{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T16:40:11Z","timestamp":1792600811420,"version":"4.5.0"},"reference-count":16,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T00:00:00Z","timestamp":1790208000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T00:00:00Z","timestamp":1790208000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Biometrical J"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Row\u2013column designs play an important role in applications where two orthogonal sources of error need to be controlled for by blocking. Field or greenhouse experiments, in which experimental units are arranged as a rectangular array of experimental units, are a prominent example. In plant breeding, the amount of seed available for the treatments to be tested may be so limited that only one experimental unit per treatment can be accommodated. In such settings, augmented designs become an interesting option, where a small set of treatments, for which sufficient seed is available, is replicated across the rectangular layout so that row and column effects, as well as the error variance, can be estimated. Here, we consider the use of an auxiliary design, also known as a contraction, to generate an augmented row\u2013column design. We make use of the fact that the efficiency factors of the contraction and the associated augmented design are closely interlinked. A major advantage of this approach is that an efficient contraction can be found by computer search at much higher computational speed than is required for direct search for an efficient augmented design. Two examples are used to illustrate the proposed method. Furthermore, we contrast the usage of our approach when the array dimensions are fixed, as is the case with microplates, to settings with more flexibility regarding the number of rows and columns of the layout, as in greenhouse and field trials.<\/jats:p>","DOI":"10.1002\/bimj.70182","type":"journal-article","created":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T06:18:46Z","timestamp":1790317126000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Rectangular Augmented Row\u2013Column Designs Generated From Contractions"],"prefix":"10.1002","volume":"68","author":[{"given":"Hans\u2010Peter","family":"Piepho","sequence":"first","affiliation":[{"name":"Biostatistics Unit Institute of Crop Science University of Hohenheim  Stuttgart Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emlyn","family":"Williams","sequence":"additional","affiliation":[{"name":"Statistical Support Network Australian National University  Canberra Australian Capital Territory Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,9,24]]},"reference":[{"key":"e_1_2_9_2_1","unstructured":"VSNi.2025.CycDesigN Version 8.0.http:\/\/www.vsni.co.uk\/software\/cycdesign\/."},{"key":"e_1_2_9_3_1","doi-asserted-by":"crossref","unstructured":"Bailey R. A. andL. M.Haines.2025. \u201cSquare Array Designs for Unreplicated Test Treatments with Replicated Controls.\u201dJournal of Agricultural Biological and Environmental Statistics.https:\/\/doi.org\/10.1007\/s13253\u2010025\u201000699\u20101.","DOI":"10.1007\/s13253-025-00699-1"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s13253-023-00570-1"},{"key":"e_1_2_9_5_1","unstructured":"Federer W. T.1956. \u201cAugmented (or Hoonuiaku) Designs.\u201dHawaiian Sugar Planters Record55 no. 2 191\u2013208."},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.2307\/2527837"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1300\/J411v14n01_04"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.2307\/2529707"},{"key":"e_1_2_9_9_1","unstructured":"Gubner J. A.2024.Block Matrix Formulas.https:\/\/gubner.ece.wisc.edu\/notes\/BlockMatrixFormulas.pdf."},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-7220-0"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.2307\/2531083"},{"key":"e_1_2_9_12_1","first-page":"147","article-title":"The Analysis of Randomized Experiments with Orthogonal Block Structure. I. Block Structure and the Null Analysis of Variance","volume":"283","author":"Nelder J. A.","year":"1965","journal-title":"Proceedings of the Royal Society London A"},{"key":"e_1_2_9_13_1","unstructured":"Patterson H. D. andE. R.Williams.1976. \u201cSome Theoretical Results on General Block Designs.\u201d InProceedings of the 5th British Combinatorial Conference.Congressus Numerantium XV Utilitas Mathematica Winnipeg:489\u2013496."},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.2134\/agronj2016.06.0325"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2020.106988"},{"key":"e_1_2_9_16_1","doi-asserted-by":"crossref","unstructured":"Williams E. R. andR. A.Bailey.2026. \u201cExtended Semi\u2010Latin Squares for Use in Field and Glasshouse Trials.\u201dJournal of Agricultural Biological and Environmental Statistics.https:\/\/doi.org\/10.1007\/s13253\u2010026\u201000729\u20106.","DOI":"10.1007\/s13253-026-00729-6"},{"key":"e_1_2_9_17_1","unstructured":"Williams E. R. andH. P.Piepho.2025.\"A Note on the Construction of Augmented Designs in Square Arrays.\"https:\/\/arxiv.org\/abs\/2501.08448."}],"container-title":["Biometrical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70182","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/bimj.70182","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70182","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T15:50:39Z","timestamp":1792597839000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/bimj.70182"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,24]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["10.1002\/bimj.70182"],"URL":"https:\/\/doi.org\/10.1002\/bimj.70182","archive":["Portico"],"relation":{},"ISSN":["0323-3847","1521-4036"],"issn-type":[{"value":"0323-3847","type":"print"},{"value":"1521-4036","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,24]]},"assertion":[{"value":"2025-11-21","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-30","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-24","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70182"}}