{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"institution":[{"name":"Authorea, Inc."}],"indexed":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T16:39:53Z","timestamp":1748968793688,"version":"3.41.0"},"posted":{"date-parts":[[2023,9,11]]},"group-title":"Preprints","reference-count":0,"publisher":"Wiley","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2023,9,11]]},"abstract":"<jats:p id=\"p1\">The models for symmetric stochastic matrices that we consider in this\nstudy are developed using the spectral analysis of the respective mean\nmatrices. The adjustment and validation of the models require the usage\nof the estimated structure vectors. The information enclosed in these\nmatrices can be condensed into the pair consisting of the estimated\nstructure vectors and the sum of squares of residuals. The results\nobtained allow for cross-sectional and longitudinal inference. For\nmodels of degree greater than one, it is also considered the possibility\nof truncating the model when eigenvalues are much higher than the\nothers. A direct consequence of the adopted methodology is the\napplication of the degree-one models to cross-product matrices and\nHilbert-Schmidt scalar product matrices. In addition to these models,\nstructured family models were also considered. The models of these\nfamilies are associated with the treatments of a base design. The action\nof the factors considered in the base design on the structure vectors is\nalso analyzed. In structured families with orthogonal base design, the\ndesigns are associated to partitions, and the hypotheses formulated are\nassociated with the spaces of these partitions. We carry out ANOVA-like\nanalysis for the action of the factors in the base design, on homolog\ncomponents on estimated structure vectors, considering that the\nestimator\u2019s structure vectors have, approximately, the same covariance\nmatrix. To apply our results, we assume the factors in the base design\nto have fixed effects and that the base design has orthogonal structure.\nThe action of factors in the base design is studied. An application is\ngiven, using a data set from a breeding program of durum wheat (\n<jats:italic>Triticum turgidum L<\/jats:italic>., Durum Group) conducted in Portugal. The\nresults show that our methodology is fully applicable to complete and\nincomplete data sets, often observed in multi-environmental trials.<\/jats:p>","DOI":"10.22541\/au.169445889.96170167\/v1","type":"posted-content","created":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T19:09:11Z","timestamp":1694459351000},"source":"Crossref","is-referenced-by-count":0,"title":["CMMSE: Joint analysis of Structured orthogonal families"],"prefix":"10.22541","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6350-5610","authenticated-orcid":true,"given":"Cristina","family":"Dias","sequence":"first","affiliation":[{"name":"Universidade Nova de Lisboa Centro de Matematica e Aplicacoes"}]},{"given":"Carla","family":"Santos","sequence":"additional","affiliation":[{"name":"Instituto Politecnico de Beja Departamento de Tecnologias e Ciencias Aplicadas"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5719-6310","authenticated-orcid":true,"given":"Nuno M.","family":"Brites","sequence":"additional","affiliation":[{"name":"ISEG Lisbon School of Economics and Management"}]},{"given":"Jo\u00e3o","family":"Mexia","sequence":"additional","affiliation":[{"name":"Universidade Nova de Lisboa Centro de Matematica e Aplicacoes"}]}],"member":"311","container-title":[],"original-title":[],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T19:09:11Z","timestamp":1694459351000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.authorea.com\/users\/661945\/articles\/664826-cmmse-joint-analysis-of-structured-orthogonal-families?commit=e7df5749a3b8878ac3a12262bbddd518f8673165"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,11]]},"references-count":0,"URL":"https:\/\/doi.org\/10.22541\/au.169445889.96170167\/v1","relation":{},"subject":[],"published":{"date-parts":[[2023,9,11]]},"subtype":"preprint"}}