# Optimal regimes with limited resources

Aaron L. Sarvet  Julien D. Laurendeau  Mats J. Stensrud  Department of
Mathematics, Ecole Polytechnique Fédérale de Lausanne, Switzerland

(May 2022)

###### Abstract

Policy-makers are often faced with the task of distributing a limited supply
of resources. To support decision-making in these settings, statisticians are
confronted with two challenges: estimands are defined by allocation strategies
that are functions of features of all individuals in a cluster; and relatedly
the observed data are neither independent nor identically distributed when
individuals compete for resources. Existing statistical approaches are
inadequate because they ignore at least one of these core features. As a
solution, we develop theory for a general policy class of dynamic regimes for
clustered data, covering existing results in classical and interference
settings as special cases. We cover policy-relevant estimands and articulate
realistic conditions compatible with resource-limited observed data. We derive
identification and inference results for settings with a finite number of
individuals in a cluster, where the observed dataset is viewed as a single
draw from a super-population of clusters. We also consider asymptotic
estimands when the number of individuals in a cluster is allowed to grow;
under explicit conditions, we recover previous results, thereby clarifying
when the use of existing methods is permitted. Our general results lay the
foundation for future research on dynamic regimes for clustered data,
including the longitudinal cluster setting.

MainReferences SMAppendix References

##  1 Introduction

Consider the canonical average treatment effect (ATE)
$\Psi\coloneqq\mathbb{E}[Y_{i}^{a=1}-Y_{i}^{a=0}]$, which is the prevailing
target of inference in applied and methodological studies. The ATE is the
expected difference between outcomes for a single individual $i$ under an
assignment to treatment ($a=1$) and control ($a=0$). Similarly, many other
common estimands are defined by a single individual’s outcome under different
treatments. For example, existing definitions of the optimal dynamic regime
$g^{\mathbf{opt}}$ are based on a function that maps an individual’s pre-
treatment features to a single treatment value, see e.g.
murphy2003optimal,robins2004optimal. In practice, the values of $\Psi$ and
$g^{\mathbf{opt}}$ are used to guide policies not just for one individual, but
for whole populations. The tacit premise is that each individual is assumed to
be exchangeable with any other, and that their potential outcomes are drawn
from a common distribution characterized by $\Psi$ and $g^{\mathbf{opt}}$.

However, in many real-world settings, this premise is immediately violated;
policy makers must use non-iid data, e.g., because of causal connections
between individuals in a cluster. Relatedly, they have interest in
interventions that depend on the features of all individuals in a cluster. For
example, many real-world policies involve regimes for prioritizing individuals
and are familiarly called waiting-list rules, triaging rules or prioritization
mechanisms, among other terms. These rules are frequently used in national
health care systems across the world lakshmi2013application,
januleviciute2013impact, mullen2003prioritising, hadorn1997new,
feek2000rationing. Today such rules are used for the provision of primary care
breton2017assessing, non-emergency surgeries australian2015national,
powers2023managing, curtis2010waiting, de2007western, valente2021new,
solans2013developing, organ transplantation OPTNSRTR20172019, palliative care
russell2020triaging, and diagnostic testing with X-rays, MRIs, or endoscopies
luigi2013oecd, among other scarce resources. To be explicit, consider the
widely cited New York Ventilator Guidelines for ventilator allocation, which
were specified by nyc2015ventilator in preparation for a possible influenza
pandemic. These guidelines describe regimes for allocating ventilators to a
cluster of individuals according to their Sequential Organ Failure Assessment
(SOFA) scores.

Existing results on dynamic regimes orellana2010dynamic, van2005history,
murphy2001cpprg ostensibly provide a framework to account for resource
limitations in causal inference. Recently, this framework has been used to
study optimal resource-constrained regimes in statistics luedtke2016optimal,
qiu2020optimal, luedtke2022individualized, canigliaestimating,
zhao2023positivity, xu2023optimal, imai2023experimental, econometrics
bhattacharya2012inferring,kitagawa2018should, mbakop2021model,
athey2021policy, sun2021treatment, pellatt2022pac, adusumilli2019dynamically,
liu2022policy, and machine learning, e.g., zhou2024optimal,
luedtke2019asymptotically, badanidiyuru2018bandits, tran2012knapsack. For
example, luedtke2016optimal and qiu2020optimal described semiparametric
efficient estimators for the optimal individualized dynamic regime (hereby,
IR) subject to fixed constraints. Later, athey2021policy described an
efficient estimation strategy using regret bounds manski2004statistical.
sun2021treatment generalize the notion of a resource constraint to one of a
fixed budget, to accomodate settings where individual-specific costs of
treatment randomly vary. canigliaestimating parameterized a marginal
structural model robins2000marginal and subsequently optimized the parameters
defining an IR under the resource constraint.

We argue that these IR proposals are ill-posed for the typical resource-
limited setting because they are restricted to individualized policies and iid
data. There is indeed a growing literature that goes beyond such a restricted
framework, which allows for interference hudgens2008toward,
sobel2006randomized or more generally causally connected individuals
van2014causal or social networks ogburn2020causal, manski2013identification.
One active strand of this literature concerns inference on classical estimands
defined by individualized policies (e.g., savje2021average). A second active
strand concerns inference on estimands commonly called “spillover” or indirect
effects hu2022average, forastiere2021identification, vanderweele2011effect.
Arguably these strands are related: either dependencies between individuals
are regarded explicitly as a nuisance that complicates inference, or interest
lies in quantifying the magnitude of this nuisance. In contrast, very few
works explicitly consider dynamic regimes in limited resource settings. The
literature on design-based inference is perhaps one exception; their
parameters are defined by regimes corresponding to treatment allocation plans
of experimental designs, which subsume, for example, Bernoulli designs,
complete randomization designs, and paired-randomization designs, see
savje2021average for a review. However these works have been formulated
exclusively for experimental data, and their experimental designs only
correspond to static stochastic regimes. In the context of observational data,
van2014causal and ogburn2020causal consider dynamic regimes in a network
setting. Similar to the exposure mappings of aronow2017estimating and
savje2024causal in experimental settings, van2014causal and ogburn2020causal’s
regimes are individualized in the sense that each individual’s treatment is
restricted to be a fixed function of at most a randomizer, their own
covariates, and the covariates of a bounded number of network ties.
Collectively, these works are not directly relevant to our setting. First, we
will consider observational data where the design, i.e., assignment of
treatment, is unknown; thus the proposed methodologies of the designed-based
literature are infeasible and their estimands are of questionable practical
relevance. Second, we will evaluate regimes that are dynamic with respect to
covariates of all individuals in a cluster; returning to the ventilator
allocation example nyc2015ventilator, whether or not an individual receives a
ventilator unit in an ICU during a pandemic is not merely a function of that
individual’s own SOFA score, but of the SOFA scores of a cluster of patients
in, say, the hospital. Such regimes have, to our knowledge, not been
considered in the interference literature.

As a solution, we work at the intersection of these two fast-growing
literatures: to consider dynamic and optimal regimes in limited resource
settings, we ground our work in frameworks traditionally applied in
interference and network settings. Specifically, we articulate assumptions
that both encode characteristic features of the resource-limited setting and
permit articulation of estimands corresponding to real-world questions.
Insodoing, we elaborate a general policy class of dynamic regimes for
clustered data, covering special cases considered by the existing literature
on interference savje2021average, van2014causal, ogburn2020causal. We further
derive identification and estimation results for settings with either a finite
or a near-infinite number of individuals in a cluster. Our elaborated results
extend to longitudinal settings wherein limited resources frequently appear,
and thereby we set the stage for future research.

As an illustrative case study, we consider the estimand from qiu2020optimal
and luedtke2016optimal, who aimed to learn an optimal regime, dynamic with
respect to covariates $L$, subject to investigator-specified constraints on
the marginal probability of treatment. While their proposed estimand and model
violate characteristic features of the resource-limited setting, we locate
their statistical parameter as a large population limiting case of a
functional identifying a more appropriate causal estimand in an elaborated
model. Thereby, we justify the use of existing statistical procedures for a
more realistic setting and also clarify when the use of such procedures would
fail.

##  2 Dynamic regimes under resource limitations

###  2.1 Data structure

Consider an observed data setting where baseline clinical features $L_{i}$,
binary treatments $A_{i}$, and outcome $Y_{i}$ are observed for $i=1,\dots,n$
individuals. In an iid setting, each vector
$O_{i}\equiv\\{L_{i},A_{i},Y_{i}\\}$ is viewed as an independent draw from a
common law $P$ with support $\mathcal{L}\times\\{0,1\\}\times\mathbb{R}$.111We
consider $L_{i}$ discrete, in order to simplify notation, but the results can
be generalized to continuous covariates. We use superscripts to denote
counterfactual (potential outcome) variables under a treatment regime. For
example, $Y_{i}^{g}$ is the counterfactual outcome that would occur under
regime $g$ for individual $i$. We use plus symbols ($+$) to distinguish
natural values robins2004effects of counterfactual intervention variables
under a regime ($A_{i}^{g}$) from assigned values of such variables
($A_{i}^{g+}$). When we consider iid settings we drop individual indices.
Except when emphasis is needed, we omit subscripts that indicate when a
function depends on a particular law, say, $P$.

###  2.2 Conventional individualized regimes

An IR can be defined as a mapping ($g$) of an individual $i$’s characteristics
$l$ and a randomizer $\delta\in[0,1]$ to a single treatment level, $a$. We
refer to such regimes as individualized, as they are a function solely of a
single individual’s features. Suppose $O^{F}_{i}\equiv\\{O_{i}^{g}\mid
g\in\Pi\\}$ collects an individual $i$’s counterfactuals across all IRs $g$ in
an unconstrained policy class $\Pi$, and that $P^{F}$ denotes the joint
distribution over these variables. To reduce notation, we omit superscripts
$F$ distinguishing $P$ vs. $P^{F}$, except as needed.

A given IR results in only a proportion of a target population receiving
treatment, $\mathbb{E}[A^{g+}]\in[0,1]$. Thus, a policy maker can consider a
class of regimes defined to satisfy the constraint,
$\Pi(\kappa)\coloneqq\\{g\in\Pi\mid\mathbb{E}[A^{g+}]\leq\kappa\\}$, and find
the regime $g^{\mathbf{opt}}$ therein that optimizes some expected utility:
$g^{\mathbf{opt}}\coloneqq\underset{g\in\Pi(\kappa)}{\arg\max}\
\mathbb{E}[Y^{g}]$. As reviewed in qiu2020optimal, determination of
$g^{\mathbf{opt}}$ admits a closed form solution dantzig1957discrete up to an
equivalence class defined by the counterfactual propensities,
$q^{g}(l)\coloneqq\mathbb{E}[A^{g+}\mid L=l]$. Letting
$\Delta(l)\coloneqq\mathbb{E}[Y^{a=1}-Y^{a=0}\mid L=l]$, and
$\eta\coloneqq\max\Big{\\{}\inf\\{c\in\mathbb{R}\mid
P(\Delta(L)>c)\leq\kappa\\},0\Big{\\}}$, then $g^{\mathbf{opt}}$ is chosen as
a regime such that

$\displaystyle
q^{\mathbf{opt}}(l)=\begin{cases}\frac{\kappa-P(\Delta(L)>\eta)}{P(\Delta(L)=\eta)}&\text{
if }\Delta(l)=\eta\text{ and }\Delta(l)>0,\\\ I(\Delta(l)>\eta)&\text{
otherwise. }\end{cases}$

Heuristically, $g^{\mathbf{opt}}$ assigns treatment to all individuals who
would experience the greatest and strictly-positive conditional ATEs (CATEs)
and who collectively constitute a proportion of the population less than or
equal to the resource constraint $\kappa$. If this proportion is less than
$\kappa$, then treatment is randomly assigned to the remaining group with the
largest strictly-positive CATE so that $q^{\mathbf{opt}}$ exactly satisfies
the constraint in expectation.

###  2.3 Why iid data models are inadequate in settings with resource
limitations

The constrained optimization approach assumes iid observed data structures.
However, if existing data arise from a setting where resources are already
limited, as will often be the case, then iid models will immediately be
violated. Consider, for example, the causal directed acyclic graph (DAG) in
Figure 1. Throughout we will use such graphs to depict Finest Fully Randomized
Causally Interpreted Structured Tree Graph (FFRCISTG) models of robins1986new,
a nonparameteric structural equation model generalizing that of
pearl2009causality. The sub-DAG involving only the nodes
$O_{j}\equiv\\{L_{j},A_{j},Y_{j}\\}$ canonically represents causal relations
between an individual’s variables for a setting without unmeasured
confounding, which is often used to justify identification of causal
parameters based on graphical criteria. Although rarely done in practice, DAGs
can depict relations between variables in a sample, $\\{O_{1},\dots,O_{n}\\}$
ogburn2014causal. Under an iid observed data model, the sub-DAG involving only
the black arrows would be a valid causal graph, wherein the graphical motif
for individual $j$ is repeated for all $n$ individuals, and there are no
causal connections between individuals. However, if a resource limitation is
operative in the observed data, then this iid sub-DAG will not faithfully
represent the causal structure: if one individual uses or occupies a scarce
treatment, such as a ventilator unit in an ICU, then this treatment is not
available to an hitherto untreated individual. We can graphically depict these
causal connections with the additional red arrows . This causal structure
contradicts independence between an arbitrary pair of individuals
$\\{O_{i},O_{j}\\}$; minimally, there is a direct causal path between one
individual’s covariates and another individual’s outcome, mediated by the
resource limitation, for example, $L_{1}\rightarrow
A_{1}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{\rightarrow}}A_{j}\rightarrow
Y_{j}$. These causal connections are related to the concept of interference
between individuals, see for example sobel2006randomized, hudgens2008toward,
savje2021average, and we provide additional remarks in Section 3. Other inter-
individual causal connections will also commonly arise in resource-limited
settings; for example, a treatment provider will often consider the clinical
features of other individuals in, say, an emergency room or military field
hospital, when deciding whether to provide a scarce treatment unit. Such
causal connections are depicted by the additional orange arrows in Figure 1.

$A_{j}$$A_{1}$$A_{n}$$L_{j}$$L_{1}$$L_{n}$$Y_{j}$$Y_{1}$$Y_{n}$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$
Figure 1: DAG for a cluster of size $n$.

###  2.4 A new policy-class for resource-limited settings

We use bold font to denote a set of variables from $i=1,..,n$, for example,
$\mathbf{O}_{n}\equiv\\{O_{1},\dots,O_{n}\\}$ denotes the factual data for all
$n$ individuals in a cluster comprising an observed data sample, and let
$\mathbb{P}_{n}$ denote its distribution. Estimands defined by
$g\in\Pi(\kappa)$ are individualized in the sense that a treatment provider
implementing this regime for a particular individual $i$ need only consider
the clinical features of that particular individual, $L_{i}$. Instead, we
consider regimes $G_{n}$ that are functions mapping all $n$ individuals’
characteristics $\mathbf{L}_{n}$ and a randomizer $\delta\in[0,1]$ to a vector
of $n$ treatments, $\mathbf{A}_{n}\in\\{0,1\\}^{n}$. These randomizers can,
for example, constitute lottery results or random-number-generators used to
break ties under equipoise. We refer to a regime $G_{n}$ as a cluster dynamic
regime (hereby CR), as they are a function of all individuals’ features in the
cluster, and we let $\Pi_{n}$ be the set of all possible CRs, analogous to
$\Pi$ for IRs. We let $\mathbf{O}_{n}^{F}\coloneqq\\{\mathbf{O}^{G_{n}}\mid
G_{n}\in\Pi_{n}\\}$ collect all $n$ individuals’ counterfactuals under all
possible CRs $G_{n}$, and we let $\mathbb{P}_{n}^{F}$ denote its joint
distribution.

In contrast to $\Pi(\kappa)$, we specifically consider the policy class

$\Pi^{F}_{n}(\kappa_{n})\equiv\Big{\\{}G_{n}:\lVert
G_{n}(\mathbf{l}_{n},\delta)\rVert\leq\kappa_{n}\text{ for all}\
\mathbf{l}_{n},\delta\Big{\\}},$

and its subclass

$\Pi_{n}(\kappa_{n})\equiv\Big{\\{}G_{n}:\lVert
G_{n}(\mathbf{l}_{n},\delta)\rVert=\kappa_{n}\text{ for all}\
\mathbf{l}_{n},\delta\Big{\\}},$

where $\kappa_{n}\in\\{0,1,\dots,n\\}$ represents the number of treatment
units available. In words, $\Pi^{F}_{n}(\kappa_{n})$ is the class of CRs where
no more than $\kappa_{n}$ individuals receive treatment and
$\Pi_{n}(\kappa_{n})$ is the sub-class where exactly $\kappa_{n}$ receive
treatment. To simplify notation, we focus on the class $\Pi_{n}(\kappa_{n})$
and defer relaxations and extensions to Appendix A.

A feature of $\Pi^{F}_{n}(\kappa_{n})$ and $\Pi_{n}(\kappa_{n})$ is that they
are not dependent on the law $\mathbb{P}_{n}^{F}$. In contrast,
$\Pi_{P^{F}}(\kappa)\coloneqq\Pi(\kappa)$ crucially depends on the law
$P^{F}$; the class of IRs guaranteed to satisfy a constraint $\kappa$ across
all laws contains only the regimes that randomly assign treatment with some
probability less than or equal to $\kappa$, including the trivial
deterministic regime that assigns $a=0$ to all individuals. In Appendix D, we
elaborate on this law dependence, and its corresponding consequences for
robustness and transportability.

####  2.4.1 Formulating cluster regimes based on rankings

In practice, regimes for limited resource settings are often formulated as
rules for assigning a rank ordering to a finite cluster of individuals, and
this rank ordering determining the assignment of the finite number of
treatment units. To accord with this formulation, we define an unmeasured
variable $\mathbf{R}_{n}$ that denotes this rank ordering. We then consider a
elaborated CR $G^{*}_{n}\equiv\\{G^{*}_{n,\mathbf{R}},G^{*}_{n,\mathbf{A}}\\}$
that involves both an intervention on $\mathbf{R}_{n}$ and $\mathbf{A}_{n}$
via $G^{*}_{n,\mathbf{R}}$, and $G^{*}_{n,\mathbf{A}}$, respectively. The
assigned ranks $\mathbf{R}_{n}^{G_{n}^{*}+}$ determine the order in which
individuals are assigned treatment, $\mathbf{A}_{n}^{G_{n}^{*}+}$, which
proceeds until all $\kappa_{n}$ treatment units are allocated. A regime that
assigned treatment whenever a resource was available,
$A_{i}^{G^{*}_{n}+}=I(\kappa_{n}\geq R_{i}^{G_{n}^{*}+})$, would guarantee
exact utilization of the available treatments. Thus each
$G_{n}\in\Pi_{n}(\kappa_{n})$ may be characterized by a rank-function (or
prioritization rule) $G^{*}_{n,\mathbf{R}}$. When policy-makers and subject
matter researchers reason about candidate regimes in terms of such
prioritization rules, then this property may facilitate sound translation of
policy questions into appropriate analyses.

##  3 Identifying resource-limited estimands

###  3.1 Structural conditions

We argued that iid statistical models for $\mathbb{P}_{n}$ are inadequate for
capturing the defining features of resource-limited settings. Here we describe
a statistical model for $\mathbb{P}_{n}$ that is coherent for these settings,
and demonstrate that relevant estimands are identifiable and estimable under
this model using available data. We do so by first articulating a general
unrestricted non-parametric structural equation model (SEM) (Pearl, 2009;
Richardson and Robins, 2013). Under an unrestricted SEM, data are generated
through recursive evaluation of the following structural functions for all
$i=1,\dots,n$:

$\displaystyle L_{i}$ $\displaystyle=f_{L_{i}}(Pa(L_{i}),\epsilon_{L_{i}}),$
$\displaystyle A_{i}$ $\displaystyle=f_{A_{i}}(Pa(A_{i}),\epsilon_{A_{i}}),$
$\displaystyle Y_{i}$ $\displaystyle=f_{Y_{i}}(Pa(Y_{i}),\epsilon_{Y_{i}}).$

For example, $A_{i}$ is determined by the structural function $f_{A_{i}}$ of
variables in the parent set $Pa(A_{i})$, and an error term,
$\epsilon_{A_{i}}$. We suppose that elements of $\mathbf{A}_{n}$ are generated
subsequent to $\mathbf{L}_{n}$ and prior to $\mathbf{Y}_{n}$. Formally, we let
$\mathbf{f}_{n}\equiv\\{f_{L_{i}},f_{A_{i}},f_{Y_{i}}\mid
i\in\\{1,\dots,n\\}\\}$ denote the set of all relevant structural equations,
and $\boldsymbol{\epsilon}_{n}\equiv\\{\epsilon_{i}\mid
i\in\\{1,\dots,n\\}\\}$ denote the set of all error terms following a joint
distribution $\mathbb{P}^{\epsilon}_{n}$, with
$\epsilon_{i}\coloneqq\\{\epsilon_{L_{i}},\epsilon_{A_{i}},\epsilon_{Y_{i}}\\}$.
Following Pearl (2009), a particular causal system $\mathbb{S}_{n}$ is then
taken to be a pairing of a set of structural equations and a distribution for
their associated error terms,
$\mathbb{S}_{n}\equiv\\{\mathbf{f}_{n},\mathbb{P}^{\epsilon}_{n}$}.

Unlike SEMs used in settings with iid observations pearl2009causality,
richardson2013single, the above structural equations allow the inclusion of
variables encoding information on individual $j\neq i$ in the structural
equations for some $V_{i}$. For example, this general SEM allows that
$L_{j}\subseteq Pa(A_{i})$, that is, it allows the covariates for individual
$j$ to have direct effects on the treatment of individual $i$. Furthermore the
structural equations are themselves indexed by $i$, reflecting that this
general SEM does not assume any invariance between structural equations for
$V_{j}$ and $V_{i}$. For example, $V_{j}$ might differ from $V_{i}$ even if
$Pa(V_{j})=Pa(V_{i})$ and $\epsilon_{V_{j}}=\epsilon_{V_{i}}$.

We then define a statistical model $\mathcal{M}_{n}$, i.e. a collection of
laws $\mathbb{P}_{n}$, through restrictions on the SEM, i.e. through
restrictions on $\mathbf{f}_{n}$ and $\mathbb{P}^{\epsilon}_{n}$. A particular
causal system $\mathbb{S}_{n}$ induces a joint distribution for
$\mathbf{O}^{F}_{n}$, that is
$\mathbb{P}_{n}\equiv\mathbb{P}_{n}(\mathbb{S}_{n})$. Thus, if we take
$\mathcal{S}_{n}$ to be a set of causal systems, then its corresponding
induced statistical model is defined as
$\mathcal{M}_{n}\equiv\\{\mathbb{P}_{n}(\mathbb{S}_{n})\mid\mathbb{S}_{n}\in\mathcal{S}_{n}\\}$.

To identify causal parameters of interest, we introduce two types of
conditions. The first type are weak structural conditions for the consistent
estimation of parameters of $\mathbf{O}_{n}$. These conditions, labelled
$A1-A3$ below, are a strict subset of those that are implicit whenever data
are described as iid, but they must be made explicit in settings where iid
models are unsatisfactory.

  * A1.

Conditional noninterference: For all $i$, $Pa(L_{i})\cup Pa(Y_{i})\subseteq
O_{i}$ and $\epsilon_{Y_{i}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\boldsymbol{\epsilon}_{n}\setminus\epsilon_{i}$.
Furthermore, $\\{\epsilon_{L_{j}}:j\in\\{1,\dots,n\\}\\}$ are mutually
independent.

  * A2.

Structural invariance: For all $i\neq j$, $f_{L_{i}}=f_{L_{j}}$ and
$f_{Y_{i}}=f_{Y_{j}}.$

  * A3.

Conditional identically-distributed errors: For all $i\neq j$, and for all
$pa(y)$, $\Big{(}\epsilon_{L_{i}}\Big{)}\sim\Big{(}\epsilon_{L_{j}}\Big{)}$,
and
$\Big{(}\epsilon_{Y_{i}}\Big{|}Pa(Y_{i})=pa(y)\Big{)}\sim\Big{(}\epsilon_{Y_{j}}\Big{|}Pa(Y_{j})=pa(y)\Big{)}$.

The second type permits identification of causal parameters of
$\mathbf{O}_{n}^{F}$ solely in terms of parameters of $\mathbf{O}_{n}$. This
condition, labelled $B$, is isomorphic to those classically assumed for
identification of individual-level parameters in the statistical causal
inference literature.

  * B.

No individual-level confounding of outcomes: For all $i$,
$Y_{i}^{a_{i}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A_{i}\mid L_{i}$.

We denote the statistical model for $\mathbb{P}_{n}$ that is induced by an
NPSEM under conditions A and B by $\mathcal{M}_{n}^{AB}$.

###  3.2 Remarks on conditions

We illustrate the structural implications of Condition A1 with DAGs in Figure
S6 of Appendix D. Condition A1 will in general be violated by a causal effect
of any variable for individual $j$, $O_{j}$, on the covariates or outcome for
individual $i$, $\\{L_{i},Y_{i}\\}$, except via paths intersected by the
treatment of individual $i$, $A_{i}$. For example, A1 does not preclude an
effect depicted by the path $L_{j}\rightarrow A_{i}\rightarrow Y_{i}$: such an
effect would be expected in a limited resource setting. However, a direct path
$A_{j}\rightarrow Y_{i}$ would violate condition A1. Furthermore, Condition A1
will in general be violated by unmeasured common causes of $Y_{i}$ and any
element in $O_{j}$, or any unmeasured common cause of $L_{i}$ and $L_{j}$.
This is encoded by restrictions on the dependence structure of the error terms
for $Y_{i}$ and $L_{i}$ with those error terms for individual $j$. However,
Condition A1 leaves unrestricted the causal and error-dependence structure
between the treatment variables of all individuals in the cluster,
$\mathbf{A}_{n}$, and between the covariates $\mathbf{L}_{n}$ and those
treatments. Thus, extensive causal connections between individuals are
permitted under this condition. Conditions A2 and A3 assert that covariate and
outcome structural equations are invariant in $i$ and that the conditional
distributions of their error terms are identical across individuals. Together,
Conditions A1-A3 provide a structure and regularity to the joint distribution
of $\mathbf{O}_{n}$ that is nevertheless markedly weaker than standard iid
models.

We graphically illustrate the structural implications of Condition B with the
DAGs in Figure S7 of Appendix D. In the context of an SEM following Conditions
A1-A3, Condition B additionally precludes classical within-individual
confounding structures, and also places additional restrictions on the causal
structure between the covariates $\mathbf{L}_{n}$ and treatments
$\mathbf{A}_{n}$. In particular, when there is an unmeasured common cause of
$L_{i}$ and $Y_{i}$, unmeasured common causes of $L_{i}$ and $A_{j}$ will
result in a variant of the classical M-bias causal structure
greenland1999causal. In this case, the M-bias arises because of the causal
connections between treatment units $A_{j}$ and $A_{i}$, characteristic of the
resource-limited setting.

By definition, $Y_{i}^{a_{i}}$ is the potential outcome of individual $i$
under an intervention solely on that individual’s own treatment (and no other
individuals). Therefore, Condition B is amenable to interrogation on
conventional graphs constructed for a single individual, for example the
single-world intervention graphs (SWIG) of richardson2013single, wherein this
condition might be contradicted by graphical structures using standard rules
for d-separation pearl2009causality, see Figure S8 for an example. However,
under Condition A1 we might immediately reformulate Condition B in terms of a
counterfactual outcome under intervention to all individuals’ treatments,
$Y_{i}^{\mathbf{a}_{n}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A_{i}\mid L_{i}$. This follows from the
restrictions on the structural equations for $Y_{i}$, whereby it follows that
$Y^{a_{i}}_{i}=Y_{i}^{\mathbf{a}_{n}}$ for all $i$, due to the exclusion of
$\mathbf{a}_{n}\setminus a_{i}$ or any of their descendants besides $A_{i}$
from the arguments of $f_{Y_{i}}$. In this sense, counterfactuals
$Y_{i}^{a_{i}}$ generated by such an SEM satisfy the stable-unit treatment
value assumption of rubin1990comment. However, other counterfactuals from this
model, defined by interventions on other variables, will in general not. For
example, an intervention on the patients’ rank-ordering for treatment
$\mathbf{R}_{n}$, without subsequent intervention on patients’ treatment
assignment would not satisfy SUTVA, i.e./ $Y^{r_{i}}_{i}\neq
Y_{i}^{\mathbf{r}_{n}}$. While we do not consider further such interventions,
they are the focus of ongoing work.

###  3.3 Identification results

Under $\mathcal{M}^{AB}_{n}$, covariates and outcomes enjoy a conditional iid
property, see Appendix C for a formal result. For example, conditional on the
measured past, the observed outcomes are independent and the probability that
any two individuals with the same covariate history $a,l$ will have the
particular outcome value $y$ is described by a common measure. We let
$Q_{Y}(y\mid a,l):=\mathbb{P}_{n}(Y_{i}=y\mid L_{i}=l,A_{i}=a)$ denote this
common measure and likewise define $Q_{L}(l):=\mathbb{P}_{n}(L_{i}=l)$. Thus
far we have not discussed any restrictions on the relations between the
parameters $q_{i}(a\mid l)\coloneqq\mathbb{P}_{n}(A_{i}=a\mid L_{i}=l)$,
across individuals $i=1,..,n$. Likewise, $q^{*}_{i}(a\mid
l)\coloneqq\mathbb{P}_{n}(A^{G_{n}+}_{i}=a\mid L_{i}=l)$ is similarly
unrestricted. There are no guarantees that $q_{i}=q_{j}$ for an arbitrary
$\mathbb{P}_{n}$ in $\mathcal{M}^{AB}_{n}$ or that $q^{*}_{i}=q^{*}_{j}$ for
some arbitrary $G_{n}\in\Pi_{n}(\kappa_{n})$. In Section 5, we discuss
conditions under which we can construct consistent estimators of individual-
level parameters $Q_{L}$ and $Q_{Y}$ from $\mathbf{O}_{n}$. When such is the
case, we will say we have identified a causal parameter whenever we express it
as a known function of at most parameters of $Q_{L}$ and $Q_{Y}$.

Analogous to classical approaches in causal inference richardson2013single, we
define an individual-level g-formula density function $f_{i}^{G_{n}}$, with
respect to an individual $i$ and $CR$ $G_{n}$,

$\displaystyle f_{i}^{G_{n}}(o)=$ $\displaystyle Q_{Y}(y\mid
a,l)q^{*}_{i}(a\mid l)Q_{L}(l),$ (1)

which will appear in our identification results. We emphasize that
$f_{i}^{G_{n}}$ is distinguished from a classical g-formula in that it will
generally differ for each individual $i$, via the intervention density
$q^{*}_{i}$.

We also define lower-dimensional summaries of $\mathbf{O}_{n}$, corresponding
to the frequency distributions of the values of $O_{i}$ in the cluster. First,
let $\mathbb{L}_{n}$ be the random vector indicating the numbers of
individuals in the cluster with each of the possible values $l\in\mathcal{L}$.
Second, let $\mathbb{L}_{n}(l)$ denote the one-dimensional random variable
indicating the number of individuals in the cluster with specific covariate
value $l$. Then $\mathbb{L}_{n}$ represents the random marginal frequency
table for the cluster with fixed $n$, and $\mathbb{L}_{n}(l)$ represents the
random value of one of its cells corresponding to value $l$. Likewise,
$\mathbb{A}_{n}$ and $\mathbb{A}_{n}(a)$ have analogous interpretations.
Elaborating this notational convention, let $B_{i}\coloneqq\\{A_{i},L_{i}\\}$
so that $\mathbb{B}_{n}$ is the random contingency table for joint frequencies
of treatment and covariates in the cluster, which we may consider conditional
on some or all of its margins, for example, conditional on
$\mathbb{L}_{n}=\mathbb{l}_{n}$. As short hand, let $\mathbb{Q}_{Y,n}$ be the
conditional density function of $\mathbb{O}_{n}$ given $\mathbb{B}_{n}$ and
let $\mathbb{Q}_{L,n}$ be the marginal density of $\mathbb{L}_{n}$.
Additionally, let $\mathbb{q}^{*}_{n}$ denote the conditional density of
$\mathbb{B}_{n}^{G_{n}+}$ given $\mathbb{L}_{n}$. See Appendix C for
identities of these compositional parameters in terms of $Q_{L}$ and $Q_{Y}$
under $\mathcal{M}^{AB}_{n}$.

We thus also define the compositional g-formula density function
$\mathbb{f}^{G_{n}}$ as

$\displaystyle\mathbb{f}^{G_{n}}(\mathbb{o}_{n})\coloneqq\mathbb{Q}_{Y,n}(\mathbb{o}_{n}\mid\mathbb{b}_{n})\mathbb{q}^{*}_{n}(\mathbb{b}_{n}\mid\mathbb{l}_{n})\mathbb{Q}_{L,n}(\mathbb{l}_{n}).$
(2)

We refer to $\mathbb{f}^{G_{n}}$ as the compositional g-formula, because it is
isomorphic to the individual-level g-formula, except with functions for
compositional parameters in place of individual-level parameters.

The following theorem links the individual-level and compositional g-formulae
with causal parameters under $G_{n}$, where we consider arbitrary real-valued
functions $h$ and $h^{\prime}$ of $\mathbb{O}_{n}$, and $O_{i}$, respectively.

######  Theorem 1.

Consider a law $\mathbb{P}_{n}\in\mathcal{M}^{AB}_{n}$ and a regime
$G_{n}\in\Pi_{n}(\kappa_{n})$. Then

$\mathbb{E}[h(\mathbb{O}_{n}^{G_{n}+})]=\sum\limits_{\mathbb{o}_{n}}h(\mathbb{o}_{n})\mathbb{f}^{G_{n}}(\mathbb{o}_{n})\
\text{and}\
\mathbb{E}[h^{\prime}(O_{i}^{G_{n}+})]=\sum\limits_{o}h^{\prime}(o)f_{i}^{G_{n}}(o),$

whenever the right hand sides of the equations are well-defined.

A proof of Theorem 1 is provided in Appendix C. We also provide a suitable
positivity condition for the identifying functionals in Theorem 1. Let
$\overline{q}_{n}(a\mid l)\coloneqq\frac{1}{n}\sum\limits_{i=1}^{n}q_{i}(a\mid
l)$ denote the average conditional density function of $A_{i}$ given $L_{i}$
over the indices $i$, and likewise define $\overline{q}^{*}_{n}$. Then, these
functionals will be well-defined when the following condition holds:

  * $C0.$

Weak finite-cluster positivity If $\overline{q}^{*}_{n}(a\mid l)Q_{L}(l)>0$
then $\overline{q}_{n}(a\mid l)>0,$ for all $a,l.$

In words, whenever a treatment and covariate history $a,l$ has positive
probability under $G_{n}$ for at least one individual $i$, then there must
exist at least one individual $j$, possibly different from $i$, that has a
positive probability of receiving that treatment level $a$ given $l$ under the
observed data law.

Analogous positivity conditions are typically considered in the causal
inference literature for g-formula identification of parameters defined by IRs
under iid models, see Expression (62) in richardson2013single. In these
settings, positivity implies the existence of an asymptotically consistent
estimator under mild regularity conditions. However, stronger conditions will
be necessary for the existence of such an estimator when a single draw from
$\mathbb{P}_{n}$ comprises the observed data.

####  3.3.1 $L$-rank-preserving regimes

Theorem 1 provides an identification result for an arbitrary CR in
$\Pi_{n}(\kappa_{n})$. As noted in Section 2.4.1, any such regime can be
characterized by a function
$G^{*}_{n,\mathbf{R}}:\mathcal{L}^{n}\times[0,1]\rightarrow
S(\\{1,\dots,n\\})$.

As an example, consider a pair of clusters with $n=4$ and $\kappa_{n}=1$ and
$\mathcal{L}\equiv\\{0,1\\}$, where in the first cluster
$\mathbf{L}_{n}=(1,1,0,0)$ and in the second cluster
$\mathbf{L}_{n}=(0,0,1,1)$. Then consider a regime $G_{n}$ such that for both
such clusters, the rank orderings of $\mathbf{r}_{n}=(1,2,3,4)$ and
$\mathbf{r}_{n}=(2,1,3,4)$ are assigned with equal probability. Each of these
clusters has the same composition: $\mathbb{L}_{n}=(2,2)$. However, in the
first cluster, individuals with $L_{i}=1$ each have a probability of 0.5 of
treatment assignment and those with $L_{i}=0$ have 0 probability of treatment
assignment, whereas in the second cluster these probabilities are exchanged.

In contrast to the preceding example, realistic policies will often correspond
to regimes that assign treatment in a way that is invariant across clusters
with the same composition. To formalize this class of regimes, we then
consider a subclass of policies,
$\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)\subset\Pi_{n}(\kappa_{n})$, defined as
follows:

######  Definition 1 ($L$-rank-preserving CRs).

The class of CRs $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ is the subset of
regimes $G_{n}$ in $\Pi_{n}(\kappa_{n})$ such that for all $i,j$ and all
$\mathbb{P}_{n}$ the following properties hold:

  1. $(1)$

There exists a function $\Lambda:\mathcal{L}_{i}\rightarrow\mathbb{R}$ where
$\Lambda(L_{i})>\Lambda(L_{j})$ implies $A_{i}^{G_{n}+}\geq A_{j}^{G_{n}+}$,
$\mathbb{P}_{n}$-almost surely; and

  2. ($2)$

$\mathbb{P}_{n}(A^{G_{n}+}_{i}=1\mid
L_{i}=l,\mathbb{L}_{n}=\mathbb{l}_{n})=p^{G_{n}}(l,\mathbb{l}_{n})\in[0,1]$
for all $l,\mathbb{l}_{n}$.

$L$-rank-preserving CRs constitute an important class of regimes. For example,
the prioritization rules deployed in national health care systems are of this
class lakshmi2013application, januleviciute2013impact, mullen2003prioritising,
hadorn1997new, feek2000rationing, as they have the feature of sorting
individuals into coarsened rank groups via some function $\Lambda$, i.e.,
property $(1)$. Furthermore, $L$-rank-preserving CRs capture some notion of
equity that is not in general satisfied by CRs, as illustrated by the previous
example; alternatively, $L$-rank-preserving CRs ensure that individuals with
the same covariate value $l$ will have the same probability of receiving
treatment, $q^{*}_{i}=q^{*}_{j}$, for all $i,j$. Finally, the optimal CR will
be an $L$-rank-preserving CR under $\mathcal{M}^{AB}_{n}$. We provide a formal
result in Section 5.

We now present an additional identification result for the expectations of
$\overline{Y}^{{G}_{n}}_{n}\coloneqq\frac{1}{n}\sum\limits_{y}y\mathbb{Y}_{n}^{{G}_{n}}(y)$,
the cluster-average potential outcome. The resulting identification formula
has a unique formulation since $\overline{Y}^{{G}_{n}}_{n}$ can be re-
expressed as a simple linear combination of individual outcomes,
$\frac{1}{n}\sum\limits Y_{i}^{{G}_{n}}$.

######  Proposition 1.

Consider a law $\mathbb{P}_{n}\in\mathcal{M}^{AB}_{n}$ and
$G_{n}\in\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$. Then

$\mathbb{E}[\overline{Y}^{{G}_{n}}_{n}]=\sum\limits_{o\in\mathcal{O}_{i}}yf_{i}^{G_{n}}(o).$

In subsequent sections, we study the asymptotic properties of these
functionals as the cluster size $n$ grows and find correspondences with
identification approaches in the constrained optimization literature. We also
consider a more general version of Proposition 1 in Appendix A, for a larger
class of CRs defined solely by property (2) of Definition 1.

####  3.3.2 Identifying intervention densities

Theorem 1 and Proposition 1 do not by themselves constitute identification
results because $q_{n}^{*}$ and $\mathbb{q}^{*}_{n}$ are by definition
counterfactual densities. Under classical IRs, this distinction is trivial
because $q_{i}^{*}$ would be known and invariant to $\mathbb{P}_{n}$,
following immediately by definition of the regime. In contrast, the
$q_{i}^{*}$ and $\mathbb{q}_{n}^{*}$ of CRs are not immediately apparent. In
Appendix A, we review identification of $\mathbb{q}^{*}_{n}$ and $q_{n}^{*}$
under general regimes $G_{n}$ and unrestricted non-parametric models, and
illustrate simplifications under regimes in special classes. We provide a
simple closed form expression for $q^{*}_{i}$ under $L$-rank preserving
regimes, $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$. This allows specification of
$G_{n}$ in terms of ${\Lambda}$, which as we argued, characterizes many
regimes of practical interest.

##  4 Large-cluster estimands

Finite cluster estimands are, in general, identified by functionals that
depend on a fixed cluster size, say $n^{*}$. In this section, we articulate
“large-cluster” estimands defined as the limits of particular sequences of
finite-cluster estimands indexed by cluster size $n$. We will show that these
large-cluster estimands can correspond to parameters defined by the IRs
already considered in the extant literature, and indeed can be identified by
identical functionals of observed data.

###  4.1 Asymptotic preliminaries

We define an arbitrary joint distribution of
$\\{\mathbf{O}^{F}_{1},\mathbf{O}^{F}_{2},\mathbf{O}^{F}_{3},\dots\\}$, the
joint vector of covariates, treatments, and outcomes for clusters of
increasing size $n$. We denote this distribution by $\mathbb{P}_{0}$, which we
call an asymptotic law. We let $\mathcal{M}_{0}^{U}$ denote the set of all
such laws that can be factorized as
$\mathbb{P}_{0}=\mathbb{P}_{1}\mathbb{P}_{2}\mathbb{P}_{3}\cdots$, and we
locate $\mathbb{P}_{n}$ as a factor of this joint law,
$\mathbb{P}_{n}\equiv\mathbb{P}_{n}(\mathbb{P}_{0})$.

For a given asymptotic law $\mathbb{P}_{0}\in\mathcal{M}_{0}^{U}$, we consider
the asymptotic properties of sequences of estimands indexed by $n$. To
facilitate consideration of these sequences, we restrict the space
$\mathcal{M}_{0}^{U}$ via the following conditions, where the primitive is the
unrestricted space of joint causal systems
$\mathcal{S}_{0}\equiv\mathcal{S}_{1}\mathcal{S}_{2}\mathcal{S}_{3}\cdots$. We
let $\mathbb{P}_{n}^{\epsilon_{L_{i}}}$ and
$\mathbb{P}_{n}^{\epsilon_{Y_{i}\mid pa(y)}}$ denote the marginal law of
$\epsilon_{L_{i}}$ and the conditional law of $\epsilon_{Y_{i}}$ given
$Pa(Y_{i})=pa(y)$ for a cluster of size $n$, respectively.

  * $D1.$

$\mathbb{P}_{n}\in\mathcal{M}_{n}^{AB}$, for all $n\in\mathbb{N}^{+}.$

  * $D2.$

$\\{f_{L_{i}},\mathbb{P}_{n}^{\epsilon_{L_{i}}},f_{Y_{i}},\mathbb{P}_{n}^{\epsilon_{Y_{i}\mid
pa(y)}}\\}\equiv\\{f_{L_{i}},\mathbb{P}_{n^{\prime}}^{\epsilon_{L_{i}}},f_{Y_{i}},\mathbb{P}_{n^{\prime}}^{\epsilon_{Y_{i}\mid
pa(y)}}\\}$ for all $n,n^{\prime}\in\mathbb{N}^{+}.$

Condition $D1$ implies that each marginal law $\mathbb{P}_{n}$ of
$\mathbb{P}_{0}$ is induced by a causal system $\mathbb{S}_{n}$ that satisfies
the conditions of $\mathcal{M}_{n}^{AB}$. Condition $D2$ states that the
structural equations for $L_{i}$ and $Y_{i}$ are invariant, as are the
marginal and conditional distributions of their error terms, respectively,
across $\mathbb{S}_{1},\mathbb{S}_{2},\mathbb{S}_{3},\dots$. A consequence of
conditions $D1$ and $D2$ is that, for a given $\mathbb{P}_{0}$, the single
functions $Q_{L}$ and $Q_{Y}$ will describe the marginal and conditional
densities of $L_{i}$ and $Y_{i}$, respectively, for all $i$, $n$. We
henceforth take limits with respect to such an asymptotic law
$\mathbb{P}_{0}$.

###  4.2 Defining and identifying large-cluster estimands

We define an arbitrary sequence of CRs, which we denote by
$\\{G_{1},G_{2},G_{3},\dots\\}$. We denote this sequence by $\mathbf{G}_{0}$,
which we call an asymptotic regime. We consider asymptotic regimes
$\mathbf{G}_{0}$ that meets the following conditions:

  * $E1.$

For each $n$, $G_{n}\in\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$.

  * $E2.$

For each $n$, $\kappa_{n}=\lfloor n\times\kappa^{*}\rfloor$, with
$\kappa^{*}\in[0,1].$

An asymptotic regime following condition $E$ is a sequence of $L$-rank-
preserving CRs with a fixed $\Lambda$, and resource constraints such that
$\frac{\kappa_{n}}{n}$ converges to a fixed proportion $\kappa^{*}$. We
discuss relaxations of condition $E1$ in Appendix A. A class of asymptotic
regimes following these conditions is of interest because it is parsimoniously
specified by two parameters, $\Lambda$ and $\kappa^{*}$, which in many
settings will sufficiently characterize an investigator’s regime of interest.
All asymptotic regimes $\mathbf{G}_{0}$ following conditions $E1$ and $E2$
with respect to the same $\Lambda$ and $\kappa^{*}$ will be equivalent under
an asymptotic law $\mathbb{P}_{0}$ following conditions $D$, in the sense that
they share a common limit of an arbitrary individual’s intervention density,
$q^{*}_{0}(a\mid l)\coloneqq\underset{n\to\infty}{\lim}q^{*}_{i}(a,l)$ for all
$i$.

We define the following large-cluster g-formula density function
$f^{\mathbf{G}_{0}}$,

$\displaystyle f^{\mathbf{G}_{0}}(o)\coloneqq$ $\displaystyle Q_{Y}(y\mid
a,l)q^{*}_{0}(a\mid l)Q_{L}(l).$ (3)

The following theorem identifies the large-cluster expected average potential
outcome

$\mathbb{E}[\overline{Y}_{0}^{\mathbf{G}_{0}}]\coloneqq\underset{n\to\infty}{\lim}\mathbb{E}_{\mathbb{P}_{n}}[\overline{Y}^{{G}_{n}}_{n}].$

######  Theorem 2.

Consider an asymptotic law $\mathbb{P}_{0}$ and an asymptotic regime
$\mathbf{G}_{0}$ following conditions $D$ and $E$, respectively. Then,
$\mathbb{E}[\overline{Y}^{\mathbf{G}_{0}}_{0}]=\sum\limits_{o\in\mathcal{O}_{i}}yf^{\mathbf{G}_{0}}(o)$
whenever the right-hand side is well-defined.

As in Section 3 we provide a suitable positivity condition for the identifying
functional in Theorem 2 to be well-defined. Define ${q}_{0}(a\mid
l)\coloneqq\underset{n\to\infty}{\lim}\overline{q}_{n}(a\mid l)$ with respect
to an asymptotic law $\mathbb{P}_{0}$. Then, this functional will be well-
defined when the following condition holds:

  * $C0^{*}.$

Weak large-cluster positivity ${q}_{0}$ is well-defined and if
$q^{*}_{0}(a\mid l)Q_{L}(l)>0$ then ${q}_{0}(a\mid l)>0$, for all $a,l$.

###  4.3 Identifying intervention densities

As with the finite-cluster parameters, the equality of Theorem 2 does not by
itself constitute an identification result because $q_{0}^{*}$ is by
definition a counterfactual density. To identify $q_{0}^{*}$, we leverage the
rank-based formulations of CRs in $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$,
where we let $\mathbb{P}_{0}(f(L_{i}))$ denote
$\sum\limits_{\mathcal{L}}I(f(l))Q(l)$, with $f(l)$ a logical proposition.

######  Proposition 2.

Consider an asymptotic law $\mathbb{P}_{0}$ and an asymptotic regime
$\mathbf{G}_{0}$ following conditions $D$ and $E$, respectively. Then,

$\displaystyle q^{*}_{0}(1\mid l)$
$\displaystyle=\begin{cases}\frac{\kappa^{*}-\mathbb{P}_{0}(\Lambda(L_{i})>\omega_{0})}{\mathbb{P}_{0}(\Lambda(L_{i})=\omega_{0})}&:\Lambda(l)=\omega_{0},\\\
I\Big{(}\Lambda(l)>\omega_{0}\Big{)}&:\text{otherwise},\end{cases}$ (4)

where
$\omega_{0}\coloneqq\textup{inf}\bigg{\\{}c:\mathbb{P}_{0}(\Lambda(L_{i})>c)\leq\kappa^{*}\bigg{\\}}.$

In Proposition 2, $\omega_{0}$ is interpretable as the almost-surely last rank
group $\Lambda(l)$ treated in a large cluster under $\mathbf{G}_{0}$, with
respect to $\mathbb{P}_{0}$. Theorem 2 and Proposition 2 taken together thus
identify $\mathbb{E}[\overline{Y}^{\mathbf{G}_{0}}_{0}]$ in terms of $Q_{L}$
and $Q_{Y}$.

###  4.4 Large-cluster optimal regimes

Let $V_{i}=c(L_{i})$ be some coarsening of $L_{i}$. Suppose we are interested
in the optimal CR that is at most a function of $\mathbf{V}_{n}$, that is the
regime $G_{n}:\\{\mathcal{V}\\}^{n}\times[0,1]\rightarrow\\{0,1\\}^{n}$ that
maximizes $\mathbb{E}[\overline{Y}_{n}^{G_{n}}]$. Then let
$\Delta(l)\coloneqq\mathbb{E}[Y_{i}^{a=1}-Y_{i}^{a=0}\mid V_{i}=c(l)]$ be the
conditional average effect of treatment for individual $i$. In a slight abuse
of notation we also let $\Delta$ denote the equivalent function with domain
$\mathcal{V}$.

######  Proposition 3 (Optimal CR).

Consider a law $\mathbb{P}_{n}\in\mathcal{M}_{n}^{AB}$. The CR
$G^{\mathbf{opt}}_{n}\equiv G_{n}\in\Pi_{n}(\kappa_{n})$ that maximizes
$\mathbb{E}[\overline{Y}_{n}^{G_{n}}]$ is the $V$-rank-preserving CR
$G_{n}\in\Pi_{n}^{d}(\kappa_{n},V_{i},\Lambda)$ characterized by
$\Lambda\equiv\Delta$.

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$, and an
asymptotic regime $\mathbf{G}_{0}$ following condition $E$. If
$G_{n^{*}}\in\mathbf{G}_{0}$ is such an optimal regime for
$\mathbb{P}_{n^{*}}(\mathbb{P}_{0})$ for some $n^{*}>1$ then
$G_{n}\in\mathbf{G}_{0}$ is the optimal such regime for
$\mathbb{P}_{n}(\mathbb{P}_{0})$ for all $n$. Let $q^{\mathbf{opt}}_{0}$
denote the intervention density under the asymptotic regime
$\mathbf{G}^{\mathbf{opt}}_{0}\equiv\Big{(}G^{\mathbf{opt}}_{1},G^{\mathbf{opt}}_{2},G^{\mathbf{opt}}_{3},\dots\Big{)}$.
Then, $\mathbb{E}[\overline{Y}_{0}^{\mathbf{G}^{\mathbf{opt}}_{0}}]$ is
identified as in Theorem 2 and its intervention density $q^{\mathbf{opt}}_{0}$
is identified as in Proposition 2 where we take
$\Lambda=\Delta_{0}\coloneqq\Delta$.

Proposition 3 illustrates that the observed data parameter identifying
$\mathbb{E}[\overline{Y}_{0}^{\mathbf{G}^{\mathbf{opt}}_{0}}]$ is isomorphic
to that parameter studied by luedtke2016optimal. We emphasize that Proposition
3 considers the optimal regime in $\Pi_{n}(\kappa_{n})$. In general, this CR
will not necessarily be the optimal regime in the unrestricted class of
$\kappa_{n}$-constrained CRs, $\Pi_{n}(\kappa_{n})$, except in special cases,
for example if we restrict model $\mathcal{M}^{AB}_{n}$ to only include laws
for which $\Delta(l)$ is positive for all $l$. We provide a straight-forward
extension of Proposition 3, for the unrestricted class of regimes
$\Pi^{F}_{n}(\kappa_{n})$ in Appendix D.

##  5 Inference

###  5.1 Asymptotics for inference on limited resource estimands

In iid settings, estimators are defined as a function of $n$ independent
observations from a common law $P$. Correspondingly, asymptotic properties of
these estimators are defined with respect to sequences of such observations
rosenblatt1956central. In limited resource settings, estimators are defined as
a function of a single draw from a some law $\mathbb{P}_{n}$. As in the
empirical processe literature van2000asymptotic, we consider asymptotic
properties in the context of sequences of such laws $\mathbb{P}_{n}$, which we
consolidate in terms of an asymptotic law $\mathbb{P}_{0}$. Let the sets of
random variables $\\{\tilde{Q}_{Y,n},\tilde{q}_{n},\tilde{Q}_{L,n}\\}$ denote
the (conditional) empirical mean analogues of $\\{Q_{Y},q_{i},Q_{L}\\}$, for
example $\tilde{Q}_{Y,n}(y\mid
a,l)\coloneqq\frac{\mathbb{O}_{n}(y,a,l)}{\mathbb{B}_{n}(a,l)}$.

Unusual challenges arise under non-iid models in the development of estimators
with desirable asymptotic properties. For example, we argued in Theorem 1 that
many cluster causal parameters are identified under $\mathcal{M}_{n}^{AB}$ and
a suitable positivity condition. The identification functional is expressed
entirely in terms of $Q_{Y}$ and $Q_{L}$, and investigator-known quantities.
We show in Appendix C that the sets $\\{L_{i}\in\mathbf{L}_{n}\\}$ and
$\\{Y_{i}\in\mathbf{Y}_{n}\mid A_{i}=a,L_{i}=l\\}$ are each correspondingly
mutually independent and identically distributed under $\mathcal{M}_{n}^{AB}$.
Thus, an apparently obvious estimator of $f^{G_{n}}_{i}(o)$, would consist of
plug-in estimators of its component factors $\\{Q_{L},Q_{Y}\\}$ using their
single-cluster empirical mean analogues
$\\{\tilde{Q}_{Y,n},\tilde{Q}_{L,n}\\}$. However, such an estimator will not
in general be asymptotically consistent (in a sense subsequently defined).

In Appendix B we give an example of an asymptotic law following conditions $D$
and weak cluster positivity $C0$, under which $\tilde{Q}_{Y,n}$ does not
converge to $Q_{Y}$. Heuristically, the observed asymptotic law
$\mathbb{P}_{0}$ may result in the convergence of $\overline{q}_{n}$, but, for
each $n$, the causal system $\mathbb{S}_{n}$ generating $\mathbb{P}_{n}$ may
be characterized by a regime $G_{n}$ equivalent to a distribution over
deterministic regimes with mass concentrated at extreme allocation strategies.
Under each such regime, $\tilde{q}_{n}(a\mid l)$ might be close to 0 or 1 for
each $(a,l)$, and so, for any given realization at any cluster size $n$,
$\Big{|}\overline{q}_{n}(a\mid l)-\tilde{q}_{n}(a\mid l)\Big{|}$ will remain
large. A similar phenomenon is highlighted in lee2021network, in their
illustration of “spurious associations” in network data. Consider a toy data
generating mechanism in which the vectors $\mathbf{L}_{n}$ and
$\mathbf{Y}_{n}$ each take the values $(1,\dots,n)$ or $(n,\dots,1)$ as
determined by independent flips of fair coins. Thus
$\mathbf{L}_{n}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathbf{Y}_{n}$, but in any given realization
(i.e. an estimator computed on $\mathbb{O}_{n}$), it will appear as if $L_{i}$
and $Y_{i}$ are perfectly correlated (negatively or positively).

###  5.2 Asymptotic consistency for finite-cluster estimands

Consider the following revised positivity condition for a given asymptotic law
$\mathbb{P}_{0}$ and regime $G_{n^{*}}$, where we let
$\overline{q}_{0}\coloneqq\underset{n\to\infty}{\lim}\tilde{q}_{n}$.

  * C1.

Strong finite-cluster positivity $\mathbb{P}_{0}$ is such that
$\overline{q}_{0}$ is well-defined and, for all $a,l$, if
$\overline{q}^{*}_{n^{*}}(a\mid l)Q_{L}(l)>0$ then $\overline{q}_{0}(a\mid
l)>0$, $\mathbb{P}_{0}-$almost surely.

To define non-parametric estimators of identifying parameters, we use that
under $\mathcal{M}^{AB}_{n^{*}}$,
$\\{\mathbb{Q}_{L,n^{*}},\mathbb{q}^{*}_{n^{*}},\mathbb{Q}_{Y,n^{*}}\\}$ may
be entirely expressed in terms of $\\{Q_{L},Q_{Y}\\}$ and known quantities. We
emphasize that we have deliberately labeled these parameters with a fixed
cluster size $n^{*}$, indicating the size defining a particular finite-cluster
estimand. Likewise define the compositional random variables
$\\{\tilde{\mathbb{Q}}_{L,n^{*}},\tilde{\mathbb{q}^{*}}_{n^{*}},\tilde{\mathbb{Q}}_{Y,n^{*}}\\}$
analogously by substituting $\\{\tilde{Q}_{L},\tilde{Q}_{Y}\\}$ for
$\\{Q_{L},Q_{Y}\\}$ wherever they appear in the corresponding parameters.
Accordingly, we construct empirical analogues of compositional and individual-
level g-formula parameters, $\tilde{\mathbb{f}}^{G_{n^{*}}}$ and
$\tilde{f}_{i}^{G_{n^{*}}}$. In Appendix B we also motivate estimators based
on an inverse-probability weighted (IPW) formulation of the individual-level
g-formula. In the following, we use the symbol $\boldsymbol{\lim}$ to denote
limits in probability with respect to a measure $\mathbb{P}_{0}$, as
$n\to\infty$.

######  Theorem 3 (Asymptotic consistency).

Consider $\mathbb{P}_{0}$ following conditions $D$ and
$G_{n^{*}}\in\Pi_{n^{\ast}}(\kappa_{n^{\ast}})$ following $C1$. Then,

$\displaystyle\boldsymbol{\lim}\sum\limits_{\mathbb{o}_{n^{\ast}}}h(\mathbb{o}_{n^{\ast}})\tilde{\mathbb{f}}^{G_{n^{\ast}}}(\mathbb{o}_{n^{*}})=\mathbb{E}[h(\mathbb{O}_{n^{\ast}}^{G_{n^{\ast}}+})],$
(5)
$\displaystyle\boldsymbol{\lim}\sum\limits_{o}h^{\prime}(o)\tilde{f}_{i}^{G_{n^{*}}}(o)=\mathbb{E}[h^{\prime}(O_{i}^{G_{n^{*}}+})].$
(6)

###  5.3 Remark on the strong positivity condition

In Appendix B, we give a stronger condition $C2$ (Sub-cluster positivity) on
the treatment assignment mechanisms $\mathbf{f}_{0,\mathbf{A}}$ in
$\mathbb{S}_{0}$ for a given asymptotic law $\mathbb{P}_{0}$, which, in
conjunction with weak positivity condition $C0$ implies the strong cluster
positivity condition $C1$, and which may recognizably capture observed data
generating mechanisms in many settings. Heuristically, this condition involves
an asymptotic law wherein, as $n$ grows, clusters may be partitioned into an
increasing number of weakly dependent sub-clusters of bounded size, each with
their own finite resource supply, along with mild regularity conditions on the
regimes implemented within each sub-cluster. As such, this condition bears
structural resemblance to the partial interference assumptions
sobel2006randomized more frequently studied in the statistical causal
inference literature. Under sub-cluster positivity ($C2$) we further have that
these empirical estimators will be regular and asymptotically linear. Under
correctly specified parametric models for $Q_{Y}$ and $\overline{q}_{0}$,
parametric g-formula or IPW estimators can be used for their corresponding
parameters and usual bootstrap procedures for iid data can be used to obtain
(1-$\alpha$)% confidence intervals.

###  5.4 Remark on inference for large-cluster estimands

Appendix B gives extensive consideration to large-cluster estimands. Therein,
we provide a strong-positivity condition analogous to $C1$. However, this
condition would not in general imply the existence of a consistent plug-in
estimator of the large-cluster g-formula, even under the conditions of Theorem
2. This follows because $\omega_{0}$ is not in general a pathwise-
differentiable parameter. Pathwise derivatives are important building blocks
in the theory of regular estimation bickel1993efficient and pathwise
differentiability is a necessary condition for the existence of locally
asymptotically unbiased or regular estimators hirano2012impossibility. We
therefore provide minimal additional conditions to generalize the results of
Theorem 3 to large-cluster estimands. Importantly, we also leverage the
results of Proposition 3 and show that the procedures of luedtke2016optimal
will inherit analogous properties when applied to a single realization from
$\mathbb{P}_{n}$: semiparametric efficiency and root-$n$ consistency when
flexible data-adaptive algorithms are used to estimate nuisance parameters. We
provide an extension of these results for the general asymptotic regime
$\mathbf{G}_{0}$, which to our knowledge has not been considered, even
heuristically, by the current literature.

Nevertheless, the regularity of large-cluster estimands will in practice be
violated whenever we consider CRs that are dynamic only with respect to a
small number of discrete covariates. Such CRs will be of interest if lower
dimensional coarsenings are more easily measured, or simpler rules are more
easily implemented, for example in intensive care triage settings. In these
cases we adapt the methods of luedtke2016statistical to motivate inferential
procedures, inspired by theory commonly used in online learning settings. We
show that these procedures allow computation of root-$n$ consistent estimators
and corresponding 95% confidence interval.

##  6 Application: Intensive Care Unit Admissions

Following harris2015delay, we study the effect of prompt ICU admission on
90-day survival. We use resampled data from a cohort study of 13011 patients
with deteriorating health referred for assessment for ICU admission in 48 UK
National Health Service (NHS) hospitals in 2010-2011. The data contain
detailed covariate information including demographic variables (age, gender),
patient clinical features, including the ICNARC physiology score, the NHS NEWS
score, SOFA score, septic diagnosis, and peri-arrest, as well as contextual
factors including indicators for admission during a winter month, and
admission out of hours (7 p.m.–7 a.m.).

ICU bed availability is a limited resource; there was an available ICU bed for
only 27.3% of the patients in the data upon their arrival to the hospital.
Patients arriving concurrently in a given time window will thus necessarily
compete for available beds. Specialist critical care outreach teams or single
ICU physicians keele2020stronger will decide on a given patient’s prompt ICU
admission on the basis of not only that patient’s own clinical features, but
of those for all such patients with indications for prompt admission. On the
basis of these facts, an appropriate model for the data could be represented
by the DAG in Figure 1 thus contradicting a conventional iid model, for
example assumed by kennedy2019survivor and mcclean2022nonparametric who
analyze these same data.

We consider the following regimes: 1) $\mathbf{G}_{0}^{\mathbf{opt}_{1}}$, the
optimal $V$-rank-preserving CR for a large cluster, where we take $V_{i}\equiv
L_{i}$, the complete set of covariates measured for each patient in the
observed data; 2) $\mathbf{G}_{0}^{\Lambda_{1}}$, a candidate $V$-rank-
preserving CR for a large cluster, where we take $V_{i}$ equivalent to a
patient’s clinical risk prediction (i.e., a simple average of their ICNARC,
NEWS, and SOFA scores), and a rank function $\Lambda_{1}$ that prioritizes
patients with higher such scores; 3) $\mathbf{G}_{0}^{\mathbf{opt}_{2}}$, the
optimal regime in a large cluster, where we take $V_{i}$ equivalent to a
6-level categorization of a patient’s clinical risk prediction; 4)
$\mathbf{G}_{0}^{\Lambda_{2}}$, a candidate $V$-rank-preserving CR for a large
cluster, inspired by a proposal to prioritize ventilator allocation during a
hypothetical influenza pandemic nyc2015ventilator using a rank function
$\Lambda_{2}$ based on a patients trichotomized SOFA-score; and 5)
${G}_{n^{*}=20}^{\Lambda_{2}}$, that rule for a finite cluster with $n=20$
patients. Note that
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{1}},\mathbf{G}_{0}^{\Lambda_{1}},\mathbf{G}_{0}^{\mathbf{opt}_{2}},\mathbf{G}_{0}^{\Lambda_{2}}\\}$
are asymptotic regimes implemented in large clusters, whereas
${G}_{n^{*}=20}^{\Lambda_{2}}$ is a regime implemented in a finite cluster;
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{1}},\mathbf{G}_{0}^{\mathbf{opt}_{2}}\\}$
are unknown optimal regimes, whereas
$\\{\mathbf{G}_{0}^{\Lambda_{1}},\mathbf{G}_{0}^{\Lambda_{2}},{G}_{n^{*}=20}^{\Lambda_{2}}\\}$
are known candidate regimes based on pre-specified rank functions,
$\Lambda_{1}$ and $\Lambda_{2}$; and
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{1}},\mathbf{G}_{0}^{\Lambda_{1}}\\}$ are
$V$-rank-preserving regimes where $V$ contains at least one continuous
covariate, whereas
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{2}},\mathbf{G}_{0}^{\Lambda_{2}},{G}_{n^{*}=20}^{\Lambda_{2}}\\}$
are regimes where $V$ is a low dimensional covariate with few categories.

For each such regime, we target the expected cluster average outcome for the
corresponding large- or finite-cluster under a range of resource limitations.
We also target additional finite-cluster parameters under regime
${G}_{n^{*}}^{\Lambda_{2}}$, identified by the compositional $g$-formula (2),
including
$\mathbb{P}_{n^{*}}\Big{(}\mathbb{Y}_{n^{*}}^{G^{\Lambda_{2}}_{n^{*}}}(1)=x\Big{)}$,
i.e. the probability that exactly $x$ of the 20 individuals in the cluster are
alive at 90-days, for each $x\in\\{0,\dots,20\\}$.

For inference on large-cluster parameters under
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{1}},\mathbf{G}_{0}^{\Lambda_{1}}\\}$, we
implement a semiparametric efficient approach adapted from luedtke2016optimal,
and we compute asymptotically-valid 95% confidence intervals based on the
empirical variance of the estimated efficient influence function.
Alternatively, when $V$ has few categories,
$\\{\mathbf{G}_{0}^{\mathbf{opt}_{2}},\mathbf{G}_{0}^{\Lambda_{2}}\\}$, we
adopt the online-learning approach adapted from luedtke2016statistical. For
these procedures, asymptotic normality of the resulting estimators is
motivated by a Martingale-based central limit theorem for triangular arrays
ganssler1978central. Finally, for inference on finite-cluster parameters under
${G}_{n^{*}}^{\Lambda_{2}}$, we compute plug-in estimators of the
compositional g-formula, and an IPW estimator of the individual-level
g-formula, where nuisance parameters are estimated using flexible parametric
models with known parametric rates. For these parameters, 95% confidence
intervals are computed based on 500 nonparametric bootstrap samples. We
provide additional details of analyses in Appendix D. Code is available at
GitHub.

$\kappa^{*}$$\mathbf{G}_{0}^{\mathbf{opt}_{1}}$$\mathbf{G}_{0}^{\Lambda_{1}}$$\mathbf{G}_{0}^{\mathbf{opt}_{2}}$$\mathbf{G}_{0}^{\Lambda_{2}}$$\widetilde{\mathbb{E}}\Big{[}\overline{Y}^{\mathbf{G}_{0}}_{0}\Big{]}$

Figure 2: Estimated values (with 95% CIs) of the expected proportion of
individuals surviving to 90-days in a large cluster, under various
prioritization regimes for prompt ICU admission.

$\kappa_{n^{*}}=5$$\kappa_{n^{*}}=20$$\kappa_{n^{*}}=5$$\kappa_{n^{*}}=20$$\widetilde{\mathbb{E}}\Big{[}\overline{Y}^{G^{\Lambda_{2}}_{n^{*}}}(1)\Big{]}$$\widetilde{\mathbb{P}}_{n^{*}}\Big{(}\mathbb{Y}_{n^{*}}^{G^{\Lambda_{2}}_{n^{*}}}(1)=x\Big{)}$

Figure 3: Estimated probabilities (with 95% CIs) that exactly $x$ individuals
survive to 90-days. Vertical lines provide the expected number of patients
surviving to 90-days.

Figure 2 presents expected large-cluster-average 90-day survival under several
large-cluster regimes. The top panel presents results using the semiparametric
efficient estimators adapted from Luedtke and van der Laan (2016). Under
$\mathbf{G}_{0}^{\mathbf{opt}_{1}}$, cluster survival is apparently maximized
when at least approximately 50% of patients can be treated, corresponding to
the estimated proportion of patients with estimated positive CATEs
(conditional on $V_{i}$) for prompt ICU admission. We illustrate the
relationship between $\kappa^{*}$, CATEs, and the expected proportion of
treated individuals in this large cluster under
$\mathbf{G}_{0}^{\mathbf{opt}_{1}}$ in Figures S3 and S4 in Appendix D. Under
the candidate regime $\mathbf{G}_{0}^{\Lambda_{1}}$, which simply admits
individuals to the ICU according to their predicted clinical risk scores,
cluster survival is consistently lower under realistic resource limitations,
suggesting that these risk scores may not optimally order individuals in terms
of their expected benefit of prompt ICU admission, nor indicate when such
admission may be harmful. The bottom panel presents results using the online-
learning-based estimators adapted from Luedtke and Van Der Laan (2016b).
Results suggest similar survival between $\mathbf{G}_{0}^{\mathbf{opt}_{2}}$
and $\mathbf{G}_{0}^{\Lambda_{2}}$ up to a resource limitation of
approximately 60%.

Figure 3 presents
$\mathbb{P}_{n^{*}}\Big{(}\mathbb{Y}_{n^{*}}^{G^{\Lambda_{2}}_{n^{*}}}(1)=x\Big{)}$
via histograms for settings with 5 ICU beds (red) vs. 20 ICU beds (blue), for
a hypothetical cluster of size $20$. When only 5 ICU beds are available, 14.1
of the 20 patients (95% CI: 13.9, 14.2) are expected to survive to 90 days,
compared to 13.3 (13.0, 13.6) when all individuals are provided an ICU bed.
Analogous results for all possible resource limitations,
$\kappa_{n}=0,\dots,20$, are presented in Table S3 in Appendix D. Apparently
worse survival among the all-treated group mirrors results for analogous
parameters in a large cluster under the prioritization mechanism
$\Lambda_{2}$, presented in the bottom panel of Figure 2. Numerical results
for Figure 2 are tabulated in Appendix D.

##  7 Future directions

The results in the main text and appendices are formulated generally. They
serve as a foundation for future research on dynamic and optimal regimes in
clustered data. The longitudinal setting will be an important extension, as
many resource-limited decisions are necessarily formulated as time-varying
dynamic regimes. For example, patients often have to wait for resources, like
ventilators and organ transplants, to become available. Furthermore, in many
healthcare settings with a fixed set of treatment units, rules for terminating
the treatment (or discharge) are important. Resources are often limited in
health-care crises, such as pandemics. For these settings, it will be
important to clarify how to validly and efficiently fuse data from crisis and
pre-crisis contexts, so as to rapidly and reliably provide statistical support
for decisions during that crisis. Finally, the results should be extended for
more complex interference and network structures.

unsrtnat-init Limited1.bib

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1

##  Appendix A Mapping the space of cluster dynamic regimes

The space of individualized dynamic regimes (IR) has expanded the last
decades. First, average treatment effects (ATE) that were once restricted to
contrasts of static interventions, for example,
$\mathbb{E}_{P^{F}}[Y^{a}-Y^{a^{\prime}}]$, were generalized to regimes that
depended on covariates alone,
$\mathbb{E}_{P^{F}}[Y^{g(L)}-Y^{g^{\prime}(L)}]$, randomizing terms,
$\mathbb{E}_{P^{F}}[Y^{g(L,\delta)}-Y^{g^{\prime}(L,\delta)}]$, and even the
natural value of treatment
$\mathbb{E}_{P^{F}}[Y^{g(L,\delta,A)}-Y^{g^{\prime}(L,\delta,A)}]$. These
classification schemes were instrumental in developing a general theory of
identification of ATEs under IRs, and clarified when conditions could be
relaxed for sub-classes of regimes. For example, in time-varying settings
sufficient and necessary exchangeability conditions for g-formula
identification of expected potential outcomes under regimes $g(\emptyset)$,
$g^{*}(\overline{L}_{k}^{g^{*}})$ and
$g^{\dagger}(\overline{L}_{k}^{g^{\dagger}},\overline{A}_{k}^{g^{\dagger}})$
are progressively stronger. See richardson2013single for a review. Elaborating
the space of these regimes also equips investigators with an expanded set of
tools to interrogate their substantive questions of interest.

In contrast, very little work has been done to map the space of cluster
dynamic regimes (CRs). A small body of work, from the perspective of design-
based inference, elaborates on a space of stochastic regimes $G_{n}(\delta)$,
subsuming concepts such as static cluster regimes, Bernoulli designs, complete
randomization designs, paired-randomization designs, and arbitrary
randomization designs, see savje2021average for a review. van2014causal and
ogburn2020causal consider dynamic treatment regimes in a network settings.
However, their regimes are individualized in the sense that each individual’s
treatment is restricted to be a function of at most a randomizer, their own
covariates, and the covariates of other individuals for whom they have ties,
as encoded by a fixed adjacency matrix. In this sense, van2014causal and
ogburn2020causal only consider regime $G_{n}$ for which each individual’s
treatment is restricted to be independent of any other individual’s treatment
conditional on their own covariates and those of their connections, and
furthermore these individualized regimes are restricted to be the same
function across all individuals in the cluster (“network”).

In this appendix we begin a general mapping of the space of CRs, with
particular focus on constrained CRs, as these are the topic of this paper. In
Section A.1 we elaborate on the generalized class of constrained CRs in which
at most (but possibly less than) $\kappa_{n}$ individuals can be assigned
treatment. We provide a rank-and-treat formulation of this class of regimes
and extend results relating stochastic and deterministic IRs to this class of
CRs. In Section A.2, we introduce a restricted class of regimes defined by the
property of $L$-equity. This property will be of interest to policy makers
whenever non-discrimination on the basis of unmeasured features (for example,
individual index $i$) is a pre-requisite for an equitable policy. We locate
$L$-rank-preserving CRs (Definition 1) as a subclass of these regimes, and
importantly one in which the optimal $L$-equitable regime is contained for
laws $\mathbb{P}_{n}^{F}$ under $\mathcal{M}^{AB}_{n}$. We also introduce
subclasses of CRs intermediate between $L$-equitable regimes and $L$-rank-
preserving regimes, which are useful for didactic purposes, but may be of
substantive interest in their own right. In Section A.3 we provide a general
result for identifying the intervention densities $q^{*}_{i}$ and
$\mathbb{q}^{*}_{n}$ comprising the individual-level and compositional
g-formulae of expressions (1) and (2), and illustrate simplified
identification results for CRs in restricted classes. In Section A.4 we
provide general identification results for compositional and individual level
parameters defined by CRs that are a function of a subset of covariates. These
results can be used to motivate estimators with significant computational
advantages over those suggested by the functionals presented in the main text.
In Section A.5 we posit the existence of an unmeasured variable indicating
sub-cluster membership, and develop a class of sub-cluster regimes. In
particular, we do so in service of articulating a class of asymptotic regimes
that imply conditions sufficient for the asymptotic consistency result of
Theorem 3, which we develop in Appendix B. Figure S1 illustrates relations
between the classes of CRs defined in this manuscript.

###  A.1 Generalized constrained CRs

In the main text, we focus results on regimes in the policy class
$\Pi_{n}(\kappa_{n})$ under which exactly $\kappa_{n}$ units are assigned
treatment. In Section 2 we illustrate alternative rank-and-treat formulations
for these regimes, where a total rank ordering is assigned according to
$G^{*}_{n,\mathbf{R}}$ and then treatment for a individual $i$ is assigned if
their rank ordering is less than or equal to the number of treatment units
$\kappa_{n}$. These regimes may be of interest particularly in settings where
$\Delta_{P^{F}}(l)>0$ for all $l$, which will often be the case for target
populations where treatment resources are scarce, for example a waiting list
for transplants, and an ICU subpopulation with clinical indications for
mechanical ventilation. However, the unknown treatment regime
$\mathbf{f}_{n,\mathbf{A}}$ implemented in the observed data is unlikely to be
equivalent to a regime $G_{n}\in\Pi_{n}(\kappa_{n})$, because in many observed
data settings, individuals or clinicians will possibly reject a treatment,
even when available, and the number of treatment units actually used by the
cluster may be less than the number of treatment units available. As examples:
a triage system may stock-pile a random proportion of their treatment units;
an individual clinician may subvert an official triaging policy in
anticipation of other individuals that they would perceive to be in greater
need; and individuals may reject an available treatment unit due to beliefs
about inferiority to momentarily unavailable treatment alternatives, religious
beliefs, or other concerns.

The space of regimes $\Pi^{F}_{n}(\kappa_{n})$ is especially important in
developing realistic conditions on the components $\mathbf{f}_{0,\mathbf{A}}$
describing the observed treatment assignment mechanisms in an asymptotic law
$\mathbb{P}_{0}$. We show in Appendix B that such conditions are generally
necessary for establishing basic asymptotic properties of estimators (for
example, asymptotic consistency), and we leverage conditions on such regimes
to establish stronger asymptotic estimation results.

####  A.1.1 Rank-based formulations

In contrast to regimes $G_{n}\in\Pi_{n}(\kappa_{n})$, an arbitrary limited
resource regime $G_{n}^{*}\in\Pi^{F}_{n}(\kappa_{n})$ would have the following
rank-and-treat formulation, where
$\\{\delta_{\mathbf{R}},\delta_{A_{1}},\dots,\delta_{A_{n}}\\}$ are randomizer
terms with some investigator-specified joint distribution. Let
$\mathfrak{A}_{i}$ be the random variable denoting the subset of indices
$\\{1,\dots,n\\}$ of those individuals who are offered treatment before
individual $i$.

$\displaystyle R_{i}^{G_{n}^{*}+}\coloneqq$ $\displaystyle
G^{*}_{n,R_{i}}(\mathbf{L}_{n},\delta_{\mathbf{R}})$ (S1) $\displaystyle
A_{i}^{G^{*}_{n}+}\coloneqq$ $\displaystyle
G^{*}_{n,A_{i}}(\mathbf{R}_{n}^{G_{n}^{*}+},\mathbf{L}_{n},\\{A_{j}^{G^{*}_{n}+}:j\in\mathfrak{A}_{i}^{G^{*}_{n}+}\\},\delta_{A_{i}})$
$\displaystyle=$ $\displaystyle
g_{i}\Big{(}\mathbf{L}_{n},\\{A_{j}^{G^{*}_{n}+}:j\in\mathfrak{A}_{i}^{G^{*}_{n}+}\\},\delta_{\mathbf{R}},\delta_{A_{i}}\Big{)}\times
I\Big{(}\kappa_{n}>\sum\limits_{\mathfrak{A}_{i}^{G^{*}_{n}+}}A_{j}^{G^{*}_{n}+}\Big{)}.$
(S2)

A rank ordering, in which available treatments are offered, is determined by
$G^{*}_{n,R_{i}}$, and subsequently in this order, each individual $i$ is
assigned treatment according to the individual rule $g_{i}$ that may vary
across individuals, until the resource supply is exhausted. $g_{i}$ represents
each individual’s individualized dynamic regime, followed in the observed
data, that determines their treatment if a unit is available (i.e.,
$I\Big{(}\kappa_{n}>\sum\limits_{\mathfrak{A}_{i}^{G^{*}_{n}+}}A_{j}^{G^{*}_{n}+}\Big{)}=1$).

Notably, a generalized resource-limited regime admits the possibility of an
assignment of strictly less than $\kappa_{n}$ treatment units, or even no
treatment units at all, within a given realization of a cluster. Any IR $g$ in
iid settings would have an analogue in this class, whereby $\kappa_{n}\geq n$,
$g_{i}=g:(l,\delta_{A_{i}})\mapsto g(l,\delta_{A_{i}})$ for all individuals,
and $\\{\delta_{A_{1}},\dots,\delta_{A_{n}}\\}$ are iid. In this sense, such
regimes are rank-static, but individually-dynamic: a single set of individual
ranks is assigned regardless of cluster configuration $\mathbf{L}_{n}$, but
individuals make decisions when treatments are available according to a non-
trivial function $g$. Alternatively, the general policy class
$\Pi_{n}(\kappa_{n})$ is the strict subset of $\Pi^{F}_{n}(\kappa_{n})$ in
which $g_{i}$ is the trivial function that maps to $1$ for all $i$. In this
sense, regimes in $\Pi_{n}(\kappa_{n})$ are rank-dynamic, but individually-
static.

An arbitrary regime in $\Pi^{F}_{n}(\kappa_{n})$ is both rank-dynamic and
individually-dynamic. Thus, we can understand this space as a product of the
space of rank regimes $G^{*}_{n,\mathbf{R}}$ and the $n$-dimensional space of
individualized treatment regimes $g_{i}$. More formally, let $\Pi$ represent
the set of possible functions $g_{i}$, and $\Pi^{n}\equiv\\{\Pi\\}^{n}$. Then
we see that there will exist a bijective mapping between
$\Pi_{n}(\kappa_{n})\times\Pi^{n}$ and $\Pi^{F}_{n}(\kappa_{n})$ because
elements $G_{n}\in\Pi_{n}(\kappa_{n})$ are uniquely characterized by their
rank function $G^{*}_{n,\mathbf{R}}$.

These rank-based formulations of generalized constrained CRs in this
manuscript will be important in future work for investigators interested in
regimes that intervene on rank alone via $G^{*}_{n,\mathbf{R}}$, or on two-
stage regimes via $G^{*}_{n,\mathbf{R}}$ and then subsequently on individual’s
treatment acceptance (conditional on treatment availability) via $g_{i}$.

####  A.1.2 Stochastic and fully deterministic CRs

In this section we draw connections between generalized CRs in
$\Pi^{F}_{n}(\kappa_{n})$ and general stochastic IRs $g:(l,\delta)\mapsto
g(l,\delta)$. General stochastic IRs despite being deterministic functions of
$L_{i}$ and $\delta$, are so-called because of their non-trivial dependence on
the exogenous randomizer term $\delta$. Heuristically, they are often
understood as two-stage regimes: first an individual will realize a value of
$\delta\sim\text{Unif}(0,1)$, which determines a deterministic IR
$g^{D}:l\mapsto g^{D}(l)$ used to assign their treatment. This interpretation
is apparent in the following formulation, where we let $\Pi^{D}$ be the set of
all deterministic IRs $g^{D}$:

$g(l,\delta)=\sum\limits_{\Pi^{D}}g^{D}(l)I\Big{(}\delta\in
q^{\dagger}_{g}(g^{D})\Big{)},$

where $q^{\dagger}_{g}$ is a bijective function characteristic of $g$ that
maps elements of $\Pi^{D}$ onto a partition of the unit interval. As such, we
have the following simple identity for the intervention density given $l$:

$P(A^{g+}=a\mid
L=l)=\sum\limits_{\Pi^{D}}I\Big{(}a=g^{D}(l)\Big{)}q^{*}_{g}(g^{D}),$

where $q^{*}_{g}$ is the function corresponding to $q^{\dagger}_{g}$ such that
$q^{*}_{g}(g^{D})$ evaluates to the width of the interval
$q^{\dagger}_{g}(g^{D})$. In the following, we extend this heuristic, and
corresponding formulation, to the class of CRs $\Pi^{F}_{n}(\kappa_{n})$.

######  Definition S1 (Fully-deterministic CRs).

The class of fully-deterministic CRs $\Pi^{D}_{n}(\kappa_{n})$ is the subset
of regimes $G_{n}$ in $\Pi^{F}_{n}(\kappa_{n})$ such that, for each
$\mathbf{l}_{n}$,
$\mathbb{E}_{\delta}[I(G_{n}(\mathbf{l}_{n},\delta)=\mathbf{a}_{\mathbf{l}_{n}})]=1$
for some
$\mathbf{a}_{\mathbf{l}_{n}}\in\\{\mathbf{a}_{n}\mid\lVert\mathbf{a}_{n}\rVert\leq\kappa_{n}\\}$.

Fully-deterministic regimes thus are characterized by an assignment mechanism
that maps a covariate vector $\mathbf{l}_{n}$ to a single treatment vector
$\mathbf{a}_{n}$ and thus we omit the randomizer term $\delta$ as an argument.

Analogous to IRs $g$, a given non-deterministic CR $G_{n}$, that is, any
$G_{n}$ that depends non-trivially on $\delta$, can be expressed as a
distribution over the set of deterministic CRs, that is

$G_{n}(\mathbf{l}_{n},\delta)=\sum\limits_{\Pi^{D}_{n}(\kappa_{n})}G_{n}^{D}(\mathbf{l}_{n})I\big{(}\delta\in
q^{\dagger}_{G_{n}}(G^{D}_{n})\big{)},$

where $q^{\dagger}_{G_{n}}$ is a bijective function characterizing $G_{n}$
that maps elements of $\Pi^{D}_{n}(\kappa_{n})$ onto increments forming a
partition of the unit interval. As such, it follows that

$\displaystyle\mathbb{P}^{F}_{n}(\mathbf{A}^{G_{n}+}_{n}=\mathbf{a}_{n}\mid\mathbf{L}_{n}=\mathbf{l}_{n})$
$\displaystyle=\sum\limits_{\Pi^{D}_{n}(\kappa_{n})}I\big{(}\mathbf{a}_{n}=G^{D}_{n}(\mathbf{l}_{n})\big{)}q^{*}_{G_{n}}(G^{D}_{n}),$

where $q^{*}_{G_{n}}$ is defined relative to $q^{\dagger}_{G_{n}}$ analogously
to $q^{*}_{g}$.

###  A.2 $L$-equitable CRs

In this section, we introduce a class of regimes under which a form of
conditional stochastic exchangeability (in the de Finetti sense) is induced
for individual treatment variables. We refer to such regimes as stochastic
$L$-equitable CRs. We call such regimes $L$-equitable because of the
impartiality with which they regard features of a cluster that depend on
individual indices $i$, with respect to covariates $L_{i}$. Articulating such
a class of regimes is important because it is defined by a feature that is
arguably necessary for any regime understood as “equitable” or “fair” in the
plain-language sense. Formally, such regimes are defined as follows:

######  Definition S2 (Stochastic $L$-equitable CRs).

The class of stochastic $L$-equitable CRs $\Pi^{*}_{n}(\kappa_{n},L_{i})$ is
the subset of regimes $G_{n}\in\Pi_{n}(\kappa_{n})$ such that for each
individual $i$, and for all laws $\mathbb{P}_{n}^{F}$, the following equality
holds for each covariate composition $\mathbb{l}_{n}$ and covariate value $l$:

$\mathbb{P}_{n}^{F}(A^{G_{n}+}_{i}=1\mid
L_{i}=l,\mathbb{L}_{n}=\mathbb{l}_{n})=p^{G_{n}}(l,\mathbb{l}_{n})\in[0,1].$

In words, $\Pi^{*}_{n}(\kappa_{n},L_{i})$ is the class of CRs for which all
individuals with the same covariate value $L_{i}=l$ in a cluster with
composition $\mathbb{l}_{n}$ have the same probability of treatment assignment
$p^{G_{n}}(l,\mathbb{l}_{n})$, regardless of the specific instantiation of
$\mathbf{L}_{n}$ or the law of the full data, $\mathbb{P}_{n}^{F}$. Under a
stochastic $L$-equitable regime, the $n$ individuals’ values of $L_{i}$ could
be arbitrarily permuted without perturbing the probability that an arbitrary
individual with covariate value $l$ would be assigned treatment.

Whenever we consider an $L$-equitable regime, $q^{*}_{i,\mathbb{l}_{n}}$ will
be fixed for all $i$. By consequence we have the following result generalizing
Proposition 1 of Section 3.

######  Proposition S1.

Consider a law $\mathbb{P}^{F}_{n}\in\mathcal{M}^{AB}_{n}$ and
$G_{n}\in\Pi^{*}_{n}(\kappa_{n},L_{i})$. Then
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[\overline{Y}^{{G}_{n}}_{n}]=\sum\limits_{o\in\mathcal{O}_{i}}yf_{i}^{G_{n}}(o).$

###### Proof.

$\displaystyle\mathbb{E}_{\mathbb{P}^{F}_{n}}[\overline{Y}^{{G}_{n}}_{n}]=\frac{1}{n}\sum\limits_{j=1}^{n}\mathbb{E}_{\mathbb{P}^{F}_{n}}[{Y}^{{G}_{n}}_{j}]=\frac{1}{n}\sum\limits_{j=1}^{n}\sum\limits_{o}yf_{j}^{G_{n}}(o)=\frac{1}{n}\sum\limits_{j=1}^{n}\sum\limits_{o}yf_{i}^{G_{n}}(o)=\sum\limits_{o}yf_{i}^{G_{n}}(o).$

The first equality follows by linearity of expectation function. The second
follows from Theorem 1. The third follows because $q_{j}^{*}=q^{*}_{i}$ for
all $j$ under regimes $G_{n}\in\Pi^{*}_{n}(\kappa_{n},L_{i})$. ∎

Note that a complete randomization design is a special case of a stochastic
$L$-equitable CRs in which $q^{*}_{i,\mathbb{l}_{n}}(1\mid
l)=\frac{\kappa_{n}}{n}$ for all $l$, $\mathbb{l}_{n}$. In contrast, a
Bernoulli design with probability $q^{*}_{i,\mathbb{l}_{n}}(1\mid
l)=\frac{\kappa_{n}}{n}$ is not an $L$-equitable CR, because under such a
design there would be a strictly positive probability that all individuals in
the cluster receive treatment, $q^{*}_{i,\mathbb{l}_{n}}(1\mid l)$. This
illustrates that the elements in $\mathbf{A}^{G_{n}+}_{n}$ are not in general
independent under $L$-equitable CRs, as observed by savje2021average in
regards to complete randomization designs.

In the following subsection we locate the $L$-rank-preserving class of regimes
$\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ considered in the main text as a
subclass of $\Pi^{*}_{n}(\kappa_{n},L_{i})$.

####  A.2.1 Deterministic, mixed-deterministic and $L$-rank-preserving CRs

We first define a subclass of stochastic $L$-equitable regimes
$\Pi^{*}_{n}(\kappa_{n},L_{i})$ closely-related to the $L$-rank-preserving
class of regimes $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ introduced in Section
2, which we refer to as deterministic CRs.

######  Definition S3 (Deterministic CRs).

The class of deterministic CRs $\Pi^{d*}_{n}(\kappa_{n},L_{i})$ is the subset
of regimes $G_{n}$ in $\Pi^{*}_{n}(\kappa_{n},L_{i})$ such that for each
$\mathbb{l}_{n}$, there exists at most one $l^{\prime}\in\mathcal{L}$ such
that $p^{G_{n}}(l^{\prime},\mathbb{l}_{n})\not\in\\{0,1\\}$.

Under such a regime, there will exist a set of functions
$\Lambda_{\mathbb{l}_{n}}:\mathcal{L}_{i}\rightarrow\mathbb{R}$ such that for
each $\mathbb{l}_{n}$
$\Lambda_{\mathbb{l}_{n}}(L_{i})>\Lambda_{\mathbb{l}_{n}}(L_{j})\implies
G^{*}_{n,R_{i}}(\mathbf{L}_{n},\delta)>G^{*}_{n,R_{j}}(\mathbf{L}_{n},\delta)$
for all $i,j$, almost surely. Note these regimes
$\Pi^{d*}_{n}(\kappa_{n},L_{i})$ differ from those in
$\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ defined in Section 2 in that coarsened
rank functions $\Lambda_{\mathbb{l}_{n}}$ may vary with a cluster’s
composition $\mathbb{l}_{n}$, and are indexed as such. Thus
$\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)\subset\Pi^{d*}_{n}(\kappa_{n},L_{i})$.

Conditional on $\mathbb{l}_{n}$, regimes in $\Pi^{d*}_{n}(\kappa_{n},L_{i})$
are static in the sense that the function $\Lambda_{\mathbb{l}_{n}}$ is fixed.
We generalize $\Pi^{d*}_{n}(\kappa_{n},L_{i})$ to a class of regimes that is
stochastic with respect to $\Lambda_{\mathbb{l}_{n}}$. Heuristically,
conditional on $\mathbb{l}_{n}$, these regimes would be implemented by first
selecting a coarsened rank function $\Lambda_{\mathbb{l}_{n}}$ using a pre-
specified distribution over a suitable set of such functions, and then
implementing the corresponding regime in $\Pi^{d*}_{n}(\kappa_{n},L_{i})$.
First, we define some additional notation. Let $\mathcal{L}_{\Lambda}$ be the
set of all coarsened rank functions, grouped by rank orderings of some
partitioning of $\mathcal{L}$. For example, if $\mathcal{L}\equiv\\{1,2,3\\}$,
$\mathcal{L}_{\Lambda}$ will have groups of coarsened rank functions defined
by the following orderings:

$\displaystyle\begin{Bmatrix}(1,2,3),&(2,1,3),&(3,2,1),\\\
(1,3,2),&(2,3,1),&(3,1,2),\\\ (\\{1,2\\},3),&(\\{1,3\\},2),&(\\{2,3\\},1),\\\
(3,\\{1,2\\}),&(2,\\{1,3\\}),&(1,\\{2,3\\}),\\\ &(\\{1,2,3\\})&\end{Bmatrix}.$

The first six rank orderings correspond to rank orderings over the elements in
$\mathcal{L}$. The second six rank orderings correspond to rank orderings over
all possible coarsenings of $\mathcal{L}$ with only 2 elements. The final rank
ordering is the trivial one in which all individuals are assigned the same
rank group. Note that $\mathcal{L}_{\Lambda}$ will have a finite number of
elements whenever $\mathcal{L}$ has a finite number of elements.

Then, let $q^{*}_{\Lambda,\mathbb{l}_{n}}$ denote some distribution over the
coarsened rank functions in $\mathcal{L}_{\Lambda}$, and define the following
policy class:

######  Definition S4 (Mixed-deterministic CRs).

The class of mixed-deterministic CRs
$\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda,\mathbb{l_{n}}})$ is the subset
of regimes $G_{n}$ in $\Pi_{n}(\kappa_{n})$ for which
$\delta_{\mathbf{R}}\equiv\\{\delta_{\Lambda},\delta^{\prime}_{\mathbf{R}}\\}$,
and the following properties hold for some set of distributions
$q^{*}_{\Lambda,\mathbb{l}_{n}}$:

  * 1.

For each $\mathbb{l}_{n}$, $G_{n}$ is in part characterized by
$q^{*}_{\Lambda,\mathbb{l_{n}}}$ that puts mass on each
$G^{*}_{n}\in\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ proportional to
$q^{*}_{\Lambda,\mathbb{l_{n}}}(\Lambda)$, as determined by
$\delta_{\Lambda}$, and where the implemented regime
$G^{D}_{n}\in\Pi^{D}_{n}(\kappa_{n})$ is determined by
$\delta^{\prime}_{\mathbf{R}}$, and

  * 2\.

$\delta_{\mathbf{R}}^{\prime}$ is independent of $\delta_{\Lambda}$.

We thus have that $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$ is the subset of
$\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda,\mathbb{l_{n}}})$ in which
$q^{*}_{\Lambda,\mathbb{l_{n}}}$ is degenerate for a single
$\Lambda_{\mathbb{l}_{n}}\equiv\Lambda$, for all $\mathbb{l}_{n}$. In Appendix
B, we consider a regime in the subclass of
$\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda,\mathbb{l_{n}}})$ in which
$q^{*}_{\Lambda,\mathbb{l_{n}}}=q^{*}_{\Lambda}$ for all $\mathbb{l}_{n}$,
i.e., the distribution over coarsened rank functions is fixed for all
compositions.

####  A.2.2 $\epsilon-L$-rank-preserving CRs

Results for large-cluster estimands in Theorem 2 are provided under a
condition $E$ whereby the regimes in $\mathbf{G}_{0}$ are restricted to a
class of $L$-rank-preserving regimes $\Pi_{n}^{d}(\kappa_{n},L_{i},\Lambda)$.
Consideration of $L$-rank-preserving regimes simplifies our consideration of
asymptotic regimes $\mathbf{G}_{0}$ to those characterized by a fixed
coarsened rank function $\Lambda$. However, an investigator interested in a
large-cluster parameter under a deterministic CR in
$\Pi_{n}^{d*}(\kappa_{n},L_{i})$ may be concerned by the seeming lack of
generality in the results of Theorem 2. To highlight the practical weakness of
restricting results to $L$-rank-preserving regimes, note that we may define a
less-restricted policy class as follows:

######  Definition S5 ($\epsilon-L$-rank-preserving deterministic CRs).

The class of $\epsilon-L$-rank-preserving CRs
$\Pi^{d\epsilon}_{n}(\kappa_{n},\Lambda,Q_{L})$ is the subset of regimes
$G_{n}$ in $\Pi^{d*}_{n}(\kappa_{n},L_{i})$ for which their exists a rank-
based formulation where, for each $\mathbb{l}_{n}$, the coarsened rank
functions are invariant, $\Lambda_{\mathbb{l}_{n}}=\Lambda$ whenever
$\sum\limits_{l\in\mathcal{L}}\Big{|}\frac{\mathbb{l}_{n}(l)}{n}-Q_{L}(l)\Big{|}<\epsilon$.

These policy classes strictly increase in size with decreasing $\epsilon$ and
approach $\Pi_{n}^{d*}(\kappa_{n},L_{i})$ as $\epsilon$ approaches 0. Note
that in large clusters under an asymptotic law $\mathbb{P}_{0}$ following
condition $D$, we have that

$\lim\limits_{n\to\infty}\mathbb{P}_{n}\Bigg{(}\Big{|}\frac{\mathbb{L}_{n}(l)}{n}-Q_{L}(l)\Big{|}>\epsilon^{\prime}\Bigg{)}=0,$

which holds for a positive $\epsilon^{\prime}$ arbitrarily close to 0. Thus,
so long as we take $\epsilon$ to be greater than $\epsilon^{\prime}$ by an
arbitrarily small amount, the probability of observing an $\mathbb{l}_{n}$ for
which $\Lambda_{\mathbb{l}_{n}}\neq\Lambda$ will approach 0 with increasing
$n$ for an arbitrarily small $\epsilon$. In other words, we may consider
policy class $\Pi^{d\epsilon}_{n}(\kappa_{n},\Lambda)$ that is only an
arbitrarily smaller subset of $\Pi^{d*}_{n}(\kappa_{n},L_{i})$ in which the
coarsened rank functions are locally invariant within an arbitrarily small
neighborhood around $\mathbb{l}_{n}$.

####  A.2.3 Practical considerations for implementing $L$-equitable regimes

Many investigators will not in general specify interest in a specific
stochastic regime. Alternatively, as has been done for stochastic IRs in the
literature robins2004effects, munoz2012population, young2014identification,
kennedy2019nonparametric, sarvet2023longitudinal, stochastic $L$-equitable CRs
may be more easily specified in terms of $q^{*}_{i,\mathbb{l}_{n}}$ directly
via $p^{G_{n}}$, but we must carefully verify that such a $p^{G_{n}}$ is
compatible with the resource limitation $\kappa_{n}$; it must be the case that
$\sum\limits_{l\in\mathcal{L}}\mathbb{l}_{n}(l)p^{G_{n}}(l,\mathbb{l}_{n})=\kappa_{n}$
for all $\mathbb{l}_{n}$. We show that for any such $p^{G_{n}}$, their will
exist a regime $G_{n}$ with intervention density
$q^{*}_{i,\mathbb{l}_{n}}(1\mid l)=p^{G_{n}}(l,\mathbb{l}_{n})$. In the
following proposition, we let $(\mathbb{b}_{n})_{\mathbb{l}_{n},\kappa_{n}}$
denote the set of compositions $\mathbb{b}_{n}$ compatible with
$\mathbb{l}_{n}$ and $\kappa_{n}$, i.e., those for which
$\sum\limits_{l}\mathbb{b}_{n}(1,l)=\kappa_{n}$ and
$\sum\limits_{a}\mathbb{b}_{n}(a,l)=\mathbb{l}_{n}$ for all $l$. We also let
$(\mathbf{l}_{n})_{\mathbb{b}_{n}}$ denote the set of configurations
$\mathbf{l}_{n}$ compatible with $\mathbb{b}_{n}$, i.e. those for which
$\sum\limits_{i=1}^{n}I\Big{(}\\{l_{i}\in\mathbf{l}_{n}\\}=l\Big{)}=\sum\limits_{a}\mathbb{b}_{n}(a,l)$
for all $l$, and we let $(G_{n}^{D})_{\mathbb{b}_{n}}$ denote the set of fully
deterministic regimes compatible with $\mathbb{b}_{n}$, i.e. those for which
$\sum\limits_{i=1}^{n}I\Big{(}\\{a_{i}\in
G^{D}_{n}(\mathbf{l}_{n})\\}=a\Big{)}I\Big{(}\\{l_{i}\in\mathbf{l}_{n}\\}=l\Big{)}=\mathbb{b}_{n}(a,l)$
for all $a,l$, for each $\mathbf{l}_{n}\in(\mathbf{l}_{n})_{\mathbb{b}_{n}}$.

######  Proposition S2.

Given a cluster with composition $\mathbb{L}_{n}=\mathbb{l}_{n}$, consider an
investigator-specified function $p^{G_{n}}$ such that
$\sum\limits_{l\in\mathcal{L}}\mathbb{l}_{n}(l)p^{G_{n}}(l,\mathbb{l}_{n})=\kappa_{n}$.
Then, there exists a stochastic $L$-equitable CR
$G_{n}\in\Pi^{*}_{n}(\kappa_{n},L_{i})$ defined by a distribution function
$q^{*}_{G_{n},\mathbb{l}_{n}}$ over $\Pi^{D}(\kappa_{n})$ such that the
following conditions hold:

  * PS2.1

For each $l$, and all $i$,
$\mathbb{P}_{n}^{F}(A^{G_{n}+}_{i}=1\mid\mathbb{L}_{n}=\mathbb{l}_{n},L_{i}=l)=p^{G_{n}}(l,\mathbb{l}_{n}).$

  * PS2.2

For each $\mathbb{b}_{n}\in(\mathbb{b}_{n})_{\mathbb{l}_{n},\kappa_{n}}$ there
exists a constant $p_{\mathbb{b}_{n}}\in[0,1]$ such that
$q^{*}_{G_{n},\mathbb{l}_{n}}(G_{n}^{D})=p_{\mathbb{b}_{n}}$ for all
$G_{n}^{D}\in(G_{n}^{D})_{\mathbb{b}_{n}}$.

Condition PS2.1 asserts the existence of a weight function characterizing a
stochastic $L$-equitable CR with $q_{i,\mathbb{l}_{n}}$ equivalent to a valid
investigator-specified $p^{G_{n}}$. Condition PS2.2 states that one such
weight function $q^{*}_{G_{n},\mathbb{l}_{n}}$ is that for which all
deterministic regimes resulting in a given composition of assigned treatment
and covariates $\mathbb{b}_{n}$ will be assigned the same weight.

The existence of such a distribution (Condition PS2.1) for any valid
$q^{*}_{i,\mathbb{l}_{n}}$ follows from an application of the Birkhoff von-
Neumann Theorem birkhoff1946three, which states that any bistochastic matrix
is decomposable as a linear combination of permutation matrices with non-
negative weights. A bistochastic matrix is an $n\times n$ matrix where each
row and each column sums to 1, and a permutation matrix is a special
bistochastic matrix where each row and column are strictly comprised of 0’s
and a single element 1. In this case, each column in the bistochastic matrix
would correspond to a single individual and each row would correspond to a
single treatment option - with the first $\kappa_{n}$ rows corresponding to
receiving treatment and the final $n-\kappa_{n}$ rows corresponding to not
receiving treatment. With these interpretations, an investigator’s valid
specification of $q^{*}_{i,\mathbb{l}_{n}}$ would correspond to the
bistochastic matrix for which the $i,j^{th}$ entry would equal $\frac{n\times
q^{*}_{i,\mathbb{l}_{n}}(1\mid L_{i})}{\kappa_{n}}$ for $j\leq\kappa_{n}$ and
$\frac{n\times q^{*}_{i,\mathbb{l}_{n}}(0\mid L_{i})}{\kappa_{n}}$ otherwise.
Each of the permutation matrices then would correspond to a specific rank
ordering in $S(\\{1,\dots,n\\})$, and its positive weight in the Birkhoff
decomposition would correspond to their probability mass in the distribution
used by the investigator to implement the CR.

A given distribution over the fully-deterministic regimes satisfying an
investigator-specified $q^{*}_{i,\mathbb{l}_{n}}$ will not in general be
unique, and furthermore, many parameters of the joint distribution of
$\mathbb{O}_{n}^{G_{n}}$ will in general vary with $G_{n}$ on the subset of
$\Pi^{*}_{n}(\kappa_{n},L_{i})$ with fixed $q^{*}_{i,\mathbb{l}_{n}}$. For
these reasons, and other pragmatic concerns related to implementation,
investigators may be interested in a particular $L$-equitable regime $G_{n}$.
Specifically, there will exist a convex combination of rank orderings whereby:
1) the only rank orderings with non-zero weights will result in
$\mathbb{b}_{n}(1,l)\in\\{\lfloor n\times q^{*}_{i,\mathbb{l}_{n}}(1\mid
l)\rfloor,\lceil n\times q^{*}_{i,\mathbb{l}_{n}}(1\mid l)\rceil\\}$ for each
$l$; and 2) if rank orderings are grouped by a common $\mathbb{b}_{n}$ under a
limitation $\kappa_{n}$, the number of such groups with positive weights will
be at most $|\mathcal{L}|$, the cardinality of $L_{i}$. These properties
follow from application of results in budish2013designing, who also provided
an algorithm for finding such groupings and corresponding weights in
computational time that is polynomial with respect to $|\mathcal{L}|$.

###  A.3 Identifying intervention densities for CRs

In this section we provide a series of results identifying intervention
densities $q^{*}_{i}$ and $\mathbb{q}^{*}_{n}$ for the class of CRs reviewed
in this Appendix. To obtain the interventional density $q^{*}_{i}$ of the
individual-level g-formula for an arbitrary regime
$G_{n}\in\Pi_{n}(\kappa_{n})$, we leverage the following identity which holds
simply by laws of probability:

$q^{*}_{i}(1,l)=\sum\limits_{\mathbb{l}_{n}}q^{*}_{i,\mathbb{l}_{n}}(1\mid
l)\mathbb{Q}_{L,n\mid l}(\mathbb{l}_{n}),$

where $\mathbb{Q}_{L,n\mid
l}(\mathbb{l}_{n})=\mathbb{P}_{n}(\mathbb{L}_{n}=\mathbb{l}_{n}\mid L_{i}=l)$.
Therefore, whenever $\mathbb{Q}_{L,n\mid l}$ and $q^{*}_{i,\mathbb{l}_{n}}$
are identified in terms of $Q_{L}$ then so will $q^{*}_{i}(1\mid l)$.

The following proposition provides identification results for
$q^{*}_{i,\mathbb{l}_{n}}$ for several classes of CRs. In the following, we
let $(\mathbf{l}_{n})_{i,l,\mathbb{l}_{n}}$ denote the set of configurations
$\mathbf{l}_{n}$ with the common entry $l_{i}=l$ and common composition
$\mathbb{l}_{n}$, and we let $(\mathbf{b}_{n})_{\mathbb{b_{n}}}$ denote the
set of configurations $\mathbf{b}_{n}$ with the common composition
$\mathbb{b_{n}}$. The identities therein follow simply by definition of the
regimes and laws of probability.

######  Proposition S3.

Consider a $\mathbb{P}_{n}^{F}\in\mathcal{M}_{n}^{U}$. We have for any
$G_{n}\in\Pi_{n}(\kappa_{n})$

$q^{*}_{i,\mathbb{l}_{n}}(a\mid
l)=\sum\limits_{(\mathbf{l}_{n})_{i,l,\mathbb{l}_{n}}}.\Bigg{[}\sum\limits_{\Pi^{D}_{n}(\kappa_{n})}I\Big{(}a=\big{\\{}a_{i}\in
G^{D}_{n}(\mathbf{l}_{n})\big{\\}}\Big{)}q^{*}_{G_{n}}(G^{D}_{n})\Bigg{]}\mathbb{P}_{n}(\mathbf{L}_{n}=\mathbf{l}_{n}\mid
L_{i}=l,\mathbb{L}_{n}=\mathbb{l}_{n}),$

and

$\displaystyle\mathbb{q}^{*}_{n}(\mathbb{b}_{n}\mid\mathbb{l}_{n})=$
$\displaystyle\sum\limits_{(\mathbf{b}_{n})_{\mathbb{b_{n}}}}\Bigg{[}\sum\limits_{\Pi^{D}_{n}(\kappa_{n})}I\Big{(}\mathbf{a}_{n}=G^{D}_{n}(\mathbf{l}_{n})\Big{)}q^{*}_{G_{n}}(G^{D}_{n})\Bigg{]}\mathbb{P}_{n}(\mathbf{L}_{n}=\mathbf{l}_{n}\mid\mathbb{L}_{n}=\mathbb{l}_{n}).$

Note that an investigator interested in identifying an individual-level
parameter $\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{G_{n}+})]$, or
linear combination thereof, for example,
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[\overline{Y}_{n}^{{G}_{n}}]$, may
sufficiently specify their regime $G_{n}$ via $p^{G_{n}}$, as opposed to the
specific functional form of $G_{n}$. While there is indeed a large class of
regimes
$G^{\prime}_{n}\in\\{\Pi_{n}^{*}(\kappa_{n},L_{i}):\mathbb{q}_{i,\mathbb{l}_{n}}^{*}(1\mid
l)=p^{G_{n}}(l,\mathbb{l}_{n})\ \forall\ \mathbb{l}_{n}\\}$ with common
$p^{G_{n}}$, i.e., stochastic $L$-equitable regimes $G^{*}_{n}$ that would
result in a particular set of assigned treatment probabilities described by
$p^{G_{n}}$, the intervention densities of the individual-level g-formula
$f_{i}^{G^{\prime}_{n}}$ will be invariant in $i$ under $\mathcal{M}_{n}^{AB}$
and identical across all such regimes $G^{\prime}_{n}$,

$q^{*}_{i}(1\mid
l)=\sum\limits_{\mathbb{l}_{n}}p^{G_{n}}(l,\mathbb{l}_{n})\mathbb{Q}_{L,n\mid
l}(\mathbb{l}_{n}),$

which is not a function of $G^{\prime}_{n}$.

However, when interest is in a compositional parameter
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h(\mathbb{O}_{n}^{G_{n}+})]$, such a
specification will not be sufficient: the intervention densities
$\mathbb{q}_{n}^{*}$ in $\mathbb{f}^{G_{n}}$ will vary across regimes with
common $p^{G_{n}}$, and computation of $\mathbb{q}_{n}^{*}$ must proceed
analogously to an arbitrary regime $G_{n}\in\Pi_{n}(\kappa_{n})$, as in
Proposition S3.

Now, we provide closed-form expressions for intervention densities
$q^{*}_{i,\mathbb{l}_{n}}$ under the set of deterministic regimes,
$\Pi^{d*}_{n}(\kappa_{n},L_{i})$, i.e., those characterized by a set of rank
functions $\Lambda_{\mathbb{l}_{n}}$. These derivations also apply to the
$L$-rank-preserving regimes, $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$, by
replacing $\Lambda_{\mathbb{l}_{n}}$ with $\Lambda$ wherever the latter
appears in subsequent expressions. Define the random variable
$\Lambda_{\mathbb{l}_{n}}(L_{i})$ as the individual’s coarsened rank under
regime $G_{n}$ in a composition with $\mathbb{l}_{n}$. Let
$S_{\Lambda_{\mathbb{l}_{n}}}(m)\coloneqq\sum\limits_{\mathcal{L}}\mathbb{l}_{n}(l)I(\Lambda_{\mathbb{l}_{n}}(l)\geq
m)$ denote the number of individuals with coarsened rank group greater than or
equal to $m$ in a cluster with composition $\mathbb{l}_{n}$, and let
$S_{\Lambda_{\mathbb{l}_{n}}}^{-}$ be a corresponding function denoting the
number of such individuals with coarsened rank group strictly greater than $m$
in such a cluster. In a cluster with composition $\mathbb{l}_{n}$, there will
exist a fixed coarsened rank representing the last group to receive at least
one treatment unit under $G_{n}$. We denote the coarsened rank for this group
by $\omega_{\mathbb{l}_{n}}$:

$\displaystyle\omega_{\mathbb{l}_{n}}\coloneqq\textup{inf}\bigg{\\{}c\in\mathbb{R}:S_{\Lambda_{\mathbb{l}_{n}}}(c)<\kappa_{n}\bigg{\\}}.$
(S3)

######  Proposition S4.

Consider a $\mathbb{P}_{n}^{F}\in\mathcal{M}_{n}^{U}$. We have for any
$G_{n}\in\Pi^{d*}_{n}(\kappa_{n},L_{i})$

$\displaystyle q^{*}_{i,\mathbb{l}_{n}}(1\mid l)$
$\displaystyle=\begin{cases}\frac{\kappa_{n}-S_{\Lambda_{\mathbb{l}_{n}}}^{-}(\omega_{\mathbb{l}_{n}})}{S_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}})-S_{\Lambda_{\mathbb{l}_{n}}}^{-}(\omega_{\mathbb{l}_{n}})}&:\Lambda_{\mathbb{l}_{n}}(l)=\omega_{\mathbb{l}_{n}},\\\
I\Big{(}\Lambda_{\mathbb{l}_{n}}(l)>\omega_{\mathbb{l}_{n}}\Big{)}&:\text{otherwise},\end{cases}$
(S4)

and for a $\mathbb{P}_{n}^{F}\in\mathcal{M}^{AB}_{n}$, we have:

$\displaystyle\mathbb{q}^{*}_{n}(\mathbb{b}_{n}\mid\mathbb{l}_{n})=\begin{cases}0&\text{if
there exists an }l\in\mathcal{L}\text{ such that:}\\\ &\ \ \ \
\Lambda_{\mathbb{l}_{n}}(l)>\omega_{\mathbb{l}_{n}}\text{ and
}\mathbb{b}_{n}(1,l)\neq\mathbb{l}_{n}(l);\text{ or }\\\ &\ \ \ \
\Lambda_{\mathbb{l}_{n}}(l)<\omega_{\mathbb{l}_{n}}\text{ and
}\mathbb{b}_{n}(1,l)\neq 0;\text{ or }\\\ &\ \ \ \
\sum\limits_{l^{\prime}:\Lambda_{\mathbb{l}_{n}}(l^{\prime})=\omega_{\mathbb{l}_{n}}}\mathbb{b}_{n}(1,l^{\prime})\neq\kappa_{n}-S^{-}_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}}),\\\
\frac{\prod\limits_{l:\Lambda_{\mathbb{l}_{n}}(l)=\omega_{\mathbb{l}_{n}}}\begin{pmatrix}[c]\mathbb{b}_{n}(1,l)\\\
\mathbb{l}_{n}(l)\end{pmatrix}}{\begin{pmatrix}[c]\kappa_{n}-S^{-}_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}})\\\
S_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}})-S^{-}_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}})\end{pmatrix}}&\text{
otherwise}.\end{cases}$ (S5)

The identity in (S4) follows by recognizing that $\omega_{\mathbb{l}_{n}}$ may
be interpreted as the last rank-group in which individuals could receive
treatment with positive probability. By construction of the regime,
individuals in rank-groups after $\omega_{\mathbb{l}_{n}}$ certainly will not
receive a treatment unit and individuals in rank-groups before
$\omega_{\mathbb{l}_{n}}$ certainly will receive a treatment unit. Individuals
in rank-group $\omega_{\mathbb{l}_{n}}$ have the same probability of receiving
treatment, by definition of the regime. As there are
$\kappa_{n}-S_{\Lambda_{\mathbb{l}_{n}}}^{-}(\omega_{\mathbb{l}_{n}})$
treatment units remaining for this group, and
$S_{\Lambda_{\mathbb{l}_{n}}}(\omega_{\mathbb{l}_{n}})-S_{\Lambda_{\mathbb{l}_{n}}}^{-}(\omega_{\mathbb{l}_{n}})$
individuals in the group, then their quotient gives the probability.

The identity in (S5) follows by recognizing that any composition that does not
have the appropriate counts of treatment within each rank group (per the
preceding arguments) will have 0 probability. When there are multiple
covariate patterns that map to the rank group $\omega_{\mathbb{l}_{n}}$, then
each of the valid compositions will be equally likely with probability equal
to their relative frequency, given by a quotient derived using simple
combinatoric identities.

######  Corollary S1.

Consider a $\mathbb{P}_{n}^{F}\in\mathcal{M}_{n}^{AB}$. We have for any
$G_{n}\in\Pi^{d*}_{n}(\kappa_{n},L_{i})$ where $\Lambda_{\mathbb{l}_{n}}$ is
bijective that
$q^{*}_{n}(\mathbb{b}_{n}\mid\mathbb{l}_{n})=I\Big{(}\mathbb{b}_{n}=\mathbb{b}_{\mathbb{l}_{n}}^{G_{n}+}\Big{)}.$

When $\Lambda$ is bijective, i.e. $L_{i}\neq
L_{j}\implies\Lambda(L_{i})\neq\Lambda(L_{j})$, then there is only a single
value $l$ such that $\Lambda(l)=\omega_{\mathbb{l_{n}}}$ and thus there is
only a single composition $\mathbb{b}_{\mathbb{l}_{n}}^{G_{n}+}$ compatible
with $G_{n}$ given a composition $\mathbb{l}_{n}$.

Under $\mathcal{M}_{n}^{AB}$, the terms $\mathbb{Q}_{L,n\mid l}$ and
$\mathbb{P}_{n}(\mathbf{L}_{n}=\mathbf{l}_{n}\mid\mathbb{L}_{n}=\mathbb{l}_{n})$
are both identified in terms of $Q_{L}$ (see Appendix C). Therefore,
$q^{*}_{i}$ and $\mathbb{q}^{*}_{n}$ are both both identified in terms of
$Q_{L}$ under $\mathcal{M}_{n}^{AB}$ in all the expressions in Propositions S3
and S4, since the remaining terms are investigator-specified or known
functions. Although identified, computations of $q_{i}^{*}$ and
$\mathbb{q}^{*}_{n}$ for generalized regimes in Proposition S3 involve
evaluating very large sums over $(\mathbf{b}_{n})_{\mathbb{b_{n}}}$ or
$(\mathbf{l}_{n})_{i,l}$ and $\Pi^{D}_{n}(\kappa_{n})$ for a general non-
deterministic regime.

Evidently, $\mathbb{q}^{*}_{n}$ may be easier to compute than $q^{*}_{i}$ for
an $L$-equitable deterministic CR. However, evaluating the g-formulae
$f_{i}^{G_{n}}$ and $\mathbb{f}^{G_{n}}$ will involve summing over the set of
possible compositions $\mathbb{l}_{n}$ in either case. Nevertheless, the
resulting computations will involve summations of orders of magnitude fewer
terms than those that require summing over $(\mathbf{b}_{n})_{\mathbb{b_{n}}}$
or $(\mathbf{l}_{n})_{i,l}$ and $\Pi^{D}_{n}(\kappa_{n})$. Furthermore,
additional computational advantages are possible under $V$-equitable regimes.

###  A.4 Computational considerations for regimes that are at most a function
of $\mathbf{V}_{n}$

When parameters for finite clusters of size $n^{*}$ are of interest, practical
concerns may arise about plug-in estimation of the compositional g-formula, or
the intervention densities for the individual-level g-formula. We re-state
these expressions here, for convenience:

$\displaystyle\mathbb{E}_{\mathbb{P}^{F}_{n^{*}}}[h(\mathbb{O}_{n^{*}}^{G_{n}+})]=$
$\displaystyle\sum\limits_{\mathbb{o}_{n^{\ast}}}h(\mathbb{o}_{n^{\ast}}){\mathbb{f}}^{G_{n^{\ast}}}(\mathbb{o}_{n^{*}})$
(S6) $\displaystyle q^{*}_{i}(1\mid l)=$
$\displaystyle\sum\limits_{\mathbb{l}_{n^{*}}}q^{*}_{i,\mathbb{l}_{n^{*}}}(1\mid
l)\mathbb{Q}_{L,n^{*}\mid l}(\mathbb{l}_{n^{*}}).$ (S7)

If we assume a binary outcome, then the number of terms in the summand of (S6)
(i.e., the cardinality of $\mathbb{o}_{n^{\ast}}$) may be formulated as the
number of weak compositions of $n^{*}$ into exactly $4|\mathcal{L}|$ parts,
and thus is obtained by the binomial coefficient
$\Big{(}^{n^{*}+4|\mathcal{L}|-1}_{n^{*}}\Big{)}$. Similarly, the number of
terms in in the summand of (S7) is obtained by
$\Big{(}^{n^{*}+|\mathcal{L}|-1}_{n^{*}}\Big{)}$.

We emphasize that the cluster size defining a parameter of interest, $n^{*}$,
is specified separately from $n$, the size of the cluster representing the
observed data on hand. For example, $n$ will often be moderately large,
whereas when interest is in a finite-cluster parameter (vs. a parameter
defined by an arbitrarily large cluster), $n^{*}$ will be relatively small,
i.e. $n^{*}<<n$. For example, our data analysis (presented in Section 6 and
Appendix D) had $n=13,011$ and considered finite-cluster parameters
corresponding to $n^{*}=20$, representing a reasonable number of beds in a
moderately-sized intensive care unit. Nevertheless, $|\mathcal{L}|$ will often
be large, reflecting an investigators concerns around adequate confounder
control. In our data analysis, we had $|\mathcal{L}|=37632$, and thus
$|\mathbb{O}_{n^{*}}|$ is prohibitively large.

Nevertheless, investigators are often interested in regimes defined by
$V_{i}=c(L_{i})$, a coarsening of the covariates used for confounder control,
such that $|\mathcal{V}|<<|\mathcal{L}|$. In other words, investigators
consider a regime
$G_{n^{*}}:\\{\mathcal{V}\\}^{n^{*}}\times[0,1]\rightarrow\\{0,1\\}^{n^{*}}.$
In our data analysis, we considered a $V$-rank-preserving regime
${G}_{n^{*}=20}^{\Lambda_{2}}$ adapted from a proposal of nyc2015ventilator,
where $V_{i}$ is the value of an individual’s trichotomized SOFA score. In the
following, we present a re-formulation of (S6) and (S7), which leverages
distinctions between $|\mathcal{V}|$ and $|\mathcal{L}|$. Let
$U_{i}\equiv\\{A_{i},V_{i}\\}$ and $W_{i}\equiv\\{Y_{i},A_{i},V_{i}\\}$.
Furthemore, let $Q_{V}$ denote the marginal densities of $V_{i}$ and $Q_{L\mid
v}$ be the conditional density of $L_{i}$ given $V_{i}=v$ under
$\\{\mathbb{P}_{n},\mathbb{P}_{n^{*}}\\}\in\mathbb{P}_{0}^{F}$ following
condition $D$. Then we consider the following definitions and subsequent
proposition:

$\displaystyle\mathbb{Q}_{V,n^{*}}(\mathbb{v}_{n^{*}})$
$\displaystyle\coloneqq\frac{n^{*}!}{\prod\limits_{l\in\mathcal{V}}\mathbb{v}_{n^{*}}(v)!}\prod\limits_{v\in\mathcal{V}}Q_{V}(v)^{\mathbb{v}_{n^{*}}(v)}$
$\displaystyle\mathbb{q}^{**}_{{n^{*}}}(\mathbb{u}_{n^{*}}\mid\mathbb{v}_{n^{*}})$
$\displaystyle\coloneqq\mathbb{P}_{n^{*}}(\mathbb{U}^{G_{n}^{*}+}_{n^{*}}=\mathbb{u}_{n^{*}}\mid\mathbb{V}_{n^{*}}=\mathbb{v}_{n^{*}})$
$\displaystyle Q^{*}_{Y}(y\mid a,v)$
$\displaystyle\coloneqq\sum\limits_{l:c(l)=v}Q_{Y}(y\mid a,l)Q_{L\mid v}(l)$
$\displaystyle\mathbb{Q}^{*}_{Y,n^{*}}(\mathbb{w}_{n^{*}}\mid\mathbb{u}_{n^{*}})$
$\displaystyle\coloneqq\prod\limits_{a,v}\frac{\mathbb{u}_{n^{*}}(a,v)!}{\prod\limits_{y\in\mathcal{Y}_{i}}\mathbb{w}_{{n^{*}}}(y,a,v)!}\prod\limits_{y\in\mathcal{Y}_{i}}Q^{*}_{Y}(y\mid
a,v)^{\mathbb{w}_{{n^{*}}}(y,a,v)}$
$\displaystyle\mathbb{f}_{red}^{G_{n^{*}}}(\mathbb{w}_{n^{*}})$
$\displaystyle\coloneqq\mathbb{Q}^{*}_{Y,n^{*}}(\mathbb{w}_{n^{*}}\mid\mathbb{u}_{n^{*}})\mathbb{q}^{**}_{n^{*}}(\mathbb{u}_{n^{*}}\mid\mathbb{v}_{n^{*}})\mathbb{Q}_{V,n^{*}}(\mathbb{v}_{n^{*}})$
(S8) $\displaystyle q^{*}_{i,\mathbb{v}_{n^{*}}}(1\mid v)$
$\displaystyle\coloneqq\mathbb{P}_{n^{*}}({A}^{G_{n}^{*}+}_{i}=a\mid
V_{i}=v,\mathbb{V}_{n^{*}}=\mathbb{v}_{n^{*}})$ (S9)

We refer to expression (S8) as the reduced compositional g-formula. Expression
(S9) is defined analogously to $q^{*}_{i,\mathbb{l}_{n^{*}}}$ except with $v$
and $\mathbb{v}_{n^{*}}$ replacing $l$ and $\mathbb{l}_{n^{*}}$. The following
proposition links this g-formula to a counterfactual parameter of interest,
where we define $\mathbb{Q}_{V,{n^{*}}\mid v}$ analogously to
$\mathbb{Q}_{L,{n^{*}}\mid l}$.

######  Proposition S5.

Consider a law $\mathbb{P}_{n}^{F}\in\mathcal{M}^{AB}_{n}$, a real-valued
function $h^{\prime\prime}$ and a regime $G_{n}\in\Pi_{n}(\kappa_{n})$, with
$G_{n}$ a function of at most $\\{\mathbf{V}_{n},\delta\\}$, i.e.,
$G_{n}:\\{\mathcal{V}\\}^{n}\times[0,1]\rightarrow\\{0,1\\}^{n}$, and with
$V_{i}\coloneqq c(L_{i})$ for all $i$. Then
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h(\mathbb{W}_{n}^{G_{n}+})]=\sum\limits_{\mathbb{w}_{n}}h^{\prime\prime}(\mathbb{w}_{n})\mathbb{f}^{G_{n}}_{red}(\mathbb{w}_{n})$
whenever the right hand side of the equation is well-defined, and also that
$q^{*}_{i}(1\mid l)=\sum\limits_{\mathbb{v}_{n}}q^{*}_{i,\mathbb{v}_{n}}(1\mid
c(l))\mathbb{Q}_{V,n\mid c(l)}(\mathbb{v}_{n}).$

For deterministic regimes in $\Pi_{n^{*}}^{d}(\kappa_{n^{*}},V_{i},\Lambda)$,
we obtain $q^{*}_{i,\mathbb{v}_{n^{*}}}$ and $\mathbb{q}^{**}_{n^{*}}$
analogously to Proposition S4. Arguments are isomorphic, so we do not repeat
them here. Proposition S5 shows us that plug-in estimation of compositional
and individual-level g-formula parameters poses far fewer computational
demands in practice, than is suggested upon inspection of their unreduced
functionals presented in Section 3.3. Additional computational gains may be
obtained for