{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:43:20Z","timestamp":1760060600884,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T00:00:00Z","timestamp":1756857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In various research domains, researchers frequently encounter multiple datasets pertaining to the same subjects, with one dataset providing explanatory variables for the others. To address this structure, we introduce the Binary 3-way PARAFAC Partial Least Squares (Bin-3-Way-PARAFAC-PLS), a novel multiway regression method. This method is specifically engineered for scenarios involving a three-way real-valued explanatory data array and a matrix of binary response data. We detail the algorithm\u2019s implementation and illustrate its practical application. Furthermore, we describe biplot representations to aid in result interpretation. The accompanying software necessary for implementing the method is also provided. Finally, the proposed method\u2019s utility in real-world problem-solving is demonstrated through its application to a psychological dataset.<\/jats:p>","DOI":"10.3390\/axioms14090678","type":"journal-article","created":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T08:04:15Z","timestamp":1756886655000},"page":"678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bin-3-Way-PARAFAC-PLS: A 3-Way Partial Least Squares for Binary Response"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7108-481X","authenticated-orcid":false,"given":"Elisa","family":"Frutos-Bernal","sequence":"first","affiliation":[{"name":"Departamento de Estad\u00edstica, Facultad de Medicina, Universidad de Salamanca, 37007 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2483-8874","authenticated-orcid":false,"given":"Laura","family":"Vicente-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica, Facultad de Medicina, Universidad de Salamanca, 37007 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2230-0587","authenticated-orcid":false,"given":"Ana Elizabeth","family":"Sipols","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Materials Science and Engineering and Electronic Technology, Universidad Rey Juan Carlos, 28933 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1137\/0905052","article-title":"The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses","volume":"5","author":"Wold","year":"1984","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/cem.1180020306","article-title":"PLS regression methods","volume":"2","year":"1988","journal-title":"J. Chemom."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0169-7439(93)85002-X","article-title":"SIMPLS\u2014An alternative approach to partial least-squares regression","volume":"18","year":"1993","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, L., Yan, S., Bakker, B.M., Hoefsloot, H., Chawes, B., Horner, D., Rasmussen, M.A., Smilde, A.K., and Acar, E. (2024). Analyzing postprandial metabolomics data using multiway models: A simulation study. BMC Bioinform., 25.","DOI":"10.1186\/s12859-024-05686-w"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6557","DOI":"10.1039\/c3ay41160e","article-title":"Fluorescence spectroscopy and multi-way techniques. PARAFAC","volume":"5","author":"Murphy","year":"2013","journal-title":"Anal. Methods"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.brainresbull.2015.05.001","article-title":"Multiscale entropy analysis of resting-state magnetoencephalogram with tensor factorisations in Alzheimer\u2019s disease","volume":"119","author":"Escudero","year":"2015","journal-title":"Brain Res. Bull."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1037\/emo0001173","article-title":"Distinguishing dimensions of emotion dynamics across 12 emotions in adolescents\u2019 daily lives","volume":"23","author":"Reitsema","year":"2023","journal-title":"Emotion"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1037\/h0047773","article-title":"A Theory of Data","volume":"67","author":"Coombs","year":"1960","journal-title":"Psychol. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Carroll, J.D., and Arabie, P. (1998). Multidimensional scaling. Measurement, Judgment and Decision Making, Elsevier.","DOI":"10.1016\/B978-012099975-0.50005-1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1080\/14786440109462720","article-title":"LIII. On lines and planes of closest fit to systems of points in space","volume":"2","author":"Pearson","year":"1901","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_11","unstructured":"Kroonenberg, P.M. (1983). Three-Mode Principal Component Analysis: Theory and Applications, DSWO."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF02289464","article-title":"Some mathematical notes on three-mode factor analysis","volume":"31","author":"Tucker","year":"1966","journal-title":"Psychometrika"},{"key":"ref_13","unstructured":"Frederiksen, N., and Gulliksen, H. (1964). The extension of factor analysis to three-dimensional matrices. Contributions to Mathematical Psychology, Holt, Rinehart and Winston."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1002\/sapm192761164","article-title":"The expression of a tensor or a polyadic as a sum of products","volume":"6","author":"Hitchcock","year":"1927","journal-title":"J. Math. Phys."},{"key":"ref_15","first-page":"1","article-title":"Foundations of the parafac procedure: Models and conditions for an explanatory multi-modal factor analysis","volume":"16","author":"Harshman","year":"1970","journal-title":"UCLA Work. Pap. Phon."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF02310791","article-title":"Analysis of individual differences in multidimensional scaling via an N-way generalization of \u201cEckart-Young\u201d decomposition","volume":"35","author":"Carroll","year":"1970","journal-title":"Psychometrika"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Smilde, A.K., Bro, R., and Geladi, P. (2005). Multi-Way Analysis: Applications in the Chemical Sciences, John Wiley & Sons.","DOI":"10.1002\/0470012110"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1002\/(SICI)1099-128X(199601)10:1<47::AID-CEM400>3.0.CO;2-C","article-title":"Multiway calibration. Multilinear PLS","volume":"10","author":"Bro","year":"1996","journal-title":"J. Chemom."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1002\/(SICI)1099-128X(199709\/10)11:5<367::AID-CEM481>3.0.CO;2-I","article-title":"Comments on multilinear PLS","volume":"11","author":"Smilde","year":"1997","journal-title":"J. Chemom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0169-7439(01)00134-4","article-title":"On the difference between low-rank and subspace approximation: Improved model for multi-linear PLS regression","volume":"58","author":"Bro","year":"2001","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.chemolab.2018.06.005","article-title":"Sparse N-way partial least squares with R package sNPLS","volume":"179","author":"Lahoz","year":"2018","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.chemolab.2019.01.004","article-title":"Sparse N-way partial least squares by L1-penalization","volume":"185","author":"Lahoz","year":"2019","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1002\/cem.785","article-title":"Partial least squares for discrimination","volume":"17","author":"Barker","year":"2003","journal-title":"J. Chemom."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.csda.2004.02.005","article-title":"PLS generalised linear regression","volume":"48","author":"Bastien","year":"2005","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Vicente-Gonzalez, L., and Vicente-Villardon, J.L. (2022). Partial Least Squares Regression for Binary Responses and Its Associated Biplot Representation. Mathematics, 10.","DOI":"10.3390\/math10152580"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Vicente-Gonzalez, L., Frutos-Bernal, E., and Vicente-Villardon, J.L. (2025). Partial Least Squares Regression for Binary Data. Mathematics, 13.","DOI":"10.3390\/math13030458"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bazzoli, C., and Lambert-Lacroix, S. (2018). Classification based on extensions of LS-PLS using logistic regression: Application to clinical and multiple genomic data. BMC Bioinform., 19.","DOI":"10.1186\/s12859-018-2311-2"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1093\/bioinformatics\/bti114","article-title":"Classification using partial least squares with penalized logistic regression","volume":"21","author":"Fort","year":"2005","journal-title":"Bioinformatics"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Vicente-Villardon, J., Galindo-Villard\u00f3n, M.P., and Blazquez-Zaballos, A. (2006). Logistic Biplots. Multiple Correspondence Analysis and Related Methods, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420011319.ch23"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2832","DOI":"10.1093\/bioinformatics\/btn552","article-title":"Identifying molecular markers associated with classification of genotypes by External Logistic Biplots","volume":"24","author":"Demey","year":"2008","journal-title":"Bioinformatics"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF02294344","article-title":"Decompositions and biplots in three-way correspondence analysis","volume":"61","author":"Carlier","year":"1996","journal-title":"Psychometrika"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1002\/1099-128X(200005\/06)14:3<105::AID-CEM582>3.0.CO;2-I","article-title":"Towards a standardized notation and terminology in multiway analysis","volume":"14","author":"Kiers","year":"2000","journal-title":"J. Chemom."},{"key":"ref_33","first-page":"601","article-title":"Generalized inverse of a matrix and its applications","volume":"Volume 1","author":"Rao","year":"1972","journal-title":"Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/71.1.1","article-title":"On the existence of maximum likelihood estimates in logistic regression models","volume":"71","author":"Albert","year":"1984","journal-title":"Biometrika"},{"key":"ref_35","first-page":"191","article-title":"Ridge estimators in logistic regression","volume":"41","author":"Cessie","year":"1992","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_36","unstructured":"R Core Team (2023). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2449","DOI":"10.1080\/02664763.2015.1043858","article-title":"The construction of a partial least-squares biplot","volume":"42","author":"Oyedele","year":"2015","journal-title":"J. Appl. Stat."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1093\/biomet\/58.3.453","article-title":"The biplot graphic display of matrices with application to principal component analysis","volume":"58","author":"Gabriel","year":"1971","journal-title":"Biometrika"},{"key":"ref_39","unstructured":"Gower, J.C., and Hand, D. (1996). Biplots, Chapman and Hall."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gower, J.C., Lubbe, S.G., and Le Roux, N.J. (2011). Understanding Biplots, John Wiley and Sons.","DOI":"10.1002\/9780470973196"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Babativa-M\u00e1rquez, J.G., and Vicente-Villard\u00f3n, J.L. (2021). Logistic Biplot by Conjugate Gradient Algorithms and Iterated SVD. Mathematics, 9.","DOI":"10.3390\/math9162015"},{"key":"ref_42","unstructured":"Vicente-Villard\u00f3n, J.L., and Hern\u00e1ndez S\u00e1nchez, J.C. (2014). Logistic Biplots for Ordinal Data with an Application to Job Satisfaction of Doctorate Degree Holders in Spain. arXiv."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11634-016-0249-7","article-title":"Logistic biplot for nominal data","volume":"11","year":"2017","journal-title":"Adv. Data Anal. Classif."},{"key":"ref_44","unstructured":"Lichman, M., and Bache, K. (2013). UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences."},{"key":"ref_45","unstructured":"Vicente-Villardon, J.L., Vicente-Gonzalez, L., and Frutos Bernal, E. (2024). MultBiplotR: Multivariate Analysis Using Biplots in R, Universidad de Salamanca."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.paid.2006.01.007","article-title":"Multidimensional individual differences in anger-related behaviors","volume":"41","author":"Ceulemans","year":"2006","journal-title":"Personal. Individ. Differ."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Barranco-Chamorro, I., and Carrillo-Garcia, R.M. (2021). Techniques to deal with off-diagonal elements in confusion matrices. Mathematics, 9.","DOI":"10.3390\/math9243233"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/678\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:38:29Z","timestamp":1760035109000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/678"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,3]]},"references-count":47,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090678"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090678","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,9,3]]}}}