{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:21:15Z","timestamp":1772119275775,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100016378","name":"Technische Universit\u00e4t Dortmund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100016378","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Classif"],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Ordinal data are frequently encountered, e.g., in the life and social sciences. Predicting ordinal outcomes can inform important decisions, e.g., in medicine or education. Two methodological streams tackle prediction of ordinal outcomes: Traditional parametric models, e.g., the proportional odds model (POM), and machine learning-based tree ensemble (TE) methods. A promising TE approach involves selecting the best performing from sets of randomly generated numeric scores assigned to ordinal response categories (ordinal forest; Hornung, 2019). We propose a new method, the ordinal score optimization algorithm, that takes a similar approach but selects scores through non-linear optimization. We compare these and other TE methods with the computationally much less expensive POM. Despite selective efforts, the literature lacks an encompassing simulation-based comparison. Aiming to fill this gap, we find that while TE approaches outperform the POM for strong non-linear effects, the latter is competitive for small sample sizes even under medium non-linear effects.<\/jats:p>","DOI":"10.1007\/s00357-024-09497-9","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T04:59:08Z","timestamp":1733720348000},"page":"364-390","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Old but Gold or New and Shiny? Comparing Tree Ensembles for Ordinal Prediction with a Classic Parametric Approach"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6980-8110","authenticated-orcid":false,"given":"Philip","family":"Buczak","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5208-0482","authenticated-orcid":false,"given":"Daniel","family":"Horn","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0976-7190","authenticated-orcid":false,"given":"Markus","family":"Pauly","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"issue":"7","key":"9497_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v034.i07","volume":"34","author":"KJ Archer","year":"2010","unstructured":"Archer, K. J. (2010). rpartOrdinal: An R package for deriving a classification tree for predicting an ordinal response. Journal of Statistical Software, 34(7), 1\u201317. https:\/\/doi.org\/10.18637\/jss.v034.i07","journal-title":"Journal of Statistical Software"},{"issue":"2","key":"9497_CR2","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.eswa.2006.10.022","volume":"34","author":"A Ben-David","year":"2008","unstructured":"Ben-David, A. (2008). Comparison of classification accuracy using Cohen\u2019s weighted kappa. Expert Systems with Applications, 34(2), 825\u2013832. https:\/\/doi.org\/10.1016\/j.eswa.2006.10.022","journal-title":"Expert Systems with Applications"},{"issue":"4","key":"9497_CR3","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.2307\/2532457","volume":"46","author":"R Brant","year":"1990","unstructured":"Brant, R. (1990). Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics, 46(4), 1171. https:\/\/doi.org\/10.2307\/2532457","journal-title":"Biometrics"},{"key":"9497_CR4","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1023\/A:1010933404324","volume":"45","author":"L Breiman","year":"2001","unstructured":"Breiman, L. (2001). Random forests. Machine Learning, 45, 123\u2013140. https:\/\/doi.org\/10.1023\/A:1010933404324","journal-title":"Machine Learning"},{"key":"9497_CR5","doi-asserted-by":"publisher","unstructured":"Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J. (1984). Classification and regression trees. New York, NY: Chapman and Hall\/CRC. https:\/\/doi.org\/10.1201\/9781315139470","DOI":"10.1201\/9781315139470"},{"issue":"2","key":"9497_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/ijb-2019-0063","volume":"16","author":"M Buri","year":"2020","unstructured":"Buri, M., & Hothorn, T. (2020). Model-based random forests for ordinal regression. The International Journal of Biostatistics, 16(2), 1\u201317. https:\/\/doi.org\/10.1515\/ijb-2019-0063","journal-title":"The International Journal of Biostatistics"},{"key":"9497_CR7","unstructured":"Christensen, R. H. B. (2022). ordinal: Regression models for ordinal data [Computer software manual]. (R package version 2022.11-16). https:\/\/CRAN.R-project.org\/package=ordinal"},{"issue":"4","key":"9497_CR8","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1037\/h0026256","volume":"70","author":"J Cohen","year":"1968","unstructured":"Cohen, J. (1968). Weighted kappa: Nominal scale agreement provision for scaled disagreement or partial credit. Psychological Bulletin, 70(4), 213\u2013220. https:\/\/doi.org\/10.1037\/h0026256","journal-title":"Psychological Bulletin"},{"key":"9497_CR9","doi-asserted-by":"publisher","unstructured":"Cortez, P. (2014). Student performance. UCI Machine Learning Repository. https:\/\/doi.org\/10.24432\/C5TG7T","DOI":"10.24432\/C5TG7T"},{"key":"9497_CR10","doi-asserted-by":"publisher","unstructured":"Cortez, P., Cerdeira, A., Almeida, F., Matos, T., & Reis, J. (2009). Wine quality. UCI Machine Learning Repository. https:\/\/doi.org\/10.24432\/C56S3T","DOI":"10.24432\/C56S3T"},{"key":"9497_CR11","doi-asserted-by":"publisher","unstructured":"Frank, E., & Hall, M. (2001). A simple approach to ordinal classification. In L. De Raedt, & P. Flach (Eds.), Lecture Notes in Computer Science: Vol. 2167. Machine learning: ECML 2001 (pp. 145\u2013156). https:\/\/doi.org\/10.1007\/3-540-44795-4_13","DOI":"10.1007\/3-540-44795-4_13"},{"issue":"1","key":"9497_CR12","doi-asserted-by":"publisher","first-page":"2200212","DOI":"10.1002\/bimj.202200212","volume":"66","author":"S Friedrich","year":"2024","unstructured":"Friedrich, S., & Friede, T. (2024). On the role of benchmarking data sets and simulations in method comparison studies. Biometrical Journal, 66(1), 2200212. https:\/\/doi.org\/10.1002\/bimj.202200212","journal-title":"Biometrical Journal"},{"issue":"10","key":"9497_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v047.i10","volume":"47","author":"G Galimberti","year":"2012","unstructured":"Galimberti, G., Soffritti, G., & Maso, M. D. (2012). Classification trees for ordinal responses in R: The rpartScore package. Journal of Statistical Software, 47(10), 1\u201325. https:\/\/doi.org\/10.18637\/jss.v047.i10","journal-title":"Journal of Statistical Software"},{"key":"9497_CR14","unstructured":"Grinsztajn, L., Oyallon, E., & Varoquaux, G. (2022). Why do tree-based models still outperform deep learning on typical tabular data? In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in Neural Information Processing Systems 35 (NeurIPS 2022)."},{"key":"9497_CR15","doi-asserted-by":"publisher","unstructured":"Hansen, N., & Ostermeier, A. (1996). Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In Proceedings of IEEE International Conference on Evolutionary Computation (pp. 312\u2013317). https:\/\/doi.org\/10.1109\/ICEC.1996.542381","DOI":"10.1109\/ICEC.1996.542381"},{"key":"9497_CR16","unstructured":"Hornung, R. (2022). ordinalForest: Ordinal forests: Prediction and variable ranking with ordinal target variables [Computer software manual]. (R package version 2.4-3). https:\/\/CRAN.R-project.org\/package=ordinalForest"},{"issue":"1","key":"9497_CR17","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/s00357-018-9302-x","volume":"37","author":"R Hornung","year":"2019","unstructured":"Hornung, R. (2019). Ordinal forests. Journal of Classification, 37(1), 4\u201317. https:\/\/doi.org\/10.1007\/s00357-018-9302-x","journal-title":"Journal of Classification"},{"key":"9497_CR18","unstructured":"Hothorn, T. (2023). TH.data: TH\u2019s data archive [Computer software manual]. (R package version 1.1-2). https:\/\/CRAN.R-project.org\/package=TH.data"},{"key":"9497_CR19","unstructured":"Hothorn, T., & Zeileis, A. (2015). partykit: A modular toolkit for recursive partitioning in R. Journal of Machine Learning Research, 16, 3905\u20133909. https:\/\/jmlr.org\/papers\/v16\/hothorn15a.html"},{"issue":"3","key":"9497_CR20","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1198\/106186006X133933","volume":"15","author":"T Hothorn","year":"2006","unstructured":"Hothorn, T., Hornik, K., & Zeileis, A. (2006). Unbiased recursive partitioning: A conditional inference framework. Journal of Computational and Graphical Statistics, 15(3), 651\u2013674. https:\/\/doi.org\/10.1198\/106186006X133933","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"9497_CR21","doi-asserted-by":"publisher","unstructured":"Immekus, J. C., Jeong, T.-s., & Yoo, J. E. (2022). Machine learning procedures for predictor variable selection for schoolwork-related anxiety: Evidence from PISA 2015 mathematics, reading, and science assessments. Large-scale Assessments in Education, 10(1). https:\/\/doi.org\/10.1186\/s40536-022-00150-8","DOI":"10.1186\/s40536-022-00150-8"},{"key":"9497_CR22","unstructured":"James, G., Witten, D., Hastie, T., & Tibshirani, R. (2021). ISLR: Data for an introduction to statistical learning with applications in R [Computer software manual]. (R package version 1.4). https:\/\/CRAN.R-project.org\/package=ISLR"},{"key":"9497_CR23","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.csda.2015.10.005","volume":"96","author":"S Janitza","year":"2016","unstructured":"Janitza, S., Tutz, G., & Boulesteix, A.-L. (2016). Random forest for ordinal responses: Prediction and variable selection. Computational Statistics & Data Analysis, 96, 57\u201373. https:\/\/doi.org\/10.1016\/j.csda.2015.10.005","journal-title":"Computational Statistics & Data Analysis"},{"key":"9497_CR24","unstructured":"Johnson, S. G. (2007). The NLopt nonlinear-optimization package [Computer software manual]. https:\/\/github.com\/stevengj\/nlopt"},{"issue":"3","key":"9497_CR25","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1093\/biomet\/33.3.239","volume":"33","author":"MG Kendall","year":"1945","unstructured":"Kendall, M. G. (1945). The treatment of ties in ranking problems. Biometrika, 33(3), 239\u2013251. https:\/\/doi.org\/10.1093\/biomet\/33.3.239","journal-title":"Biometrika"},{"key":"9497_CR26","unstructured":"Kendall, M. G. (1948). Rank correlation methods. London, UK: Griffin."},{"key":"9497_CR27","doi-asserted-by":"crossref","unstructured":"Kleiber, C., & Zeileis, A. (2008). AER: Applied econometrics with R [Computer software manual]. https:\/\/CRAN.Rproject.org\/package=AER","DOI":"10.32614\/CRAN.package.AER"},{"key":"9497_CR28","doi-asserted-by":"publisher","unstructured":"Kramer, S., Widmer, G., Pfahringer, B., & de Groeve, M. (2000). Prediction of Ordinal Classes Using Regression Trees. In Z. W. Ra\u015b, & S. Ohsuga (Eds.), Lecture Notes in Computer Science: Vol. 1932. Foundations of Intelligent Systems. ISMIS 2000 (pp. 426\u2013434). https:\/\/doi.org\/10.1007\/3-540-39963-1_45","DOI":"10.1007\/3-540-39963-1_45"},{"key":"9497_CR29","unstructured":"Leisch, F., & Dimitriadou, E. (2021). mlbench: Machine learning benchmark problems [Computer software manual]. (R package version 2.1-3.1)"},{"key":"9497_CR30","unstructured":"Liaw, A., & Wiener, M. (2002). Classification and regression by randomForest. R News, 2(3), 18\u201322. https:\/\/CRAN.R-project.org\/doc\/Rnews\/"},{"issue":"2","key":"9497_CR31","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1111\/j.2517-6161.1980.tb01109.x","volume":"42","author":"P McCullagh","year":"1980","unstructured":"McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society: Series B (Methodological), 42(2), 109\u2013127. https:\/\/doi.org\/10.1111\/j.2517-6161.1980.tb01109.x","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"key":"9497_CR32","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","volume":"7","author":"JA Nelder","year":"1965","unstructured":"Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7, 308\u2013313. https:\/\/doi.org\/10.1093\/comjnl\/7.4.308","journal-title":"The Computer Journal"},{"issue":"2","key":"9497_CR33","doi-asserted-by":"publisher","first-page":"205","DOI":"10.2307\/2347760","volume":"39","author":"B Peterson","year":"1990","unstructured":"Peterson, B., & Harrell, F. E. (1990). Partial proportional odds models for ordinal response variables. Applied Statistics, 39(2), 205. https:\/\/doi.org\/10.2307\/2347760","journal-title":"Applied Statistics"},{"key":"9497_CR34","doi-asserted-by":"crossref","unstructured":"P\u00e9trowski, A., & Ben-Hamida, S. (2017). Evolutionary algorithms. Hoboken, NJ: Wiley.","DOI":"10.1002\/9781119136378"},{"issue":"3","key":"9497_CR35","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/s00180-007-0077-5","volume":"23","author":"R Piccarreta","year":"2007","unstructured":"Piccarreta, R. (2007). Classification trees for ordinal variables. Computational Statistics, 23(3), 407\u2013427. https:\/\/doi.org\/10.1007\/s00180-007-0077-5","journal-title":"Computational Statistics"},{"issue":"3","key":"9497_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/widm.1301","volume":"9","author":"P Probst","year":"2019","unstructured":"Probst, P., Wright, M. N., & Boulesteix, A. (2019). Hyperparameters and tuning strategies for random forest. WIREs Data Mining and Knowledge Discovery, 9(3), 1\u201315. https:\/\/doi.org\/10.1002\/widm.1301","journal-title":"WIREs Data Mining and Knowledge Discovery"},{"key":"9497_CR37","unstructured":"R Core Team (2023). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https:\/\/www.R-project.org\/"},{"key":"9497_CR38","unstructured":"Rowan, T. H. (1990). Functional stability analysis of numerical algorithms (Doctoral dissertation). Retrieved from ProQuest Dissertations and Theses database. (UMI No. 9031702)"},{"issue":"5","key":"9497_CR39","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1037\/a0023322","volume":"96","author":"W Shen","year":"2011","unstructured":"Shen, W., Kiger, T. B., Davies, S. E., Rasch, R. L., Simon, K. M., & Ones, D. S. (2011). Samples in applied psychology: Over a decade of research in review. Journal of Applied Psychology, 96(5), 1055\u20131064. https:\/\/doi.org\/10.1037\/a0023322","journal-title":"Journal of Applied Psychology"},{"issue":"2","key":"9497_CR40","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/s00357-021-09406-4","volume":"39","author":"G Tutz","year":"2021","unstructured":"Tutz, G. (2021). Ordinal trees and random forests: Score-free recursive partitioning and improved ensembles. Journal of Classification, 39(2), 241\u2013263. https:\/\/doi.org\/10.1007\/s00357-021-09406-4","journal-title":"Journal of Classification"},{"issue":"2","key":"9497_CR41","doi-asserted-by":"publisher","DOI":"10.1002\/wics.1545","volume":"14","author":"G Tutz","year":"2022","unstructured":"Tutz, G. (2022). Ordinal regression: A review and a taxonomy of models. WIREs Computational Statistics, 14(2), e1545. https:\/\/doi.org\/10.1002\/wics.1545","journal-title":"WIREs Computational Statistics"},{"issue":"2","key":"9497_CR42","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1111\/insr.12484","volume":"90","author":"G Tutz","year":"2022","unstructured":"Tutz, G., & Berger, M. (2022). Sparser ordinal regression models based on parametric and additive location-shift approaches. International Statistical Review, 90(2), 306\u2013327. https:\/\/doi.org\/10.1111\/insr.12484","journal-title":"International Statistical Review"},{"issue":"1","key":"9497_CR43","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v077.i01","volume":"77","author":"MN Wright","year":"2017","unstructured":"Wright, M. N., & Ziegler, A. (2017). ranger: A fast implementation of random forests for high dimensional data in C++ and R. Journal of Statistical Software, 77(1), 1\u201317. https:\/\/doi.org\/10.18637\/jss.v077.i01","journal-title":"Journal of Statistical Software"},{"issue":"1","key":"9497_CR44","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1002\/1097-0142(1950)3:1<32::aid-cncr2820030106>3.0.co;2-3","volume":"3","author":"WJ Youden","year":"1950","unstructured":"Youden, W. J. (1950). Index for rating diagnostic tests. Cancer, 3(1), 32\u201335. https:\/\/doi.org\/10.1002\/1097-0142(1950)3:1<32::aid-cncr2820030106>3.0.co;2-3","journal-title":"Cancer"},{"issue":"3","key":"9497_CR45","doi-asserted-by":"publisher","first-page":"388","DOI":"10.1111\/insr.12032","volume":"81","author":"FM Zahid","year":"2013","unstructured":"Zahid, F. M., & Tutz, G. (2013). Proportional odds models with high-dimensional data structure. International Statistical Review, 81(3), 388\u2013406. https:\/\/doi.org\/10.1111\/insr.12032","journal-title":"International Statistical Review"}],"container-title":["Journal of Classification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-024-09497-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00357-024-09497-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-024-09497-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T07:43:35Z","timestamp":1753429415000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00357-024-09497-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,9]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["9497"],"URL":"https:\/\/doi.org\/10.1007\/s00357-024-09497-9","relation":{"has-preprint":[{"id-type":"doi","id":"10.31219\/osf.io\/v7bcf","asserted-by":"object"}]},"ISSN":["0176-4268","1432-1343"],"issn-type":[{"value":"0176-4268","type":"print"},{"value":"1432-1343","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,9]]},"assertion":[{"value":"20 November 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}