145 N. Riverside Dr., Iowa City, IA 52242, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sarah M.","family":"Bird","sequence":"additional","affiliation":[{"name":"Department of Biostatistics and Informatics, University of Colorado\u2013Anschutz Medical Campus, 13001 E. 17 Ave., Aurora, CO 80031, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8562-9534","authenticated-orcid":false,"given":"Logan M.","family":"Harris","sequence":"additional","affiliation":[{"name":"Department of Biostatistics and Informatics, University of Colorado\u2013Anschutz Medical Campus, 13001 E. 17 Ave., Aurora, CO 80031, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0650-1119","authenticated-orcid":false,"given":"Patrick J.","family":"Breheny","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, College of Public Health, University of Iowa, 145 N. Riverside Dr., Iowa City, IA 52242, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0514-7664","authenticated-orcid":false,"given":"Joseph E.","family":"Cavanaugh","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, College of Public Health, University of Iowa, 145 N. Riverside Dr., Iowa City, IA 52242, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the Dimension of a Model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A New Look at the Statistical Model Identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Harrell, F.E. (2015). Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-19425-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1146\/annurev-statistics-041715-033413","article-title":"Statistical Model Choice","volume":"3","author":"Claeskens","year":"2016","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/0005-1098(78)90005-5","article-title":"Modeling by shortest data description","volume":"14","author":"Rissanen","year":"1978","journal-title":"Automatica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1093\/biomet\/asn034","article-title":"Extended Bayesian information criteria for model selection with large model spaces","volume":"95","author":"Chen","year":"2008","journal-title":"Biometrika"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0304-4076(94)01701-8","article-title":"Information Criteria for Selecting Possibly Misspecified Parametric Models","volume":"71","author":"Sin","year":"1996","journal-title":"J. Econom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1093\/biomet\/68.1.45","article-title":"An Optimal Selection of Regression Variables","volume":"68","author":"Shibata","year":"1981","journal-title":"Biometrika"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1214\/aos\/1176344897","article-title":"Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process","volume":"8","author":"Shibata","year":"1980","journal-title":"Ann. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1093\/biomet\/76.2.297","article-title":"Regression and time series model selection in small samples","volume":"76","author":"Hurvich","year":"1989","journal-title":"Biometrika"},{"key":"ref_11","first-page":"661","article-title":"Some Comments on Cp","volume":"15","author":"Mallows","year":"1973","journal-title":"Technometrics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression Shrinkage and Selection via the Lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s10182-021-00431-7","article-title":"Ranked sparsity: A cogent regularization framework for selecting and estimating feature interactions and polynomials","volume":"106","author":"Peterson","year":"2022","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"e13172","DOI":"10.1111\/ijpo.13172","article-title":"The predictive utility of the in utero exposome for childhood adiposity in independent and integrated frameworks","volume":"19","author":"VanHawkins","year":"2024","journal-title":"Pediatr. Obes."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","article-title":"Model selection and estimation in regression with grouped variables","volume":"68","author":"Yuan","year":"2006","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/10618600.2012.681250","article-title":"A sparse-group lasso","volume":"22","author":"Simon","year":"2013","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"7691937","DOI":"10.1155\/2017\/7691937","article-title":"IPF-LASSO: Integrative-penalized regression with penalty factors for prediction based on multi-omics data","volume":"2017","author":"Boulesteix","year":"2017","journal-title":"Comput. Math. Methods Med."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.2307\/3315687","article-title":"Bayesian Variable Selection with Related Predictors","volume":"24","author":"Chipman","year":"1996","journal-title":"Can. J. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1198\/jasa.2010.tm08281","article-title":"Variable Selection with the Strong Heredity Constraint and Its Oracle Property","volume":"105","author":"Choi","year":"2010","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1080\/10618600.2014.938812","article-title":"Learning Interactions via Hierarchical Group-Lasso Regularization","volume":"24","author":"Lim","year":"2015","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1080\/01621459.2016.1264956","article-title":"Model Selection for High-Dimensional Quadratic Regression via Regularization","volume":"113","author":"Hao","year":"2018","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1214\/13-AOS1096","article-title":"A lasso for hierarchical interactions","volume":"41","author":"Bien","year":"2013","journal-title":"Ann. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1080\/24754269.2019.1611145","article-title":"Discussion of prior-based Bayesian information criterion (PBIC) by M.J. Bayarria, James O. Berger, Woncheol Jang, Surajit Ray, Luis R. Pericchi, and Ingmar Visser","volume":"3","author":"Peterson","year":"2019","journal-title":"Stat. Theory Relat. Fields"},{"key":"ref_24","unstructured":"Lumley, T., and Miller, A. (Leaps: Regression Subset Selection, 2017). Leaps: Regression Subset Selection, R package version 3.0."},{"key":"ref_25","unstructured":"McLeod, A., and Xu, C. (Bestglm: Best Subset GLM and Regression Utilities, 2018). Bestglm: Best Subset GLM and Regression Utilities, R package version 0.37."},{"key":"ref_26","unstructured":"Peterson, R.A. (2026, June 16). ICtoolkit: Information Criteria for Penalized and Stepwise Model Selection, Available online: https:\/\/github.com\/petersonR\/ICtoolkit."},{"key":"ref_27","unstructured":"Friedman, J., Hastie, T., Tibshirani, R., Narasimhan, B., Tay, K., Simon, N., and Yang, J. (Glmnet: Lasso and Elastic-Net Regularized Generalized Linear Models, 2023). Glmnet: Lasso and Elastic-Net Regularized Generalized Linear Models, R package version 4.1-8."},{"key":"ref_28","unstructured":"Breheny, P. (Ncvreg: Regularization Paths for SCAD and MCP Penalized Regression Models, 2024). Ncvreg: Regularization Paths for SCAD and MCP Penalized Regression Models, R package version 3.14-1."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Venables, W.N., and Ripley, B.D. (2002). Modern Applied Statistics with S, Springer. [4th ed.].","DOI":"10.1007\/978-0-387-21706-2"},{"key":"ref_30","first-page":"1","article-title":"Multivariate adaptive regression splines","volume":"19","author":"Friedman","year":"1991","journal-title":"Ann. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1001\/jamainternmed.2022.1118","article-title":"Stress-Related Disorders of Family Members of Patients Admitted to the Intensive Care Unit With COVID-19","volume":"182","author":"Amass","year":"2022","journal-title":"JAMA Intern. Med."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1038\/nm.1790","article-title":"Gene expression-based survival prediction in lung adenocarcinoma: A multi-site, blinded validation study","volume":"14","author":"Shedden","year":"2008","journal-title":"Nat. Med."},{"key":"ref_33","unstructured":"Seedorff, J. (BranchGLM: Efficient Best Subset Selection for GLMs via Branch and Bound Algorithms, 2024). BranchGLM: Efficient Best Subset Selection for GLMs via Branch and Bound Algorithms, R package version 3.0.1."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1111\/j.1541-0420.2008.00989.x","article-title":"Extending the Modified Bayesian Information Criterion (mBIC) to dense markers and multiple interval mapping","volume":"64","author":"Bogdan","year":"2008","journal-title":"Biometrics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/943\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:53:09Z","timestamp":1787583189000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/943"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,22]]},"references-count":34,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["e28090943"],"URL":"https:\/\/doi.org\/10.3390\/e28090943","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,22]]}}}