{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T03:58:55Z","timestamp":1790308735494,"version":"4.1.0"},"reference-count":13,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:00:00Z","timestamp":1782259200000},"content-version":"vor","delay-in-days":174,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,7,6]]},"abstract":"<jats:title>SUMMARY<\/jats:title>\n                  <jats:p>This paper addresses a long-standing open problem in crossed random-effects models under unbalanced designs: how to find an analytic expression for the inverse of $ {V} $, the covariance matrix of the observed response. For unbalanced crossed designs, $ {V} $ is dense, and the lack of a closed-form representation for $ {V}^{-1} $ has, until now, made the use of likelihood-based methods computationally challenging and difficult to analyse mathematically. We use the Khatri\u2013Rao product to represent $ {V} $ and then construct a modified covariance matrix whose inverse admits an exact spectral decomposition. Building on this construction, we obtain an elegant and simple approximation to $ {V}^{-1} $ for asymptotic unbalanced designs. For nonasymptotic settings, we derive an accurate and interpretable approximation under mildly unbalanced data and establish an exact inverse representation as a low-rank correction to this approximation, applicable to arbitrary degrees of unbalance. Simulations demonstrate the framework\u2019s accuracy, stability and tractability.<\/jats:p>","DOI":"10.1093\/biomet\/asag041","type":"journal-article","created":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T11:42:54Z","timestamp":1782214974000},"source":"Crossref","is-referenced-by-count":0,"title":["On the inverse of covariance matrices for unbalanced crossed designs"],"prefix":"10.1093","volume":"113","author":[{"given":"Ziyang","family":"Lyu","sequence":"first","affiliation":[{"name":"UNSW Data Science Hub, School of Mathematics and Statistics, University of New South Wales ,Sydney, New South Wales 2052,","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S A","family":"Sisson","sequence":"additional","affiliation":[{"name":"UNSW Data Science Hub, School of Mathematics and Statistics, University of New South Wales ,Sydney, New South Wales 2052, \u00a0 scott.sisson@unsw.edu.au","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A H","family":"Welsh","sequence":"additional","affiliation":[{"name":"Research School of Finance, Actuarial Studies and Statistics, Australian National University , Kingsley Street , Canberra, Australian Capital Territory 2600, \u00a0 alan.welsh@anu.edu.au","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,6,24]]},"reference":[{"key":"2026092423170802500_asag041-B1","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1214\/17-EJS1236","article-title":"Efficient moment calculations for variance components in large unbalanced crossed random effects models","volume":"11","author":"Gao","year":"2017","journal-title":"Electron. J. Statist."},{"key":"2026092423170802500_asag041-B2","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1214\/21-AOS2121","article-title":"Backfitting for large scale crossed random effects regressions","volume":"50","author":"Ghosh","year":"2022","journal-title":"Ann. Statist."},{"key":"2026092423170802500_asag041-B3","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1137\/1023004","article-title":"On deriving the inverse of a sum of matrices","volume":"23","author":"Henderson","year":"1981","journal-title":"SIAM Rev."},{"key":"2026092423170802500_asag041-B4","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.laa.2003.12.011","article-title":"Dispersion matrix in balanced mixed ANOVA models","volume":"382","author":"Jiang","year":"2004","journal-title":"Lin. Algeb. Appl"},{"key":"2026092423170802500_asag041-B5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-91695-4","volume-title":"Large Sample Techniques for Statistics","author":"Jiang","year":"2022"},{"key":"2026092423170802500_asag041-B6","doi-asserted-by":"crossref","first-page":"1298","DOI":"10.1214\/25-AOS2504","article-title":"Asymptotic distribution of maximum likelihood estimator in generalized linear mixed models with crossed random effects","volume":"53","author":"Jiang","year":"2025","journal-title":"Ann. Statist."},{"key":"2026092423170802500_asag041-B7","first-page":"167","article-title":"Solutions to some functional equations and their applications to characterization of probability distributions","volume":"30","author":"Khatri","year":"1968","journal-title":"Sankhya A"},{"key":"2026092423170802500_asag041-B8","doi-asserted-by":"crossref","first-page":"2956","DOI":"10.1214\/24-AOS2469","article-title":"Increasing dimension asymptotics for two-way crossed mixed effect models","volume":"52","author":"Lyu","year":"2024","journal-title":"Ann. Statist."},{"key":"2026092423170802500_asag041-B9","author":"Lyu","year":"2025"},{"key":"2026092423170802500_asag041-B10","first-page":"465","article-title":"Dispersion matrices for variance components models","volume":"74","author":"Searle","year":"1979","journal-title":"J. Am. Statist. Assoc"},{"key":"2026092423170802500_asag041-B11","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1080\/03610927308827062","article-title":"Wanted: an inverse matrix","volume":"2","author":"Searle","year":"1973","journal-title":"Commun. Statist"},{"key":"2026092423170802500_asag041-B12","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1038\/s41597-024-04332-7","article-title":"Multi-environment field trials for wheat yield, stability and breeding progress in Germany","volume":"12","year":"2025","journal-title":"Sci. Data"},{"key":"2026092423170802500_asag041-B13","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1080\/03610918208812276","article-title":"Another approach to inverting a covariance matrix when data are unbalanced","volume":"11","author":"Wansbeek","year":"1982","journal-title":"Commun. Statist. B"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag041\/68605589\/asag041.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/3\/asag041\/68605589\/asag041.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/3\/asag041\/68605589\/asag041.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T03:17:15Z","timestamp":1790306235000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asag041\/8715499"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,7,6]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asag041","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026]]},"published":{"date-parts":[[2026]]},"article-number":"asag041"}}