{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T08:18:09Z","timestamp":1793089089625,"version":"4.5.0"},"reference-count":38,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T00:00:00Z","timestamp":1688688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"NSF Grants","award":["DMS-1612456"],"award-info":[{"award-number":["DMS-1612456"]}]},{"name":"NSF Grants","award":["DMS-1916378"],"award-info":[{"award-number":["DMS-1916378"]}]},{"name":"ARO Grants","award":["W911NF-15-1-0423"],"award-info":[{"award-number":["W911NF-15-1-0423"]}]},{"name":"ARO Grants","award":["W911NF-20-1-0051"],"award-info":[{"award-number":["W911NF-20-1-0051"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,9,29]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In the 1930s, Psychologists began developing Multiple-Factor Analysis to decompose multivariate data into a small number of interpretable factors without any a priori knowledge about those factors. In this form of factor analysis, the Varimax factor rotation redraws the axes through the multi-dimensional factors to make them sparse and thus make them more interpretable. Charles Spearman and many others objected to factor rotations because the factors seem to be rotationally invariant. Despite the controversy, factor rotations have remained widely popular among people analyzing data. Reversing nearly a century of statistical thinking on the topic, we show that the rotation makes the factors easier to interpret because the Varimax performs statistical inference; in particular, principal components analysis (PCA) with a Varimax rotation provides a unified spectral estimation strategy for a broad class of semi-parametric factor models, including the Stochastic Blockmodel and a natural variation of Latent Dirichlet Allocation. In addition, we show that Thurstone\u2019s widely employed sparsity diagnostics implicitly assess a key leptokurtic condition that makes the axes statistically identifiable in these models. PCA with Varimax is fast, stable, and practical. Combined with Thurstone\u2019s straightforward diagnostics, this vintage approach is suitable for a wide array of modern applications.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad029","type":"journal-article","created":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T15:04:21Z","timestamp":1688742261000},"page":"1037-1060","source":"Crossref","is-referenced-by-count":47,"title":["Vintage factor analysis with Varimax performs statistical inference"],"prefix":"10.1093","volume":"85","author":[{"given":"Karl","family":"Rohe","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muzhe","family":"Zeng","sequence":"additional","affiliation":[{"name":"UW Madison Department of Statistics, 1300 University Ave , Madison, WI 53706 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,7,7]]},"reference":[{"issue":"4","key":"2023110616083214500_qkad029-B2","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1214\/13-AOS1138","article-title":"Pseudo-likelihood methods for community detection in large sparse networks","volume":"41","author":"Amini","year":"2013","journal-title":"The Annals of Statistics"},{"key":"2023110616083214500_qkad029-B3","author":"Ammar","year":"2018"},{"issue":"1","key":"2023110616083214500_qkad029-B4","first-page":"2773","article-title":"Tensor decompositions for learning latent variable models","volume":"15","author":"Anandkumar","year":"2014","journal-title":"The Journal of Machine Learning Research"},{"key":"2023110616083214500_qkad029-B5","author":"Anderson","year":"1956"},{"key":"2023110616083214500_qkad029-B6","doi-asserted-by":"crossref","DOI":"10.1002\/9781119970583","volume-title":"Latent variable models and factor analysis: A unified approach","author":"Bartholomew","year":"2011"},{"issue":"1","key":"2023110616083214500_qkad029-B8","first-page":"907","article-title":"Certifiably optimal low rank factor analysis","volume":"18","author":"Bertsimas","year":"2017","journal-title":"The Journal of Machine Learning Research"},{"issue":"50","key":"2023110616083214500_qkad029-B9","doi-asserted-by":"crossref","first-page":"21068","DOI":"10.1073\/pnas.0907096106","article-title":"A nonparametric view of network models and Newman\u2013Girvan and other modularities","volume":"106","author":"Bickel","year":"2009","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2023110616083214500_qkad029-B13","author":"Chaudhuri","year":"2012"},{"key":"2023110616083214500_qkad029-B16","author":"Chen","year":"2021"},{"key":"2023110616083214500_qkad029-B17","author":"Chen","year":"2020"},{"key":"2023110616083214500_qkad029-B21","volume-title":"An introduction to probability theory and its applications","author":"Feller","year":"1971"},{"issue":"1","key":"2023110616083214500_qkad029-B22","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1111\/j.1751-5823.2009.00076.x","article-title":"Karl Pearson and the origin of kurtosis","volume":"77","author":"Fiori","year":"2009","journal-title":"International Statistical Review"},{"issue":"4","key":"2023110616083214500_qkad029-B24","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF02289601","article-title":"Oblique factor analytic solutions by orthogonal transformations","volume":"29","author":"Harris","year":"1964","journal-title":"Psychometrika"},{"issue":"2","key":"2023110616083214500_qkad029-B25","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","article-title":"Stochastic blockmodels: First steps","volume":"5","author":"Holland","year":"1983","journal-title":"Social Networks"},{"key":"2023110616083214500_qkad029-B26","volume-title":"Independent component analysis","author":"Hyv\u00e4rinen","year":"2004"},{"key":"2023110616083214500_qkad029-B28","author":"Jin","year":"2017"},{"key":"2023110616083214500_qkad029-B29","volume-title":"Applied multivariate statistical analysis","author":"Johnson","year":"2007"},{"key":"2023110616083214500_qkad029-B30","volume-title":"Principal component analysis","author":"Jolliffe","year":"2002"},{"issue":"3","key":"2023110616083214500_qkad029-B31","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02289233","article-title":"The Varimax criterion for analytic rotation in factor analysis","volume":"23","author":"Kaiser","year":"1958","journal-title":"Psychometrika"},{"issue":"4","key":"2023110616083214500_qkad029-B32","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/BF02291817","article-title":"A second generation Little Jiffy","volume":"35","author":"Kaiser","year":"1970","journal-title":"Psychometrika"},{"issue":"1","key":"2023110616083214500_qkad029-B33","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1177\/001316447403400115","article-title":"Little Jiffy, Mark IV","volume":"34","author":"Kaiser","year":"1974","journal-title":"Educational and Psychological Measurement"},{"issue":"3","key":"2023110616083214500_qkad029-B38","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1002\/rsa.20713","article-title":"Concentration and regularization of random graphs","volume":"51","author":"Le","year":"2017","journal-title":"Random Structures & Algorithms"},{"key":"2023110616083214500_qkad029-B39","author":"Lei","year":"2020"},{"key":"2023110616083214500_qkad029-B41","volume-title":"Multivariate analysis","author":"Mardia","year":"1979"},{"issue":"124","key":"2023110616083214500_qkad029-B42","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/14786446008642818","article-title":"V. Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres","volume":"19","author":"Maxwell","year":"1860","journal-title":"The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science"},{"key":"2023110616083214500_qkad029-B47","doi-asserted-by":"crossref","DOI":"10.1007\/b98888","volume-title":"Functional data analysis","author":"Ramsay","year":"2005"},{"key":"2023110616083214500_qkad029-B48","volume-title":"Applied functional data analysis: Methods and case studies","author":"Ramsay","year":"2007"},{"key":"2023110616083214500_qkad029-B49","author":"Revelle","year":"2017"},{"issue":"45","key":"2023110616083214500_qkad029-B52","doi-asserted-by":"crossref","first-page":"12679","DOI":"10.1073\/pnas.1525793113","article-title":"Co-clustering directed graphs to discover asymmetries and directional communities","volume":"113","author":"Rohe","year":"2016","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2023110616083214500_qkad029-B53","author":"Salcianu","year":"2020"},{"issue":"3","key":"2023110616083214500_qkad029-B56","doi-asserted-by":"crossref","first-page":"37","DOI":"10.21105\/joss.00037","article-title":"tidytext: Text mining and analysis using tidy data principles in R","volume":"1","author":"Silge","year":"2016","journal-title":"The Journal of Open Source Software"},{"key":"2023110616083214500_qkad029-B58","doi-asserted-by":"crossref","DOI":"10.1037\/10018-000","volume-title":"The vectors of mind: Multiple-factor analysis for the isolation of primary traits","author":"Thurstone","year":"1935"},{"key":"2023110616083214500_qkad029-B59","volume-title":"Multiple factor analysis","author":"Thurstone","year":"1947"},{"issue":"6","key":"2023110616083214500_qkad029-B62","doi-asserted-by":"crossref","first-page":"2905","DOI":"10.1214\/13-AOS1151","article-title":"Minimax sparse principal subspace estimation in high dimensions","volume":"41","author":"Vu","year":"2013","journal-title":"The Annals of Statistics"},{"issue":"4","key":"2023110616083214500_qkad029-B63","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1214\/16-AOAS977","article-title":"Coauthorship and citation networks for statisticians: Comment","volume":"10","author":"Wang","year":"2016","journal-title":"The Annals of Applied Statistics"},{"key":"2023110616083214500_qkad029-B65","author":"Zhang","year":"2018"},{"key":"2023110616083214500_qkad029-B66","author":"Zhang","year":"2014"},{"key":"2023110616083214500_qkad029-B67","author":"Zhang","year":"2018"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkad029\/50840092\/qkad029.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/4\/1037\/52714936\/qkad029.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/4\/1037\/52714936\/qkad029.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,6]],"date-time":"2023-11-06T11:09:12Z","timestamp":1699268952000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/85\/4\/1037\/7221295"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,7]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,7,7]]},"published-print":{"date-parts":[[2023,9,29]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkad029","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,9]]},"published":{"date-parts":[[2023,7,7]]}}}