{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T21:19:28Z","timestamp":1790025568580,"version":"4.0.1"},"reference-count":20,"publisher":"The Econometric Society","issue":"4","license":[{"start":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T00:00:00Z","timestamp":1546300800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QE"],"abstract":"<jats:p>This paper studies inference in randomized controlled trials with covariate\u2010adaptive randomization when there are multiple treatments. More specifically, we study in this setting inference about the average effect of one or more treatments relative to other treatments or a control. As in Bugni, Canay, and Shaikh (2018), covariate\u2010adaptive randomization refers to randomization schemes that first stratify according to baseline covariates and then assign treatment status so as to achieve \u201cbalance\u201d within each stratum. Importantly, in contrast to Bugni, Canay, and Shaikh (2018), we not only allow for multiple treatments, but further allow for the proportion of units being assigned to each of the treatments to vary across strata. We first study the properties of estimators derived from a \u201cfully saturated\u201d linear regression, that is, a linear regression of the outcome on all interactions between indicators for each of the treatments and indicators for each of the strata. We show that tests based on these estimators using the usual heteroskedasticity\u2010consistent estimator of the asymptotic variance are invalid in the sense that they may have limiting rejection probability under the null hypothesis strictly greater than the nominal level; on the other hand, tests based on these estimators and suitable estimators of the asymptotic variance that we provide are exact in the sense that they have limiting rejection probability under the null hypothesis equal to the nominal level. For the special case in which the target proportion of units being assigned to each of the treatments does not vary across strata, we additionally consider tests based on estimators derived from a linear regression with \u201cstrata fixed effects,\u201d that is, a linear regression of the outcome on indicators for each of the treatments and indicators for each of the strata. We show that tests based on these estimators using the usual heteroskedasticity\u2010consistent estimator of the asymptotic variance are conservative in the sense that they have limiting rejection probability under the null hypothesis no greater than and typically strictly less than the nominal level, but tests based on these estimators and suitable estimators of the asymptotic variance that we provide are exact, thereby generalizing results in Bugni, Canay, and Shaikh (2018) for the case of a single treatment to multiple treatments. A simulation study and an empirical application illustrate the practical relevance of our theoretical results.<\/jats:p>","DOI":"10.3982\/qe1150","type":"journal-article","created":{"date-parts":[[2019,11,21]],"date-time":"2019-11-21T09:04:39Z","timestamp":1574327079000},"page":"1747-1785","source":"Crossref","is-referenced-by-count":68,"title":["Inference under covariate\u2010adaptive randomization with multiple treatments"],"prefix":"10.3982","volume":"10","author":[{"given":"Federico A.","family":"Bugni","sequence":"first","affiliation":[{"name":"Department of Economics, Duke University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ivan A.","family":"Canay","sequence":"additional","affiliation":[{"name":"Department of Economics, Northwestern University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Azeem M.","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Economics, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2359","published-online":{"date-parts":[[2019]]},"reference":[{"key":"quan200011-bib-0001","unstructured":"Bai, Y. (2018), \u201cOn optimal stratification in randomized controlled trials.\u201d Manuscript, The University of Chicago."},{"key":"quan200011-bib-0002","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.worlddev.2017.09.011","volume":"102","author":"Berry","year":"2018","journal-title":"World Development"},{"key":"quan200011-bib-0003","first-page":"200","volume":"1","author":"Bruhn","year":"2009","journal-title":"American Economic Journal: Applied Economics"},{"key":"quan200011-bib-0004","doi-asserted-by":"crossref","unstructured":"Bugni, F. A., I. A. Canay, and A. Shaikh (2019), \u201cSupplement to \u2018Inference under covariate-adaptive randomization with multiple treatments\u2019.\u201d Quantitative Economics Supplemental Material, 10, https:\/\/doi.org\/10.3982\/QE1150.","DOI":"10.3982\/QE1150"},{"key":"quan200011-bib-0005","doi-asserted-by":"crossref","unstructured":"Bugni, F. A., I. A. Canay, and A. M. Shaikh (2018), \u201cInference under covariate-adaptive randomization.\u201d Journal of the American Statistical Association, forthcoming.","DOI":"10.2139\/ssrn.3334011"},{"key":"quan200011-bib-0006","unstructured":"Callen, M., S. Gulzar, A. Hasanain, Y. Khan, and A. Rezaee (2019), \u201cPersonalities and public sector performance: Evidence from a health experiment in Pakistan.\u201d NBER Working Paper No. 21180."},{"key":"quan200011-bib-0007","first-page":"222","volume":"8","author":"Chong","year":"2016","journal-title":"American Economic Journal: Applied Economics"},{"key":"quan200011-bib-0008","doi-asserted-by":"crossref","unstructured":"Dizon-Ross, R. (2018), \u201cParents' beliefs about their children's academic ability: Implications for educational investments.\u201d Manuscript, University of Chicago Booth School of Business.","DOI":"10.3386\/w24610"},{"key":"quan200011-bib-0009","doi-asserted-by":"crossref","first-page":"2757","DOI":"10.1257\/aer.20121607","volume":"105","author":"Duflo","year":"2015","journal-title":"American Economics Review"},{"key":"quan200011-bib-0010","doi-asserted-by":"crossref","first-page":"3895","DOI":"10.1016\/S1573-4471(07)04061-2","volume":"4","author":"Duflo","year":"2007","journal-title":"Handbook of development economics"},{"key":"quan200011-bib-0011","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1093\/biomet\/58.3.403","volume":"58","author":"Efron","year":"1971","journal-title":"Biometrika"},{"key":"quan200011-bib-0012","doi-asserted-by":"crossref","unstructured":"Hu, Y. and F. Hu (2012), \u201cAsymptotic properties of covariate-adaptive randomization.\u201d Annals of Statistics. 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