{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,12]],"date-time":"2025-07-12T22:56:01Z","timestamp":1752360961386},"reference-count":49,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2022,10,31]],"date-time":"2022-10-31T00:00:00Z","timestamp":1667174400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,11,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Blocked randomized designs are used to improve the precision of treatment effect estimates compared to a completely randomized design. A block is a set of units that are relatively homogeneous and consequently would tend to produce relatively similar outcomes if the treatment had no effect. The problem of finding the optimal blocking of the units into equal sized blocks of any given size larger than two is known to be a difficult problem\u2014there is no polynomial time method guaranteed to find the optimal blocking. All available methods to solve the problem are heuristic methods. We propose methods that run in polynomial time and guarantee a blocking that is provably close to the optimal blocking. In all our simulation studies, the proposed methods perform better, create better homogeneous blocks, compared with the existing methods. Our blocking method aims to minimize the maximum of all pairwise differences of units in the same block. We show that bounding this maximum difference ensures that the error in the average treatment effect estimate is similarly bounded for all treatment assignments. In contrast, if the blocking bounds the average or sum of these differences, the error in the average treatment effect estimate can still be large in several treatment assignments.<\/jats:p>","DOI":"10.1111\/rssb.12545","type":"journal-article","created":{"date-parts":[[2022,10,31]],"date-time":"2022-10-31T23:55:41Z","timestamp":1667260541000},"page":"1726-1750","source":"Crossref","is-referenced-by-count":2,"title":["An Approximation Algorithm for Blocking of an Experimental Design"],"prefix":"10.1093","volume":"84","author":[{"given":"Bikram","family":"Karmakar","sequence":"first","affiliation":[{"name":"Department of Statistics , , Gainesville , Florida , USA"},{"name":"University of Florida , , Gainesville , Florida , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2022,10,31]]},"reference":[{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1002\/pst.433","article-title":"Optimal caliper widths for propensity-score matching when estimating differences in means and differences in proportions in observational studies","volume":"10","author":"Austin","year":"2011","journal-title":"Pharmaceutical Statistics"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1080\/01621459.1987.10478487","article-title":"Restricted randomization: a practical example","volume":"82","author":"Bailey","year":"1987","journal-title":"Journal of the American Statistical Association"},{"key":"2023030906295427600_","volume-title":"nbpMatching: functions for optimal non-bipartite matching","author":"Beck","year":"2016"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-349-03521-2","volume-title":"Graph theory with applications","author":"Bondy","year":"1976"},{"key":"2023030906295427600_","first-page":"311","article-title":"The 2K\u2212p fractional factorial designs","volume":"3","author":"Box","year":"1961","journal-title":"Technometrics"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1214\/aos\/1009210551","article-title":"Blocked regular fractional factorial designs with maximum estimation capacity","volume":"29","author":"Cheng","year":"2001","journal-title":"Annals of Statistics"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"234","DOI":"10.2307\/2344179","article-title":"The planning of observational studies of human populations","volume":"128","author":"Cochran","year":"1965","journal-title":"Journal of the Royal Statistical Society: Series A"},{"key":"2023030906295427600_","volume-title":"Planning of experiments","author":"Cox","year":"1958"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","DOI":"10.1201\/9781420035834","volume-title":"The theory of the design of experiments","author":"Cox","year":"2000"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF02240182","article-title":"Alternate strategies for solving bottleneck assignment problems\u2014analysis and computational results","volume":"33","author":"Derigs","year":"1984","journal-title":"Computing"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/BF02288324","article-title":"Solving nonbipartite matching problems via shortest path techniques","volume":"13","author":"Derigs","year":"1988","journal-title":"Annals of Operations Research"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"1830","DOI":"10.2105\/AJPH.93.11.1830","article-title":"New inroads in preventing adolescent drug use: results from a large-scale trial of Project ALERT in middle schools","volume":"93","author":"Ellickson","year":"2003","journal-title":"American Journal of Public Health"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","DOI":"10.1002\/0470108584","volume-title":"Variations on split plot and split block experiment designs","author":"Federer","year":"2007"},{"key":"2023030906295427600_","volume-title":"The design of experiments","author":"Fisher","year":"1935"},{"key":"2023030906295427600_","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/0091-7435(88)90053-9","article-title":"The television, school and family smoking prevention and cessation project. 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