{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T19:54:42Z","timestamp":1765828482174,"version":"3.43.0"},"reference-count":26,"publisher":"Wiley","issue":"13-14","license":[{"start":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T00:00:00Z","timestamp":1750636800000},"content-version":"vor","delay-in-days":22,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000272","name":"National Institute for Health and Care Research","doi-asserted-by":"publisher","award":["NIHR 204294","NIHR158242","NIHR200132"],"award-info":[{"award-number":["NIHR 204294","NIHR158242","NIHR200132"]}],"id":[{"id":"10.13039\/501100000272","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","award":["MR\/V038591\/1"],"award-info":[{"award-number":["MR\/V038591\/1"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>ABSTRACT<\/jats:title><jats:p>Stepped\u2010wedge studies usually entail regular sampling of clusters over time. Yet the precision of the treatment effect estimator can sometimes be improved if the regular sampling scheme is replaced by one with preferential allocation of observations to particular time\u2010epochs within each cluster. We present some exact results for optimizing the allocation for a general experimental layout under a mixed effects model with a time\u2010varying cluster\u2010autocorrelation structure, together with an algorithm for generating optimal allocations. An index of cluster variation is introduced, an increasing function of both the intra\u2010class correlation and the total sample size, which encapsulates the influence of cluster\u2010level variation on the optimal allocation. For any specified layout there is a sampling scheme (the \u2018best natural allocation\u2019) that solves the optimization problem for all values of this index up to a threshold value which depends only on the cluster autocorrelations. Under such a scheme the treatment effect estimator is equal to a simple difference between the means of the treated and control observations. Best natural allocations stand alongside conventional parallel and cross\u2010over designs in giving equal weight to observations from all participants, even under stepped\u2010wedge layouts with irreversible interventions. When applied to a recent study of primary care training programmes in low\u2010 and middle\u2010 income countries (The REaCH study), the results lead to substantial reductions in total sample size, without loss of precision. For stepped\u2010wedge layouts with block\u2010exchangeable or time\u2010decaying cluster autocorrelations, we present explicit conditions for the optimality of staircase\u2010type sampling schemes, which can arise as best natural allocations in such cases.<\/jats:p>","DOI":"10.1002\/sim.70100","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T10:30:51Z","timestamp":1750674651000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Allocation of Observations in Stepped\u2010Wedge and Other Cluster Studies With Correlated Cluster\u2010Period Effects"],"prefix":"10.1002","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2022-045X","authenticated-orcid":false,"given":"Alan J.","family":"Girling","sequence":"first","affiliation":[{"name":"Institute of Applied Health Research University of Birmingham  Birmingham UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8972-769X","authenticated-orcid":false,"given":"Samuel I.","family":"Watson","sequence":"additional","affiliation":[{"name":"Institute of Applied Health Research University of Birmingham  Birmingham UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,6,23]]},"reference":[{"key":"e_1_2_11_2_1","doi-asserted-by":"publisher","DOI":"10.1136\/bmj.h391"},{"key":"e_1_2_11_3_1","doi-asserted-by":"publisher","DOI":"10.1136\/bmj.k1614"},{"key":"e_1_2_11_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cct.2006.05.007"},{"key":"e_1_2_11_5_1","doi-asserted-by":"publisher","DOI":"10.1177\/09622802231202379"},{"key":"e_1_2_11_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.6850"},{"key":"e_1_2_11_7_1","doi-asserted-by":"publisher","DOI":"10.1177\/1740774519873888"},{"key":"e_1_2_11_8_1","doi-asserted-by":"publisher","DOI":"10.1177\/1740774517723921"},{"key":"e_1_2_11_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2015.01.024"},{"key":"e_1_2_11_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.7853"},{"key":"e_1_2_11_11_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.12959"},{"key":"e_1_2_11_12_1","doi-asserted-by":"publisher","DOI":"10.1186\/s12874-023-01969-7"},{"key":"e_1_2_11_13_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.7028"},{"key":"e_1_2_11_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11222\u2010023\u201010280\u2010w"},{"key":"e_1_2_11_15_1","doi-asserted-by":"publisher","DOI":"10.1186\/s12874\u2010020\u201001155\u2010z"},{"key":"e_1_2_11_16_1","doi-asserted-by":"publisher","DOI":"10.1177\/0962280217734981"},{"key":"e_1_2_11_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.8089"},{"key":"e_1_2_11_18_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0370164600014346"},{"key":"e_1_2_11_19_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/79.4.763"},{"key":"e_1_2_11_20_1","unstructured":"S. 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