{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T14:02:29Z","timestamp":1791986549814,"version":"4.4.0"},"reference-count":45,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T00:00:00Z","timestamp":1674777600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100001352","name":"National University of Singapore","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001352","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000001","name":"U.S. National Science Foundation","doi-asserted-by":"crossref","award":["1945136"],"award-info":[{"award-number":["1945136"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,2,27]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Randomized experiments are the gold standard for causal inference and enable unbiased estimation of treatment effects. Regression adjustment provides a convenient way to incorporate covariate information for additional efficiency. This article provides a unified account of its utility for improving estimation efficiency in multiarmed experiments. We start with the commonly used additive and fully interacted models for regression adjustment in estimating average treatment effects (ATE), and clarify the trade-offs between the resulting ordinary least squares (OLS) estimators in terms of finite sample performance and asymptotic efficiency. We then move on to regression adjustment based on restricted least squares (RLS), and establish for the first time its properties for inferring ATE from the design-based perspective. The resulting inference has multiple guarantees. First, it is asymptotically efficient when the restriction is correctly specified. Second, it remains consistent as long as the restriction on the coefficients of the treatment indicators, if any, is correctly specified and separate from that on the coefficients of the treatment-covariate interactions. Third, it can have better finite sample performance than the unrestricted counterpart even when the restriction is moderately misspecified. It is thus our recommendation when the OLS fit of the fully interacted regression risks large finite sample variability in case of many covariates, many treatments, yet a moderate sample size. In addition, the newly established theory of RLS also provides a unified way of studying OLS-based inference from general regression specifications. As an illustration, we demonstrate its value for studying OLS-based regression adjustment in factorial experiments. Importantly, although we analyse inferential procedures that are motivated by OLS, we do not invoke any assumptions required by the underlying linear models.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkac003","type":"journal-article","created":{"date-parts":[[2023,1,28]],"date-time":"2023-01-28T09:54:55Z","timestamp":1674899695000},"page":"1-23","source":"Crossref","is-referenced-by-count":13,"title":["Covariate adjustment in multiarmed, possibly factorial experiments"],"prefix":"10.1093","volume":"85","author":[{"given":"Anqi","family":"Zhao","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, National University of Singapore , Singapore , Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2704-2353","authenticated-orcid":false,"given":"Peng","family":"Ding","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California , Berkeley, CA , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,1,27]]},"reference":[{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"265","DOI":"10.3982\/ECTA12675","article-title":"Sampling-based versus design-based uncertainty in regression analysis","volume":"88","author":"Abadie","year":"2020","journal-title":"Econometrica"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1080\/01621459.2021.1978468","article-title":"Noncompliance and instrumental variables for 2K factorial experiments","author":"Blackwell","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"27","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"7383","DOI":"10.1073\/pnas.1510506113","article-title":"Lasso adjustments of treatment effect estimates in randomized experiments","volume":"113","author":"Bloniarz","year":"2016","journal-title":"Proceedings of the National Academy of Sciences of the United States of America"},{"key":"2023022804421646500_","volume-title":"Statistics for experimenters: Design, innovation, and discovery","author":"Box","year":"2005","edition":"2nd ed."},{"issue":"4","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1958","DOI":"10.1214\/16-AOAS959","article-title":"Improving covariate balance in 2K factorial designs via rerandomization with an application to a New York City Department of Education High School Study","volume":"10","author":"Branson","year":"2016","journal-title":"The Annals of Applied Statistics"},{"issue":"524","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1784","DOI":"10.1080\/01621459.2017.1375934","article-title":"Inference under covariate-adaptive randomization","volume":"113","author":"Bugni","year":"2018","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1747","DOI":"10.3982\/QE1150","article-title":"Inference under covariate-adaptive randomization with multiple treatments","volume":"10","author":"Bugni","year":"2019","journal-title":"Quantitative Economics"},{"issue":"21","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"2687","DOI":"10.1002\/sim.3643","article-title":"Developing multicomponent interventions using fractional factorial designs","volume":"28","author":"Chakraborty","year":"2009","journal-title":"Statistics in Medicine"},{"issue":"3","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1037\/a0015826","article-title":"Design of experiments with multiple independent variables: A resource management perspective on complete and reduced factorial designs","volume":"14","author":"Collins","year":"2009","journal-title":"Psychological Methods"},{"issue":"4","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1111\/rssb.12085","article-title":"Causal inference from 2K factorial designs by using potential outcomes","volume":"77","author":"Dasgupta","year":"2015","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"525","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1080\/01621459.2017.1407322","article-title":"Decomposing treatment effect variation","volume":"114","author":"Ding","year":"2019","journal-title":"Journal of the American Statistical Association"},{"key":"2023022804421646500_","first-page":"3895","volume-title":"Handbook of development economics","author":"Duflo","year":"2007"},{"issue":"526","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1080\/01621459.2018.1476246","article-title":"Causal interaction in factorial experiments: Application to conjoint analysis","volume":"114","author":"Egami","year":"2019","journal-title":"Journal of the American Statistical Association"},{"key":"2023022804421646500_","volume-title":"The design of experiments","author":"Fisher","year":"1935","edition":"1st ed."},{"issue":"4","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"994","DOI":"10.1093\/biomet\/asy034","article-title":"Regression-assisted inference for the average treatment effect in paired experiments","volume":"105","author":"Fogarty","year":"2018","journal-title":"Biometrika"},{"issue":"2","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.aam.2006.12.003","article-title":"On regression adjustments to experimental data","volume":"40","author":"Freedman","year":"2008","journal-title":"Advances in Applied Mathematics"},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1214\/07-AOAS143","article-title":"On regression adjustments in experiments with several treatments","volume":"2","author":"Freedman","year":"2008","journal-title":"Annals of Applied Statistics"},{"issue":"3","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"563","DOI":"10.2307\/2109587","article-title":"The restricted least squares estimator: A pedagogical note","volume":"73","author":"Greene","year":"1991","journal-title":"The Review of Economics and Statistics"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1080\/01621459.2021.1941053","article-title":"The generalized Oaxaca\u2013Blinder estimator","author":"Guo","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/pan\/mpt024","article-title":"Causal inference in conjoint analysis: Understanding multidimensional choices via stated preference experiments","volume":"22","author":"Hainmueller","year":"2014","journal-title":"Political Analysis"},{"key":"2023022804421646500_","volume-title":"Design and analysis of experiments: Introduction to experimental design","author":"Hinkelmann","year":"2008","edition":"2nd ed."},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139025751","volume-title":"Causal inference for statistics, social, and biomedical sciences: An introduction","author":"Imbens","year":"2015"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1097\/00010694-195205000-00012","volume-title":"The design and analysis of experiments","author":"Kempthorne","year":"1952"},{"issue":"520","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1080\/01621459.2017.1295865","article-title":"General forms of finite population central limit theorems with applications to causal inference","volume":"112","author":"Li","year":"2017","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1111\/rssb.12353","article-title":"Rerandomization and regression adjustment","volume":"82","author":"Li","year":"2020","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1214\/18-AOS1790","article-title":"Rerandomization in 2K factorial experiments","volume":"48","author":"Li","year":"2020","journal-title":"The Annals of Statistics"},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1214\/12-AOAS583","article-title":"Agnostic notes on regression adjustments to experimental data: Reexamining Freedman\u2019s critique","volume":"7","author":"Lin","year":"2013","journal-title":"The Annals of Applied Statistics"},{"issue":"4","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1093\/biomet\/asaa038","article-title":"Regression-adjusted average treatment effect estimates in stratified randomized experiments","volume":"107","author":"Liu","year":"2020","journal-title":"Biometrika"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.spl.2016.01.010","article-title":"On randomization-based and regression-based inferences for 2K factorial designs","volume":"112","author":"Lu","year":"2016","journal-title":"Statistics and Probability Letters"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.spl.2016.07.010","article-title":"Covariate adjustment in randomization-based causal inference for 2K factorial designs","volume":"119","author":"Lu","year":"2016","journal-title":"Statistics and Probability Letters"},{"issue":"2","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1111\/j.1467-9868.2012.01048.x","article-title":"Adjusting treatment effect estimates by post-stratification in randomized experiments","volume":"75","author":"Miratrix","year":"2013","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"2","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1214\/12-AOS1008","article-title":"Rerandomization to improve covariate balance in experiments","volume":"40","author":"Morgan","year":"2012","journal-title":"The Annals of Statistics"},{"issue":"522","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1080\/01621459.2017.1294076","article-title":"Using standard tools from finite population sampling to improve causal inference for complex experiments","volume":"113","author":"Mukerjee","year":"2018","journal-title":"Journal of the American Statistical Association"},{"issue":"5","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1080\/07474938.2020.1824732","article-title":"Revisiting regression adjustment in experiments with heterogeneous treatment effects","volume":"40","author":"Negi","year":"2021","journal-title":"Econometric Reviews"},{"issue":"4","key":"2023022804421646500_","first-page":"465","article-title":"On the application of probability theory to agricultural experiments","volume":"5","author":"Neyman","year":"1923","journal-title":"Statistical Science"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1002\/cjs.11734","article-title":"Causal inference for multiple treatments using fractional factorial designs","author":"Pashley","year":"2022","journal-title":"The Canadian Journal of Statistics"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316436","volume-title":"Linear statistical inference and its applications","author":"Rao","year":"1973","edition":"2nd ed."},{"issue":"1","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.jspi.2009.07.008","article-title":"Is regression adjustment supported by the Neyman model for causal inference?","volume":"140","author":"Schochet","year":"2010","journal-title":"Journal of Statistical Planning and Inference"},{"issue":"5","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"568","DOI":"10.3102\/1076998618786968","article-title":"Design-based estimators for average treatment effects for multi-armed RCTs","volume":"43","author":"Schochet","year":"2018","journal-title":"Journal of Educational and Behavioral Statistics"},{"key":"2023022804421646500_","volume-title":"Principles of econometrics","author":"Theil","year":"1971"},{"issue":"23","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"4658","DOI":"10.1002\/sim.3113","article-title":"Covariate adjustment for two-sample treatment comparisons in randomized clinical trials: A principled yet flexible approach","volume":"27","author":"Tsiatis","year":"2008","journal-title":"Statistics in Medicine"},{"key":"2023022804421646500_","volume-title":"Experiments: Planning, analysis, and optimization","author":"Wu","year":"2009"},{"key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/01621459.2022.2049278","article-title":"Toward better practice of covariate adjustment in analyzing randomized clinical trials","author":"Ye","year":"2022","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1093\/biomet\/asab051","article-title":"Regression-based causal inference with factorial experiments: estimands, model specifications and design-based properties","volume":"109","author":"Zhao","year":"2022","journal-title":"Biometrika"},{"issue":"2","key":"2023022804421646500_","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1214\/21-AOS2144","article-title":"Reconciling design-based and model-based causal inferences for split-plot experiments","volume":"50","author":"Zhao","year":"2022","journal-title":"The Annals of Statistics"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/1\/1\/49347859\/qkac003.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/1\/1\/49347859\/qkac003.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T23:06:26Z","timestamp":1679526386000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/85\/1\/1\/7008637"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,27]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1,27]]},"published-print":{"date-parts":[[2023,2,27]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkac003","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,2,1]]},"published":{"date-parts":[[2023,1,27]]}}}