# Single-World Intervention Graphs for Defining, Identifying, and
Communicating Estimands in Clinical Trials

Alex Ocampo  Jemar R. Bather  Department of Biostatistics, Harvard T.H. Chan
School of Public Health, Boston, MA

###### Abstract

Confusion often arises when attempting to articulate target estimand(s) of a
clinical trial in plain language. We aim to rectify this confusion by using a
type of causal graph called the Single-World Intervention Graph (SWIG) to
provide a visual representation of the estimand that can be effectively
communicated to interdisciplinary stakeholders. These graphs not only display
estimands, but also illustrate the assumptions under which a causal estimand
is identifiable by presenting the graphical relationships between the
treatment, intercurrent events, and clinical outcomes. To demonstrate its
usefulness in pharmaceutical research, we present examples of SWIGs for
various intercurrent event strategies specified in the ICH E9(R1) addendum, as
well as an example from a real-world clinical trial for chronic pain. Latex
code to generate all the SWIGs shown is this paper is made available. We
advocate clinical trialists adopt the use of SWIGs in their estimand
discussions during the planning stages of their studies.

Keywords: SWIG,Estimands, Clinical Trials, Causal Inference, Potential
Outcomes

##  1 Introduction

The ICH E9(R1) addendum [1] was a pivotal achievement in aligning all
stakeholders (e.g., statisticians, clinicians, regulators) on target treatment
effect(s) of randomized clinical trials (RCTs). This guidance came at an ideal
time in statistical history as it aligned with work developed by pioneers in
causal inference. Their advances include methods to define, identify, and
communicate causal treatment effects from both observational data and
randomized trials [2, 3, 4, 5, 6, 7, 8]. Causal inferences has allowed us to
define when we can step beyond the mantra, "correlation does not imply
causation". This is pertinent to the pharmaceutical industry, where
demonstrating causation of a new treatment is the primary objective.

While the tools of causal inference (e.g., potential outcomes, causal graphs,
etc.) are not explicitly mentioned in the addendum, the language of causal
inference appears throughout. For example, Section A.3 states that research
questions such as “how the outcome of treatment compares to what would have
happened to the same subjects under alternative treatment” are pivotal for
drug development and licensing. Another example can be found in Section A.3.2:
“A scenario is envisaged in which the intercurrent event would not occur.”
These types of research questions and hypothetical scenarios can be defined
using potential outcomes, prompting the application of causal inference
methods. As a result, recent research has mathematically translated estimands
from the ICH E9(R1) addendum into the causal inference framework [9, 10]. This
paper continues this trend by providing stakeholders with an accessible
tutorial of how to characterize various estimands described in the addendum
using causal graphs.

Causal graphs were popularized by Pearl [11, 12] in the form of Directed
Acyclic Graphs (DAGs). Figure 1 shows a classic example of a DAG where a
treatment $A$ has a causal effect on the intercurrent event $M$ and outcome
$Y$. Here $M$ also has a causal effect on $Y$. Through a graphical
representation, DAGs encode the research team’s assumptions about which
variables causally affect one another. DAGs make explicit the independencies
(and conditional independencies) between variables, which can be read off of
the graph using the $d$-separation criterion [13]. For these reasons, clinical
trialists can use DAGs to effectively communicate the relationships among
different variables of interest in the study. As a result, DAGs can serve as
an invaluable tool to inform decision-making prior to performing statistical
analyses. They also have the potential advantage of uncovering novel research
questions within established trials, thus mitigating trial expenses while
expanding therapeutic research.

$A$$M$$Y$ Figure 1: An example of a DAG with a treatment $A$, intercurrent
event $M$, and outcome $Y$

However, DAGs are not without limitation. Their primary shortcoming is that
they do not visualize potential outcomes [14], which are needed to define the
estimands outlined in the ICH-E9(R1) addendum. In fact, as we will show in
this paper, the DAG in Figure 1 could be associated with multiple estimands in
a RCT, depending on the strategy used for handling intercurrent events.
Luckily, Single-World Intervention Graphs (SWIGs) overcome this limitation by
allowing DAGs to incorporate potential outcomes. As with DAGs, independencies
among variables can be identified using $d$-separation. Using these
properties, we demonstrate how to graphically represent common target
estimands in clinical trials for various intercurrent event strategies using
SWIGs.

This paper proceeds as follows. Section 2 provides a brief overview of SWIGs.
This gives the reader the prerequisites to digest the example SWIGs in the
subsequent sections. Section 3 provides examples of SWIGs for various
intercurrent event strategies outlined in the ICH E9 addendum. Section 4
provides a case study of a SWIG in the context of a clinical trial for chronic
pain. Lastly, Section 5 concludes with a discussion.

##  2 Overview of SWIGs

In 2013, SWIGs were introduced by Richardson & Robins to unite the potential
outcome and graphical approaches to causal inference [15, 16]. Their paper
makes clear why SWIGs are necessary to do so, by considering the simplest DAG
where treatment $A$ has a causal effect on clinical outcome $Y$,

$A$$Y$ Figure 2: Simplest DAG

where interest lies in the estimand for the average treatment effect:

$\Delta=E[Y(a=1)]-E[Y(a=0)].$

Where $Y(a=1)$ is the potential outcome if experimental treatment were taken
and $Y(a=0)$ is the potential outcome if reference treatment were taken. Note
that all people in the population have both potential outcomes $Y(a=1)$ and
$Y(a=0)$ despite that typically only one of them being observed in practice.
For a review of potential outcomes framework, see [17, 18, 19, 20]. Clearly
there is a disconnect between the DAG and the estimand, as the potential
outcomes $Y(a=1)$ and $Y(a=0)$ do not appear on the DAG. This is where SWIGs
provide added value. To see how, consider the following SWIGs associated with
the DAG in Figure 2:

$A$$a=1$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=1})$
(a) SWIG for Experimental Treatment

$A$$a=0$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=0})$
(b) SWIG for Reference Treatment

Figure 3: Two possible SWIGs for DAG in Figure 2

The SWIGs clearly display the potential outcomes $Y(a=1)$ and $Y(a=0)$. This
is due to the node-splitting transformation that took variable $A$ on the DAG
and split it into two components: $A$ and $a$. Splitting the node represents
asking the same "what if" questions that the potential outcomes pose: "What
would the outcome be in a world where everyone took experimental treatment
($a=1$) or reference treatment ($a=0$)?". Therefore, splitting the node on the
DAG results in changing the downstream variable from the observed outcome $Y$
to its corresponding potential outcomes $Y(a=1)$ and $Y(a=0)$. Since in this
scenario there are two possible potential outcomes, there are two possible
SWIGs. In practice, we often collapse the possible SWIGs into one general SWIG
for any arbitrary $a$, which is shown in Figure 4.

$A$$a$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$
Figure 4: General graph representing the two SWIGs in Figure 3

Most importantly, we can apply the same $d$-separation criterion used for DAGs
to read independencies (and conditional independencies) between the variables
from the SWIGs as well. In doing so we see that $A$ is independent of both
$Y(a=1)$ and $Y(a=0)$ \- i.e. $A\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(a)$. This is because there is no backdoor
path between $A$ and the potential outcomes on the SWIG. Also, note that the
node-splitting blocks the forward path between $A$ and $Y(a)$. This is because
we have now postulated a single-world where everyone took treatment $a$, so
that the $A$ we observed in practice no longer effects $Y(a)$.

Given these independencies, we can identify the estimand of the average
treatment effect $\Delta$ from the observed data as follows:

$\displaystyle\Delta$ $\displaystyle=E[Y(a=1)]-E[Y(a=0)]$
$\displaystyle=E[Y(a=1)|A=1]-E[Y(a=0)|A=0]\hskip
21.33955pt(Y(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$ $\displaystyle=E[Y|A=1]-E[Y|A=0]\hskip
51.21495pt(\text{consistency})$

The independency $Y(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A$, which was given from our assumed SWIG,
allowed us to condition on $A$ in the expectation in line 2 of the equation.
This is intuitive because if $A$ is independent of $Y(a)$ then we would expect
the mean of $Y(a)$ to be the same in any subgroup of $A$ as in the whole
population. In addition, we invoked the consistency assumption in the final
line to change the potential outcome $Y(a)$ to the observed $Y$. The
consistency assumption states that $Y=Y(a=1)A+Y(a=0)(1-A)$ [21]. In layman’s
terms the consistency assumption states that the observed outcome under
observed treatment is consistent with the underlying potential outcome for
that same treatment. See, Table 1 for an example of consistency between
potential outcomes and the observed data for five subjects - data that only
God could see. Additionally, the usual causal assumptions of SUVTA [22] and
positivity [20] apply as well.

__ id $Y(a=0)$ $Y(a=1)$ $A$ $Y$ 1 60 52 1 52 2 45 37 0 45 3 46 38 1 38 4 75 67
1 67 5 21 15 0 21

Table 1: God’s Table

Like DAGs, SWIGs encode one’s assumptions about the process on which we are
collecting data. It may be possible that two statisticians are involved in the
same trial, but draw different SWIGs because they have different causal
assumptions about the real world process. There may be differences in two
statisticians choice on which nodes to split as well. However, the node-
splitting decisions reflect interest in different estimands (or clinical
questions) rather than causal assumptions about variables. This motivates the
following section, which demonstrates SWIGs in the context of the ICH E9(R1)
Addendum.

##  3 Intercurrent Event Strategies from the ICH-E9(R1) Addendum in SWIGs

We now explore visualizing various strategies for handling intercurrent events
(IEs) using SWIGs. There are infinite ways to explore such estimands.
Important work has started taking place in recent years to characterize
specific clinical scenarios using SWIGs that incorporate time, multiple
outcome measurements, and competing risks on the graph [23, 24, 25, 26]. We
have chosen to focus on simple cases in this section for pedagogical reasons
in hopes that readers can afterwards extend the basic principles detailed in
this paper to their own trial work.

In all subsequent examples, $A$ is a binary treatment indicator that can be
set to $a\in\\{0,1\\}$. We will consider a world where we intervene on $A$;
consequently, this node will always be split in all subsequent SWIGs and all
descendants of $a$ will take on their potential outcome representations. Also,
because we are operating in the context of a randomized trial, $A$ will have
no parents in all SWIGs presented; treatment is purely consequence of
randomization and nothing else. The potential outcome $M(a)$ represents the
value of the intercurrent event in a world where treatment $a$ is taken. We
consider $M(a)=1$ if the intercurrent event occurs and $M(a)=0$ if not.
Lastly, $Y(a)$ is the clinical outcome of interest in world where treatment
$a$ is taken.

###  3.1 Treatment Policy

$A$$a$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$
Figure 5: SWIG for an ITT Estimand

In the treatment policy, or intention to treat (ITT), strategy the
intercurrent event is ignored. The corresponding SWIG (Figure 2) admits to
this, as there is no node-splitting taking place on $M(a)$. The treatment
effect (i.e. estimand) of interest measures a combination of the direct effect
of the drug on the outcome and the indirect effect mediated through the
intercurrent event. In the SWIG above, we can see that $Y(a)$ is independent
of $A$, i.e. there are no paths between $A$ and $Y(a)$. Therefore, we can
condition on $A$ in the expectations and then use the consistency assumption
to identify the estimand from the observed data as follows:

$\displaystyle\Delta_{ITT}$ $\displaystyle=E[Y(1)]-E[Y(0)]$
$\displaystyle=E[Y(1)|A=1]-E[Y(0)|A=0]\hskip
21.33955pt(Y(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$ $\displaystyle=E[Y|A=1]-E[Y|A=0]\hskip
51.21495pt(\text{consistency})$

Identification is complete when the estimand of interest can be expressed as a
function of the observed data alone [27, 28] \- i.e., no potential outcome
variables remain in the expression. Note that for the ITT estimand above, the
equivalence $E[Y(1)]-E[Y(0)]=E[Y(1,M(1))]-E[Y(0,M(0))]$ exists, which clearly
demonstrates that the intercurrent event is ignored in this context. Due to
this fact, it would actually be possible to collapse this SWIG into one
identical to Figure 4 and retain the necessary independencies for
identification; however, we recommend including even those IE that are ignored
on the graph because this helps communicate that an ITT strategy has been
adopted.

###  3.2 Hypothetical

The hypothetical strategy envisages a counterfactual scenario - e.g., one in
which the intercurrent event would not occur. A SWIG can make this explicit by
splitting the node on the intercurrent event. Ergo, the "single world" that
the SWIG depicts is one where, in addition to treatment being set to $a$, the
intercurrent event is set to $m=0$. As a consequence of the node-splitting,
the potential outcomes of interest are now $Y(a,m=0)$. This hypothetical
estimand can be written mathematically as:

$\displaystyle\Delta_{hypo}$ $\displaystyle=E[Y(a=1,m=0)]-E[Y(a=0,m=0)]$

We present two examples of SWIGs for a hypothetical estimand. In this first
example (Figure 6), there is an unobserved confounder $U$ between $M(a)$ and
$Y(a,m)$. We indicate that this variable is unobserved by shading in the node.

$A$$a$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$m$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}m})$$U$
Figure 6: SWIG for a hypothetical estimand with an unobserved confounder

As we can see, $Y(a,m)\not\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}M(a)$ because of the backdoor path through
$U$. In other words, when two variables share a common cause they are
inherently related. Therefore,

$\displaystyle E[Y(a=1,m=0)]-E[Y(a=0,m=0)]\neq E[Y|A=1,M=0]-E[Y|A=0,M=0]$

Our attempts to identify the estimand fail because we cannot condition on the
observed $M$ in the expectations. Consequently, we cannot express this causal
contrast of potential outcomes as a function of the observed data. This
example highlights the extra layer of information that the SWIG adds: the
causal assumptions that determine whether identification of the estimand using
the observed data is possible. SWIGs go one step beyond defining estimands in
plain language or with tables.

In the second example, we assume to have observed a rich enough set of
confounders between $M(a)$ and $Y(a,m)$ to obtain the required independence to
identify the causal effect, i.e. the estimand, of interest. We denote
conditioning on a variable, or blocking the causal pathway, by drawing a
square box around this variable on the graph (Figure 4).

$A$$a$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$m$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}m})$$C$
Figure 7: SWIG for a hypothetical estimand with no unobserved confounding

The SWIG shows that we have observed and controlled for $C$ and can therefore
induce the conditional independence $Y(a,m)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}M(a)|C$. This allows us to reach
identification by stratifying, or integrating, across levels of $C$:

$\displaystyle E[Y(a,m)]$ $\displaystyle=\sum_{c}E[Y(a,m)|C=c]P(C=c)$
$\displaystyle=\sum_{c}E[Y(a,m)|C=c,M(a)=m]P(C=c)\hskip
56.9055pt(Y(a,m)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}M(a)|C)$
$\displaystyle=\sum_{c}E[Y(a,m)|C=c,M(a)=m,A=a]P(C=c)\hskip
21.33955pt(Y(a,m)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$
$\displaystyle=\sum_{c}E[Y|C=c,M=m,A=a]P(C=c)\hskip
68.2866pt(\text{consistency})$

Substituting this into $\Delta_{hypo}$:

$\displaystyle\Delta_{hypo}$ $\displaystyle=E[Y(a=1,m=0)]-E[Y(a=0,m=0)]$
$\displaystyle=\sum_{c}E[Y|C=c,M=0,A=1]P(C=c)-\sum_{c}E[Y|C=c,M=0,A=0]P(C=c)$

Therefore, we’ve demonstrated that the hypothetical estimand can be identified
from the observed data under the scenario defined by the SWIG in Figure 7. In
this example, and in most hypothetical scenarios, we are making the assumption
of no unobserved confounding between the IE and outcome. This is why,
oftentimes, the sister assumption, missing at random (MAR)[29], is utilized
for imputing these "missing" potential outcomes using an imputation model that
leverages baseline covariates $C$. Of course, these are untestable
assumptions, which highlight the importance of conducting a sensitivity
analyses when considering a hypothetical estimand. Lastly, those readers
familiar with the mediation literature in causal inference will note that many
hypothetical estimands are examples of controlled direct effects [30].

Of course, it must be plausible that one could in theory imagine such a world.
It is possible to postulate a world in which everyone in your RCT population
either receives treatment or does not receive treatment. However, the ability
to conceive of a world in which the IE never happens will also depend on the
specific IE at hand. Depending on the clinical context, it may be plausible to
consider a world in which rescue medication is not taken or treatment
discontinuation due to administrative dropout did not occur. In the event of
active drug side effects, this may be less plausible. In the case of death,
this is highly implausible.

###  3.3 Composite

Composite estimands incorporate the IE into the variable definition. In the
simplest case of a binary outcome $Y(a)\in\\{0,1\\}$ the composite variable is
defined as:

$U(a)=\begin{cases}Y(a)&\text{if }M(a)=0\\\ 0&\text{if }M(a)=1\\\
\end{cases}\vspace{3mm}$

Which corresponds to treating occurrence of an IE as a failure. Moreover,
there exist strategies for creating composite estimands for continuous
endpoints [31, 32]. The composite strategy is often employed in the case where
the IE $M(a)$ under consideration is death, and hence the potential outcome
$Y(a)$ doesn’t exist when the IE occurs (i.e. $M(a)=1$). We characterize such
an example in Figure 8, which assumes no causal effect of the IE on outcome
and omit the arrow from $M(a)$ to $Y(a)$. An example of a SWIG for such a
composite estimand could then be:

$A$$a$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$U({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$
Figure 8: SWIG for a composite estimand

It is straightforward to see on the SWIG that $U(a)$ is a combination of the
outcome $M(a)$ and $Y(a)$. It is also clear from the SWIG that
$A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U(a)$, allowing us to identify the composite
estimand:

$\displaystyle\Delta_{C}$ $\displaystyle=E[U(1)]-E[U(0)]$
$\displaystyle=E[U(1)|A=1]-E[U(0)|A=0]\hskip
21.33955pt(U(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$ $\displaystyle=E[U|A=1]-E[U|A=0]\hskip
42.67912pt(\text{by consistency})$

In the above example, we do not assume that the IE has a causal effect on the
outcome - i.e. no arrow from $M(a)$ to $Y(a)$. This would be the case if the
IE were death for example. There are of course situations where interest lies
in a composite estimand, but there is an arrow from $M(a)$ to $Y(a)$. For
example, this would unarguably be the case where $M$ is treatment
discontinuation. Under such scenarios, the composite estimand remains
identifiable. In fact, much like the ITT estimand it would remain identifiable
even if there are unobserved confounders between $M(a)$ and $Y(a)$ because no
matter what, the required independence $U(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A$ remains.

###  3.4 Principal Stratum

In the Principal Stratum (PS) strategy, interest lies in a sub-population of
patients defined by whether or not an IE would occur under a particular
treatment. We consider one such example: the population of patients who would
not have the IE under active treatment, i.e. $M(a=1)=0$. This causal contrast,
or estimand, of interest is therefore:

$\displaystyle\Delta_{PS}=E[Y(a=1)|M(a=1)=0]-E[Y(a=0)|M(a=1)=0]$

Consider first the following SWIG, that considers a world in which every
patient took the treatment and conditions on the strata of interest.

$A$$a=1$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=1})=0$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=1})$
Figure 9: SWIG where active treatment is given for those who would not have
the IE under treatment (i.e. conditioning on $M(a=1)=0$)

The SWIG reveals two important facts. Firstly, as is true in all randomized
trials $A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(a)$. Secondly, and most importantly, this
makes clear that we are interested in the direct effect of treatment on
outcome, as sub-setting to the principal strata $M(a=1)=0$ blocks the effect
of treatment mediated through the IE. Using this SWIG, we can identify the
expected value of $Y(a=1)$ in our principal strata, i.e. the first term in
$\Delta_{PS}$:

$\displaystyle E[Y(a=1)|M(a=1)=0]$ $\displaystyle=E[Y(a=1)|M(a=1)=0,A=1]\hskip
7.11317pt(Y(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$ $\displaystyle=E[Y|M=0,A=1]\hskip
14.22636pt(\text{by consistency})$

Which is straightforward as before. The difficulty in PS estimands comes into
play when trying to identify the second term in $\Delta_{PS}$ from the
observed data, i.e. $E[Y(a=0)|M(a=1)=0]$. If we consider the SWIG for the
untreated world

$A$$a=0$$M({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=0})$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a=0})$
Figure 10: SWIG where reference treatment is given for those who would not
have the IE under treatment $M(a=1)=0$

it becomes evident that identifying the second term requires a cross-world
assumption. We want to know the expected value of $Y(a=0)$ in the strata of
$M(a=1)=0$, but these variables do not appear on the same SWIG, or in the same
world. This is a fundamental challenge of PS estimands and SWIGs help to make
this challenge explicit when communicating the estimand. Of course, this
should not downplay the clinical importance of trying to understand PS
estimands and has not hindered statisticians attempts to estimate them [33].
The SWIG helps makes explicit the assumptions needed to do so, and then allows
us to begin discussing how plausible PS assumptions may be.

Given the independencies presented on the two SWIGs, consider trying to
identify the second term of this principal stratum estimand from the observed
data:

$\displaystyle E[Y(a=0)|M(a=1)=0]$ $\displaystyle=E[Y(a=0)|M(a=1)=0,A=0]\hskip
7.11317pt(Y(a)\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A)$ $\displaystyle=E[Y|M(a=1)=0,A=0]\hskip
14.22636pt(\text{by consistency})$

The challenge arises in identifying the stratum of patients in the reference
arm of the trial who would not have had the IE had they taken experimental
treatment, something we never see in practice. The SWIG can help us to
communicate this.

##  4 Example from a Clinical Trial for Chronic Pain

We now outline an example of how one can use a SWIG to characterize an
estimand in a Phase II clinical trial for chronic pain. Callegari et al. [34]
provide a detailed exposition of the various attributes necessary to define a
slew of clinically relevant target estimands in this context. Herein, we focus
on the primary estimand outlined in their paper. We demonstrate how a SWIG can
be used to define this estimand. This SWIG then allows the independencies
between variables to be read off this graph to demonstrate identification of
the estimand. This case study illustrates how a SWIG can aid in communicating
a target estimand from a clinical trial.

To construct the graph, consider the following notation for the relevant
variables used to characterize the estimand so that we can concisely place
them on the graph. As above, we denote $A$ as the random variable representing
treatment assignment. In the context of a chronic pain outlined in Callegari
et al, this variable can take on two values: $A=1$ for active treatment and
$A=0$ for placebo. However, because the SWIGs are equivalent under either
treatment, we only need to draw one SWIG wherein we intervene on $A$ and set
the value to arbitrary $a\in\\{0,1\\}$. We denote the relevant intercurrent
events as:

  * •

$M_{1}$ = Intake of short acting pain relief medication

  * •

$M_{2}$ = Treatment discontinuation due to Adverse Event, Loss of Efficacy, or
intake of prohibited medications

  * •

$M_{3}$ = Change of dose of allowed concomitant medication for pain

  * •

$M_{4}$ = Treatment discontinuation due to Administrative or Other reasons

Their primary estimand definition outlines the IE strategy used for each of
the intercurrent events. That is, $M_{1}$ and $M_{2}$ a treatment policy
strategy is adapted. That is, the data post IE is deemed relevant for these IE
and collected and used in the analysis. For IEs $M_{3}$ and $M_{4}$, a
hypothetical strategy is most relevant. Data post IE, even if available, are
discarded and treated as missing. For these IEs, what would have happened had
the IEs not occured (i.e. $M_{3}=M_{4}=0$) is most clinically relevant. The
potential outcome of interest is $Y(a,m_{1}=0,m_{4}=0)$, representing weekly
mean pain score change from baseline using the 11-point numerical rating scale
that would be observed had treatment $a$ been taken, and had intercurrent
events $M_{1}$ and $M_{4}$ not occurred. Thus, the estimand of interest is:

$\Delta_{RCT}=E[Y(a=1,m_{1}=0,m_{4}=0)]-E[Y(a=0,m_{1}=0,m_{4}=0)]$

$A$$a$$M_{1}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$M_{2}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$M_{3}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a})$$m_{3}$$M_{4}$$m_{4}$$Y({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}m_{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}m_{4}})$C
Figure 11: SWIG for a Clinical Trial in Chronic Pain

It is clear from the SWIG that $A$, $M_{1}$, and $M_{4}$ are d-separated from
the potential outcome $Y(a,m_{1},m_{4})$ given that we can condition on $C$.
These are the necessary independencies for identification. The mathematics for
identifying this estimand are shown in supplementary information 1.

##  5 Discussion

It is easy to get lost in the jargon of estimands; SWIGs provide an escape.
These causal graphs allow a quick view of the clinically relevant treatment
effect. Such visuals may appeal to many, considering that roughly 65% of the
population are visual learners [35]. Time is of the essence to get effective
treatments to those in need; tools like SWIGs help hasten estimand
discussions. Additionally, SWIGs can limit mistakes and misalignment between
stakeholders as they are succinct and clear.

SWIGs encode clinical assumptions about how treatment, IE, confounders, and
outcomes all casually interact with one another. As seen from the examples in
this paper, these assumptions govern whether or not an estimand is
identifiable, and thus estimable from the observed data. IE often embed
observational studies within the RCT, and SWIGs help statisticians understand
how to account for the biases. We recommend clinical trialists adopt SWIGs in
preliminary planning discussions when defining estimands. The visuals are
succinct and the rules of identification are simple to explain to
interdisciplinary teams.

Furthermore, defining estimands explicitly using the mathematical notation of
potential outcomes removes ambiguity. For instance, in the case of the
clinical trial example in section 4, the proposed analysis leads to confusion
about what potential outcome is being targeted. The authors consider treatment
discontinuation due to adverse events, loss of efficacy, and intake of
prohibited medications as unfavorable events in their estimand definition. In
these scenarios, they propose using retrieved drop out (RDO) data in the
analysis, which is aligned with a treatment policy strategy. That would imply
interest in potential outcome $Y(a=1,m_{1}=0,m_{4}=0)$, which is to ignore
$M_{3}$ as we did in section 4. However, should RDO data be unavailable, the
authors suggest imputing outcomes based on the placebo data, which rather
implies interest in $Y(a=0,m_{1}=0,m_{4}=0)$ among the principal stratum
$M_{3}(a=1)=1$. Perhaps it is believed that these potential outcome
distributions are equivalent. Nevertheless, the clinically relevant estimand
needs to be made clear. Furthermore, their primary estimand definition written
in section 3.1 does not align with their characterization of the estimand in
Fig 1. This is because the latter proposes a different IE strategy for $M_{2}$
in the placebo arm, which was not mentioned in section 3.1. The SWIG removes
these ambiguities from the estimand definition by clearly showing the
potential outcome of interest on the graph.

Our work is limited in a number of ways. Firstly, we ignore the "while on
treatment" IE strategy in our presentation of SWIGs because of our own bias
that this type of estimand is difficult to interpret causally. Additionally,
previous work detailing causal diagrams in the estimand framework has added
another mediating variable between the IE and outcome representing the
intervention take post IE [9], but we find this redundant as their
relationship is often deterministic and can be considered jointly in one
variable. This may be a simplification on our part. Should the distinction
prove valuable, this or any other variables should be added to the graph. Our
main goal herein is to show the utility of SWIGs in characterizing estimands.
In practice, many nuances specific to the trial will reveal themselves, making
the SWIGs more complicated than those presented in this paper. Lastly, we used
differences in expectations throughout to define our estimands for simplicity,
but causal estimands can be contrasts (e.g. ratios, etc.) of any marginal
statistical functional (e.g. hazards, odds, etc.).

It is important for clinical trialists–especially statisticians–to learn to
speak the language of causal inference in order to adapt the vast
methodological progress made in this area over the last few decades. We
believe incorporating SWIGs into estimand discussions is one such step in the
right direction. To enable our readers to start using SWIGs, all LaTeX code to
generate the graphs in this paper in included in supporting information 2.

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Appendix: Identification of Estimand in Clinical Trial Example

As stated in section 4, the causal estimand of interest is:

$\Delta_{RCT}=E[Y(a=1,m_{3}=0,m_{4}=0)]-E[Y(a=0,m_{3}=0,m_{4}=0)]$

Using the independencies on the SWIG in Figure X we can identify the estimand.
Since the procedure for identifying the two expectations of potential outcomes
in the contrast above are identical, we show identification of arbitrary
$E[Y(a,m_{3}=0,m_{4}=0)]$:

$\displaystyle E[Y(a,m_{3}=0,m_{4}=0)]$
$\displaystyle=E[Y(a,m_{3}=0,m_{4}=0)|A=a]$
$\displaystyle=\sum_{c}E[Y(a,m_{3}=0,m_{4}=0)|A=a,C=c]Pr(C=c)$
$\displaystyle=\sum_{c}E[Y(a,m_{3}=0,m_{4}=0)|A=a,C=c,M_{3}=0,M_{4}=0]Pr(C=c)\hskip
7.11317pt(1)$ $\displaystyle=\sum_{c}E[Y|A=a,C=c,M_{3}=0,M_{4}=0]Pr(C=c)\hskip
7.11317pt(2)$

Which leads to identification of the estimand. We use the fact that
$\\{M_{3},M_{4}\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(a,m_{3}=0,m_{4}=0)|C$ in line (1) to invoke
the consistency assumption in line (2) and identify the causal effect from the
observed data.