  This paper studies inference in randomized controlled trials with
covariate-adaptive randomization when there are multiple treatments. More
specifically, we study inference about the average effect of one or more
treatments relative to other treatments or a control. As in Bugni et al.
(2018), covariate-adaptive randomization refers to randomization schemes that
first stratify according to baseline covariates and then assign treatment
status so as to achieve balance within each stratum. In contrast to Bugni et
al. (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 fully
saturated linear regression, i.e., 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-consistent estimator of the asymptotic variance are
invalid; on the other hand, tests based on these estimators and suitable
estimators of the asymptotic variance that we provide are exact. 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 strata fixed effects,
i.e., 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-consistent estimator of the
asymptotic variance are conservative, but tests based on these estimators and
suitable estimators of the asymptotic variance that we provide are exact. A
simulation study illustrates the practical relevance of our theoretical
results.
