Optimal designs for crossover model with partial interactions

IF 0.7 4区 数学 Q3 STATISTICS & PROBABILITY Stat Pub Date : 2024-03-07 DOI:10.1002/sta4.668
Futao Zhang, Pierre Druilhet, Xiangshun Kong
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We find that for a fixed number of treatments, say <mjx-container aria-label=\"t\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"t\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c3669d78-641d-4172-958e-37ddc1934825/sta4668-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"t\" data-semantic-type=\"identifier\">t</mi></mrow>$$ t $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, the number of distinct treatments appearing in the support sequences increases with the increase of the number of periods, <mjx-container aria-label=\"k\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"k\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c09a2ac1-1512-49b7-8baa-c3acf0ec7390/sta4668-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"k\" data-semantic-type=\"identifier\">k</mi></mrow>$$ k $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, until <mjx-container aria-label=\"k greater than or equals t squared\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,4\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"k greater than or equals t squared\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,≥\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"2,3\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/7fd889b0-645a-4233-b6e1-cac91f2f92b4/sta4668-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,4\" data-semantic-content=\"1\" data-semantic-role=\"inequality\" data-semantic-speech=\"k greater than or equals t squared\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">k</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,≥\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">≥</mo><msup data-semantic-=\"\" data-semantic-children=\"2,3\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow></msup></mrow>$$ k\\ge {t}^2 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, in which case all <mjx-container aria-label=\"t\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"t\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4b2da8a5-5d23-4833-9a71-556a4d4dd7b4/sta4668-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"t\" data-semantic-type=\"identifier\">t</mi></mrow>$$ t $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> treatments appear. The optimal design can be constructed from up to two representative sequences, within which each treatment appears in consecutive periods with equal or almost equal numbers of replications.","PeriodicalId":56159,"journal":{"name":"Stat","volume":"134 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stat","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/sta4.668","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

This paper studies the universally optimal designs for estimating total effects under crossover models with partial interactions. We provide necessary and sufficient conditions for a symmetric design to be universally optimal, based on which algorithms can be used to derive optimal symmetric designs under any form of the within-block covariance matrix. To cope with the computational complexity of algorithms when the experimental scale is too large, we provide the analytical form of optimal designs under the type-H covariance matrix. We find that for a fixed number of treatments, say , the number of distinct treatments appearing in the support sequences increases with the increase of the number of periods, , until , in which case all treatments appear. The optimal design can be constructed from up to two representative sequences, within which each treatment appears in consecutive periods with equal or almost equal numbers of replications.
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具有部分交互作用的交叉模型的优化设计
本文研究了在具有部分交互作用的交叉模型下估计总效应的普遍最优设计。我们提供了对称设计成为普遍最优设计的必要条件和充分条件,在此基础上,可以使用算法推导出任何形式的块内协方差矩阵下的最优对称设计。为了应对实验规模过大时算法的计算复杂性,我们提供了 H 型协方差矩阵下最优设计的解析形式。我们发现,对于固定数量的处理(例如 t$$ t $$),支持序列中出现的不同处理的数量会随着周期数 k$$ k $$的增加而增加,直到 k≥t2$$ k\ge {t}^2 $$,在这种情况下,所有 t$$ t $$的处理都会出现。最佳设计最多可由两个代表性序列构成,其中每个处理在重复次数相等或几乎相等的连续时期内出现。
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来源期刊
Stat
Stat Decision Sciences-Statistics, Probability and Uncertainty
CiteScore
1.10
自引率
0.00%
发文量
85
期刊介绍: Stat is an innovative electronic journal for the rapid publication of novel and topical research results, publishing compact articles of the highest quality in all areas of statistical endeavour. Its purpose is to provide a means of rapid sharing of important new theoretical, methodological and applied research. Stat is a joint venture between the International Statistical Institute and Wiley-Blackwell. Stat is characterised by: • Speed - a high-quality review process that aims to reach a decision within 20 days of submission. • Concision - a maximum article length of 10 pages of text, not including references. • Supporting materials - inclusion of electronic supporting materials including graphs, video, software, data and images. • Scope - addresses all areas of statistics and interdisciplinary areas. Stat is a scientific journal for the international community of statisticians and researchers and practitioners in allied quantitative disciplines.
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