# Conditionally Affinely Invariant Rerandomization and its Admissibility

Zhen Zhong
Yau Mathematical Sciences Center, Tsinghua University

Donald B. Rubin
Yau Mathematical Sciences Center, Tsinghua University
and Fox School of Business, Temple University

###### Abstract

Rerandomization utilizes modern computing ability to search for covariate
balance improved experimental design while adhering to the randomization
principle originally advocated by RA Fisher. Conditionally affinely invariant
rerandomization has the “Equal Percent Variance Reducing” property on subsets
of conditionally ellipsoidally symmetric covariates. It is suitable to deal
with covariates of varying importance or mixed types and usually produces
multiple balance scores. “Unified” and “ intersection” methods are common ways
of deciding on multiple scores. In general, “ intersection” methods are
computationally more efficient but asymptotically inadmissible. As
computational cost is not a major concern in experimental design, we recommend
“unified” methods to build admissible criteria for rerandomization.

Keywords: Randomized experiments; ellipsoidal; decision theory; Mahalanobis
distance

##  1 Introduction

Randomized controlled trials (RCTs) are the “gold standard” for studying
causal relationships, yet can be expensive to implement. tukey1993tightening
also describes the “platinum standard” for RCTs, where probability statements,
such as statistical significance depend only on exactly how the trial was
conducted. Extensive covariates are now commonly available at the design
stages of RCTs. According to a survey by bruhn2009pursuit, researchers tend to
rerandomize when imbalance in covariates across treatments occurs but the
exact rerandomization procedures are generally not described in detail. Any
subsequent analysis that ignores the design stage does not adhere to Tukey’s
“platinum standard”.

morgan2012rerandomization (MR, 2012) formalizes rerandomization by specifying
a balance criterion before associating treatments, accepting a randomized
assignment only after the criterion is met, and then analyzing results using
Fisher’s randomization tests. Transparent in its procedures, rerandomization
ensures the validity of randomization-based inference. Moreover, it leads to
increased precision of estimated causal effects and narrower confidence
intervals when covariates are correlated with outcomes (li2018asymptotic). MR
focuses on rerandomization methods with affinely invariant balance criteria.
Using symmetry arguments, MR obtains the “Equal Percent Variance Reducing”
(EPVR) property over ellipsoidally distributed covariates. In practice, affine
invariance can be relaxed to apply to tiers of covariates (e.g.
morgan2015rerandomization) or special discrete covariates (e.g.
wang2021rerandomization, johansson2022rerandomization). A general theory for
such designs is our first major topic. We make extensions to the affinely
invariant balance criteria and obtain the conditionally EPVR property over
conditionally ellipsoidally distributed covariates parallel to
rubin1992affinely. We then discuss how this theory can be applied to the
covariate balance improved RCTs mentioned above.

In contrast, other researchers seek optimally balanced designs (e.g.,
bertsimas2015power, kallus2018optimal). But the sets of optimally balanced
assignments are too restricted to allow valid randomization-based inference
(johansson2021optimal). Therefore, it’s necessary to keep a minimal acceptance
probability so that meaningful randomization-based inference can be drawn.
Inspired by the statistical decision theory (wald1949statistical,
berger1985statistical), we give a formal definition of admissible criteria and
characterize them with sufficient and necessary conditions in general
scenarios. “Unified” methods and “intersection” methods are two common ways of
building criteria that involve multiple balance scores. We identify a class of
admissible methods as “unified”. Moreover, under mild conditions, a class of
“intersection” methods are found to be inadmissible and can be dominated by
some “unified” methods, at least asymptotically.

We organize the rest materials as follows. Section 2 reviews affinely
invariant rerandomization methods. Section 3 builds the theory of
conditionally affinely invariant rerandomization. Section 4 applies the theory
to two cases of covariate balance improved RCTs. Section 5 studies admissible
criteria and compares “unified” methods with “intersection” methods. Section 6
makes further discussion and concludes.

##  2 Affinely Invariant Rerandomization

Suppose a randomized experiment is conducted with $n$ units. Let
$\boldsymbol{W}$ denote the treatment assignment vector of dimension $n$ whose
$i$th value equals $1$ if unit $i$ ($i=1,2,\ldots,n$) is treated and equals
$0$ if not. Under complete randomization, a treatment assignment is randomly
drawn from
$\mathcal{W}=\\{\boldsymbol{W}\in\\{0,1\\}^{n}|\sum{}W_{i}=n_{\text{1}}\\}$,
where $n_{\text{1}}$ is the treatment group size;
$n_{\text{0}}=n-n_{\text{1}}$ is the control group size.

When experimental units are enrolled in a study, they typically include their
baseline covariates such as age and gender, that could potentially affect
experimental outcomes. Let $\boldsymbol{x}$ be an $n\times{}k$ real-valued
matrix in which the $i$’th row $\boldsymbol{x}_{i}$ is the covariate vector
for unit $i$. Under complete randomization, a random draw of treatment
assignment from $\mathcal{W}$ can be highly correlated with certain linear
combination of covariates. It’s not a good idea to conduct the experiment with
such an ”unlucky” assignment because covariates are not balanced between
treatment and control groups.

Rerandomization rejects a random draw from $\mathcal{W}$ if some pre-
determined balance criterion is not satisfied. The criterion can be formulated
as an indicator function $\varphi(\boldsymbol{x},\boldsymbol{W})$, where a
decision is made to accept $\boldsymbol{W}$ if $\varphi=1$ and if $\varphi=0$
to reject $\boldsymbol{W}$ and rerandomize. A practical implementation given
by morgan2012rerandomization is based on the squared Mahalanobis distance
between treatment and control groups:

$D\equiv
np_{w}(1-p_{w})(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})\boldsymbol{S}^{-1}(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})^{\prime}$

where $p_{w}=n_{\text{1}}/n$ is the propensity score, $\boldsymbol{S}$ is the
sample covariance matrix, and
$\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}$ is
the difference in treatment and control group means of covariates. Given some
prespecified threshold $\alpha$, the balance criterion is defined as

$\varphi^{*}(\boldsymbol{x},\boldsymbol{W})=\begin{cases}1,&D\leq\alpha,\\\
0,&D>\alpha.\end{cases}$

Consider an affine transformation on covariates given by
$\mathcal{A}(\boldsymbol{x})=\boldsymbol{x}A+\boldsymbol{b}$, where $A$ is an
invertible matrix and $\boldsymbol{b}$ a constant vector. A balance criterion
is said to be affinely invariant if it satisfies

$\varphi\left(\mathcal{A}(\boldsymbol{x}),\boldsymbol{W}\right)=\varphi\left(\boldsymbol{x},\boldsymbol{W}\right)$

for any affine transformation $\mathcal{A}$. One can easily verify that
$\varphi^{*}$ is affinely invariant. To study the theoretical properties of
affinely invariant rerandomization methods, both $\boldsymbol{x}$ and
$\boldsymbol{W}$ are viewed as random variables. Under random sampling of
units from the super-population and randomized assignment, using symmetry
arguments, morgan2012rerandomization show that affinely invariant
rerandomization methods are “Equal Percent Variance Reducing” (EPVR).

######  Theorem 2.1 (morgan2012rerandomization).

If $\varphi$ is affinely invariant and if $\boldsymbol{x}$ is ellipsoidally
symmetric, then

$\displaystyle\mathbb{E}\left[\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}\mid\varphi=1\right]$
$\displaystyle=\mathbb{E}\left[\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}\right]=\boldsymbol{0},$
(1) $\displaystyle\text{and
}\operatorname{cov}\left(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}\mid\varphi=1\right)$
$\displaystyle\propto\operatorname{cov}\left(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}\right).$
(2)

Equation (2) indicates that using $\varphi$, the reduction in variance for any
covariate and any linear function of covariates is the same, namely EPVR. In
mathematical formulation, let $\boldsymbol{Y}$ be any linear combination of
covariates. For the difference in treatment and control group means
$\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}$,
we have

$\frac{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)-\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\mid\varphi=1\right)}{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)}=1-\nu,$

where $\nu$ is the ratio of covariance matrices in (2).

##  3 Conditionally Affinely Invariant Rerandomization

Although affinely invariant rerandomization procedures equally balance all
covariates, in practice some covariates can be more predictive of outcomes
than other covariates. When covariates vary in a priori importance, better
balance should be required for more important covariates
(morgan2015rerandomization). Another common situation is that covariates are a
mixture of categorical and continuous variables. It is shown by
johansson2022rerandomization that stratification on binary covariates followed
by rerandomization on continuous covariates is more efficient than
rerandomization on all covariates at the same time. Parallel to the
conditionally affinely invariant matching literature (rubin1992affinely), we
develop a general framework, conditionally affinely invariant rerandomization
methods, for covariates balanced to different extents.

###  3.1 Conditionally Ellipsoidal Distributions

We first adopt an algebraic definition of ellipsoidal distributions from
cambanis1981theory. Suppose a covariate vector $\boldsymbol{X}$ has a
distribution on the $k$-dimensional Euclidean space $\mathbb{R}^{k}$. For some
vector $\boldsymbol{\mu}\in\mathbb{R}^{k}$ and some $k\times{}k$ positive
semi-definite matrix $\boldsymbol{\Sigma}$, if the characteristic function of
$\boldsymbol{X}-\boldsymbol{\mu}$ can be written as
$\psi(\boldsymbol{h}\boldsymbol{\Sigma}\boldsymbol{h}^{\prime})$ for
$\boldsymbol{h}\in\mathbb{R}^{k}$, we say that $\boldsymbol{X}$ has an
ellipsoidal distribution with parameters $\boldsymbol{\mu}$,
$\boldsymbol{\Sigma}$ and $\psi$, and write
$\boldsymbol{X}\sim{}EC_{k}(\boldsymbol{\mu},\boldsymbol{\Sigma},\psi)$. When
$\boldsymbol{\Sigma}$ is proportional to the identity matrix, we say that
$\boldsymbol{X}$ has a spherical distribution.

We assume the parameter $\boldsymbol{\Sigma}$ to be non-singular. Otherwise,
there are covariates linearly depend on the rest (cambanis1981theory) and we
just select a basis of covariates and conduct rerandomization on them. Let
$\stackrel{{\scriptstyle d}}{{=}}$ represent the identically distributed
relationship between random variables. Next lemma introduces a geometric
definition of the ellipsoidally distributed random variable that originates
from dempster1969elements.

######  Lemma 3.0.1.

Given a positive definite $\boldsymbol{\Sigma}$, the following two statements
are equivalent:

  1. 1.

$\boldsymbol{X}\sim{}EC_{k}(\boldsymbol{\mu},\boldsymbol{\Sigma},\psi)$.

  2. 2.

$(\boldsymbol{X}-\boldsymbol{\mu})\boldsymbol{LQ}\stackrel{{\scriptstyle
d}}{{=}}(\boldsymbol{X}-\boldsymbol{\mu})\boldsymbol{L},\text{ for any
orthogonal }k\times{}k\text{ matrix }\boldsymbol{Q}$,

where $\boldsymbol{L}$ satisfies the Cholesky decomposition
$\boldsymbol{\Sigma}^{-1}=\boldsymbol{L}\boldsymbol{L}^{\prime}$.

###### Proof.

When $\boldsymbol{X}\sim{}EC_{k}(\boldsymbol{\mu},\boldsymbol{\Sigma},\psi)$,
it follows from Theorem 1 of cambanis1981theory that

$\boldsymbol{X}\stackrel{{\scriptstyle
d}}{{=}}\boldsymbol{\mu}+R\boldsymbol{U}\boldsymbol{L}^{\prime}$

where $\boldsymbol{U}$ is uniformly distributed on the unit sphere in
$\mathbb{R}^{k}$ and $R\geq 0$ is independent of $\boldsymbol{U}$. For any
orthogonal $k\times{}k$ matrix $\boldsymbol{Q}$, we have
$R\boldsymbol{U}\boldsymbol{Q}\stackrel{{\scriptstyle
d}}{{=}}R\boldsymbol{U}$, which indicates
$(\boldsymbol{X}-\boldsymbol{\mu})\boldsymbol{LQ}\stackrel{{\scriptstyle
d}}{{=}}(\boldsymbol{X}-\boldsymbol{\mu})\boldsymbol{L}$. Conversely, if (ii)
holds for an arbitrary random vector $\boldsymbol{X}$, then the characteristic
function $\psi$ of $(\boldsymbol{X}-\boldsymbol{\mu})\boldsymbol{L}$ is a
function of $\boldsymbol{h}\boldsymbol{h}^{\prime}$. It follows that the
characteristic function of $\boldsymbol{X}-\boldsymbol{\mu}$ is
$\psi(\boldsymbol{h}\boldsymbol{\Sigma}\boldsymbol{h}^{\prime})$ and
$\boldsymbol{X}\sim{}EC_{k}(\boldsymbol{\mu},\boldsymbol{\Sigma},\psi)$. ∎

The family of ellipsoidal distributions can be extended by removing the
rotation symmetry in some dimensions. Suppose a covariate vector
$\boldsymbol{X}$ contains $k_{s}$ special covariates and $k_{r}$ remaining
covariates. Without loss of generality, we write
$\boldsymbol{X}=\left(\boldsymbol{X}^{(r)},\boldsymbol{X}^{(s)}\right)$. If
there exists a common positive definite matrix $\boldsymbol{\Sigma}$ such that
$\boldsymbol{X}^{(r)}\mid\boldsymbol{X}^{(s)}\sim{}EC_{r}\left(\boldsymbol{\mu}_{\boldsymbol{X}^{(s)}},\boldsymbol{\Sigma},\psi_{\boldsymbol{X}^{(s)}}\right)$,
we say that $\boldsymbol{X}$ is conditionally ellipsoidally distributed.

The original definition of conditionally ellipsoidal distributions is given by
rubin1992affinely. Our definition is more general to include a wider variety
of distributions. For example, if $\boldsymbol{X}$ follows a multivariate
student t-distribution, $\boldsymbol{X}^{(r)}\mid\boldsymbol{X}^{(s)}$ is
still multivariate student t-distributed, where
$\boldsymbol{\mu}_{\boldsymbol{X}^{(s)}}$ is an affine transformation of
$\boldsymbol{X}^{(s)}$ and
$\operatorname{cov}\left(\boldsymbol{X}^{(r)}\mid\boldsymbol{X}^{(s)}\right)$
is proportionate to
$\operatorname{cov}\left(\boldsymbol{X}^{(r)}\mid\boldsymbol{0}\right)$ but
not a constant matrix (see e.g. cornish1954multivariate). Another example is
the general location model, introduced by olkin1961multivariate and further
generalized by liu1998ellipsoidally, where $\boldsymbol{X}^{(s)}$ has a
multinomial distribution and $\boldsymbol{X}^{(r)}$ is conditionally normally
or student-t distributed with proportional covariance matrices.

Following rubin1992affinely, it’s useful to introduce the canonical form for
conditionally ellipsoidal distributions in the generalized sense. Let
$\boldsymbol{\Sigma}^{-1}=\boldsymbol{L}\boldsymbol{L}^{\prime}$ be the
Cholesky decomposition. We define the canonical $\boldsymbol{X}^{(r)}$ as
$\boldsymbol{Z}^{(r)}=\left(\boldsymbol{X}^{(r)}-\boldsymbol{\mu}_{\boldsymbol{X}^{(s)}}\right)\boldsymbol{L}$.
When transformed into this canonical form,
$\boldsymbol{Z}^{(r)}\mid\boldsymbol{X}^{(s)}$ is spherically distributed.
Moreover, components of $\boldsymbol{Z}^{(r)}$ are exchangeable in both the
conditional distribution $\boldsymbol{Z}^{(r)}\mid\boldsymbol{X}^{(s)}$ and
the joint distribution of
$\left(\boldsymbol{Z}^{(r)},\boldsymbol{X}^{(s)}\right)$.

###  3.2 Conditionally Affinely Invariant Balance Criteria

We also give a stochastic extension of rerandomization balance criteria. For
any measurable function $\varphi(\boldsymbol{x},\boldsymbol{W})$ with a range
in $[0,1]$, let $\delta(\boldsymbol{x},\boldsymbol{W})$ be a Bernoulli random
variable with success probability $\varphi(\boldsymbol{x},\boldsymbol{W})$.
Conditional on $\boldsymbol{x}$ and $\boldsymbol{W}$, a decision is made to
accept $\boldsymbol{W}$ when $\delta=1$ and reject $\boldsymbol{W}$ and
rerandomize when $\delta=0$. Note that it coincides with the deterministic
balance criteria in morgan2012rerandomization if $\varphi$ takes only binary
values. An example of non-deterministic balance criteria is $K$-randomized
experiments proposed by banerjee2020theory, where one out of $K$ independently
randomized assignments is selected based on the maximization of a predefined
utility function.

Suppose covariates are divided into two subsets according to their prognostic
importance. Without loss of generality, set
$\boldsymbol{x}=\left(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)}\right)$ where
$\boldsymbol{x}^{(s)}$ contains $k_{s}$ special covariates and
$\boldsymbol{x}^{(r)}$ contains $k_{r}$ remaining covariates. A stochastic
balance criterion $\varphi$ is said to be conditionally affinely invariant if

$\varphi\left(\left(\mathcal{A}_{r}\left(\boldsymbol{x}^{(r)}\right),\boldsymbol{x}^{(s)}\right),\boldsymbol{W}\right)=\varphi\left(\left(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)}\right),\boldsymbol{W}\right)$

holds for any affine transformation $\mathcal{A}_{r}$.

###  3.3 The Conditionally EPVR Property

Results in this subsection are based on the canonical form of conditionally
ellipsoidally symmetric covariates. Let
$\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}$
and
$\overline{\boldsymbol{x}}^{(s)}_{\text{1}}-\overline{\boldsymbol{x}}^{(s)}_{\text{0}}$
be the differences in treatment and control group means of remaining and
special covariates, respectively. Under both random sampling of units from the
super-population and randomized assignment, we have the following results.

######  Theorem 3.1.

If $\varphi$ is conditionally affinely invariant on $\boldsymbol{x}^{(r)}$
given $\boldsymbol{x}^{(s)}$ and if
$\left(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)}\right)$ is conditionally
ellipsoidally symmetric in canonical form, then

$\displaystyle\quad\mathbb{E}\left[\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\mid\delta=1\right]=\mathbb{E}\left[\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\right]=\boldsymbol{0},$
(3)
$\displaystyle\quad\mathbb{E}\left[\left(\overline{\boldsymbol{x}}^{(s)}_{\text{1}}-\overline{\boldsymbol{x}}^{(s)}_{\text{0}}\right)^{\prime}\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\right)\mid\delta=1\right]=\boldsymbol{0},$
(4)
$\displaystyle\operatorname{cov}\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\mid\delta=1\right)\propto\boldsymbol{I},$
(5)

where $\boldsymbol{I}$ is a $k_{r}$ dimensional identity matrix.

###### Proof.

Let $\boldsymbol{Q}$ be a $k_{r}\times{}k_{r}$ orthogonal matrix. Since
$\varphi$ is conditionally affinely invariant on $\boldsymbol{x}^{(r)}$ given
$\boldsymbol{x}^{(s)}$, we have

$\varphi\left(\left(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)}\right),\boldsymbol{W}\right)=\varphi\left(\left(\boldsymbol{x}^{(r)}\boldsymbol{Q},\boldsymbol{x}^{(s)}\right),\boldsymbol{W}\right).$

Since units are random samples from the super-population and the covariate
distribution is conditionally ellipsoidally symmetric in the canonical form,
we have

$\boldsymbol{x}^{(r)},\delta\mid\boldsymbol{x}^{(s)}\stackrel{{\scriptstyle
d}}{{=}}\boldsymbol{x}^{(r)}\boldsymbol{Q},\delta\mid\boldsymbol{x}^{(s)}.$

For a statistic like the difference in means, it follows that

$\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\mid{}\delta=1,\boldsymbol{x}^{(s)}\stackrel{{\scriptstyle
d}}{{=}}\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\right)\boldsymbol{Q}\mid{}\delta=1,\boldsymbol{x}^{(s)}.$

Given $\boldsymbol{Q}$ is an arbitrary orthogonal matrix, we know from Lemma
3.0.1 that
$\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\right)\mid{}\delta=1,\boldsymbol{x}^{(s)}$
has a spherical distribution and center at $\boldsymbol{0}$. We then have the
moment calculation

$\displaystyle\quad\mathbb{E}\left[\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right]=\boldsymbol{0},$
(6)
$\displaystyle\quad\mathbb{E}\left[\left(\overline{\boldsymbol{x}}^{(s)}_{\text{1}}-\overline{\boldsymbol{x}}^{(s)}_{\text{0}}\right)^{\prime}\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\right)\mid{}\delta=1,\boldsymbol{x}^{(s)}\right]=\boldsymbol{0},$
(7)
$\displaystyle\operatorname{cov}\left(\overline{\boldsymbol{x}}^{(r)}_{\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\text{0}}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right)\propto\boldsymbol{I}.$
(8)

Equations (3) - (5) hold by taking expectations over $\boldsymbol{x}^{(s)}$ in
(6) - (8). ∎

Extending the EPVR property of affinely invariant rerandomization methods, a
rerandomization method is said to be conditionally EPVR if the percent
reduction in variance (PRIV) is the same for any linear combination of a
subset of covariates.

######  Corollary 3.1.

If $\varphi$ is conditionally affinely invariant on $\boldsymbol{x}^{(r)}$
given $\boldsymbol{x}^{(s)}$ and if
$\boldsymbol{x}=(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)})$ is conditionally
ellipsoidally symmetric in canonical form, then rerandomization is
conditionally EPVR for $\boldsymbol{x}^{(r)}$ and any linear function of
$\boldsymbol{x}^{(r)}$.

##  4 Application to Covariate Balance Improved RCTs

###  4.1 Balancing on Tiers of Covariates

When high dimensional covariates with varying importance are collected at the
design stage, it can be helpful to partition the covariates into $T>1$ tiers,
where the higher tier contains more predictive covariates, and could be
handled with more stringent balance criteria based on the Mahalanobis
distance, as discussed by morgan2015rerandomization. The finest division is
one covariate in each tier, which is rerandomization based on the weighted
Euclidean distance (lu2023design).

For $t=1,2,\ldots,T$, denote $\boldsymbol{x}^{(t)}$ as the $n\times{}k_{t}$
dimensional covariate matrix of tier $t$ and $\boldsymbol{x}^{(-t)}$ of the
remaining tiers. Without loss of generality, write
$\boldsymbol{x}=\left(\boldsymbol{x}^{(1)},\ldots,\boldsymbol{x}^{(T)}\right)$.
Let tier $1$ contain the most important covariates and subsequent tiers
contain increasingly less important covariates. morgan2015rerandomization
recommend balancing the portion in each tier that are orthogonal to the
covariates that have already been balanced in previous tiers. So we firstly
pre-process covariates with block-wise Gram-Schmit orthogonalization.

Suppose covariates follow an ellipsoidal symmetrical distribution
$EC_{k}(\boldsymbol{\mu},\boldsymbol{\Sigma},\psi)$. According to
cambanis1981theory, there exists a regular conditional distribution
$\boldsymbol{x}_{i}^{(t)}\mid\boldsymbol{x}_{i}^{(-t)}$ that is ellipsoidal
and admits a representation

$\boldsymbol{x}_{i}^{(t)}\mid\boldsymbol{x}_{i}^{(-t)}\sim{}EC_{k_{t}}\left(\boldsymbol{\mu}_{t}+\boldsymbol{x}_{i}^{(-t)}\boldsymbol{B}_{t},\boldsymbol{\Sigma}_{t},\psi_{\boldsymbol{x}_{i}^{(-t)}}\right),$

where $\left(\boldsymbol{\mu}_{t},\boldsymbol{B}_{t}\right)$ is the projection
coefficient vector of $\boldsymbol{x}_{i}^{(t)}$ on
$\left(1,\boldsymbol{x}_{i}^{(-t)}\right)$. By replacing
$\boldsymbol{x}_{i}^{(t)}$ with its projection residual
$\boldsymbol{z}_{i}^{(t)}=\boldsymbol{x}_{i}^{(t)}-\boldsymbol{\mu}_{t}-\boldsymbol{x}_{i}^{(-t)}\boldsymbol{B}_{t}$,
we find $\boldsymbol{z}_{i}^{(t)}$ and $\boldsymbol{x}_{i}^{(-t)}$ becomes
orthogonal and the joint distribution of
$\left(\boldsymbol{z}_{i}^{(t)},\boldsymbol{x}_{i}^{(-t)}\right)$ is still
ellipsoidally symmetric.

We repeat the projection and replacement to sequentially obtain
$\boldsymbol{z}_{i}^{(T)},\ldots,\boldsymbol{z}_{i}^{(1)}$. Because projection
residuals of hier tiers are orthogonal with covariates of lower tiers, in each
step $t$ it’s equivalent to just project $\boldsymbol{x}_{i}^{(t)}$ over
$\left(1,\boldsymbol{x}_{i}^{(1)},\ldots,\boldsymbol{x}_{i}^{(t-1)}\right)$.
As a result, a new covariate matrix
$\boldsymbol{z}=\left(\boldsymbol{z}^{(1)},\ldots,\boldsymbol{z}^{(T)}\right)$
is obtained with $i$th row being
$\boldsymbol{z}_{i}=\left(\boldsymbol{z}_{i}^{(1)},\ldots,\boldsymbol{z}_{i}^{(T)}\right)$.
After the block-wise Gram-Schmit orthogonalization, components of
$\boldsymbol{z}_{i}$ of different tiers become orthogonal.

Consider block-wise Gram-Schmit orthogonalization on tiers of covariates
followed by subspace-conditionally affinely invariant rerandomization, where
the acceptance criterion satisfies

$\varphi\left(\left(\mathcal{A}_{t}\left(\boldsymbol{z}^{(t)}\right),\boldsymbol{z}^{(-t)}\right),\boldsymbol{W}\right)=\varphi\left(\left(\boldsymbol{z}^{(t)},\boldsymbol{z}^{(-t)}\right),\boldsymbol{W}\right)$

for any affine transformation $\mathcal{A}_{t}$ applied to tier $t$
($t=1,2,\ldots,T$). This class of methods allows for prior knowledge of
covariates’ importance to be reflected. For $t=1,2,\ldots,T$, let
$\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}$
be the difference in treatment and control group means of orthogonalized
covariates in tier $t$. Under both random sampling of units from the super-
population and randomized assignment, we have the following results.

######  Theorem 4.1.

If $\varphi$ is subspace-conditionally affinely invariant on $\boldsymbol{z}$
and if $\boldsymbol{z}$ is obtained from block-wise Gram-Schmidt
orthogonalization of $\boldsymbol{x}$ that is ellipsoidally symmetric, then

$\displaystyle\quad\mathbb{E}\left[\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\mid{}\delta=1\right]=\mathbb{E}\left[\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right]=\boldsymbol{0},$
(9)
$\displaystyle\quad\mathbb{E}\left[\left(\overline{\boldsymbol{z}}^{(s)}_{\text{1}}-\overline{\boldsymbol{z}}^{(s)}_{\text{0}}\right)^{\prime}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\mid{}\delta=1\right]=\boldsymbol{0},$
(10)
$\displaystyle\operatorname{cov}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\mid{}\delta=1\right)\propto\operatorname{cov}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)$
(11)

for $t,s=1,2,\ldots,T$ and $t\neq s$.

###### Proof.

For $t=1,2,\ldots,T$, let $\boldsymbol{z}_{i}^{(-t)}$ be the remaining
covariates besides $\boldsymbol{z}_{i}^{(t)}$. According to
cambanis1981theory, there exists a regular conditional distribution
$\boldsymbol{z}_{i}^{(t)}\mid\boldsymbol{z}_{i}^{(-t)}$ that is ellipsoidal
and admits a representation

$\boldsymbol{z}_{i}^{(t)}\mid\boldsymbol{z}_{i}^{(-t)}\sim{}EC_{k_{t}}\left(\boldsymbol{0},\boldsymbol{\Sigma}_{t},\psi_{\boldsymbol{z}_{i}^{(-t)}}\right),$

where $\boldsymbol{0}$ is a zero vector because $\boldsymbol{z}_{i}^{(t)}$ is
mean zero and is uncorrelated with $\boldsymbol{z}_{i}^{(-t)}$. Let
$\boldsymbol{\Sigma}_{t}^{-1}=\boldsymbol{L}_{t}\boldsymbol{L}_{t}^{\prime}$
be the Cholesky decomposition. According to our definition,
$\boldsymbol{z}_{i}^{(t)}\boldsymbol{L}_{t}\mid\boldsymbol{z}_{i}^{(-t)}$ is
in the canonical form. We can apply Theorem 3.1 to obtain that for
$t,s=1,2,\ldots,T$ and $t\neq s$,

$\displaystyle\quad\mathbb{E}\left[\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{L}_{t}\mid{}\delta=1\right]=\mathbb{E}\left[\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{L}_{t}\right]=\boldsymbol{0},$
$\displaystyle\quad\mathbb{E}\left[\boldsymbol{L}_{t}^{\prime}\left(\overline{\boldsymbol{z}}^{(s)}_{\text{1}}-\overline{\boldsymbol{z}}^{(s)}_{\text{0}}\right)^{\prime}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{L}_{t}\mid{}\delta=1\right]=\boldsymbol{0},$
$\displaystyle\operatorname{cov}\left(\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{L}_{t}\mid{}\delta=1\right)\propto\boldsymbol{I}_{t},$

where $\boldsymbol{I}_{t}$ is the $k_{t}$ dimensional identity matrix.
Transforming back completes the proof. ∎

Let $\boldsymbol{Y}$ be a linear combination of covariates. Under subspace-
conditionally affinely invariant rerandomization, the reduction in variance of
$\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}$
can be decomposed into those of projections on tiers of orthogonalized
covariates.

######  Corollary 4.1.

Under the same assumptions as Theorem 4.1,

$\frac{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)-\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\mid{}\delta=1\right)}{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)}=1-\sum_{t=1}^{T}R^{2}_{t}\nu_{t},$

where $R^{2}_{t}$ is the squared multiple correlation coefficient between
$\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}$
and
$\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}$
and $\nu_{t}$ is the proportional ratio in (11) for $t=1,2,\ldots,T$.

###### Proof.

Since
$\boldsymbol{z}=\left(\boldsymbol{z}^{(1)},\ldots,\boldsymbol{z}^{(T)}\right)$
is a linear transformation of $\boldsymbol{x}$, we can write
$\boldsymbol{Y}=\boldsymbol{z}^{(1)}\boldsymbol{\gamma}_{1}+\ldots+\boldsymbol{z}^{(T)}\boldsymbol{\gamma}_{T}$
for some constant vectors
$\boldsymbol{\gamma}_{1},\ldots,\boldsymbol{\gamma}_{T}$. From (10) and (11),
we have

$\displaystyle\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\mid{}\delta=1\right)$
$\displaystyle=\sum_{t=1}^{T}\boldsymbol{\gamma}_{t}^{\prime}\operatorname{cov}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\mid\delta=1\right)\boldsymbol{\gamma}_{t}$
$\displaystyle=\sum_{t=1}^{T}\nu_{t}\boldsymbol{\gamma}_{t}^{\prime}\operatorname{cov}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{\gamma}_{t}.$

As components of $\boldsymbol{z}_{i}$ of different tiers are uncorrelated, we
further obtain

$\boldsymbol{\gamma}_{t}^{\prime}\operatorname{cov}\left(\overline{\boldsymbol{z}}^{(t)}_{\text{1}}-\overline{\boldsymbol{z}}^{(t)}_{\text{0}}\right)\boldsymbol{\gamma}_{t}=R_{t}^{2}\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right),\quad\text{for
}t=1,2,\ldots,T$

and $\sum_{t=1}^{T}R_{t}^{2}=1$. Combining above results, we have

$\frac{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)-\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\mid{}\delta=1\right)}{\operatorname{var}\left(\overline{\boldsymbol{Y}}_{\text{1}}-\overline{\boldsymbol{Y}}_{\text{0}}\right)}=1-\sum_{t=1}^{T}R_{t}^{2}\nu_{t}.$

∎

###  4.2 Stratified Rerandomization

Stratification is a common way to avoid chance imbalance of covariates in a
randomized experiment. Suppose that $\boldsymbol{x}^{(s)}$ consists of
discrete covariates taking values in
$\\{\boldsymbol{c}(1),\ldots,\boldsymbol{c}(J)\\}$. A stratified randomized
experiment is to divide units into $J$ strata according to the values of
$\boldsymbol{x}^{(s)}$ and then completely randomize treatment assignment
within each stratum. Conceptually, it is equivalent to rerandomization with
the acceptance criterion

$\varphi_{S}\left(\boldsymbol{x}^{(s)},\boldsymbol{W}\right)=\left\\{\begin{aligned}
&1,\ \sum_{\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j)}(W_{i}-p_{j})=0,\
\forall{}j=1,2,\ldots,J\\\ &0,\ \text{else},\end{aligned}\right.$

where $\boldsymbol{x}^{(s)}_{i}$ is the discrete covariate vector of unit $i$
and $p_{j}$ is the stratum-specific propensity score taking the value in
$(0,1)$.

Stratifying on several continuous variables is usually impractical for reasons
related to the “curse of dimensionality”. As a substitute,
johansson2022rerandomization suggest first stratifying on a set of binary
covariates and then rerandomizing on remaining covariates using the
Mahalanobis distance. wang2021rerandomization extend this framework to general
discrete covariates and further introduce balance criteria with overall or
stratum-specific Mahalanobis distances. Next, we explore the theoretical
implications of stratified rerandomization with conditionally affinely
invariant balance criteria that include above cases.

Suppose that, after stratification on $\boldsymbol{x}^{(s)}$, affinely
invariant rerandomization is conducted on continuous covariates
$\boldsymbol{x}^{(r)}$ for additional reduction in variance. Denote
$\varphi_{R}$ to be a stochastic balance criterion that is affinely invariant
on $\boldsymbol{x}^{(r)}$. The whole balance criterion can be represented as

$\varphi\left(\boldsymbol{x},\boldsymbol{W}\right)=\varphi_{R}\left(\boldsymbol{x}^{(r)},\boldsymbol{W}\right)\times\varphi_{S}\left(\boldsymbol{x}^{(s)},\boldsymbol{W}\right),$

which is conditionally affinely invariant on $\boldsymbol{x}^{(r)}$ given
$\boldsymbol{x}^{(s)}$.

For $j=1,2,\ldots,J$, let
$\overline{\boldsymbol{x}}^{(r)}_{j\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{j\text{0}}$
be the difference in treatment and control group means of
$\boldsymbol{x}^{(r)}_{i}$ given $\boldsymbol{x}_{i}^{(s)}=\boldsymbol{c}(j)$.
In stratified randomized experiments, there are various kinds of statistics
combined from within-stratum comparisons (see e.g. imbens2015causal, Ch.9).
Therefore, we consider
$\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)=\sum_{j=1}^{J}\lambda_{j}\left(\overline{\boldsymbol{x}}^{(r)}_{j\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{j\text{0}}\right)$
for a general $J$-component vector $\boldsymbol{\lambda}$, to which we refer
as the $\boldsymbol{\lambda}$-comparison in treatment and control group means
of remaining covariates. Under both random sampling of units from the super-
population and stratified randomized assignment, we have the following
results.

######  Theorem 4.2.

If
$\varphi\left(\boldsymbol{x},\boldsymbol{W}\right)=\varphi_{R}\left(\boldsymbol{x}^{(r)},\boldsymbol{W}\right)\times\varphi_{S}\left(\boldsymbol{x}^{(s)},\boldsymbol{W}\right)$
with $\varphi_{R}$ affinely invariant on $\boldsymbol{z}^{(r)}$ and
$\varphi_{S}$ representing stratification on $\boldsymbol{x}^{(s)}$, and if
$(\boldsymbol{x}^{(r)},\boldsymbol{x}^{(s)})$ is conditionally ellipsoidally
symmetric, then

$\displaystyle\quad\mathbb{E}\left[\overline{\boldsymbol{z}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right]=\mathbb{E}\left[\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\mid\boldsymbol{x}^{(s)}\right]=\boldsymbol{0},$
(12)
$\displaystyle\operatorname{cov}\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right)\propto\operatorname{cov}\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\mid\boldsymbol{x}^{(s)}\right).$
(13)

###### Proof.

By assumption, we could write
$\boldsymbol{x}^{(r)}_{i}\mid\boldsymbol{x}^{(s)}_{i}\sim{}EC_{r}\left(\boldsymbol{\mu}_{\boldsymbol{x}^{(s)}_{i}},\boldsymbol{\Sigma},\psi_{\boldsymbol{x}^{(s)}_{i}}\right)$.
Let $\boldsymbol{\Sigma}^{-1}=\boldsymbol{L}\boldsymbol{L}^{\prime}$ be the
Cholesky decomposition. Similar to the proof of Theorem 3.1, we know that
$\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\boldsymbol{L}\mid{}\delta=1,\boldsymbol{x}^{(s)}$
has spherical distribution and center $\boldsymbol{0}$. We then calculate its
moments to have

$\displaystyle\quad\mathbb{E}\left[\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\boldsymbol{L}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right]=\mathbb{E}\left[\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\boldsymbol{L}\mid\boldsymbol{x}^{(s)}\right]=\boldsymbol{0},$
$\displaystyle\operatorname{cov}\left(\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\boldsymbol{L}\mid{}\delta=1,\boldsymbol{x}^{(s)}\right)\propto\boldsymbol{I}.$

Transforming back completes the proof. ∎

Fixing a $J$-component vector $\boldsymbol{\lambda}$, a stratified
rerandomization method is said to be EPVR if the PRIV is the same for the
$\boldsymbol{\lambda}$-comparison of any linear combination of covariates.

######  Corollary 4.2.

Under the same assumptions as Theorem 4.2, stratified rerandomization is EPVR
for $\boldsymbol{x}$ and any linear function of $\boldsymbol{x}$.

###### Proof.

Let $\boldsymbol{Y}$ be any linear combination of covariates. Given
$\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j)$, we can write the $i$th component
of $\boldsymbol{Y}$ as
$Y_{i}=\boldsymbol{c}(j)\boldsymbol{\gamma}_{0}+\boldsymbol{x}^{(r)}_{i}\boldsymbol{\gamma}$.
For $j=1,2,\ldots,J$, let
$\overline{\boldsymbol{Y}}_{j\text{1}}-\overline{\boldsymbol{Y}}_{j\text{0}}$
be the difference in treatment and control means of $Y_{i}$ within stratum
$j$. It follows that

$\displaystyle\operatorname{var}\left(\sum_{j=1}^{J}\lambda_{j}\left(\overline{\boldsymbol{Y}}^{(r)}_{j\text{1}}-\overline{\boldsymbol{Y}}^{(r)}_{j\text{0}}\right)\mid{}\delta=1,\boldsymbol{x}^{(s)}\right)=$
$\displaystyle\boldsymbol{\gamma}^{\prime}\operatorname{cov}\left(\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\mid\delta=1,\boldsymbol{x}^{(s)}\right)\boldsymbol{\gamma}$
$\displaystyle=$
$\displaystyle\nu\boldsymbol{\gamma}^{\prime}\operatorname{cov}\left(\left(\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{1}}-\overline{\boldsymbol{x}}^{(r)}_{\boldsymbol{\lambda}\text{0}}\right)\mid\boldsymbol{x}^{(s)}\right)\boldsymbol{\gamma}$
$\displaystyle=$
$\displaystyle\nu\operatorname{cov}\left(\sum_{j=1}^{J}\lambda_{j}\left(\overline{\boldsymbol{Y}}^{(r)}_{j\text{1}}-\overline{\boldsymbol{Y}}^{(r)}_{j\text{0}}\right)\mid\boldsymbol{x}^{(s)}\right),$

where $\nu$ is the ratio of covariance matrices in (13). ∎

##  5 Admissible criteria

In this section, we turn to determination of balance criteria within certain
families. This is closely related to determining rejection regions in
statistical hypothesis testing problems. The theory was originally developed
by neyman1933ix and further generalized by wald1949statistical to a branch
named the statistical decision theory. Previous literature has focused on
certain class of risk functions under randomized decision rules. To address
our goal, this line of theory is extended with focus on conditional
expectations of score functions under stochastic criteria.

Given a probability space $(\Omega,\mathcal{F},P)$, let $\mathcal{C}$ be the
set of measurable functions whose ranges are in $[0,1]$ and
$\mathcal{C}_{0}=\\{\varphi\in\mathcal{C}:\mathbb{E}[\varphi]>0\\}$ be the
subset of $\mathcal{C}$ whose elements have non-zero acceptance probabilities.
Each $\varphi\in\mathcal{C}_{0}$ defines a stochastic criterion, where a
sample $\omega\in\Omega$ is accepted if and only if a Bernoulli trial
$\delta(\omega)$ with success probability $\varphi(\omega)$ turns out to be
$1$. A score function $h$ is defined as a real-valued integrable function on
$\Omega$. We are interested in the average score on accepted samples. With the
law of total expectation,

$\displaystyle\mathbb{E}[h\delta]$
$\displaystyle=\mathbb{E}[h\delta\mid{}\delta=1]\Pr(\delta=1)+\mathbb{E}[h\delta\mid{}\delta=0]\Pr(\delta=0)$
$\displaystyle=\mathbb{E}[h\mid{}\delta=1]\Pr(\delta=1)$

It follows that

$\mathbb{E}[h\mid{}\delta=1]=\frac{\mathbb{E}[h\delta]}{\Pr(\delta=1)}=\frac{\mathbb{E}[h\varphi]}{\mathbb{E}[\varphi]}:=\mathbb{E}[h\mid{}\varphi].$
(14)

Equation (14) serves as the definition of the average score on accepted
samples under a stochastic criterion.

Given a class of score functions $\\{h_{\theta}:\theta\in\Theta\\}$, a
criterion $\varphi\in\mathcal{C}_{0}$ is said to be inadmissible if it can be
dominated by another $\varphi^{\prime}\in\mathcal{C}_{0}$ with
$\mathbb{E}[\varphi^{\prime}]\geq\mathbb{E}[\varphi]$ and

$\mathbb{E}[{h}_{\theta}\mid\varphi^{\prime}]\leq\mathbb{E}[{h}_{\theta}\mid\varphi],\quad\theta\in\Theta$

where at least one inequality indexed by $\theta$ is strict. This means we can
simultaneously improve the average scores under $\varphi$ without reducing the
acceptance probability. A criterion is said to be admissible if there exists
no other criterion in $\mathcal{C}_{0}$ that dominates it. These concepts are
parallel to classical definitions of risk functions (see e.g.
berger1985statistical). However, as the functional space
$\\{\mathbb{E}[{h}_{\theta}\mid\varphi]:\varphi\in\mathcal{C}_{0}\\}$ is
generally non-convex, a new theorem is needed to study its property with
respect to a finite number of score functions.

######  Theorem 5.1.

Suppose that $\Theta=\\{1,2,\ldots,M\\}$.

$(\mathrm{i})$ A sufficient condition for $\varphi\in\mathcal{C}_{0}$ to be
admissible is the existence of positive constants $\pi_{1},\ldots,\pi_{M}$ and
$\alpha\in\mathbb{R}$ such that

$\varphi(\omega)=\begin{cases}1,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)<\alpha,\\\
0,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)>\alpha.\end{cases}$ (15)

$(\mathrm{ii})$ A necessary condition for $\varphi\in\mathcal{C}_{0}$ to be
admissible is the existence of non-negative constants $\pi_{1},\ldots,\pi_{m}$
and $\alpha\in\mathbb{R}$ such that (15) holds almost surely.

We can choose a version of (15) that is measurable on the $\sigma$-algebra
$\sigma_{M}$ generated by $h_{1}$, $\ldots$, $h_{M}$ as

$\varphi^{*}(\omega)=\begin{cases}1,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)<\alpha,\\\
p,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)=\alpha,\\\
0,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)>\alpha,\end{cases}$ (16)

where $p\in[0,1]$. This means we make a decision with $\varphi^{*}$ by
flipping a coin with success probability $p$ on the set
$\\{\omega\mid\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)=\alpha\\}\in\sigma_{M}$. A
procedure that makes use of all the $M$ score functions for building a
$\sigma_{M}$-measurable criterion is called the “unified” method. Although
Theorem 5.1 gives a admissible class of “unified” methods, it is not always
preferable because the cost of inquiring all $M$ score functions can be high.
An alternative popular method is the “intersection” method, which divides the
procedure into $M$ stages in order. In the $m$th stage ($m=1,2,\ldots,M$), a
stochastic criterion $\varphi_{m}$ invokes the first $m$ score functions
$h_{1}$, $\ldots$, $h_{m}$ and is measurable on the $\sigma$-algebra generated
by them. Let $\delta_{1},\ldots,\delta_{M}$, be the corresponding Bernoulli
trials. For a decision process, we iterate $m$ from $1$ to $M$ and reject the
sample when $\delta_{m}(\omega)=0$. Conceptually, it is equivalent to a
stochastic criterion that makes a decision based on
$\prod_{m=1}^{M}\delta_{m}$. However, if there is a chance that we decide
before the final step, the cost of making full inquiries of all score
functions can be reduced.

To be specific, for $m=1,2,\ldots,M$ and $\omega\in\Omega$, let
$c_{m}(\omega)$ be the cost of inquiring the value $h_{m}(\omega)$. We assume
$c_{m}$ to be positive and integrable. For simplicity, we also ignore the cost
of storing and reusing inquiries, because it is often negligible compared with
the inquiry cost. Given a “unified” method that routinely inquiries all $M$
score functions, the expected cost for a random sample is
$\sum_{m=1}^{M}\mathbb{E}[c_{m}]$. We show in the next theorem that an
“intersection” method can reduce it.

######  Theorem 5.2.

If an “intersection” method has $M>1$ stages in order with the acceptance
probability in the final stage larger than that of overall stages, then the
expected cost for a random sample is less than
$\sum_{m=1}^{M}\mathbb{E}[c_{m}]$.

Consider a relatively broad class of “intersection” methods that decide on
$\xi_{m}=\sum_{\ell=1}^{m}\pi_{m\ell}h_{\ell}$ in each $m$th stage, where
$\pi_{m1},\ldots,\pi_{mm}$ are real-valued constants with $\pi_{mm}$
normalized to be $1$ for $m=1,2,\ldots,M$. Of course, all constants except for
$\pi_{mm}$ ($m=1,2,\ldots,M$) can be zero, reflecting the simplest case where
separate bounds are placed on each score function. In the next proposition, we
give sufficient conditions for such “intersection” methods to be inadmissible
and provide a direct improvement with “unified” methods.

######  Proposition 5.2.1.

Suppose that $\varphi_{1:M}\in\mathcal{C}_{0}$ is intersected by $M$ separate
criteria:

$\varphi_{m}(\omega)=\begin{cases}1,&\xi_{m}<\alpha_{m},\\\
p_{m},&\xi_{m}=\alpha_{m},\\\ 0,&\xi_{m}>\alpha_{m},\end{cases}\quad
p_{m}\in[0,1],\ \alpha_{m}\in\mathbb{R}.$ (17)

If $\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{M})$ has a density that is positive
in a neighborhood of $(\alpha_{1},\ldots,\alpha_{M})$, then $\varphi_{1:M}$ is
inadmissible and can be dominated by some $\varphi^{*}\in\mathcal{C}_{0}$ in
the form of (16) with non-negative weights.

######  Remark 1.

Proposition 5.2.1 is sometimes applied to limiting distributions that admit
probability density functions. Consider a sequence of score functions
$\boldsymbol{h}_{n}=\left(h_{1,n},\ldots,h_{M,n}\right)$, $n=1,2,\ldots$, that
converge in distribution to
$\boldsymbol{h}_{\infty}=\left(h_{1,\infty},\ldots,h_{M,\infty}\right)$ as
$n\rightarrow\infty$. If $\varphi_{1:M}$ is inadmissible and dominated by
$\varphi^{*}$ with respect to $\boldsymbol{h}_{\infty}$, then $\varphi_{1:M}$
is said to be asymptotically inadmissible and asymptotically dominated by
$\varphi^{*}$.

As an intriguing example, we first apply our theory to an economic problem.

######  Example 1.

Suppose that $N_{0}$ members are to be selected out of a pool of $N$
candidates, each with $M$ scores, and our goal is to find a subgroup whose $M$
average scores cannot be dominated. Reformulated with abstract notations,
$\Omega$ is the pool of candidates, $\mathcal{F}$ is the collection of all
subgroups and $P$ is the uniform distribution. The criterion space
$\mathcal{C}_{0}$ consists of all indicator functions of subgroups that have
at least $N_{0}$ members. For $m=1,2,\ldots,M$, the score function $h_{m}$
corresponds to the mapping from candidates to their $m$th scores.

Applying the sufficient condition of Theorem 5.1, we know that for positive
constants $\pi_{1},\ldots,\pi_{M}$, a subgroup of members with the lowest
$N_{0}$ weighted scores $\sum_{m=1}^{M}\pi_{m}h_{m}$ is admissible. However,
it’s sometimes not practical to select memebers according to their weighted
scores. For example, if a company plans to hire a group of people from a large
pool of candidates, it usually divides the whole procedure into $M$ stages in
order. Candidates who fail in some stage will not enter the following stages.
Suppose decisions in the $m$th stage are made upon the weighted scores
$\sum_{\ell=1}^{m}\pi_{\ell}h_{\ell}$ of remaining candidates. That is, the
$i$th remaining candidate can pass this stage if and only if
$\sum_{\ell=1}^{m}\pi_{\ell}h_{\ell}(i)\leq\alpha_{m}$ for some prespecified
constant $\alpha_{m}$. As an “intersection” method, if the number of
candidates kept until the final stage is less than $N$, the inquiry cost will
be reduced compared with that of the “unified” method.

Let $\boldsymbol{h}_{\infty}=\left(h_{1,\infty},\ldots,h_{M,\infty}\right)$ be
the super-population score functions. If candidates are random samples from
the super-population, and if
$\boldsymbol{\xi}_{\infty}=\left(\xi_{1,\infty},\ldots,\xi_{M,\infty}\right)$
satisfies the conditions of Proposition 5.2.1, where
$\xi_{m,\infty}=\sum_{\ell=1}^{m}\pi_{\ell}/\pi_{m}h_{\ell,\infty}$,
$m=1,2,\ldots,M$, then candidates’ scores converges in distribution to
$\boldsymbol{h}_{\infty}$ as $N\rightarrow\infty$ and the “intersection”
method is asymptotically inadmissible. In conclusion, there is a trade-off
between admissibility and cost. How to find a balanced strategy remains an
open problem.

In experimental design, however, the cost of rerandomization routines is not a
major concern because it is the computational cost of balance scores. On the
other hand, if the performance of rerandomization routines is measured by
average balance scores of accepted assignments, it’s important to identify
inadmissible balance criteria and find ways to improve them.

######  Example 2.

raab2001balance suggest a class of balance criteria for clustered randomized
experiments. For RCTs, their criterion can be simplified as placing a
threshold on a weighted sum of squared Euclidean distances
$\sum_{\ell=1}^{k}\pi_{\ell}\left(\overline{\boldsymbol{x}}^{[\ell]}_{\text{1}}-\overline{\boldsymbol{x}}^{[\ell]}_{\text{0}}\right)^{2}$,
where
$\overline{\boldsymbol{x}}^{[\ell]}_{\text{1}}-\overline{\boldsymbol{x}}^{[\ell]}_{\text{0}}$
is the difference of between treatment and control group means of the $\ell$th
covariate. A simple choice of weights would be the inverses of covariate
variances, which is equivalent to standardizing covariates and using equal
weights.

moulton2004covariate suggests placing separate bounds on the squared
difference in means of each standardized covariate and intersecting them to
obtain a balance criterion. It is equivalent to a popular way that checks the
p-value of a two-sample t-test for each covariate (see e.g. bruhn2009pursuit).

For $\ell=1,2,\ldots,k$, let $S_{\ell\ell}$ be the sample variance of the
$\ell$th covariate. In Appendix B, we show that under complete randomization
and certain regularity assumptions, as $n\rightarrow\infty$,

$\left(\frac{\overline{\boldsymbol{x}}^{[1]}_{\text{1}}-\overline{\boldsymbol{x}}^{[1]}_{\text{0}}}{\sqrt{S_{11}/n}},\ldots,\frac{\overline{\boldsymbol{x}}^{[k]}_{\text{1}}-\overline{\boldsymbol{x}}^{[k]}_{\text{0}}}{\sqrt{S_{kk}/n}}\right)$

converges in distribution to a multivariate normal distribution with a non-
singular covariance. Let
$h_{\ell}=n\left(\overline{\boldsymbol{x}}^{[\ell]}_{\text{1}}-\overline{\boldsymbol{x}}^{[\ell]}_{\text{0}}\right)^{2}/S_{\ell\ell}$
be the $\ell$th balance score. The limiting distribution of
$(h_{1},\ldots,h_{k})$ has a positive density in
$\mathbb{R}_{+}^{k}=\\{(y_{1},\ldots,y_{k})\in\mathbb{R}^{k}\mid
y_{1}>0,\ldots,y_{k}>0\\}$. According to Proposition 5.2.1, if positive bounds
are placed on each $h_{\ell}$, such an “intersection” balance criterion turns
out to be asymptotically inadmissible, which can be asymptotically dominated
by some “unified” method that places a single bound on a weighted sum of
$h_{\ell}$’s. Moreover, if all weights are chosen to be positive, then the
“unified” method is admissible according to (i) of Theorem 5.1. Therefore, we
recommend raab2001balance’s criterion instead of moulton2004covariate’s for
improved balance in covariates.

The correlation between covariates is not taken into consideration in the last
example, and it can be invoked by conditionally affinely invariant balance
criteria.

######  Example 3.

Continuing the discussion of Section 4.1, we describe
morgan2015rerandomization’s implementation based on tiers of Mahalanobis
distances. Given the covariate matrix
$\boldsymbol{x}=\left(\boldsymbol{x}^{(1)},\ldots,\boldsymbol{x}^{(T)}\right)$
and for $t>1$, denote $e_{ij}^{(t)}$ as the residual of $x_{ij}^{(t)}$ after
regressing the $j$th covariate in tier $t$ on covariates in the first $t-1$
tiers:

$x_{ij}^{(t)}=\beta_{0}+\sum_{z=1}^{t-1}\beta_{z}^{\top}\boldsymbol{x}_{i}^{(z)}+e_{ij}^{(t)}.$
(18)

Let $\boldsymbol{e}^{(t)}=\left(e_{ij}^{(t)}\right)$ be the matrix of
covariate residuals in tier $t>1$. For notation brevity, we also define
$\boldsymbol{e}^{(1)}=\boldsymbol{x}^{(1)}$. Let $\boldsymbol{E}_{(t,t)}$ be
the sample covariance matrix and
$\overline{\boldsymbol{e}}_{\text{1}}^{(t)}-\overline{\boldsymbol{e}}_{\text{0}}^{(t)}$
be the difference in treatment and control group means of
$\boldsymbol{e}^{(t)}$; the balance scores are given by tiers of Mahalanobis
distances

$h_{t}=np_{w}(1-p_{w})\left(\overline{\boldsymbol{e}}_{\text{1}}^{(t)}-\overline{\boldsymbol{e}}_{\text{0}}^{(t)}\right)\boldsymbol{E}_{(t,t)}^{-1}\left(\overline{\boldsymbol{e}}_{\text{1}}^{(t)}-\overline{\boldsymbol{e}}_{\text{0}}^{(t)}\right)^{\prime},\quad
t=1,2,\ldots,T.$ (19)

Since each $h_{t}$ remains invariant if an affine transformation is applied to
$\boldsymbol{e}^{(t)}$, any balance criterion that based on
$\boldsymbol{h}=(h_{1},\ldots,h_{T})$ is subspace-conditionally affinely
invariant on
$\boldsymbol{e}=\left(\boldsymbol{e}^{(1)},\ldots,\boldsymbol{e}^{(T)}\right)$.
To achieve computational efficiency, morgan2015rerandomization recommend a
balance criterion $\varphi_{1:T}$ that is intersected by

$\varphi_{t}(\boldsymbol{x},\boldsymbol{W})=\begin{cases}1,&h_{t}\leq\alpha_{t},\\\
0,&h_{t}>\alpha_{t},\end{cases}\quad\alpha_{t}\in\mathbb{R}_{+},\
t=1,2,\ldots,T.$ (20)

In Appendix B we show that, under complete randomization and certain
regularity assumptions, $\boldsymbol{h}$ converges in distribution to
$\boldsymbol{\chi}^{2}$, a vector of mutually independent chi-square random
variables. Because $(\alpha_{1},\ldots,\alpha_{T})$ lies in
$\mathbb{R}_{+}^{T}=\\{(y_{1},\ldots,y_{T})\in\mathbb{R}^{T}\mid
y_{1}>0,\ldots,y_{T}>0\\}$, on which $\boldsymbol{\chi}^{2}$ has a positive
density, according to Proposition 5.2.1, $\varphi_{1:T}$ is asymptotically
inadmissible and can be asymptotically dominated by some “unified” method that
places a single bound on a weighted sum $\sum_{t=1}^{T}\pi_{t}h_{t}$.

lu2023design compare a criterion that places a bound on optimally weighted sum
of Euclidean distances of orthogonalized covariates with $\varphi_{1:T}$ and
show that the former is superior to the latter. However, lu2023design’s
optimal weights contain practically unknown quantities, leaving a question
whether their criterion is still superior with possibly misspecified weights.
Our theory confirms that $\varphi_{1:T}$ is inferior because it can be
asymptotically dominated in terms of average balance scores on accepted
assignments. In Appendix C, we link this fact with reduction in the asymptotic
variance of the difference-in-means estimator and suggest a way of building
admissible balance criteria with vague prior information of covariate
importance.

######  Example 4.

Continuing the discussion of Subsection 4.2, we describe one of
wang2021rerandomization’s implementations based on stratum-specific
Mahalanobis distances. Let
$\overline{\boldsymbol{x}}_{j\text{1}}^{(r)}-\overline{\boldsymbol{x}}_{j\text{0}}^{(r)}$
and $\boldsymbol{S}_{j}^{(r)}$ be the stratum-specific difference in treatment
and control group means and sample covariance of continuous covariates; the
balance scores are given by stratum-specific Mahalanobis distances:

$h_{j}=np_{j}(1-p_{j})\left(\overline{\boldsymbol{x}}_{j\text{1}}^{(r)}-\overline{\boldsymbol{x}}_{j\text{0}}^{(r)}\right)\boldsymbol{S}_{j}^{(r)-1}\left(\overline{\boldsymbol{x}}_{j\text{1}}^{(r)}-\overline{\boldsymbol{x}}_{j\text{0}}^{(r)}\right)^{\prime},\quad
j=1,2,\ldots,J.$ (21)

Since each $h_{j}$ remains invariant if an affine transformation is applied to
$\boldsymbol{x}^{(r)}$, any balance criterion based on
$\boldsymbol{h}=(h_{1},\ldots,h_{J})$ is conditionally affinely invariant. In
Appendix B, we show that, under stratified randomization and certain
regularity assumptions, $\boldsymbol{h}$ converges in probability to
$\boldsymbol{\chi}^{2}$, again a vector of mutually independent chi-square
random variables. Therefore, wang2021rerandomization’s stratified
rerandomization that places separate positive bounds on each $h_{j}$ is
asymptotically inadmissible, and can be asymptotically dominated by some
“unified” method that places a single bound on a weighted sum
$\sum_{j=1}^{J}\pi_{j}h_{j}$. In Appendix C, we link this fact with reduction
in the asymptotic variances of general difference-in-means estimators and
suggest a way of building admissible balance criterion with vague prior
information of heteroscedasticity in continuous covariates across $J$ strata.

##  6 Conclusion

When covariates are regarded as random draws from a distribution that exhibits
symmetry on some dimensions, balance criteria that are conditionally affinely
invariant on these dimensions would be conditionally EPVR. Two important
instances, balancing on tiers of covariates and stratified rerandomization,
are studied under this framework. Each instance involves multiple balance
scores in practical implementation. “Intersection” methods are frequently
adopted for computational efficiency or simplicity and these methods turn out
to be asymptotically inadmissible. Since the computational cost is not the
major concern in designing RCTs, our final recommendation is to stick to the
“unified” methods and rerandomize when a weighted sum of balance scores
exceeds a pre-specified threshold.

## References

  * (1)
  * Banerjee et al. (2020) Banerjee, A. V., Chassang, S., Montero, S. & Snowberg, E. (2020), ‘A theory of experimenters: Robustness, randomization, and balance’, American Economic Review 110(4), 1206–1230.
  * Berger (1985) Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, Springer Science & Business Media.
  * Bertsimas et al. (2015) Bertsimas, D., Johnson, M. & Kallus, N. (2015), ‘The power of optimization over randomization in designing experiments involving small samples’, Operations Research 63(4), 868–876.
  * Bruhn & McKenzie (2009) Bruhn, M. & McKenzie, D. (2009), ‘In pursuit of balance: Randomization in practice in development field experiments’, American economic journal: applied economics 1(4), 200–232.
  * Cambanis et al. (1981) Cambanis, S., Huang, S. & Simons, G. (1981), ‘On the theory of elliptically contoured distributions’, Journal of Multivariate Analysis 11(3), 368–385.
  * Cornish (1954) Cornish, E. A. (1954), ‘The multivariate t-distribution associated with a set of normal sample deviates’, Australian Journal of Physics, vol. 7, p. 531 7, 531.
  * Dempster (1969) Dempster, A. P. (1969), Elements of continuous multivariate analysis, Addison Wesley.
  * Dvoretzky et al. (1951) Dvoretzky, A., Wald, A. & Wolfowitz, J. (1951), ‘Relations among certain ranges of vector measures.’.
  * Imbens & Rubin (2015) Imbens, G. W. & Rubin, D. B. (2015), Causal inference in statistics, social, and biomedical sciences, Cambridge University Press.
  * Johansson et al. (2021) Johansson, P., Rubin, D. B. & Schultzberg, M. (2021), ‘On optimal rerandomization designs’, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 83(2), 395–403.
  * Johansson & Schultzberg (2022) Johansson, P. & Schultzberg, M. (2022), ‘Rerandomization: A complement or substitute for stratification in randomized experiments?’, Journal of Statistical Planning and Inference 218, 43–58.
  * Kallus (2018) Kallus, N. (2018), ‘Optimal a priori balance in the design of controlled experiments’, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80(1), 85–112.
  * Li & Ding (2017) Li, X. & Ding, P. (2017), ‘General forms of finite population central limit theorems with applications to causal inference’, Journal of the American Statistical Association 112(520), 1759–1769.
  * Li et al. (2018) Li, X., Ding, P. & Rubin, D. B. (2018), ‘Asymptotic theory of rerandomization in treatment–control experiments’, Proceedings of the National Academy of Sciences 115(37), 9157–9162.
  * Liu & Rubin (1998) Liu, C. & Rubin, D. B. (1998), ‘Ellipsoidally symmetric extensions of the general location model for mixed categorical and continuous data’, Biometrika 85(3), 673–688.
  * Lu et al. (2023) Lu, X., Liu, T., Liu, H. & Ding, P. (2023), ‘Design-based theory for cluster rerandomization’, Biometrika 110(2), 467–483.
  * Morgan & Rubin (2012) Morgan, K. L. & Rubin, D. B. (2012), ‘Rerandomization to improve covariate balance in experiments’, The Annals of Statistics 40(2), 1263–1282.
  * Morgan & Rubin (2015) Morgan, K. L. & Rubin, D. B. (2015), ‘Rerandomization to balance tiers of covariates’, Journal of the American Statistical Association 110(512), 1412–1421.
  * Moulton (2004) Moulton, L. H. (2004), ‘Covariate-based constrained randomization of group-randomized trials’, Clinical trials 1(3), 297–305.
  * Neyman et al. (1990) Neyman, J., Dabrowska, D. M. & Speed, T. P. (1990), ‘On the application of probability theory to agricultural experiments. essay on principles. section 9.’, Statistical Science pp. 465–472.
  * Neyman & Pearson (1933) Neyman, J. & Pearson, E. S. (1933), ‘Ix. on the problem of the most efficient tests of statistical hypotheses’, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 231(694-706), 289–337.
  * Olkin & Tate (1961) Olkin, I. & Tate, R. F. (1961), ‘Multivariate correlation models with mixed discrete and continuous variables’, The Annals of Mathematical Statistics pp. 448–465.
  * Raab & Butcher (2001) Raab, G. M. & Butcher, I. (2001), ‘Balance in cluster randomized trials’, Statistics in medicine 20(3), 351–365.
  * Rubin (1974) Rubin, D. B. (1974), ‘Estimating causal effects of treatments in randomized and nonrandomized studies.’, Journal of educational Psychology 66(5), 688.
  * Rubin (1980) Rubin, D. B. (1980), ‘Randomization analysis of experimental data: The fisher randomization test comment’, Journal of the American statistical association 75(371), 591–593.
  * Rubin & Thomas (1992) Rubin, D. B. & Thomas, N. (1992), ‘Affinely invariant matching methods with ellipsoidal distributions’, The Annals of Statistics pp. 1079–1093.
  * Tukey (1993) Tukey, J. W. (1993), ‘Tightening the clinical trial’, Controlled clinical trials 14(4), 266–285.
  * Wald (1949) Wald, A. (1949), ‘Statistical decision functions’, The Annals of Mathematical Statistics pp. 165–205.
  * Wang et al. (2021) Wang, X., Wang, T. & Liu, H. (2021), ‘Rerandomization in stratified randomized experiments’, Journal of the American Statistical Association pp. 1–10.

##  Appendix A Proofs of Theorems and Propositions

###### Proof of Theorem 5.1.

$(\mathrm{i})$ Firstly, we prove the case of $M=1$. Let $\varphi^{\prime}$ be
another criterion in $\mathcal{C}_{0}$ with
$\mathbb{E}[\varphi^{\prime}]\geq\mathbb{E}[\varphi]$. Denote by $S^{+}$ and
$S^{-}$ the sets in $\Omega$ where $\varphi-\varphi^{\prime}>0$ and $<0$
respectively. If $\omega$ in $S^{+}$, $\varphi(\omega)$ must be positive and
$a\geq{}\pi_{1}h_{1}(\omega)$. In a similar way $a\leq{}\pi_{1}h_{1}(\omega)$
for all $\omega$ in $S^{-}$. Hence,

$\int\left(a-\pi_{1}h_{1}\right)\left(\varphi-\varphi^{\prime}\right)dP=\int_{S^{+}\cup
S^{-}}\left(a-\pi_{1}h_{1}\right)\left(\varphi-\varphi^{\prime}\right)dP\geq
0.$

It follows that $\int{}h_{1}\varphi
dP-\int{}h_{1}\varphi^{\prime}dP\leq\alpha/\pi_{1}\left(\int\varphi{}dP-\int\varphi^{\prime}dP\right)$
and that

$\displaystyle\mathbb{E}[h_{1}\mid\varphi]-\mathbb{E}[h_{1}\mid\varphi^{\prime}]$
$\displaystyle=\frac{\int{}h_{1}\varphi{}dP}{\int\varphi{}dP}-\frac{\int{}h_{1}\varphi^{\prime}dP}{\int\varphi^{\prime}dP}$
$\displaystyle=\frac{\int{}h_{1}\varphi{}dP+\mathbb{E}[h_{1}\mid\varphi]\left(\int\varphi^{\prime}dP-\int\varphi{}dP\right)}{\int\varphi^{\prime}dP}-\frac{\int{}h_{1}\varphi^{\prime}dP}{\int\varphi^{\prime}dP}$
$\displaystyle\leq\frac{\left(\mathbb{E}[h_{1}\mid\varphi]-a/\pi_{1}\right)\left(\int\varphi^{\prime}dP-\int\varphi{}dP\right)}{\int\varphi^{\prime}dP}$
$\displaystyle\leq 0,$

where the last inequality is because $a\geq\pi_{1}h_{1}$ on $\varphi>0$ and
$\mathbb{E}[\varphi^{\prime}]\geq\mathbb{E}[\varphi]$.

Secondly, we prove the $M>1$ case by contradiction. Denote by
$\boldsymbol{\pi}\cdot\boldsymbol{h}=\sum_{m=1}^{M}\pi_{m}{}h_{m}$. Suppose
there exists $\varphi^{\prime}\in\mathcal{C}_{0}$ that dominates $\varphi$.
Summing these inequalities with respect to the positive vector
$\boldsymbol{\pi}$, we have
$\mathbb{E}\left[\boldsymbol{\pi}\cdot\boldsymbol{h}\mid\varphi^{\prime}\right]<\mathbb{E}\left[\boldsymbol{\pi}\cdot\boldsymbol{h}\mid\varphi\right]$.
However, as shown in the $M=1$ case,
$\mathbb{E}\left[\boldsymbol{\pi}\cdot\boldsymbol{h}\mid\varphi^{\prime}\right]\geq\mathbb{E}\left[\boldsymbol{\pi}\cdot\boldsymbol{h}\mid\varphi\right]$.
This contradiction means that $\varphi^{\prime}$ does not exist and $\varphi$
is an admissible criterion.

$(\mathrm{ii})$ For all $\varphi$ in $\mathcal{C}$, let
$U\subset\mathbb{R}^{M+1}$ be the set of points whose first $M$ coordinates
are $\int h_{1}\varphi dP,\ldots,\int h_{M}\varphi dP$ and the last is $\int
1-\varphi dP$. Note that $U$ is convex because it is the image of a linear
transformation on the convex set $\mathcal{C}$. Suppose an admissible
criterion $\varphi\in\mathcal{C}_{0}$ is mapped to some
$u=\left(u_{1},\ldots,u_{M+1}\right)\in{}U$. Let $V$ be the set of all points
whose first $M$ coordinates are strictly lower than those of $u$ for fixed
$u_{M+1}$. We state that $U\cap{}V=\emptyset$. Otherwise, we can find a point
in $U$ mapped from $\varphi^{\prime}$ such that
$\mathbb{E}[\varphi^{\prime}]=u_{M+1}$ and $\int h_{m}\varphi^{\prime}dP<\int
h_{m}\varphi dP$, $m=1,2,\ldots,M$, which means $\varphi$ is no longer
admissible, a contradiction.

By the separation theorem there exists a hyperplane $\Pi$ that separates two
disjoint convex sets $U$ and $V$. An immediate observation is $u\in\Pi$
because $u\in{}U\cap\overline{V}$. Let the equation of $\Pi$ evaluated at $u$
be $\pi_{0}+\sum_{m=1}^{M}\pi_{m}{}u_{m}+au_{M+1}=0$. According to the
analytical form of the separation theorem,

$\displaystyle\sum_{m=1}^{M}\pi_{m}\left(v_{m}-u_{m}\right)\leq
0,\quad\forall(v_{1},\ldots,v_{m},u_{M+1})\in V,$ (22)
$\displaystyle\int\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\varphi^{\prime\prime}dP\geq\int\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\varphi
dP,\quad\forall\varphi^{\prime\prime}\in\mathcal{C}.$ (23)

It can be directly proved that $\pi_{m}\geq 0,m=1,2,\ldots,M$ by letting each
coordinate tend to $-\infty$ with the rest fixed in (22).

Consider $\varphi^{*}\in\mathcal{C}$ such that $\varphi^{*}(\omega)=1$ when
$\boldsymbol{\pi}\cdot\boldsymbol{h}(\omega)\leq\alpha$ and
$\varphi^{*}(\omega)=0$ when
$\boldsymbol{\pi}\cdot\boldsymbol{h}(\omega)>\alpha$. Then $\varphi^{*}$
attains the lower bound of (23). Let $S$ be the intersection of the set, on
which $\varphi^{*}$ and $\varphi$ differs, with
$\\{\omega\in\Omega\mid{}\boldsymbol{\pi}\cdot\boldsymbol{h}(\omega)\neq{}a\\}$.
Since
$\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\left(\varphi^{*}-\varphi\right)>0$
on $S$ and equals $0$ on $\Omega\backslash{}S$, we have

$\displaystyle 0$
$\displaystyle\leq\int_{S}\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\left(\varphi^{*}-\varphi\right)dP$
$\displaystyle=\int\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\left(\varphi^{*}-\varphi\right)dP=0$

Thus,
$\int_{S}\left(\boldsymbol{\pi}\cdot\boldsymbol{h}-a\right)\left(\varphi^{*}-\varphi\right)dP=0$
which implies that $P(S)=0$ as was to be proved. ∎

###### Proof of Theorem 5.2.

For a random sample $\omega\in\Omega$, if
$\prod_{m=1}^{M-1}\delta_{m}(\omega)=1$, then we have to inquiry the $M$th
score function for final decision. Otherwise we can terminate before the $M$th
stage and have at most $M-1$ inquiries of score functions. Under the
assumption
$\mathbb{E}[\varphi_{M}]>\mathbb{E}\left[\prod_{m=1}^{M}\varphi_{m}\right]$,

$\displaystyle\Pr\left(\prod_{m=1}^{M}\delta_{m}=1\right)=\Pr(\delta_{M}=1)-\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0,\delta_{M}=1\right)$
$\displaystyle>$
$\displaystyle\Pr\left(\prod_{m=1}^{M}\delta_{m}=1\right)-\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0\right).$

It follows that $\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0\right)>0$. In other
words, there is a positive chance that a rejection is made before the final
stage. Let $N(\omega)$ be the random number of inquiries until termination.
The cost of the “intersection” method is

$\displaystyle\mathbb{E}\left[\sum_{m=1}^{N}c_{m}\right]=$
$\displaystyle\mathbb{E}\left[\sum_{m=1}^{N}c_{m}\mid\prod_{m=1}^{M-1}\delta_{m}=1\right]\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=1\right)$
$\displaystyle+\mathbb{E}\left[\sum_{m=1}^{N}c_{m}\mid\prod_{m=1}^{M-1}\delta_{m}=0\right]\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0\right)$
$\displaystyle\leq$
$\displaystyle\sum_{m=1}^{M}\mathbb{E}[c_{m}]\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=1\right)+\sum_{m=1}^{M-1}\mathbb{E}[c_{m}]\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0\right)$
$\displaystyle\leq$
$\displaystyle\sum_{m=1}^{M}\mathbb{E}[c_{m}]-\mathbb{E}[c_{M}]\Pr\left(\prod_{m=1}^{M-1}\delta_{m}=0\right)<\sum_{m=1}^{M}\mathbb{E}[c_{m}],$

where the last inequality is because $\mathbb{E}[c_{M}]>0$ by assumption. ∎

###### Proof of Proposition 5.2.1.

Suppose that, on the contrary, $\varphi_{1:M}$ is admissible. According to
(ii) of Theorem 5.1, there exists a hyperplane that separates
$\\{\boldsymbol{h}(\omega)\mid\varphi_{1:M}(\omega)=1\\}$ and
$\\{\boldsymbol{h}(\omega)\mid\varphi_{1:M}(\omega)\neq 1\\}$ up to a set of
probability zero. Since $\boldsymbol{h}$ is a linear transform of
$\boldsymbol{\xi}$, similar hyperplane exists for $\boldsymbol{\xi}$. In other
words, up to a set of probability zero, there exists a closed half plane $H$
that contains $\\{\boldsymbol{\xi}(\omega)\mid\varphi_{1:M}(\omega)=1\\}$, and
its interior $H^{o}$ is disjoint with
$\\{\boldsymbol{\xi}(\omega)\mid\varphi_{1:M}(\omega)\neq 1\\}$. Consider an
open ball $B$ with center
$\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{M})$, on which
$\boldsymbol{\xi}$ has a positive density, and a closed convex cone
$C=\\{(y_{1},\ldots,y_{M})\mid
y_{1}\leq\alpha_{1},\ldots,y_{M}\leq\alpha_{M}\\}$. By definition,

$\begin{cases}\boldsymbol{\xi}(\omega)\in B\cap C,\text{ if
}\varphi_{1:M}(\omega)=1\\\ \boldsymbol{\xi}(\omega)\in B\cap C^{c},\text{ if
}\varphi_{1:M}(\omega)\neq 1\end{cases},$

where $C^{c}$ is the complement of $C$. It follows that $B\cap C\subset H$ and
$B\cap C^{c}\cap H^{o}=\emptyset$. By the geometry of these sets, $H$ does not
exist, a contradiction.

For all $\varphi\in\mathcal{C}$, let $U_{\text{ext}}\subset\mathbb{R}^{2M+2}$
be the set of points whose coordinates are

$\int h_{1}\varphi dP,\ldots,\int h_{M}\varphi dP,\int\varphi dP,\int
h_{1}(1-\varphi)dP,\ldots,\int h_{M}(1-\varphi)dP,\int 1-\varphi dP.$

Applying Theorem 1 of dvoretzky1951relations, $U_{\text{ext}}$ is convex and
compact. By coordinate projection over $U_{\text{ext}}$, we obtain $U$
represented as

$U=\left\\{\left(\int h_{1}\varphi dP,\ldots,\int h_{M}\varphi dP,\int
1-\varphi dP\right):\varphi\in\mathcal{C}\right\\}.$

Note that $U$ is still convex and compact. Consider a family of convex sets
$V(q)=\\{(y_{1},\ldots,y_{M+1})\in\mathbb{R}^{M+1}\mid y_{1}\leq
u_{1}-q,\ldots,y_{M}\leq u_{M}-q,u_{M+1}\\}$, where

$u=(u_{1},\ldots,u_{M},u_{M+1})=\left(\int h_{1}\varphi_{1:M}dP,\ldots,\int
h_{M}\varphi_{1:M}dP,\int 1-\varphi_{1:M}dP\right).$

Define $q_{0}=\inf\left\\{q\in\mathbb{R}:V(q)\cap U=\emptyset\right\\}$.
Because $U$ is bounded from below in each coordinate, we know that
$q_{0}<\infty$.

We state $V(q_{0})\cap U\neq\emptyset$. If, on the contrary, $V(q_{0})\cap
U=\emptyset$, because $V(q_{0})$ is convex and closed, there exists a
hyperplane that strictly separates them. It indicates the existence of
$q_{1}<q_{0}$ such that $V(q_{1})\cap U=\emptyset$, which contradicts with the
definition of $q_{0}$.

If $q_{0}=0$, because $u\in V(0)\cap U$, $u$ is a boundary point of $U$.
Repeating the proof of Theorem 5.1, we know that a necessary condition for
$\varphi_{1:M}$ is the existence of a hyperplane that separates
$\\{\boldsymbol{h}(\omega)\mid\varphi_{1:M}(\omega)=1\\}$ and
$\\{\boldsymbol{h}(\omega)\mid\varphi_{1:M}(\omega)\neq 1\\}$ up to a set of
probability zero. However, it has been disproved at the beginning of the
proof. Therefore, there exists $u^{*}\in V(q_{0})\cap U$ for $q_{0}>0$, where

$u^{*}=\left(u^{*}_{1},\ldots,u^{*}_{M},u^{*}_{M+1}\right)=\left(\int
h_{1}\varphi^{*}dP,\ldots,\int h_{M}\varphi^{*}dP,\int 1-\varphi^{*}dP\right)$

satisfies $\mathbb{E}[\varphi^{*}]=\mathbb{E}[\varphi_{1:M}]$ and
$\mathbb{E}[h_{m}\mid\varphi^{*}]<\int\mathbb{E}[h_{m}\mid\varphi_{1:M}]$ for
$m=1,2,\ldots,M$. And because $u^{*}$ is a boundary point of $U$, we know from
the proof of Theorem 5.1 that

$\varphi^{*}(\omega)=\begin{cases}1,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)<\alpha,\\\
0,&\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)>\alpha\end{cases}$

almost surely for some $\alpha\in\mathbb{R}$ and non-negative constants
$\pi_{1},\ldots,\pi_{M}$. By assumption, $\boldsymbol{h}$ has a probabilistic
density function transformed from that of $\boldsymbol{\xi}$. Therefore, the
critical set
$C=\\{\omega\in\Omega\mid\sum_{m=1}^{M}\pi_{m}{}h_{m}(\omega)=\alpha\\}$ has a
probability measure $0$. Setting $\varphi^{*}=1$ over $C$ completes the proof.
∎

##  Appendix B More Details on Finite Population Asymptotics

Appendix B provides more details on asymptotic distributions of balance scores
involved in above examples, where the only source of randomness comes from
repeated assignment. By $X_{1}\dot{\ \sim\ }X_{2}$ we mean random variables
$X_{1}$ and $X_{2}$ converges weakly to the same distribution as
$n\rightarrow\infty$. Following li2017general, a set of regularity assumptions
for completely randomized experiments is introduced as follows.

######  Assumption B.1.

As $n\rightarrow\infty$, (i) the propensity score $p_{w}=n_{\text{1}}/n$ has a
limit in $(0,1)$; (ii) the sample covariance of covariates, $\boldsymbol{S}$,
has a limiting value that is non-singular; (iii)
$\max_{1\leq{}i\leq{}n}\|\boldsymbol{x}_{i}-\overline{\boldsymbol{x}}\|_{2}/n\rightarrow
0$, where $\overline{\boldsymbol{x}}$ is the sample mean vector of covariates
and $\|\cdot\|_{2}$ is the Euclidean distance in $\mathbb{R}^{k}$.

Under Assumption B.1 and complete randomization, apply Theorem 5 of
li2017general and we have
$\sqrt{n}(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})\dot{\
\sim\ }N\left(\boldsymbol{0},(p_{w}(1-p_{w}))^{-1}\boldsymbol{S}\right)$. Let
$\operatorname{diag}(\boldsymbol{S})$ be the matrices of diagonal elements of
$\boldsymbol{S}$. By assumption (ii) of Assumption B.1, as
$n\rightarrow\infty$, $\operatorname{diag}(\boldsymbol{S})$ has a limiting
value, implying

$\left(\frac{\overline{\boldsymbol{x}}^{[1]}_{\text{1}}-\overline{\boldsymbol{x}}^{[1]}_{\text{0}}}{\sqrt{S_{11}/n}},\ldots,\frac{\overline{\boldsymbol{x}}^{[k]}_{\text{1}}-\overline{\boldsymbol{x}}^{[k]}_{\text{0}}}{\sqrt{S_{kk}/n}}\right)\dot{\
\sim\
}N\left(\boldsymbol{0},\operatorname{diag}(p_{w}(1-p_{w})\boldsymbol{S})^{-1}\boldsymbol{S}\right).$

Hence, the finite population asymptotics in Example 2 is justified.

For $t\geq 1$, let $\boldsymbol{S}_{(\overline{t},\overline{t})}$ be the
sample covariance of covariates in the first $t$ tiers; let
$\boldsymbol{S}_{(t,t)}$ be that of covariates in tier $t$. For $t>1$, let
$\boldsymbol{S}_{(\overline{t-1},t)}$ be the sample covariance between
covariates in tier $t$ and the first $t-1$ tiers. For $t>1$, we also introduce
the projection matrix
$\boldsymbol{P}_{t}=\boldsymbol{S}_{(\overline{t-1},\overline{t-1})}^{(-1)}\boldsymbol{S}_{(\overline{t-1},t)}$.
If $1<t<T$, we extend $\boldsymbol{P}_{t}$ to a $k\times k_{t}$ dimensional
matrix $\widetilde{\boldsymbol{P}}_{t}$ by appending $0$’s below it; if $t=1$,
we define $\widetilde{\boldsymbol{P}}_{1}$ as a $k\times k_{1}$ dimensional
zero matrix; and if $t=T$, we just write
$\widetilde{\boldsymbol{P}}_{T}=\boldsymbol{P}_{T}$. Denote by
$\overline{\boldsymbol{e}}_{\text{1}}-\overline{\boldsymbol{e}}_{\text{0}}=\left(\overline{\boldsymbol{e}}_{\text{1}}^{(1)}-\overline{\boldsymbol{e}}_{\text{0}}^{(1)},\ldots,\overline{\boldsymbol{e}}_{\text{1}}^{(T)}-\overline{\boldsymbol{e}}_{\text{0}}^{(T)}\right)$.
According to (18),
$\overline{\boldsymbol{e}}_{\text{1}}-\overline{\boldsymbol{e}}_{\text{0}}=(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)$,
where $\boldsymbol{I}$ is a $k$-dimensional identity matrix and
$\widetilde{\boldsymbol{P}}=\left(\widetilde{\boldsymbol{P}}_{1},\ldots,\widetilde{\boldsymbol{P}}_{T}\right)$.
Under Assumption B.1, as $n\rightarrow\infty$, $\widetilde{\boldsymbol{P}}$
has a finite limit. It follows that

$\sqrt{n}\left(\overline{\boldsymbol{e}}_{\text{1}}-\overline{\boldsymbol{e}}_{\text{0}}\right)\dot{\
\sim\
}\sqrt{n}(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)\dot{\
\sim\ }\boldsymbol{g},$ (24)

where
$\boldsymbol{g}=\left(\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{T}\right)$ and
each $\boldsymbol{g}_{t}$ represents the limiting distribution of
$\sqrt{n}\left(\overline{\boldsymbol{e}}_{\text{1}}^{(t)}-\overline{\boldsymbol{e}}_{\text{0}}^{(t)}\right)$.
Since
$\sqrt{n}(\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}})\dot{\
\sim\ }N\left(\boldsymbol{0},(p_{w}(1-p_{w}))^{-1}\boldsymbol{S}\right)$, we
have

$\boldsymbol{g}\sim
N\left(\boldsymbol{0},(p_{w}(1-p_{w}))^{-1}\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)^{\prime}\boldsymbol{S}\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)\right).$
(25)

Further calculation reveals that $\boldsymbol{g}_{t}$, $t\geq 1$, are mutually
uncorrelated with

$\displaystyle\boldsymbol{g}_{1}\sim
N\left(\boldsymbol{0},(p_{w}(1-p_{w}))^{-1}\boldsymbol{S}_{(1,1)}\right)$ (26)
$\displaystyle\boldsymbol{g}_{t}\sim
N\left(\boldsymbol{0},(p_{w}(1-p_{w}))^{-1}\left(\boldsymbol{S}_{(t,t)}-\boldsymbol{S}_{(\overline{t-1},t)}^{\prime}\boldsymbol{S}_{(\overline{t-1},\overline{t-1})}^{(-1)}\boldsymbol{S}_{(\overline{t-1},t)}\right)\right),\quad
t>1.$

Since uncorrelation implies independence for multivariate normal
distributions, we know that $\boldsymbol{g}_{t}$, $t\geq 1$, are mutually
independent. Moreover, the sample covariance of $\boldsymbol{e}^{(t)}$ denoted
by $\boldsymbol{E}_{(t,t)}$ satisfies

$\displaystyle\boldsymbol{E}_{(1,1)}=\boldsymbol{S}_{(1,1)}$ (27)
$\displaystyle\boldsymbol{E}_{(t,t)}=\boldsymbol{S}_{(t,t)}-\boldsymbol{S}_{(\overline{t-1},t)}^{\prime}\boldsymbol{S}_{(\overline{t-1},\overline{t-1})}^{(-1)}\boldsymbol{S}_{(\overline{t-1},t)},\quad
t>1$

It follows that the balance scores defined by (19) satisfy

$h_{t}\dot{\ \sim\
}\boldsymbol{g}_{t}\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\boldsymbol{g}^{(t)^{\prime}}{\
\sim\ }\chi^{2}_{k_{t}},\quad t=1,2,\ldots,T,$

where each $\chi^{2}_{k_{t}}$ is a chi-square random variable of $k_{t}$
degrees of freedom. Hence, the finite population asymptotics in Example 3 is
justified.

We then consider finite population asymptotics under stratified randomization.
Following wang2021rerandomization, a set of regularity assumptions for
stratified randomized experiments is introduced as follows.

######  Assumption B.2.

For $j=1,2,\ldots,J$, as $n_{j}\rightarrow\infty$, (i) the stratum-specific
propensity score $p_{j}$ has a limit in $(0,1)$; (ii) the stratum-specific
sample covariance, $\boldsymbol{S}_{j}^{(r)}$, has a limiting value that is
non-singular; (iii)
$\max_{\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j)}\left\|\boldsymbol{x}^{(r)}_{i}-\overline{\boldsymbol{x}}^{(r)}_{j}\right\|_{2}/n\rightarrow
0$, where $\overline{\boldsymbol{x}}^{(r)}_{j}$ is the sample mean vector of
covariates in stratum $j$.

Under Assumption B.2 and stratified randomization, apply Theorem 5 of
li2017general and we have
$\sqrt{n_{j}}\left(\overline{\boldsymbol{x}}_{j\text{1}}-\overline{\boldsymbol{x}}_{j\text{0}}\right)\dot{\
\sim\
}N\left(\boldsymbol{0},(p_{j}(1-p_{j}))^{-1}\boldsymbol{S}_{j}^{(r)}\right)$
as $n_{j}\rightarrow\infty$, $j=1,2,\ldots,J$. Under stratified randomization,
treatment assignments of $J$ strata are mutually independent, indicating the
same holds for $J$ balance scores defined by (21). Moreover, these balance
scores satisfy

$h_{j}=\sqrt{n_{j}}\left(\overline{\boldsymbol{x}}_{j\text{1}}^{(r)}-\overline{\boldsymbol{x}}_{j\text{0}}^{(r)}\right)\left((p_{j}(1-p_{j}))^{-1}\boldsymbol{S}_{j}^{(r)}\right)^{-1}\sqrt{n_{j}}\left(\overline{\boldsymbol{x}}_{j\text{1}}^{(r)}-\overline{\boldsymbol{x}}_{j\text{0}}^{(r)}\right)^{\prime}\dot{\
\sim\ }\chi^{2}_{j},\quad j=1,2,\ldots,J,$

where each $\chi^{2}_{j}$ is a chi-square random variable of $k_{r}$ degrees
of freedom. Hence, the finite population asymptotics in Example 4 is
justified.

##  Appendix C Choice of Weights

Appendix C focuses on choice of weights for the “unified” methods involved in
conditionally affinely invariant rerandomization. We first review the Neyman-
Rubin causal model (splawa1990application, rubin1974estimating) and then
introduce additional regularity assumptions necessary to our analysis.

###  C.1 Balancing on Tiers of Covariates

Suppose that the SUTVA (rubin1980randomization) holds. Let $Y_{i}(0)$ and
$Y_{i}(1)$ be potential outcomes of unit $i$ under control and treatment. A
common estimand in RCT is the average treatment effect
$\tau=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}(1)-Y_{i}(0))$. With realized treatment
assignment $\boldsymbol{W}$, let $Y_{i}=Y_{i}(1)W_{i}+Y_{i}(0)(1-W_{i})$ be
the observed outcome of unit $i$. Under complete randomization, $\tau$ can be
estimated by the difference-in-means estimator
$\widehat{\tau}=\frac{1}{n_{\text{1}}}\sum_{W_{i}=1}Y_{i}-\frac{1}{n_{\text{0}}}\sum_{W_{i}=0}Y_{i}$.

Suppose that the “unified” balance criterion in Example 3 is defined as

$\varphi^{*}(\boldsymbol{x},\boldsymbol{W})=\begin{cases}1,&\sum_{t=1}^{T}\pi_{t}h_{t}\leq\alpha,\\\
0,&\sum_{t=1}^{T}\pi_{t}h_{t}>\alpha,\end{cases}$ (28)

where each $h_{t}$ is given by (19). For fixed $\boldsymbol{x}$, let
$\mathcal{M}^{*}=\left\\{\boldsymbol{W}\in\mathcal{W}\mid\varphi^{*}\left(\boldsymbol{x},\boldsymbol{W}\right)=1\right\\}$
be set of the accepted assignments under rerandomization with (28). To keep a
minimal acceptance probability at least asymptotically, we first introduce a
requirement as follows.

######  Assumption C.1.

Fixing a positive acceptance probability $p_{\alpha}$, the threshold in (28)
satisfies
$\Pr\left(\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right)=p_{\alpha}$,
where $\chi_{k_{1}}^{2},\ldots,\chi_{k_{T}}^{2}$ are mutually independent chi-
square random variables, each of $k_{t}$ degrees of freedom.

Within treatment arm $w=0,1$, let $\overline{Y}(w)$ be the sample mean of
potential outcomes; let $S^{2}_{Y}(w)$ be the sample variance of potential
outcomes; let $\boldsymbol{S}_{Y,\boldsymbol{x}}(w)$ be the sample covariance
between potential outcomes and covariates; also, let $S^{2}_{\tau}$ be the
sample variance of $Y_{i}(1)-Y_{i}(0)$. We introduce additional regularity
assumptions originated from li2018asymptotic.

######  Assumption C.2.

As $n\rightarrow\infty$, for $w=0,1$, (i) $S^{2}_{Y}(w)$,
$\boldsymbol{S}_{Y,\boldsymbol{x}}(w)$ and $S^{2}_{\tau}$ have finite limiting
values; (ii) $\max_{1\leq i\leq N}|Y_{i}(w)-\overline{Y}(w)|/n\rightarrow 0$.

Under Assumptions B.1, C.2 and complete randomization, apply Theorem 4 of
li2017general and we have

$\sqrt{n}\left(\widehat{\tau}-\tau,\overline{\boldsymbol{x}}_{\text{1}}-\overline{\boldsymbol{x}}_{\text{0}}\right)\dot{\
\sim\
}N\left(\boldsymbol{0},\begin{pmatrix}\frac{S^{2}_{Y}(1)}{p_{w}}+\frac{S^{2}_{Y}(0)}{1-p_{w}}-S^{2}_{\tau},&\frac{\boldsymbol{S}_{Y,\boldsymbol{x}}(1)}{p_{w}}+\frac{\boldsymbol{S}_{Y,\boldsymbol{x}}(0)}{1-p_{w}}\\\
\frac{\boldsymbol{S}_{Y,\boldsymbol{x}}^{\prime}(1)}{p_{w}}+\frac{\boldsymbol{S}_{Y,\boldsymbol{x}}^{\prime}(0)}{1-p_{w}},&(p_{w}(1-p_{w}))^{-1}\boldsymbol{S}\\\
\end{pmatrix}\right).$

For $w=0,1$, denote by
$\boldsymbol{S}_{Y,\boldsymbol{e}}(w)=\boldsymbol{S}_{Y,\boldsymbol{x}}(w)\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)$.
It follows from (24) and (25) that

$\sqrt{n}\left(\widehat{\tau}-\tau,\overline{\boldsymbol{e}}_{\text{1}}-\overline{\boldsymbol{e}}_{\text{0}}\right)\dot{\
\sim\ }(g_{0},\boldsymbol{g})\sim
N\left(\boldsymbol{0},\begin{pmatrix}\frac{S^{2}_{Y}(1)}{p_{w}}+\frac{S^{2}_{Y}(0)}{1-p_{w}}-S^{2}_{\tau},&\frac{\boldsymbol{S}_{Y,\boldsymbol{e}}(1)}{p_{w}}+\frac{\boldsymbol{S}_{Y,\boldsymbol{e}}(0)}{1-p_{w}}\\\
\frac{\boldsymbol{S}_{Y,\boldsymbol{e}}^{\prime}(1)}{p_{w}}+\frac{\boldsymbol{S}_{Y,\boldsymbol{e}}^{\prime}(0)}{1-p_{w}},&\frac{\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)^{\prime}\boldsymbol{S}\left(\boldsymbol{I}-\widetilde{\boldsymbol{P}}\right)}{p_{w}(1-p_{w})}\\\
\end{pmatrix}\right).$

Moreover, under additional Assumptions C.1 and rerandomization with (28),
apply Corollary A1 of li2018asymptotic and we have

$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}^{*}\dot{\
\sim\ }g_{0}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha,\text{ for
}\chi_{k_{t}}^{2}=\boldsymbol{g}_{t}\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\boldsymbol{g}_{t}^{\prime},\
t=1,2,\ldots,T.$

We could project $g_{0}$ over
$\boldsymbol{g}=(\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{T})$ to obtain
$g_{0}=\sum_{t=1}^{T}\boldsymbol{g}_{t}\boldsymbol{\gamma}_{t}+\epsilon$,
where $\boldsymbol{\gamma}_{1},\ldots,\boldsymbol{\gamma}_{T}$ are projection
coefficients and $\epsilon$ is the normally distributed projection residual
independent of $\boldsymbol{g}$. For $t=1,2,\ldots,T$, let
$\boldsymbol{g}_{-t}$ contain all vectors of $\boldsymbol{g}$ except
$\boldsymbol{g}_{t}$. Because the asymptotic balance criterion is subspace-
conditionally affinely invariant on $\boldsymbol{g}_{t}$, we apply Lemma 3.0.1
to have
$\boldsymbol{g}_{t}\mid\boldsymbol{g}_{-t},\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha$
ellipsoidally distributed with the center $\boldsymbol{0}$ and the covariance
proportional to $\operatorname{cov}\left(\boldsymbol{g}_{t}\right)$. It
follows that

$\displaystyle\operatorname{var}\left(g_{0}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right)$
$\displaystyle=\sum_{t=1}^{T}\operatorname{var}\left(\boldsymbol{g}_{t}\boldsymbol{\gamma}_{t}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right)+\operatorname{var}(\epsilon)$
$\displaystyle=\sum_{t=1}^{T}\boldsymbol{\gamma}_{t}^{\prime}\operatorname{cov}\left(\boldsymbol{g}_{t}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right)\boldsymbol{\gamma}_{t}+\operatorname{var}(\epsilon)$
$\displaystyle=\sum_{t=1}^{T}\nu_{t}\boldsymbol{\gamma}_{t}^{\prime}\operatorname{cov}\left(\boldsymbol{g}_{t}\right)\boldsymbol{\gamma}_{t}+\operatorname{var}(\epsilon)$
$\displaystyle=\operatorname{var}\left(g_{0}\right)\sum_{t=1}^{T}\nu_{t}R_{t}^{2}+\operatorname{var}\left(g_{0}\right)\left(1-\sum_{t=1}^{T}R_{t}^{2}\right),$

where
$R_{t}^{2}=\operatorname{cov}\left(g_{0},\boldsymbol{g}_{t}\right)\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\operatorname{cov}\left(g_{0},\boldsymbol{g}_{t}\right)^{\prime}/\operatorname{var}\left(g_{0}\right)$,
and

$\displaystyle\nu_{t}$
$\displaystyle=\frac{1}{k_{t}}\operatorname{trace}\left(\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\operatorname{cov}\left(\boldsymbol{g}_{t}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right)\right)$
$\displaystyle=\frac{1}{k_{t}}\mathbb{E}\left[\operatorname{trace}\left(\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\boldsymbol{g}_{t}^{\prime}\boldsymbol{g}_{t}\right)\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right]$
$\displaystyle=\frac{1}{k_{t}}\mathbb{E}\left[\boldsymbol{g}_{t}\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\boldsymbol{g}_{t}^{\prime}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right]$
$\displaystyle=\frac{1}{k_{t}}\mathbb{E}\left[\chi_{k_{t}}^{2}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right].$

Therefore, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}^{*}$
is the limit of

$\operatorname{var}\left(g_{0}\right)\sum_{t=1}^{T}\frac{R_{t}^{2}}{k_{t}}\mathbb{E}\left[\chi^{2}_{k_{t}}\mid\sum_{t=1}^{T}\pi_{t}\chi_{k_{t}}^{2}\leq\alpha\right]+\operatorname{var}\left(g_{0}\right)\left(1-\sum_{t=1}^{T}R_{t}^{2}\right).$
(29)

On the other hand, li2018asymptotic study the rerandomization method wtih
$\varphi_{1:T}$ given by (20). For fixed $\boldsymbol{x}$, define the set of
accepted assignments as
$\mathcal{M}_{1:T}=\left\\{\boldsymbol{W}\in\mathcal{W}\mid\varphi_{1:T}\left(\boldsymbol{x},\boldsymbol{W}\right)=1\right\\}$.
According to Corollary 4 of li2018asymptotic, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}_{1:T}$
is the limit of

$\operatorname{var}\left(g_{0}\right)\left(1-\sum_{t=1}^{T}(1-v_{k_{t},\alpha_{t}})R_{t}^{2}\right),$
(30)

where

$v_{k_{t},\alpha_{t}}=\frac{\Pr(\chi_{k_{t}+2}^{2}\leq\alpha_{t})}{\Pr(\chi_{k_{t}}^{2}\leq\alpha_{t})}=\frac{\int_{0}^{\alpha_{t}}\frac{y^{k_{t}/2+1}e^{-y}}{2^{k_{t}/2+1}\Gamma(k_{t}/2+1)}dy}{\int_{0}^{\alpha_{t}}\frac{y^{k_{t}/2}e^{-y}}{2^{k_{t}/2}\Gamma(k_{t}/2)}dy}=\frac{1}{k_{t}}\mathbb{E}\left[\chi^{2}_{k_{t}}\mid\chi_{k_{t}}^{2}\leq{}\alpha_{t}\right].$
(31)

Substituting (31) into (30), the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}_{1:T}$
is the limit of

$\operatorname{var}\left(g_{0}\right)\sum_{t=1}^{T}\frac{R_{t}^{2}}{k_{t}}\mathbb{E}\left[\chi^{2}_{k_{t}}\mid\chi_{k_{t}}^{2}\leq{}\alpha_{t}\right]+\operatorname{var}\left(g_{0}\right)\left(1-\sum_{t=1}^{T}R_{t}^{2}\right).$
(32)

Comparing (32) with (29), if $\varphi^{*}$ asymptotically dominates
$\varphi_{1:T}$, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}^{*}$
is less than that of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}_{1:T}$,
provided that each limit of $\operatorname{var}\left(g_{0}\right)R_{t}^{2}$ is
positive.

Suppose that we choose $\pi_{t}^{*}=R_{t}^{2}/k_{t}$ ($t=1,2,\ldots,T$) in
(28) and set the threshold $\alpha$ according to Assumption C.1. Applying (i)
of Theorem 5.1 with the $m=1$ case, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}-\tau\right)\mid\boldsymbol{W}\in\mathcal{M}^{*}$
is minimized among all possible choices of $\\{\pi_{t}\\}_{t=1}^{T}$. We next
give a heuristic interpretation of this set of weights.

Let $Y_{i}(p_{w})=(1-p_{w})Y_{i}(1)+p_{w}Y_{i}(0)$ be the “center” potential
outcome of unit $i$ ($i=1,2,\ldots,n$). It’s direct to check that

$\widehat{\tau}-\tau=\overline{Y}_{\text{1}}(p_{w})-\overline{Y}_{\text{0}}(p_{w})$
(33)

where the right hand side is the difference in means of $Y_{i}(p_{w})$ between
treatment and control groups. By construction,
$\operatorname{var}\left(g_{0}\right)=p_{w}^{-1}S^{2}_{Y}(1)+(1-p_{w})^{-1}S^{2}_{Y}(0)-S^{2}_{\tau}$,
which is the variance of $\sqrt{n}(\widehat{\tau}-\tau)$ under complete
randomization. Combined with (33), we know that
$\operatorname{var}\left(g_{0}\right)=(p_{w}(1-p_{w}))^{-1}S_{Y}^{2}(p_{w})$,
where $S_{Y}^{2}(p_{w})$ is the sample variance of “center” potential
outcomes. Using similar arguments, we can also verify that, for
$t=1,2,\ldots,T$,
$\operatorname{cov}\left(g_{0},\boldsymbol{g}_{t}\right)=(p_{w}(1-p_{w}))^{-1}\boldsymbol{S}_{Y,\boldsymbol{e}_{t}}(p_{w})$,
where $\boldsymbol{S}_{Y,\boldsymbol{e}_{t}}(p_{w})$ is the sample covariance
between the “center” potential outcome and covariates in tier $t$ after
preprocessed with (18). It follows from (26), (27) and the definition of
$R_{t}^{2}$ that

$\displaystyle R_{t}^{2}$
$\displaystyle=\operatorname{cov}\left(g_{0},\boldsymbol{g}_{t}\right)\operatorname{cov}\left(\boldsymbol{g}_{t}\right)^{-1}\operatorname{cov}\left(g_{0},\boldsymbol{g}_{t}\right)^{\prime}/\operatorname{var}\left(g_{0}\right)$
$\displaystyle=\boldsymbol{S}_{Y,\boldsymbol{e}_{t}}(p_{w})\boldsymbol{E}_{(t,t)}^{-1}\boldsymbol{S}_{Y,\boldsymbol{e}_{t}}(p_{w})^{\prime}/S_{Y}^{2}(p_{w}),\quad
t=1,2,\ldots,T.$

This implies each $\pi_{t}^{*}$ numerically equals the average squared
multiple correlation coefficient between the “center” potential outcome and
covariates in tier $t$ after preprocessed with (18). As this set of optimal
weights contains practically unknown quantities, we could just select
$\\{\pi_{t}\\}_{t=1}^{T}$ to be positive and in decreasing order from tier 1
to tier $T$, reflecting our prior knowledge of decreasing importance in tiers
of covariates. According to (i) of Theorem 5.1, it also ensures admissibility
of $\varphi^{*}$ in terms of average balance scores of accepted assignments.

###  C.2 Stratified Rerandomization

In stratified randomized experiments, there are various kinds of estimands
combined from $J$ stratum-specific average treatment effects
$\tau_{j}=\sum_{\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j)}(Y_{i}(1)-Y_{i}(0))$,
$j=1,2,\ldots,J$ (see e.g. imbens2015causal Ch. 8). Therefore, we consider
$\tau_{\boldsymbol{\lambda}}=\sum_{j=1}^{J}\lambda_{j}\tau_{j}$ defined by a
general $J$-component vector $\boldsymbol{\lambda}$ and use
$\widehat{\tau}_{\boldsymbol{\lambda}}=\sum_{j=1}^{J}\lambda_{j}\widehat{\tau}_{j}$
to estimate $\tau_{\boldsymbol{\lambda}}$, where

$\widehat{\tau}_{j}=\frac{1}{n_{j}p_{j}}\sum_{\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j),W_{i}=1}Y_{i}-\frac{1}{n_{j}(1-p_{j})}\sum_{\boldsymbol{x}^{(s)}_{i}=\boldsymbol{c}(j),W_{i}=0}Y_{i},\quad
j=1,2,\ldots,J.$

Suppose that the “unified” balance criterion in Example 4 is defined as

$\varphi^{*}(\boldsymbol{x}^{(r)},\boldsymbol{W})=\begin{cases}1,&\sum_{j=1}^{J}\pi_{j}h_{j}\leq\alpha,\\\
0,&\sum_{j=1}^{J}\pi_{j}h_{j}>\alpha,\end{cases}$ (34)

where each $h_{j}$ is given by (21). For fixed $\boldsymbol{x}$, let

$\mathcal{S}^{*}=\left\\{\boldsymbol{W}\in\mathcal{W}\mid\varphi_{S}\left(\boldsymbol{x}^{(s)},\boldsymbol{W}\right)\cdot\varphi^{*}\left(\boldsymbol{x}^{(r)},\boldsymbol{W}\right)=1\right\\}$

be set of the accepted assignments under stratified rerandomization with (34).
To keep a minimal acceptance probability at least asymptotically, we first
introduce a requirement as follows.

######  Assumption C.3.

Fixing a positive acceptance probability $p_{\alpha}$, the threshold in (34)
satisfies
$\Pr\left(\sum_{j=1}^{J}\pi_{j}\chi_{j}^{2}\leq\alpha\right)=p_{\alpha}$,
where $\chi_{1}^{2},\ldots,\chi_{J}^{2}$ are mutually independent chi-square
random variables, each of $k_{r}$ degrees of freedom.

Within stratum $j=1,2,\ldots,J$ and treatment arm $w=0,1$, let
$\overline{Y}_{j}(w)$ be the sample mean of potential outcomes; let
$S^{2}_{Y_{j}}(w)$ be the sample variance of potential outcomes; let
$\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(w)$ be the sample covariance
between potential outcomes and continuous covariates; also, let
$S^{2}_{\tau_{j}}$ be the sample variance of $Y_{i}(1)-Y_{i}(0)$ given
$\boldsymbol{x}_{i}^{(s)}=\boldsymbol{c}(j)$. We introduce additional
regularity assumptions originated from wang2021rerandomization.

######  Assumption C.4.

For $j=1,2,\ldots,J$ and $w=0,1$, as $n\rightarrow\infty$, (i) $n_{j}/n$,
$S^{2}_{Y_{j}}(w)$, $\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(w)$ and
$S^{2}_{\tau_{j}}$ have finite limiting values; (ii)
$\max_{\boldsymbol{x}_{i}^{(s)}=\boldsymbol{c}(j)}|Y_{i}(w)-\overline{Y}_{j}(w)|/n_{j}\rightarrow
0$.

Under Assumptions B.2, C.4 and stratified randomization, apply Theorem 4 of
li2017general and we have that, for $j=1,2,\ldots,J$,

$\sqrt{n_{j}}\left(\widehat{\tau}_{j}-\tau_{j},\overline{\boldsymbol{x}}_{j\text{1}}-\overline{\boldsymbol{x}}_{j\text{0}}\right)\dot{\
\sim\ }(g_{0j},\boldsymbol{g}_{j}),$

where

$(g_{0j},\boldsymbol{g}_{j})\sim
N\left(\boldsymbol{0},\begin{pmatrix}\frac{S^{2}_{Y_{j}}(1)}{p_{j}}+\frac{S^{2}_{Y_{j}}(0)}{1-p_{j}}-S^{2}_{\tau_{j}},&\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(1)}{p_{j}}+\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(0)}{1-p_{j}}\\\
\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)\prime}(1)}{p_{j}}+\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)\prime}(0)}{1-p_{j}},&(p_{j}(1-p_{j}))^{-1}\boldsymbol{S}_{j}^{(r)}\\\
\end{pmatrix}\right).$

Because treatment assignments of $J$ strata are mutually independent, we know
that

$\left(\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right),\sqrt{n_{1}}\left(\overline{\boldsymbol{x}}_{1\text{1}}-\overline{\boldsymbol{x}}_{1\text{0}}\right),\ldots,\sqrt{n_{J}}\left(\overline{\boldsymbol{x}}_{J\text{1}}-\overline{\boldsymbol{x}}_{J\text{0}}\right)\right)\dot{\
\sim\ }\left(g_{0},\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{J}\right),$

where $\left(g_{0},\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{J}\right)$ is
multivariate normally distributed with

$\displaystyle\operatorname{var}\left(g_{0}\right)=\sum_{j=1}^{J}\zeta_{j}\operatorname{var}\left(g_{0j}\right)=\sum_{j=1}^{J}\zeta_{j}\left(\frac{S^{2}_{Y_{j}}(1)}{p_{j}}+\frac{S^{2}_{Y_{j}}(0)}{1-p_{j}}-S^{2}_{\tau_{j}}\right)$
$\displaystyle\operatorname{cov}\left(g_{0},\boldsymbol{g}_{j}\right)=\sqrt{\zeta_{j}}\operatorname{cov}\left(g_{0j},\boldsymbol{g}_{j}\right)=\sqrt{\zeta_{j}}\left(\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(1)}{p_{j}}+\frac{\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(0)}{1-p_{j}}\right)$
$\displaystyle\operatorname{cov}\left(\boldsymbol{g}_{j},\boldsymbol{g}_{j^{\prime}}\right)=\left\\{\begin{array}[]{l}\operatorname{cov}\left(\boldsymbol{g}_{j}\right)=(p_{j}(1-p_{j}))^{-1}\boldsymbol{S}_{j}^{(r)},\quad
j=j^{\prime}\\\ \boldsymbol{0},\quad j\neq j^{\prime}\end{array}\right.$

for $\zeta_{j}=\lambda_{j}^{2}n/n_{j}$, $j,j^{\prime}=1,2,\ldots,J$. Moreover,
under additional Assumption C.3, we can repeat the proof of Proposition A1 of
li2018asymptotic to obtain that

$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}^{*}\dot{\
\sim\ }g_{0}\mid\sum_{j=1}^{J}\pi_{j}\chi_{j}^{2}\leq\alpha,\text{ for
}\chi_{j}^{2}=\boldsymbol{g}_{j}\operatorname{cov}\left(\boldsymbol{g}_{j}\right)^{-1}\boldsymbol{g}_{j}^{\prime},\
j=1,2,\ldots,J.$

Similar to the last example, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}^{*}$
is the limit of

$\operatorname{var}\left(g_{0}\right)\sum_{j=1}^{J}\frac{R_{j}^{2}}{k_{j}}\mathbb{E}\left[\chi^{2}_{j}\mid\sum_{j=1}^{J}\pi_{j}\chi_{j}^{2}\leq\alpha\right]+\operatorname{var}\left(g_{0}\right)\left(1-\sum_{j=1}^{J}R_{j}^{2}\right),$
(35)

where
$R_{j}^{2}=\operatorname{cov}\left(g_{0},\boldsymbol{g}_{j}\right)\operatorname{cov}\left(\boldsymbol{g}_{j}\right)^{-1}\operatorname{cov}\left(g_{0},\boldsymbol{g}_{j}\right)^{\prime}/\operatorname{var}\left(g_{0}\right)$
for $j=1,2,\ldots,J$.

On the other hand, wang2021rerandomization’s balance criterion for SRRsM,
denoted by $\varphi_{1:J}$, is intersected by

$\varphi_{j}(\boldsymbol{x}^{(r)},\boldsymbol{W})=\begin{cases}1,&h_{j}\leq\alpha_{j},\\\
0,&h_{j}>\alpha_{j},\end{cases}\quad\alpha_{j}\in\mathbb{R}_{+},\
j=1,2,\ldots,J,$ (36)

where each $h_{j}$ is given in (21). For fixed $\boldsymbol{x}$, let

$\mathcal{S}_{1:J}=\left\\{\boldsymbol{W}\in\mathcal{W}\mid\varphi_{S}\left(\boldsymbol{x}^{(s)},\boldsymbol{W}\right)\cdot\prod_{j=1}^{J}\varphi_{j}\left(\boldsymbol{x}^{(r)},\boldsymbol{W}\right)=1\right\\}$

be set of the accepted assignments under stratified rerandomization with (36).
Under SRRsM, treatment assignments of $J$ strata are mutually independent
given $\varphi_{1:J}$ and covariates, implying that the same holds for $J$
asymptotic distributions of
$\sqrt{n_{j}}\left(\widehat{\tau}_{j}-\tau_{j}\right)\mid\boldsymbol{W}\in\mathcal{S}_{1:J}$.
Therefore, under Assumptions B.2 and C.4, apply Corollary 2 of
li2018asymptotic and we obtain the asymptotic variance of
$\sqrt{n_{j}}\left(\widehat{\tau}_{j}-\tau_{j}\right)\mid\boldsymbol{W}\in\mathcal{S}_{1:J}$
as the limit of

$\operatorname{var}\left(g_{0j}\right)\left(1-\left(1-v_{k_{r},\alpha_{j}}\right)R^{2}_{0j}\right),$

where
$R_{0j}^{2}=\operatorname{cov}\left(g_{0j},\boldsymbol{g}_{j}\right)\operatorname{cov}\left(\boldsymbol{g}_{j}\right)^{-1}\operatorname{cov}\left(g_{0j},\boldsymbol{g}_{j}\right)^{\prime}/\operatorname{var}\left(g_{0j}\right)=\zeta_{j}^{-1}\operatorname{var}\left(g_{0}\right)/\operatorname{var}\left(g_{0j}\right)R_{j}^{2}$
for $j=1,2,\ldots,J$. Again, by independent treatment assignments of $J$
strata under stratified randomization and rerandomization with
$\varphi_{1:J}$, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}_{1:J}$
has a limit

$\displaystyle\sum_{j=1}^{J}\zeta_{j}\operatorname{var}\left(g_{0j}\right)\left(1-\left(1-v_{k_{r},\alpha_{j}}\right)R^{2}_{0j}\right)$
(37) $\displaystyle=$
$\displaystyle\operatorname{var}\left(g_{0}\right)-\sum_{j=1}^{J}\operatorname{var}\left(g_{0}\right)\left(1-v_{k_{r},\alpha_{j}}\right)R^{2}_{j}$
$\displaystyle=$
$\displaystyle\operatorname{var}\left(g_{0}\right)\sum_{j=1}^{J}\frac{R_{j}^{2}}{k_{r}}\mathbb{E}\left[\chi_{j}^{2}\mid\chi_{j}^{2}\leq\alpha_{j}\right]+\operatorname{var}\left(g_{0}\right)\left(1-\sum_{j=1}^{J}R_{j}^{2}\right)$

Comparing (37) with (35), if $\varphi^{*}$ asymptotically dominates
$\varphi_{1:J}$, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}^{*}$
is less than that of
$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}_{1:J}$,
provided that each limit of $\operatorname{var}\left(g_{0}\right)R_{j}^{2}$ is
positive.

Suppose that we choose $\pi_{j}^{*}=R_{j}^{2}/k_{r}$ ($j=1,2,\ldots,J$) in
(34) and set the threshold $\alpha$ according to Assumption C.3. Applying (i)
of Theorem 5.1 with the $m=1$ case, the asymptotic variance of
$\sqrt{n}\left(\widehat{\tau}_{\boldsymbol{\lambda}}-\tau_{\boldsymbol{\lambda}}\right)\mid\boldsymbol{W}\in\mathcal{S}^{*}$
is minimized among all possible choices of $\\{\pi_{j}\\}_{j=1}^{J}$. By
definition, for $j=1,2,\ldots,J$,

$\displaystyle\pi_{j}^{*}$
$\displaystyle=\zeta_{j}\operatorname{var}\left(g_{0j}\right)/\operatorname{var}\left(g_{0}\right)R_{0j}^{2}/k_{r}$
(38)
$\displaystyle\propto\zeta_{j}\operatorname{cov}\left(g_{0j},\boldsymbol{g}_{j}\right)\operatorname{cov}\left(\boldsymbol{g}_{j}\right)^{-1}\operatorname{cov}\left(g_{0j},\boldsymbol{g}_{j}\right)^{\prime}$
$\displaystyle\propto\zeta_{j}(p_{j}(1-p_{j}))^{-1}\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(p_{j})\boldsymbol{S}_{j}^{(r)-1}\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)\prime}(p_{j}),$

where
$\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(p_{j})=(1-p_{j})\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(1)+p_{j}\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(0)$
is the sample covariance between the “center” potential outcome and continuous
covariates of stratum $j$. Since
$\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)}(p_{j})\boldsymbol{S}_{j}^{(r)-1}\boldsymbol{S}_{Y_{j},\boldsymbol{x}_{j}}^{(r)\prime}(p_{j})$
equals the practically unknown variance of the “center” potential outcome
projected onto continuous covariates of stratum $j$, we could replace it in
(38) with the first eigenvalue of $\boldsymbol{S}_{j}^{(r)}$, i.e. the largest
sample variance among all linear combinations of continuous covariates of
stratum $j$. A common choice of $\boldsymbol{\lambda}$ would be
$\lambda_{j}=n_{j}/n$ ($j=1,2,\ldots,J$). According to (i) of Theorem 5.1,
with $\\{\pi_{j}\\}_{j=1}^{J}$ specified positive in the aforementioned way,
it also ensures admissibility of $\varphi^{*}$ in terms of average balance
scores of accepted assignments.