  Applications of CAR for balancing continuous covariates remain comparatively
rare, especially in multi-treatment clinical trials, and the theoretical
properties of multi-treatment CAR have remained largely elusive for decades. In
this paper, we consider a general framework of CAR procedures for
multi-treatment clinal trials which can balance general covariate features,
such as quadratic and interaction terms which can be discrete, continuous, and
mixing. We show that under widely satisfied conditions the proposed procedures
have superior balancing properties; in particular, the convergence rate of
imbalance vectors can attain the best rate $O_P(1)$ for discrete covariates,
continuous covariates, or combinations of both discrete and continuous
covariates, and at the same time, the convergence rate of the imbalance of
unobserved covariates is $O_P(\sqrt n)$, where $n$ is the sample size. The
general framework unifies many existing methods and related theories,
introduces a much broader class of new and useful CAR procedures, and provides
new insights and a complete picture of the properties of CAR procedures. The
favorable balancing properties lead to the precision of the treatment effect
test in the presence of a heteroscedastic linear model with dependent covariate
features. As an application, the properties of the test of treatment effect
with unobserved covariates are studied under the CAR procedures, and consistent
tests are proposed so that the test has an asymptotic precise type I error even
if the working model is wrong and covariates are unobserved in the analysis.
