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We assume that the first moment of the predictor matrix given the response can be decomposed into a<jats:italic>row<\/jats:italic>and<jats:italic>column<\/jats:italic>component via a Kronecker product structure. We obtain least squares and maximum likelihood estimates of the sufficient reductions in the matrix predictors, derive statistical properties of the resulting estimates and present fast computational algorithms with assured convergence. The performance of the proposed approaches in regression and classification is compared in simulations.We illustrate the methods on two examples, using longitudinally measured serum biomarker and neuroimaging data.<\/jats:p>","DOI":"10.1007\/s41060-020-00228-y","type":"journal-article","created":{"date-parts":[[2020,8,4]],"date-time":"2020-08-04T12:02:36Z","timestamp":1596542556000},"page":"11-26","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Least squares and maximum likelihood estimation of sufficient reductions in regressions with matrix-valued predictors"],"prefix":"10.1007","volume":"11","author":[{"given":"Ruth M.","family":"Pfeiffer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel B.","family":"Kapla","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Efstathia","family":"Bura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,4]]},"reference":[{"key":"228_CR1","volume-title":"An Introduction to Multivariate Statistical Analysis","author":"TW Anderson","year":"2003","unstructured":"Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. 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