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# Effects of Causes and Causes of Effects

A. Philip Dawid1 and Monica Musio2 1Statistical Laboratory, University of
Cambridge, UK; email<EMAIL_ADDRESS>2Dipartimento di Matematica ed
Informatica, Università degli Studi di Cagliari, Italy; email<EMAIL_ADDRESS>
###### Abstract

We describe and contrast two distinct problem areas for statistical causality:
studying the likely effects of an intervention (“effects of causes”), and
studying whether there is a causal link between the observed exposure and
outcome in an individual case (“causes of effects”). For each of these, we
introduce and compare various formal frameworks that have been proposed for
that purpose, including the decision-theoretic approach, structural equations,
structural and stochastic causal models, and potential outcomes. It is argued
that counterfactual concepts are unnecessary for studying effects of causes,
but are needed for analysing causes of effects. They are however subject to a
degree of arbitrariness, which can be reduced, though not in general
eliminated, by taking account of additional structure in the problem.

###### doi:

10.1146/((please add article doi))

###### keywords:

causal Bayesian network, counterfactual, decision theoretic causality,
directed acyclic graph, instrumental variable, interval of ambiguity,
potential outcome, probability of causation, statistical causality, stochastic
causal model, structural causal model, structural equation model, twin network

††journal: Xxxx. Xxx. Xxx. Xxx.

## INTRODUCTION

###  1 Overview

The enterprise of “statistical causality” has seen much activity in recent
years, both in its foundational and theoretical aspects, and in applications.
However it remains rare to draw the distinction (recognised by Mill (1843))
between two different problem areas within it: assessing (in individual cases,
or in general) the likely effects of applied or considered interventions—the
problem of “effects of causes”, EoC; and assessing, in an individual case,
whether or not an observed outcome was caused by an earlier intervention or
exposure—the problem of “causes of effects”, CoE. Where this distinction is
made, it is typically assumed that both problems can be represented and
addressed using a common theoretical framework, such as the structural causal
model of Pearl (2009).

The purpose of the current article is to emphasise the important logical and
technical differences between EoC and CoE problems, and to explore and compare
the various ways in which problems of each kind can be and have been
formulated. In particular it is argued that different tools are appropriate
for the two different purposes.

In § 2 we introduce the variety of concerns to be addressed, in the context of
a specific law suit. In Part I we introduce and compare a variety of
formalisms that have been proposed to address “Effects of Causes”. Section 4
briefly summarises some philosophical and implementational issues. Section 5
introduces, with examples, the problem of inference in the presence of an
instrumental variable, which is then used throughout Part I as a hook on which
to hang the general discussion. Section 6 describes purely probabilistic
aspects. Then § 7 introduces the decision-theoretic approach to EoC, § 8 an
approach based on linear models, § 9 a nonparametric generalisation of that,
and § 10 the approach based on potential outcomes.

We turn to address “Causes of Effects”in Part II, for problems similar to
those of § 2. Section 11 points to the need for counterfactual inference,
which can not however totally resolve the ambiguities inherent in such
problems. Two ways of conducting counterfactual modelling are described in §
12, which can both be subsumed in the potential outcome approach of § 13. In §
14 we consider how empirical data can be used to inform CoE analysis, but can
not totally resolve the inherent ambiguities. In § 15 we apply this to address
the legal CoE issues of § 2, showing how the basic ambiguity, expressed by
interval bounds on the “probability of causation”, can be refined when we can
observe other variables in the problem. Section 16 indicates just how limited
our CoE analyses have been, and what difficulties might attend further
extension.

Our concluding remarks summarise some of the lessons to learned from this
review of the different approaches to EoC and CoE.

###  2 A law suit

In 2014 a class action (“multidistrict litigation”, MDL) was brought in the
United States by more than three thousand women who sued the pharmaecutical
company Pfizer, claiming that they developed (type 2) diabetes as a result of
taking its drug Lipitor (Atorvastatin Calcium) (Case Report, 2015). The
plaintiffs identified two “bellwether cases” of women making such a claim for
closer attention.

In order to succeed in such a suit, the plaintiffs would have to demonstrate,
in succession, to the Law’s satisfaction, two points (Dawid et al., 2014):

General causation:

Can Lipitor cause diabetes?

Specific causation:

In the individual cases, did Lipitor cause their diabetes?

The eventual judgment was in favour of Pfizer. It was judged that general
causation had not been established for doses 10mg, 20mg and 40mg of the drug,
but could be considered for the 80mg dose. And with regard to the bellwether
cases, it was judged that specific causation could not be established.

The distinction the court made between the two varieties of causal question,
general and specific, is fundamental, and occurs in many contexts. It has
various descriptions. Philosophers talk of “type” and “token” causation. Legal
scholars talk of “group” and “individual” causation, and have coined the
expression G2i (“Group to individual”) for the task of arguing from one to the
other (Faigman et al., 2014). In statistical contexts we may talk of inference
about “the effects of causes” (EoC), and “the causes of effects” (CoE), which
are the designations we mostly use here.

###  3 Statistical and causal questions

Questions about individual cases can usefully be organised in a fourfold
classification. We exemplify these for the bellwether case of Juanita, who is
55 years old, has a total cholesterol of 250 mg/dL, LDL of 175 mg/dL, HDL of
46 mg/dL, triglycerides of 142 mg/dL, weighs 176 lbs, and has a body mass
index (BMI) of 26.37.

Forecasting:

Juanita has started taking a 80mg dose of Lipitor daily. Is she likely to
develop diabetes?

Backcasting:

Juanita has developed diabetes. Did she take Lipitor, and if so in what dose
and for how long?

Decision:

Juanita is considering whether to take Lipitor, but is worried about
developing diabetes. What should she do?

Attribution:

Juanita took Lipitor 80mg daily for 3 years, and developed diabetes. Was that
because she took Lipitor?

While Forecasting and Backcasting are fundamentally purely statistical
exercises, Decision and Attribution can be classified as “causal”
questions—the former addressing “Effects of Causes” (EoC), and the latter,
“Causes of Effects” (CoE).

Forecasting. This is an apparently straightforward statistical task, at least
conceptually: we gather high quality data on individuals sufficiently like
Juanita, taking the same treatment, and observe the proportion going on to
develop diabetes. In practice this simple recipe will be complicated by non-
random sampling of cases, differences in background characteristics,
difficulties associated with long-term follow-up, censoring by death, etc.,
etc. Handling such complications has been a prime focus of statistical
research over many decades, and though the issues raised are very far from
trivial, they raise no new issues of principle. But we would also need to
argue that the proportion, estimated from the data, of individuals developing
diabetes can be identified with Juanita’s “individual risk”. While this does
raise some subtle philosophical issues (Dawid, 2017), they can largely be
ignored for practical purposes.

Backcasting. This refers to the task of “predicting” uncertain past events on
the basis of later observations. In a statistical context, this is most
typically performed by application of Bayes’s theorem. Suppose we do not know
whether or not Juanita took the Lipitor, but, as above, have estimated the two
“forward” forecast probabilities, under each scenario. We would also need to
assign a prior probability to the event that she did, in fact, take the drug.
Bayes’s theorem supplies the machinery for combining these ingredients to
produce the required “backward” probability that she indeed took Lipitor, on
the basis of her having developed diabetes. Although such Bayesian inferences
have, from the very beginning, often been described as estimating the
“probabilities of causes”, use of the term “cause” here is not really
appropriate, since even if we can conclude that Juanita had taken Lipitor,
that might not have been the cause of her diabetes.

Applying Bayes’s theorem is not the only way to conduct backcasting. More
straightforwardly, we could simply collect a sample of individuals
sufficiently like Juanita, confine attention to those who develop diabetes,
and use the proportion of these who had taken Lipitor to estimate the desired
probability for Juanita. Indeed, there are circumstances where this simple
approach may be preferable to the Bayesian route (Dawid, 1976).

Decision. Forecasting is of fundamental importance in decision analysis.
Suppose Juanita has not yet started taking Lipitor, and is considering whether
or not to do so. One of her concerns is whether she will develop diabetes. She
should thus consider, and compare, how probable this event is under two
possible scenarios: that she does, or that she does not, take the drug. This
would requires two separate forecasting exercises, and correspondingly data
from two different sets of individuals, according as they do or do not take
Lipitor.

But new difficulties now arise in gathering and using such data. In
particular, the very treatment desired by such an individual might be related
to her overall health status, and thus affect her risk of developing
diabetes—even were she not to receive that desired treatment. In such as case
it becomes problematic to disentangle the effects of desire for treatment and
of application of treatment. This is an example of the problem of
“confounding”, which requires careful attention in such cases.

Attribution. Questions of forecasting, backcasting and decision, although
beset with many practical difficulties, can all, in principle at least, be
answered directly by means of probabilities attached to unknown events of
interest, probabilities that can be estimated given suitable data. However, a
question of attribution—such as “did taking Lipitor cause Juanita’s
diabetes”—is not so readily resolved. For what is it now that is unknown? We
know that Juanita took Lipitor, and we know that she developed diabetes. There
is no unknown event about which we require inference. Rather, it the
relationship between these events that is uncertain—was it causal, or not?
Even to understand what we might mean by such a question is problematic.

We shall consider how to formalise such questions, and explore just what can
be concluded from data about them, in Part II below.

##  Part I Effects of Causes

###  4 Introduction

####  4.1 Causality and agency

Philosophers have debated causality for millennia, and have propounded a large
variety of conceptions and approaches. Statisticians, on the other hand, had
traditionally been reluctant to imbue their inferences with causal meaning.
But in recent years much more attention has been given to what we can now term
“statistical causality”. Particularly influential have been the contributions
of Rubin (1974), who promoted a formulation based on “potential outcomes”, and
of Pearl (2009), based on graphical representations.

Implicit in both these approaches is the idea of a cause as an intervention
applied to a system, in line with the “agency” interpretation of causality
(Reichenbach, 1956, Price, 1991, Hausman, 1998, Woodward, 2003, 2016). A main
task for statistical causality is to make inference about the effects of such
interventions—that is, understanding the “effects of causes”—on the basis of
data. When making use of data, it is important to distinguish between data
generated through experimentation and purely observational data.

####  4.2 Experiment

In an experiment, interventions are made on experimental units according to
some known protocol, often involving randomisation, and their responses
measured. To the extent that the experimental units and interventions can be
regarded as representative of future interventions on new units, it is in
principle straightforward to infer what effects those interventions will have
in future. “Design and analysis of experiments” is a major enterprise within
modern statistics, involving many subtle and technical considerations, but no
special issues of principle arise.

####  4.3 Observation

Things are not so straightforward when the data available are purely
observational, and the process whereby treatment interventions were applied to
units is not known. For example, when choosing between two treatments, a
doctor may have given one preferentially to those patients he considers
sicker. Then a simple comparison of the outcomes in the two treatment groups
will be misleading, since even if there is no difference between the
treatments, a difference in outcomes may be seen because of the difference in
general health of the two treatment groups. This is the problem of
“confounding”, which prevents us from taking the observational data at face
value. In such a case it may or may not be possible to assess, by more
sophisticated means, genuine causal effects, depending on what is observed and
what assumptions can reasonably be made. If we know or can reasonably assume
how the doctor behaved, and have data on the patient characteristics that the
doctor used, then we can make meaningful comparisons and extract causal
conclusions; but—in the absence of further structure or assumptions—this will
not be the case if either of the conditions fails.

Much of the modern enterprise of statistical causality is focused on
addressing this issue of extracting causal conclusions from observational
data. In order to do so, it will invariably be necessary to make assumptions,
generally untestable in practice, about the relationship between the
behaviours of the “idle” observational system, which generates the observed
data, and the same system under a specified intervention—which is what it
wanted, but is not directly observed. Such assumptions are sometimes made
explicit, and so open to reasoned scrutiny and debate, but sometimes they
remain implicit and hidden, being taken for granted without critical
examination. The kind of relationships required can typically be expressed,
explicitly or implicitly, as asserting the equality of certain ingredients in
both idle and interventional circumstances. While such invariance properties
have sometimes been taken as the very definition of causality (Bühlmann,
2020), they can be applied without any such philosophical commitment. (Our own
philosophical standpoint remains that based on agency.)

The “do-calculus” (Pearl (2009, §3.4); see also Dawid (2015, §9.7)) applies to
problems that can be modelled by means of a directed acyclic graph,
representing both assumed conditional independence properties of the
observational regime, and assumed relationships between the observational and
interventional regimes. For such a case it supplies a complete method for
determining whether a causal estimand of interest can be identified from
observational data, and if so how.

###  5 Instrumental variable

Below we shall introduce, compare and contrast some of the different
statistical formalisms that have been used to model effects of causes. To be
concrete, we shall consider, for each formalism, how it might model an
instrumental variable problem (Bowden & Turkington, 1984). This involves, in
addition to the treatment variable $X$ and response variable $Y$, a further
observed variable $Z$ (the instrument), and an unobserved variable $U$—all
defined for individuals in a study or larger population. Typically $Z$ is
binary, $X$ and $Y$ are binary or continuous, and $U$ is multivariate. Note
that in this problem it is not possible, without imposing still further
structure, to identify the causal effect of $X$ on $Y$ from observational
data.

We suppose:

  1. (a).

$U$ is a set of pre-existing characteristics of the individual

  2. (b).

$Z$ is associated with $X$, but not with $U$

  3. (c).

While $X$ could in principle be assigned externally, in the study it was not

  4. (d).

“Exclusion restriction”: Given $X$, and the individual characteristics $U$,
the response $Y$ is unaffected by $Z$. (This vague requirement will be
clarified further below).

###### Example 1

Encouragement trial
In an encouragement trial (Holland, 1988), students are randomly assigned to
receive encouragement to study. However, a student may not or may not respond
to the encouragement. Here $Z$ is a binary assignment indicator, taking value
1 for encouragement, 0 for no encouragement; $X$ is the number of hours the
student actually studies; $Y$ is the student’s score in the final test; and
$U$ comprises individual characteristics of the student, that may affect both
$X$ and $Y$; because of randomisation, $Z$ is independent of $U$. We are
interested in how a student’s choice of study hours affects their test score.
$\Box$

###### Example 2

Incomplete compliance
In a medical trial, each patient is randomly assigned to take either active
treatment ($Z=1$) or placebo ($Z=0$). However, the patient may not comply with
the assignment, so that the treatment actually taken, $X=1$ or $0$, may differ
from $Z$. Finally we observe whether the patient recovers ($Y=1$) or not. We
allow for possible dependence of both $X$ and $Y$ on further unobserved
patient characteristics, $U$. Again, randomisation ensures that $Z$ is
independent of $U$. We are principally interested in the effect of taking the
treatment on recovery. $\Box$

###### Example 3

Availability trial
In a variation of Example 2, $Z=1$ means the active treatment is made
available to the patient, $Z=0$ that it is not; $X=1$ if the treatment is
taken, $X=0$ if not. It is supposed that if the treatment is unavailable
($Z=0$) it can not be taken ($X=0$) (though it need not be taken when it is
available). $\Box$

###### Example 4

Mendelian randomisation (Katan, 1986)
Low serum cholesterol level ($X=1$) is thought to be a risk factor for cancer
($Y=1$). Both serum cholesterol and cancer may be affected by indicators of
lifestyle ($U$). Possession of the E2 allele ($Z=1$) of the apolipoprotein E
(APOE) gene is known to be associated with low serum cholesterol level: this
relationship need not be causal, but may arise because APOE is in linkage
disequilibrium with the actual causative gene. Since “Nature” randomises the
APOE allele at birth, and its level is not thought to affect lifestyle, $U$
should not be associated with $Z$. We are interested in whether intervening to
raise serum cholesterol could lower the risk of cancer. $\Box$

There are a number of questions we could ask (but not necessarily be able to
answer) in such examples. In Example 2, these might include:

  1. (i).

“What is the probability of recovery for a patient who is assigned to active
treatment (irrespective of the treatment actually taken)?”

  2. (ii).

“What is probability of recovery for a patient who (irrespective of assigned
treatment) in fact took active treatment?”

  3. (iii).

“What is the probability that a patient who recovered complied with the
assignment?”

  4. (iv).

“What is the effect on recovery of assignment to treatment?”

  5. (v).

“What is the causal effect of taking the treatment on recovery?”

Questions (i), (ii) and (iii) inhabit the lowest rung, “seeing”, of Pearl’s
“ladder of causation” (Pearl & Mackenzie, 2018), the first two being instances
of “forecasting”, and the last of “backcasting”. Questions (iv) and (v) are on
the second rung of the ladder, “doing”, being instances of “decision”. In the
sequel we shall mainly be interested in (v). Since $Z$ has been randomised, we
could argue that the “intention to treat” question (iv) is essentially the
same as (i), which can be straightforwardly addressed from the study data on
$(Z,Y)$. However, (v) can not readily be answered in the same way as (ii) (an
“as treated” analysis), since $X$ has not been randomised, and any observed
association between $X$ and $Y$ might be due to their common dependence on
$U$.

###  6 Seeing: Conditional independence

We first consider how to express, formally, purely probabilistic properties of
the observational joint distribution of $(X,Y,Z,U)$. This is all that is
required to address forecasting and backcasting questions such as (i)–(iii).
However, it will not be possible to formulate, let alone solve, causal queries
such as (v) in this setting: these live on the second rung, “doing”.

Specifically, (b) implies that $Z$ is independent of $U$: using the notation
of Dawid (1979), we write this as

$\mbox{$Z\,\mbox{$\perp\\!\\!\\!\perp$}\,U$}.$ (1)

Here (d) is interpreted as asserting the probabilistic independence of $Y$
from $Z$, conditional on $X$ and $U$:

$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}\,Z\mid(X,U)$}.$ (2)

####  6.1 Graphical representation

Figure 1: Instrumental variable: seeing

It is often convenient to display such conditional independence properties by
means of a directed acyclic graph (DAG). Each node in the DAG represents a
variable in the problem, and missing arrows represent assumed properties of
conditional independence in their joint distribution—see Dawid (2015, Section
6) for full details. The DAG is a partial description, displaying only
qualitative aspects, of the joint distribution.

The DAG representing (1) and (2) looks like Figure 1 (the dotted outline of
$U$ is non-essential, merely a reminder that $U$ is unobserved). The absence
of an arrow between $Z$ and $U$ represents their independence, (1), while the
missing arrow from $Z$ to $Y$ represents their conditional independence, given
the “parents” of $Y$, namely $X$ and $U$, (2). In general in a DAG
representation, any variable is conditionally independent of its non-
descendants, given its parents. Further conditional independence properties
implied by these can be read off the DAG, using the $d$-separation (Verma &
Pearl, 1990) or equivalent moralisation (Lauritzen et al., 1990) criteria, as
described in Dawid (2015).

The qualitative DAG representation of a joint distribution can be expanded to
a full quantitative description by specifying, for each variable, its
conditional distribution given its parents in the DAG. (This would be required
to encode the condition in (b) that $X$ is not independent of $Z$). Elegant
algorithms exist, taking advantage of the DAG structure, for streamlining
quantitative computation of joint and conditional probabilities (Cowell et
al., 1999). Such probabilities are what is needed to address questions of
forecasting and backcasting.

###  7 Doing: Decision-theoretic causality

The decision-theoretic (DT) approach to causality has been described in this
journal (Dawid, 2015); its foundational underpinnings are examined in Dawid
(2021a).

We have several regimes of interest. For each possible value $x$ of $X$ we
have an interventional regime, where treatment value $x$ is forced on an
individual (that is, $X$ is “set” to $x$, which we notate as $X\leftarrow x$).
We also have an “idle” regime, in which the treatment $X$ is merely observed,
and any value may occur. It is helpful to introduce a non-stochastic “regime
indicator” variable $F_{X}$, where $F_{X}=x$ labels the interventional regime
with $X\leftarrow x$, and $F_{X}=\emptyset$ labels the idle regime. The
response variable $Y$ may have different distributions in the different
regimes. The object of causal inference will usually be some contrast between
the response distributions in the various interventional regimes—this is what
is required to address the decision problem of choosing which value to set $X$
to. For example, when $X$ is binary, interest typically centres on the
difference in the expected response between the two interventional regimes,
${\mbox{E}}(Y\mid F_{X}=1)-{\mbox{E}}(Y\mid F_{X}=0)$, which is termed the
“average causal effect”, ACE.

But in the cases to be considered there will be no data available directly
relevant to the interventional settings of interest, and we shall want to make
use of observational data collected under the idle regime, $F_{X}=\emptyset$,
to make inferences about what would happen in interventional settings. This
may or may not be possible. At the least, it will be necessary to make, and
justify, relationships between the idle and the interventional settings. DT
studies when and how such relationships can be used to support causal
inference from observational data.

For example, we might be willing to assume, in addition to (1) and (2), that,
no matter whether $X$ is merely observed ($F_{X}=\mbox{$\emptyset$}$), or is
set by external intervention ($F_{X}=0$ or $1$), the following ingredients
will be the same:

  1. (e).

the distribution of $Z$

  2. (f).

the distribution of $U$, with $U$ independent of $Z$

  3. (g).

the conditional distribution of $Y$ given $(Z,X,U)$ (which would then in all
cases depend only on $(X,U)$, since this is so under regime
$F_{X}=\mbox{$\emptyset$}$, by (d).)

These properties do not follow logically from (1) and (2) (Dawid, 2021b), and
if they are to be applied they need additional argument, such as described in
Dawid (2021a).

We can interpret assumptions (e)–(g) as conditional independence properties:

$\displaystyle Z$ $\perp\\!\\!\\!\perp$ $\displaystyle F_{X}$ (3)
$\displaystyle U$ $\perp\\!\\!\\!\perp$ $\displaystyle(F_{X},Z)$ (4)
$\displaystyle Y$ $\perp\\!\\!\\!\perp$ $\displaystyle(F_{X},Z)\mid{(X,U)}.$
(5)

Even though $F_{X}$ is a non-stochastic indicator of regime
(observational/interventional), these intuitively meaningful “extended
conditional independence” expressions can be manipulated essentially just as
if $F_{X}$ were a random variable (Constantinou & Dawid, 2017).

In this approach, the conditions (e)–(g), or equivalently (3)–(5), which
relate to behaviour under possible intervention at $X$, are the full “causal
ingredients” of our model.

####  7.1 Graphical representation

We can augment Figure 1 with an additional node for $F_{X}$ (square to
indicate it is non-stochastic). We obtain Figure 2.

Figure 2: Instrumental variable: doing

This DAG represents (in exactly the same way as before) the assumed
conditional independence assumptions (1)–(5), which fully embody our causal
assumptions. (The dashed arrows from $Z$ and $U$ to $X$ are there to indicate
that they are absent under an interventional regime $F_{X}=x$, since then we
have $X=x$, trivially independent of $(Z,U)$). Note in particular that the
arrow from $Z$ to $X$ in Figure 2 does not encode a causal effect of $Z$ on
$X$, since (e)–(g) are fully consistent with cases, such as Example 4, where
$Z$ and $X$ are merely associated (Dawid, 2010, §10).

####  7.2 Causal Bayesian Network

Pearl (2009) uses the same causal semantics as described above to construct
what he terms a causal Bayesian network (CBN). The difference is that he would
normally consider the possibility of intervention on every observable
variable—which in our case would mean adding further intervention indicator
nodes $F_{Z}$, $F_{Y}$ to Figure 2, parents, respectively, of $Z$ and $Y$. In
such a case the presence of all the intervention nodes is usually taken for
granted, and omitted from the augmented DAG—so rendering it visually
indistinguishable from an unaugmented DAG, here Figure 1. However there are
clear advantages to retaining explicit intervention nodes in the figure:

  1. (i).

This eliminates the possibility of confusion between rung 1 (seeing) and rung
2 (doing) interpretations of apparently identical DAGs.

  2. (ii).

The “causal” links assumed between regimes are fully represented by
$d$-separation properties of the augmented DAG.

  3. (iii).

It will (as above) often be appropriate to consider interventions on only some
of the variables. In particular, there will then be no need to impose the
additional cross-regime causal constraints associated with further,
inessential, intervention indicators.

####  7.3 Estimation

Even after assuming links, as above, between the observational and
interventional regimes, it does not follow that we have enough structure to
enable us to use the observational data to estimate, say, the causal effect,
ACE, of $X$ on $Y$. And indeed, in this example, further structure must be
imposed to support such causal inference. For instance, in Example 1 we might
require that $Y$ has a linear regression on $(X,U)$ (this being the same in
all regimes, by (5)):

${\mbox{E}}(Y\mid X,U,F_{X})=W+\beta X$ (6)

where $W$ is a function of $U$. Since then ${\mbox{E}}(Y\mid
F_{X}=x)=w_{0}+\beta x$, where $w_{0}={\mbox{E}}(W)$, $\beta$ has a clear
causal interpretation. Also, restricting attention to the observational regime
$F_{X}=\mbox{$\emptyset$}$, we obtain ${\mbox{E}}(Y\mid
Z)={\mbox{E}}\\{{\mbox{E}}(Y\mid X,U,Z)\mid Z\\}={\mbox{E}}\\{{\mbox{E}}(Y\mid
X,U)\mid Z\\}$, by (5), $=w_{0}+\beta{\mbox{E}}(X\mid Z)$, by (4). This
implies that we can estimate $\beta$, from the observational data, as the
ratio of the coefficients of $Z$ in the sample linear regressions of $Y$ on
$Z$ and of $X$ on $Z$.

In cases such as Example 2 with binary $X$, (6) is equivalent to

$\mbox{\rm SCE}(u)=\beta$ (7)

a constant for all $u$, where $\mbox{\rm SCE}(u)={\mbox{E}}(Y\mid
U=u,F_{X}=1)-{\mbox{E}}(Y\mid U=u,F_{X}=0)$ is the specific causal effect of
$X$ on $Y$, relevant to the subpopulation having $U=u$. That is to say, the
specific causal effect is required to be non-random, the same in all
subpopulations. Then $\mbox{\rm ACE}={\mbox{E}}\\{\mbox{\rm SCE}(U)\\}=\beta$
also, and so is estimable as above. Alternatively, when all variables are
binary, without making any modelling assumptions we can determine bounds on
ACE from the data (Balke & Pearl, 1997, Dawid, 2003).

###  8 Linear structural equation model (SEM)

Linear structural equation modelling, closely related to path analysis
(Wright, 1921), is perhaps the earliest approach to instrumental variable
problems—and much else besides. It can be considered as an extension of linear
regression modelling. In the context of the encouragement trial of Example 1,
we might express the relationship between $Z,X,Y$ by the pair of regression-
like equations:

$\displaystyle X$ $\displaystyle=$ $\displaystyle\alpha_{0}+\alpha_{1}Z+U_{X}$
(8) $\displaystyle Y$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X+U_{Y}.$ (9)

(Such a system would often be completed with a further equation for $Z$, which
here would simply be $Z=U_{Z}$. However we omit this on account of its
triviality). Here $U_{X}$, $U_{Y}$ are zero-mean “residual error” terms. In
this problem it would be assumed that $U_{X}$ and $U_{Y}$ are uncorrelated
with $Z$, but not necessarily with each other. The absence of $Z$ in (9)
embodies the “exclusion restriction”.

This model can be rendered graphically as in Figure 3.

Figure 3: Structural equation graph

This may be compared with Figure 1, identifying $U=(U_{X},U_{Y})$.

As discussed by Pearl (2009, §5.1.2), the intended interpretation—in
particular, the causal interpretation—of SEM has often been unclear. Pearl’s
suggestion is as follows.

  1. (i).

In the system (8)–(9), $X$ is functionally determined by $(Z,U_{X})$, and $Y$
is functionally determined by $(X,U_{Y})$. So we can solve for $(X,Y)$ in
terms of $(Z,U_{X},U_{Y})$. If we have a joint distribution for
$(Z,U_{X},U_{Y})$, this determines a joint distribution for $(Z,X,Y)$—which
can be regarded as representing the undisturbed system.

  2. (ii).

If, alternatively, we intervene to set $X$ to $x$, it is assumed that we can
replace (8) by $X=x$, but retain (9) essentially as is, so that
$Y=\beta_{0}+\beta_{1}x+U_{Y}$, with the distribution of $U_{Y}$ unchanged.

This approach gives a causal semantics to a SEM, relating the observational
regime with possible interventional regimes. As with any such assumed
relationship, it is not to be taken for granted, but argued for in the context
of each particular problem. When we can assume (ii), $\beta_{1}$ has a clear
causal interpretation, being the rate of change of ${\mbox{E}}(Y\mid
X\leftarrow x)$ with respect to the value set, $x$. However, since $U_{Y}$ is
correlated with $U_{X}$, and hence in general with $X$, $\beta$ will not be
the coefficient of $X$ in the observational regression of $Y$ on $X$ in (i),
so can not be identified from that. Instead we can argue that, since $U_{X}$
is uncorrelated with $Z$, ${\mbox{E}}(X\mid Z)=\alpha_{0}+\alpha_{1}Z$; and,
since $U_{Y}$ is uncorrelated with $Z$, ${\mbox{E}}(Y\mid
Z)=\beta_{0}+\beta_{1}{\mbox{E}}(X\mid
Z)=\beta_{0}+\alpha_{0}\beta_{1}+\alpha_{1}\beta_{1}Z$. It again follows, as
in § 7.3, that $\beta_{1}$ can be identified as the ratio of the coefficients
of $Z$ in the observational regressions of $Y$ on $Z$ and $X$ on $Z$.

###  9 Structural Causal Model (SCM)

We can generalise the system (8)–(9) by dropping the linearity requirement,
yielding

$\displaystyle X$ $\displaystyle=$ $\displaystyle f_{X}(Z,U_{X})$ (10)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle f_{Y}(X,U_{Y})$ (11)

where $f_{X}$, $f_{Y}$ are specified general functions of their arguments, and
$(U_{X},U_{Y})$ have a specified joint distribution, typically not being
independent of each other, but being jointly independent of $Z$. The absence
of $Z$ as an argument of $f_{Y}$ embodies the exclusion restriction. Pearl &
Mackenzie (2018) refer to such a nonparametric structural equation model as a
Structural Causal Model (SCM), and we use this designation in the sequel.

Again, this system can be represented graphically by Figure 3. But with no
loss of generality we can use $U$ instead of the pair $(U_{x},U_{Y})$, and
write the system as

$\displaystyle X$ $\displaystyle=$ $\displaystyle f_{X}(Z,U)$ (12)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle f_{Y}(X,U)$ (13)

with $U$ independent of $Z$. This system again determines a joint
observational distribution for $(X,Y,Z)$, which is represented by Figure 1. We
might again imbue this structural equation system with causal semantics (whose
relevance in a real life context will need justification): assume that, under
an intervention $X\leftarrow x$, we can replace (12) by $X=x$, and (13) by
$Y=f_{Y}(x,U)$, where $U$ is supposed to retain its original distribution. The
extended structure is then again represented by Figure 2. In particular, $Y$
will be independent of $Z$ in an interventional regime, where the dotted
arrows in Figure 2 are absent. In contrast, in the observational regime, the
distribution of $Y$, given $X=x,Z=z$, is that of $Y=f_{Y}(x,U)$ given
$f_{X}(z,U)=x$; because of this conditioning, the value of $Z$ will typically
make a difference to the distribution of $Y$.

In using Figure 1 (and, implicitly or explicitly, Figure 2) as representations
of the SCM system (12)–(13), we are supplying these figures with yet another
semantic interpretation, where the dependence of $X$ and $Y$ on their parents
is taken as deterministic, not stochastic. This is to be contrasted with the
CBN interpretation of § 7.2, in which all relationships are allowed to be
stochastic.

It can be be shown that, by suitable choice for the distribution of its $U$
(which distribution is, however, not uniquely determined), the SCM model can
fully reproduce the joint distribution of $(X,Y,Z)$, in all regimes, implied
by a given fully stochastic DT model. This property holds in general for any
problem represented by a DAG. For identifying effects of causes we gain
nothing by replacing a stochastic DT model with a deterministic SCM
model.111See however § 10.1, and especially § 10.1.1, for further discussion
of this point. Also, while Balke & Pearl (1997) make essential use of the
deterministic functional relationships in the SCM to derive estimable bounds
on ACE in the case of binary variables, Dawid (2003) showed how the same
bounds can be obtained from the purely stochastic DT model. In particular, we
again can not identify the causal effect of $X$ on $Y$ without further
assumptions, such as linearity in (13), as for (9) (with $U_{Y}$ some function
of $U$).

####  9.1 A comment

If, in a SEM or SCM, $U$ is regarded as a persistent attribute of an
individual, the assumed determinism would mean that we would get the same
output each time we applied the same intervention to that individual. That
would be an unreasonable assumption in most contexts. Consequently we should
normally consider $U$ as also incorporating information specific to the
occasion of application (including, perhaps, “random error”), varying from
occasion to occasion. Nevertheless, assuming the distribution of $U$ does not
change, average causal effects will still be constant, and so be meaningful,
across occasions.

###  10 Potential outcomes

In the potential outcomes (PO) approach to statistical causality (Rubin,
1974), for each possible value $x$ of the treatment $X$, we conceive of a
version $Y(x)$ of the outcome variable $Y$, all of these versions co-existing,
even before application of treatment. It is supposed that $Y(x)$ (or, to be
more explicit, $Y(X=x)$) is the outcome that would be observed in the
interventional regime $F_{X}=x$. Typically it is further assumed
(“consistency”) that in the idle regime $F_{X}=\emptyset$ also, whenever
$X=x$, the outcome will be $Y=Y(x)$ (which is why we don’t distinguish between
$Y(X=x)$ and $Y(X\leftarrow x)$). Consistency is required to relate the
observational and interventional regimes.

In the special but common and important case of binary $X$,222Of course,
similar considerations apply more generally. intervening to set $X$ to $1$
would reveal the value of $Y(1)$, while $Y(0)$ would remain unobserved; and
similarly on interchanging 1 and 0\. In this approach, the single response $Y$
is replaced by a bivariate quantity $\mbox{$\mathbf{Y}$}=(Y(0),Y(1))$, which
must thus be endowed with a bivariate distribution. The fundamental causal
contrast, comparing the effects of the two interventions, is considered to be
the individual causal effect, $\mbox{\rm ICE}=Y(1)-Y(0)$. However, direct
inference about ICE is complicated by the fact (termed “the fundamental
problem of causal inference” by Holland (1986)) that, because it is logically
impossible to intervene on the same individual in two mutually exclusive ways
simultaneously, we can never observe ICE, or estimate its distribution. For
this reason, it is customary to divert attention to the expected individual
causal effect, ${\mbox{E}}(\mbox{\rm ICE})$. By linearity of expectation, this
is ${\mbox{E}}\\{Y(1)\\}-{\mbox{E}}\\{Y(0)\\}$, each term of which involves
only one intervention. This then is the PO version of ACE, as introduced in §
7, with essentially the same interpretation. Note however that there is no
analogue of ICE in the DT approach; neither is there any DT analogue of, say,
${\mbox{var}}(\mbox{\rm ICE})$, which involves the correlation between $Y(0)$
and $Y(1)$—a correlation that can never be estimated, on account of the
fundamental problem of causal inference.

If we start with a SCM representation of a system, we can use it to construct
associated potential outcomes. For example, starting from (13), just define,
for each $z$, $X(Z=z)=f_{X}(z,U)$, and for each $x$, $Y(X=x)=f_{Y}(x,U)$.
Under the SCM causal semantics, $X(Z=z)=f_{X}(z,U)$ is assumed to supply the
value of $Y$, when $X=x$, whether or not there are interventions at $X$ or
anywhere else in the system (except at $Y$ itselvf); this corresponds to the
PO consistency property. This construction of POs makes them all functions of
$U$, whose distribution thus generates a joint distribution for all POs.

Typically, however, a PO analysis would not make explicit use of an exogenous
variable such as $U$, and might not want to require that there exist any real-
world variable or set of variables $U$ with the properties assumed in § 9 (for
example, that $Y$ is fully determined by $X$ and $U$). Instead, one starts by
introducing, as primitives, jointly distributed stochastic potential outcomes,
$X(Z=z)$, $Y(X=x)$, $Y(Z=z)$, for all possible values of $x$ and $z$, and work
directly with them. The exclusion restriction now becomes
$Y(Z=z)=Y\\{X=X(Z=z)\\}$.

Introduce now a new variable $V$, which is simply the collection of all
$X(Z=z)$’s and $Y(X=x)$’s, as $z$ and $x$ vary. Then $X$ is fully determined
by $(Z,V)$: when $Z=z$, we simply select the relevant element $X(Z=z)$ of $V$
(this being valid in all regimes, by consistency); similarly $Y$ is determined
by $(X,V)$. We thus recover a formal333In a genuine SCM, $U$ is regarded as a
set of unobserved real-world background variables, with an appropriate, in
principle knowable, distribution, that, together with $X$, would determine
$Y$. But is is hard to conceive of such a real-world interpretation of $V$.
identity with the SCM of (12) and (13), with $V$ substituting for $U$—so long
as we have $V$ independent of $Z$. If we were starting from a SCM, as above,
the independence of $U$ and $Z$, and thus of $V$ and $Z$, would be easy to
justify, since $U$ represents pre-existing characteristics of the individual,
and $Z$ is randomised. To make a similar argument for $V$ when taking POs as
primitive is more problematic, since $V$ does not correspond to any real-world
quantity (in particular, on account of “the fundamental problem of causal
inference”, certain elements of $V$, e.g. $X(Z=1)$ and $X(Z=2)$, are not
simultaneously observable). Nevertheless, the typical assumption is that it is
indeed meaningful to consider the collection $V$ of all possible potential
responses as a pre-existing (albeit unobservable) characteristic of the
individual, and thus argue that $V$ is independent of the randomised variable
$Z$. In this case, we recover a purely formal identity with an SCM model. Now
the linearity condition (6) is equivalent to
$Y(X=x)-Y(X=x^{\prime})=\beta(x-x^{\prime})$—which is thus being required to
be non-random.

####  10.1 A variation

The above specifications can be considered simply as more detailed ways of
realising the CBN structure of § 7, which is more general since in a CBN we
need not assume the existence of potential outcomes, which can not be derived
from its stochastic form. And indeed, for estimating the average causal
effect, the extra structure imposed beyond that of a CBN does not offer any
improvement. But a SCM or PO approach does allow us to formulate, and purports
to solve, other causal questions. We consider one such in the context of
Example 2, where $Z$, $X$, $Y$, are all binary (Imbens & Angrist, 1994,
Angrist et al., 1996).

Let ${\bf X}$ denote the pair $(X(Z=0),X(Z=1))$, and $\bf Y$ the pair
$(Y(X=0),(Y(X=1))$. We shall assume consistency, the exclusion restriction
$Y(Z=z)=Y\\{X=X(Z=z)\\}$, and $\mbox{$Z\,\mbox{$\perp\\!\\!\\!\perp$}\,({\bf
X},{\bf Y})$}.$ It is easy to see that

$Y(Z=z)=Y(X=1)X(Z=z)+Y(X=0)\\{1-X(Z=z)\\}.$ (14)

We can define the following “individual causal effects”:

$\displaystyle\mbox{\rm ICE}_{Z\rightarrow X}$ $\displaystyle=$ $\displaystyle
X(Z=1)-X(Z=0)$ (15) $\displaystyle\mbox{\rm ICE}_{Z\rightarrow Y}$
$\displaystyle=$ $\displaystyle Y(Z=1)-Y(Z=0)$ (16) $\displaystyle\mbox{\rm
ICE}_{X\rightarrow Y}$ $\displaystyle=$ $\displaystyle Y(X=1)-Y(X=0)$ (17)

and deduce from (14) that

$\mbox{\rm ICE}_{Z\rightarrow Y}=\mbox{\rm ICE}_{Z\rightarrow
X}\times\mbox{\rm ICE}_{X\rightarrow Y}.$ (18)

We note that, since $Z$ is randomised, $\mbox{\rm ACE}_{Z\rightarrow
X}={\mbox{E}}(\mbox{\rm ICE}_{Z\rightarrow X})={\mbox{E}}(X\mid
Z=1)-{\mbox{E}}(X\mid Z=0)$ is readily estimable from the observational data,
and so likewise is $\mbox{\rm ACE}_{Z\rightarrow Y}={\mbox{E}}(Y\mid
Z=1)-{\mbox{E}}(Y\mid Z=0)$. However, there is no immediate parallel for
$\mbox{\rm ACE}_{X\rightarrow Y}$, since $X$ has not been randomised.

If we could replace each ICE term in (18) by its expectation ACE, we would
have

$\mbox{\rm ACE}_{X\rightarrow Y}=\mbox{\rm ACE}_{Z\rightarrow Y}/\mbox{\rm
ACE}_{Z\rightarrow X}$ (19)

where the right hand-side of (19) is estimable from the observational data (it
is assumed that $Z$ has a causal effect on $X$, so that $\mbox{\rm
ACE}_{Z\rightarrow X}\neq 0$).

When we can assume (7), $\mbox{\rm ICE}_{X\rightarrow Y}=\beta$ is non-random,
we can take expectations in (18), and (19) does indeed hold, allowing
estimation of $\beta$. But more generally ${\bf X}$ and ${\bf Y}$ are not
independent of each other (when constructed from a SCM, they both involve the
same variable $U$), and so neither are $\mbox{\rm ICE}_{Z\rightarrow X}$ and
$\mbox{\rm ICE}_{X\rightarrow Y}$. So we can not just take expectations of all
terms in (18), and (19) is typically not valid.

To make further progress, other assumptions must be imposed, in particular,
monotonicity:

$X(Z=1)\geq X(Z=0).$ (20)

That is to say, we do not have any “defiers”, for which both $X(Z=0)=1$
(treatment would be taken when not assigned) and $X(Z=1)=0$ (treatment would
not be taken when assigned).

Even monotonicity is not sufficient to allow estimation of $\mbox{\rm
ACE}_{X\rightarrow Y}$. However, it does allow a new interpretation of the
right-hand side of (19). For it implies that the individual causal effect
$\mbox{\rm ICE}_{Z\rightarrow X}$ of (15) is either 1 or 0. Thus, from (18),

$\displaystyle\mbox{\rm ACE}_{Z\rightarrow Y}$ $\displaystyle=$
$\displaystyle{\mbox{E}}(\mbox{\rm ICE}_{X\rightarrow Y}\mid\mbox{\rm
ICE}_{Z\rightarrow X}=1)\times\Pr(\mbox{\rm ICE}_{Z\rightarrow X}=1)$
$\displaystyle=$ $\displaystyle{\mbox{E}}(\mbox{\rm ICE}_{X\rightarrow
Y}\mid\mbox{\rm ICE}_{Z\rightarrow X}=1)\times{\mbox{E}}(\mbox{\rm
ICE}_{Z\rightarrow X}).$

It follows that

$\mbox{\rm ACE}_{Z\rightarrow Y}/\mbox{\rm ACE}_{Z\rightarrow
X}={\mbox{E}}(\mbox{\rm ICE}_{X\rightarrow Y}\mid\mbox{\rm ICE}_{Z\rightarrow
X}=1).$ (21)

The right-hand side of (21) is termed the “local average treatment effect”,
LATE. Under monotonicity, LATE is estimable from the data, since the left-hand
side of (21) is.

#####  10.1.1 Critical comments

  1. (i).

Considerations similar to § 9.1 suggest that it would typically be appropriate
to regard potential outcomes, and so individual causal effects, as varying
from one occasion to another, only their expectations remaining constant.

  2. (ii).

In general the monotonicity assumption is untestable, since (under the
assumptions of (i) above) $X(Z=1)$ and $X(Z=0)$ can not both be observed on
the same occasion. However, it must hold in the case of an availability trial
as in Example 3, where necessarily $X(Z=0)=0$. Another extreme case where it
can be inferred is in the presence of a variable $W$, a complete mediator
between $Z$ and $X$ (so that $X(Z)=X\\{W(Z)\\}$), where we have empirical
evidence that, with probability 1, $W(Z=1)=1$ and $X(W=0)=0$. If $X(Z=0)=1$,
then we deduce $W(Z=0)=1=W(Z=1)$, and so $X(Z=1)=X(Z=0)=1$, and we have no
defiers.

  3. (iii).

LATE is an average causal effect in a subgroup of the population: those for
whom both $X(Z=0)=0$ and $X(Z=1)=1$. These are termed “compliers”, since they
would take the treatment if assigned to do so, and not take it if not assigned
(in an availability setting, they are those who would take the treatment if
assigned to do so). However it is impossible to tell who belongs to this
subpopulation by knowing only what treatment was assigned and what treatment
was taken (in an availability trial, an individual who was assigned treatment
and took it must be a complier, but we still can not tell the status of an
individual who was not assigned treatment). Indeed, assuming as in (i) that
${\bf X}$ will vary from occasion to occasion, so too will the group of
compliers. So the relevance of LATE in practice is debatable. Even its
definition, relying as it does on a “cross-world” comparison of potential
outcomes under both $Z=1$ and $Z=0$, can be criticised as essentially
metaphysical and unscientific (Dawid & Didelez, 2012).

  4. (iv).

In cases such as Example 4 where $Z$ is not directly causal for $X$, the
notation $X(Z=z)$ is meaningless, and the above analysis can not even get
started.

##  Part II CAUSES OF EFFECTS

###  11 Introduction

Let us consider again the initial attribution example: Juanita took Lipitor
80mg daily for 3 years and developed diabetes. Was that because she took
Lipitor?

One way of formulating this CoE question is through what the courts sometimes
refer to as the “but for” test: is it the case that, but for her having taking
Lipitor, the diabetes would not have developed? This immediately plunges us
into counterfactual considerations. We know that, in the actual world, the
Lipitor was taken and diabetes developed, and are asked to contrast this with
the outcome in a counterfactual world, in which (counter to the known facts)
the Lipitor had not been taken. The problem of course is that the
counterfactual world is, by definition, unobservable, and even its
existence—certainly its uniqueness—are questionable.

Even in deciding on the exact question, choices have to be made. Juanita took
Lipitor 80mg daily for 3 years. Did she develop diabetes because she took the
80mg dose (the only one for which the court accepted general causation),
rather than 40mg? Did it develop because she took it for 3 years, rather than
2 years? Each such choice conjures up a different counterfactual world for
comparison with this one. We also have choice over what was the observed
response: that she developed diabetes at some point?; that she developed
diabetes within 1 year of stopping Lipitor? Detailed specification is
obviously important in cases where the response is death: since death is
certain, even in a counterfactual world, we can never say that an individual
would never have died, but for some exposure.444Even in EoC cases,
specification of the outcome matters. In March 2021, a few cases of blood
clots among individuals who had received the Astra-Zeneca vaccine against
Covid-19 were observed, and concerns were raised about a possible causal
connexion, leading to a pause in the vaccine’s roll-out in some countries.
When it was pointed out that the rate of such clots was in fact lower than in
the general population, attention turned to the few cases of a specific rare
presentation of the blood clot, cerebral venous sinus thrombosis, and whether
the vaccine could cause that. (For this outcome too, there was no evidence for
causation).

Under the “but for” criterion, “causation” is understood as the case that, in
the appropriate counterfactual world, where Juanita did not take the Lipitor
(in the same way that she in fact did), she did not develop diabetes (in the
relevant time-frame). This is appropriate when the response is all or nothing.
We can also consider cases with a continuous response, such as time to death,
but then it is not so clear what the focus of our attention should be. We
might ask, for example, does death occur later, in the relevant counterfactual
world, than it actually did in this world (Greenland, 1999)?

Even when our variables have been carefully specified, and the relevant
counterfactual question formulated, it remains unclear just how to conceive of
and structure the counterfactual world of interest. Lewis (1973) develops an
approach based on the “closest possible world” to this one, save only for the
change to the exposure; but this only shifts, not solves, the problem. There
appears to be an unresolvable ambiguity about our counterfactual contrast.

Clearly there are deep philosophical problems, as well as technical
specification issues, besetting any approach to formulating a CoE problem. In
the sequel we deal only with the case of binary exposure and outcome
variables, denoted by $X$ and $Y$ respectively, assuming the above
specification problems have been addressed. But there will still remain some
ambiguity about the relevant counterfactual world, which will be reflected in
ambiguity about the answer to the CoE question.

In § 12 we introduce two approaches to relating the actual and counterfactual
worlds: SCM and CIM. The former is essentially deterministic, while the latter
allows some stochastic elements. However both make assumptions that might be
regarded as over-strong, leading to misleadingly precise answers to the CoE
question. In § 13 we show how each of these models can be reformulated in
terms of “potential outcomes”. In § 14 we explain how, taking full account of
real-world data on exposure and outcome, this approach can handle and quantify
the remaining ambiguities, by supplying an appropriate “interval of ambiguity”
for the probability of causation, PC.

We can narrow the interval of ambiguity for an individual case by deeper
understanding of the mechanisms and processes involved (Beyea & Greenland,
1999)—even when we can’t access the specific details of these for the
individual case at hand. We develop this theme in the remaining sections,
showing how information about additional variables can tighten the bounds on
PC.

###  12 Counterfactual constructions

####  12.1 SCM

The approach of Pearl (2009) (see also Pearl (2015), Dawid et al. (2015)) to
CoE is based on SCMs. In the case of Juanita, this would involve the
introduction of an unobserved exogenous “background” variable $U$, and the
assumption that Juanita’s diabetes status $Y$ is fully determined by her
Lipitor status $X$ and $U$: $Y=f_{Y}(X,U)$ (this requires a conception of $U$
as comprising all other pre-existing quantities that, together with $X$, would
totally determine $Y$—a collection that may not be easy to comprehend, let
alone specify). In some contexts it might be appropriate (“ignorability”) to
regard $X$ as independent of $U$ in the observational regime, as would happen,
if, for example, $X$ is generated by a randomising device. We do not impose
this throughout, and will specify where we do assume it.

The same functional relationship $Y=f_{Y}(X,U)$ is assumed to hold
(“consistency”) whether or not $X$ is imposed by external intervention. To
this invariance requirement, familiar from “effects of causes” analysis, we
add another, specific to “causes of effects”: that the value of the background
variable $U$ be the same in both the factual world, and in the counterfactual
world that we wish to contrast with this one.

To start with, we assume that the function $f_{Y}$ and the joint distribution
of $(X,U)$ are known. These unrealistic requirements are removed in § 14
below.

In the factual world we have observed $X=1$ and $Y=1$, i.e. $f_{Y}(1,U)=1$. We
can express the resulting uncertainty about the value of $U$ by means of its
conditional distribution, given $X=1,f_{Y}(1,U)=1$ (or, under ignorability,
given only $f_{Y}(1,U)=1$). We now turn to consider the counterfactual world.
Although $U$ is supposed to be the same in both worlds (and thus endowed with
the above conditional distribution), $X$ and $Y$ need not be. We introduce
“mirror variables”, $X^{\prime}$ and $Y^{\prime}$, as their counterfactual
counterparts. We retain the general structure across worlds, so that
$Y^{\prime}=f_{Y}(X^{\prime},U)$, both in observational and in interventional
counterfactual regimes.

In the counterfactual world, we now consider the effect of an intervention
$X^{\prime}\leftarrow 0$. The value of $Y^{\prime}$ will be $f_{Y}(0,U)$.
Using the previous conditional distribution of $U$, we obtain the
counterfactual distribution for $Y^{\prime}$, given the hypothetical
intervention $X^{\prime}\leftarrow 0$ and the factual knowledge $X=Y=1$. We
can thus evaluate the “probability of causation” as the probability that, in
this distribution, $Y^{\prime}=0$. That is,

$\mbox{\rm PC}=\Pr(Y^{\prime}=0\mid X=1,Y=1,X^{\prime}\leftarrow 0).$ (22)

####  12.2 Stochastic Causal Model (StCM)

A generalisation of the above model, which does not require deterministic
functional relationships, was suggested by Dawid (2000, §12), although it has
not been developed in detail. We again assume that the variable $U$ retains
its identity across the parallel worlds, and introduce the mirror variables
$X^{\prime}$, $Y^{\prime}$. But we now allow the dependence of $Y$ on $(X,U)$
to be given by a known conditional probability distribution: this allows a
more liberal attitude to the nature of $U$, which can be a perfectly normal,
in principle observable (though typically not observed), variable.

We assume that the same stochastic relationship also governs the dependence of
$Y^{\prime}$ on $(X^{\prime},U)$. We again require consistency to relate
observational and interventional regimes: the value of $Y$ (resp.,
$Y^{\prime}$) would be the same, whether $X$ (resp., $X^{\prime}$) arose by
intervention or not. In order to complete the specification, we regard
$Y^{\prime}$ and $Y$ as conditionally independent, given $(X,X^{\prime},U)$
(for example, we might consider them as involving “random noise”, operating
independently across worlds).

Having now a joint distribution for all variables in all worlds, we can again
compute $\mbox{\rm PC}=\Pr(Y^{\prime}=0\mid X=1,Y=1,X^{\prime}\leftarrow 0)$.
Specifically, by conditional independence,

$\displaystyle\Pr(Y^{\prime}=0\mid X=1,Y=1,X^{\prime}\leftarrow 0,U)$
$\displaystyle=$ $\displaystyle\Pr(Y^{\prime}=0\mid X^{\prime}\leftarrow 0,U)$
$\displaystyle=$ $\displaystyle\Pr(Y^{\prime}=0\mid X^{\prime}=0,U)$
$\displaystyle=$ $\displaystyle\Pr(Y=0\mid X=0,U).$

Then PC is the expectation of this, in the conditional distribution of $U$
given $X=1,Y=1,X^{\prime}\leftarrow 0$. But setting $X^{\prime}$ does not
affect the joint distribution of $(X,Y,U)$, so we just compute, from Bayes’s
theorem, the posterior density

$p(u\mid X=1,Y=1)\propto\Pr(X=1\mid u)\,\Pr(Y=1\mid X=1,u)\,p(u),$

where $p(u)$ is the prior density of $U$ (and the first term on the right-hand
side can be omitted under ignorability). Finally,

$\mbox{\rm PC}=\frac{\int\Pr(Y=0\mid X=0,u)\,\Pr(X=1\mid u)\,\Pr(Y=1\mid
X=1,u)\,p(u)\,du}{\int\Pr(X=1\mid u)\,\Pr(Y=1\mid X=1,u)\,p(u)\,du}.$ (23)

####  12.3 Twin network

In both the SCM and StCM approaches, the required computation can be automated
by building a “twin network” representation of the problem (Pearl, 2009,
§7.1.4), as in Figure 4, and making use of probability propogation algorithms
(Cowell et al., 1999) as implemented in software systems such as
Hugin.555https://www.hugin.com This is useful in more complex problems with
more variables.

Figure 4: Twin network. Under ignorability, the arrow from $U$ to $X$ can be
removed. That from $U$ to $X^{\prime}$ is absent because we are considering
$X^{\prime}$ as set by external intervention, taking no account of $U$.

The factual information $X=1,Y=1$ and the counterfactual intervention
$F_{X^{\prime}}=0$ (implying $X^{\prime}=0$) are entered at the relevant
nodes, and propagated through the network to obtain the appropriate
conditional distribution for $Y^{\prime}$.

####  12.4 More general StCMs

The StCM approach involves nominating some of the variables in a problem as
shared across worlds, while the others are allowed to differ. The associated
twin network will have a single copy of the shared variables and mirror copies
of the others, with the original DAG replicated and stitched together through
the shared variables. As discussed in Dawid (2000), the choice of which
variables are to be regarded as shared666Mackie (1980) refers to these as the
“causal field”, and notes the rôle of our own choice in its specification. is
a matter of imagination rather than science, and should relate to the specific
problem of interest—there can be no context-free right answer. For example,
there have been law suits by various states against tobacco companies,
claiming that if they had publicised their knowledge of the dangers of smoking
when they first knew of them, many lives could have been saved. Damages are
sought for the additional costs placed on health services—meaning the excess
cost in the actual world, over that of an imagined world in which they had
made their knowledge public. But how should we imagine that world? One could
reasonably argue that, in such a world, by giving up smoking, people would
have lived longer than they actually did. Then the actual (non-)actions of the
tobacco companies might well have increased the cost to the health services.
But what seems to be required for the case at hand is to imagine a world where
people had the same lifetimes, but were healthier, i.e., to regard lifetimes
as shared across parallel worlds—and this even though lifetimes can be
considered an effect of the companies’ decisions. It is not clear how such
considerations could be accommodated in a SCM.

###  13 Potential outcomes

As observed in § 10, the SCM approach produces implied potential outcomes,
$Y(1)=f_{Y}(1,U)$, $Y(0)=f_{Y}(0,U)$. The pair
$\mbox{$\mathbf{Y}$}=(Y(1),Y(0))$ is a function of $U$, with a bivariate
distribution induced by that of $U$. And in fact $\mathbf{Y}$ is all that
needs to be retained of $U$ to fully describe the problem: we can replace $U$
by $\mathbf{Y}$, with the functional dependence of $Y$ on
$(X,\mbox{$\mathbf{Y}$})$ given simply by $Y=Y(X)$. The problem can thus be
more concisely expressed in terms of $(X,\mbox{$\mathbf{Y}$})$, these having a
joint distribution. Then (22) becomes

$\mbox{\rm PC}=\Pr(Y(0)=0\mid X=1,Y(1)=1).$ (24)

Note that under ignorability $X$ is independent of $\mathbf{Y}$ (this is
indeed the very definition of ignorability in the PO framework), and then the
conditioning on $X=1$ in (24) can be removed.

For the StCM approach we proceed as follows. The stochastic dependence of $Y$
on $(X,U)$ can be modelled by introducing a further unobserved “noise”
variable $V$, independent of $(X,U)$, and representing $Y=f_{Y}(X,U,V)$ for a
suitable function $f_{Y}$. This can be done in many ways. One possible way
uses the probability integral transformation: if $Y$ is a univariate variable
whose conditional distribution function $F_{x,u}(y)$, given $X=x,U=u$, is
strictly increasing, take $V$ to be uniform on $[0,1]$, and
$f_{Y}(x,u,v)=F^{-1}_{x,u}(v)$.

There will be a counterfactual mirror $V^{\prime}$ of $V$, with
$Y^{\prime}=f_{Y}(X,U,V^{\prime})$. We now define potential outcomes
$Y(1)=f_{Y}(1,U,V)$, $Y(0)=f_{Y}(0,U,V^{\prime})$, having a joint distribution
induced by that of $(U,V,V^{\prime})$. Although the variables so constructed
will depend on the specific choices made for the noise variable $V$ and the
function $f_{Y}$, it is easy to see that their joint distribution will in all
cases be that of $(Y,Y^{\prime})$, given interventions $X\leftarrow 1$,
$X^{\prime}\leftarrow 0$. And since $X=1\Rightarrow Y=Y(1)$, etc., (24) again
holds.

We thus see that, in all cases, we can ignore the finer details, and represent
the problem by means of a joint distribution for $(X,\mbox{$\mathbf{Y}$})$,
with PC given by (24).

###  14 Empirical information

We have so far supposed that the full probabilistic structure of the model,
with its variables $U,X,Y$, is known. In a StCM, we can take $U$ to be a
specified potentially observable variable, and then this assumption is not
unreasonable. However it is typically implausible for a SCM, where, in order
to achieve the required deterministic dependence of $Y$ on $(X,U)$, we would
have to conceive of a fantastically rich $U$. Alternatively, we can re-express
the problem in terms of the pair $\mathbf{Y}$ of potential responses, with a
joint distribution for $(X,\mbox{$\mathbf{Y}$})$. Without making further
assumptions on the originating SCM or StCM, there are no constraints on this
joint distribution (other than independence of $X$ and $\mathbf{Y}$ under
ignorability). We can however gather empirical data to constrain it, and thus
hope to estimate PC by (24).

In the sequel we proceed on this basis, and consider what can indeed be
estimated. We initially assume that we can only observe $X$ and $Y$, in
interventional and/or observational circumstances. We can estimate
$\Pr\\{Y(x)=1\\}=\Pr(Y=1\mid X\leftarrow x)$ from interventional studies. When
we can not assume ignorability, we can also estimate, from observational data,
$\Pr\\{Y(1)=1\mid X=1\\}=\Pr(Y=1\mid X=1)$ (by consistency), and
$\Pr\\{Y(0)=1\mid X=0\\}=\Pr(Y=1\mid X=0)$, as well as the marginal
distribution of $X$. For this general case it might initially seem problematic
to estimate, say, $\Pr\\{Y(1)=1\mid X=0\\}$, since this involves non-
coexisting worlds, one with $X=1$ and the other with $X=0$; but we can in fact
solve for it, using $\Pr\\{Y(1)=1\\}=\Pr\\{Y(1)=1\mid
X=1\\}\,\Pr(X=1)+\Pr\\{Y(1)=1\mid X=0\\}\,\Pr(X=0)$, where all other terms are
estimable.

We can thus estimate the bivariate distribution of $(X,Y(1))$, and likewise
that of $(X,Y(0))$. However, the full trivariate distribution of
$(X,Y(1),Y(0))$ is not estimable: since we can never observe both $Y(0)$ and
$Y(1)$ simultaneously, no data can tell us directly about the dependence
between $Y(0)$ and $Y(1)$, either marginally or conditionally on $X$. And
since (24) requires such information, typically PC is not identifiable from
empirical data.

Nevertheless, the estimable bivariate distributions do impose constraints on
the possible values of PC. Moreover, these constraints can often be tightened
still further when we can observe other, related, variables in the problem. We
now turn to investigate such constraints, in a number of contexts.

###  15 Analysis of the probability of causation

Let us consider the initial attribution example: Juanita took Lipitor 80mg
daily for 3 years ($X=1$) and developed diabetes ($Y=1$). Was that because she
took Lipitor? We wish to address this question and assess the probability of
causation, PC, for Juanita’s case, using data collected on other individuals.
To this end we assume

Exchangeability

Juanita is similar to the population from which probabilities have been
computed, so that those probabilities apply to her.

Exchangeability may require restriction of the data considered to individuals
deemed sufficiently like Juanita.

Except where relaxed in § 15.3 below, we shall also assume

Ignorability

The fact that Juanita chose to take the drug is not informative about her
response to it, either factually or counterfactually. Formally, we require
independence, $X\,\mbox{$\perp\\!\\!\\!\perp$}\,\mbox{$\mathbf{Y}$}$, between
$X$ and the pair of potential responses $\mbox{$\mathbf{Y}$}=(Y(1),Y(0))$.

Ignorability is a strong assumption, and will often be inappropriate. When it
can be assumed, we can use data from either experimental or observational
studies; otherwise we need data from both of these.

Under ignorability, the target (24) becomes

$\mbox{\rm PC}=\Pr(Y(0)=0\mid Y(1)=1).$ (25)

We proceed to assess this using the general potential outcome framework of §
13 and § 14, where no assumptions are imposed on the joint distribution of
$Y(0)$ and $Y(1)$ beyond those that can be informed by the empirical data.
Further details may be found in Dawid et al. (2017), Dawid & Musio (2021).

####  15.1 Basic inequalities

Suppose we have access to (observational or experimental) data, supplying
values for

$\Pr\\{Y(x)=y\\}=\Pr(Y=y\mid X=x)\quad(x=0,1\quad y=0,1).$ (26)

Define

$\displaystyle\tau$ $\displaystyle:=$ $\displaystyle\Pr(Y=1\mid
X=1)-\Pr(Y=1\mid X=0)$ $\displaystyle\rho$ $\displaystyle:=$
$\displaystyle\Pr(Y=1\mid X=1)-\Pr(Y=0\mid X=0).$

The joint distribution of $(Y(0),Y(1))$ must have the form of Table 1, where
the marginal probabilities are given by (26), re-expressed in terms of $\tau$
and $\rho$, and where the unidentified “slack” quantity $\xi$ embodies the
residual ambiguity in the distribution.

| $Y(1)=0$ | $Y(1)=1$ |
---|---|---|---
$Y(0)=0$ | $\frac{1}{2}(1-\rho-\xi)$ | $\frac{1}{2}(\xi+\tau)$ | $\frac{1}{2}(1+\tau-\rho)$
$Y(0)=1$ | $\frac{1}{2}(\xi-\tau)$ | $\frac{1}{2}(1+\rho-\xi)$ | $\frac{1}{2}(1-\tau+\rho)$
| $\frac{1}{2}(1-\tau-\rho)$ | $\frac{1}{2}(1+\tau+\rho)$ | 1
Table 1: Joint distribution of $Y(0)$ and $Y(1)$

For all the entries of Table 1 to be non-negative, we require

$|\tau|\leq\xi\leq 1-|\rho|.$ (27)

The probability of causation (25) is

$\mbox{\rm PC}=\frac{\xi+\tau}{1+\tau+\rho}.$ (28)

On using (27), we obtain the following interval bounds for PC:

$l:=\max\left\\{0,\frac{2\tau}{1+\tau+\rho}\right\\}\leq\mbox{PC}\leq\min\left\\{1,\frac{1+\tau-\rho}{1+\tau+\rho}\right\\}=:u,$
(29)

or equivalently

$l=\max\left\\{0,1-\frac{1}{\mbox{RR}}\right\\}\leq\mbox{PC}\leq\min\left\\{1,\frac{\Pr(Y=0\mid
X=0)}{\Pr(Y=1\mid X=1)}\right\\}=u,$ (30)

where

$\mbox{RR}=\frac{\Pr(Y=1\mid X=1)}{\Pr(Y=1\mid X=0)}$ (31)

is the risk ratio (Robins & Greenland, 1989).

In the absence of additional information or assumptions, these bounds
constitute the best available inference regarding PC. In particular,
$\mbox{RR}>2$, “doubling the risk”, implies that $\mbox{PC}>0.5$. In a civil
legal case, causality might then be concluded “on the balance of
probabilities”. However, because of the remaining ambiguity, expressed by the
inequalities in (30), finding that RR falls short of 2 does not imply that
$\mbox{PC}<0.5$.

####  15.2 Refining the inequalities: Covariates

When we have additional information we may be able to refine our inferences
about PC (Kuroki & Cai, 2011).

Thus suppose that we also have information on a covariate $S$, a pretreatment
individual characteristic that can vary from person to person and can have an
effect on both $X$ and $Y$. The relevant potential responses are now
$\mbox{$\mathbf{X}$}:=(X(s):s\in S)$, $\mbox{$\mathbf{Y}$}:=(Y(s,x):s\in
S,x=0\mbox{ or }1)$ and the relationship between potential and actual
responses is $X=X(S)$, $Y=Y(S,X)$. We again assume exchangeability and
ignorability, the latter now being formalised as mutual independence between
$S$, $\mathbf{X}$ and $\mathbf{Y}$.

For simplicity we suppose that $S$ is discrete and that we can estimate from
the data the full joint distribution of $(S,X,Y)$.

In the case that we are also able to measure $S$ for Juanita, say $S=s$, we
can simply restrict the experimental subjects to those having the same
covariate value (who are thus like Juanita in all relevant respects). The
probability of causation is now

$\mbox{\rm PC}(s)=\Pr\left(Y(s,0)=0\mid Y(s,1)=1\right).$

We can bound this just as in (30), but with all probabilities now conditioned
on $S=s$, obtaining $l(s)\leq\mbox{\rm PC}(s)\leq u(s)$.

More interesting is the case in which we don’t observe $S$ for Juanita. We
have to consider what would have been the response if, counterfactually,
Juanita’s exposure had been $X=0$. We assume that this is the minimal change
made between the factual and the counterfactual worlds, so that, in
particular, there is no change to the value or distribution of $S$.

The probability of causation is now:

PC $\displaystyle=$ $\displaystyle\Pr\left\\{Y(S,0)=0\mid
X(S)=1,Y(S,1)=1\right\\}$ (32) $\displaystyle=$
$\displaystyle\sum_{s}\mbox{PC}(s)\times\Pr(S=s\mid X=1,Y=1).$

There are no logical relationships between the distributions of
$(Y(s,0),Y(s,1))$ for different values of $S$. So by independently varying the
values taken by the slack variables in the joint distribution of these
potential responses, all the lower bounds $l(s)$ for $\mbox{PC}(s)$ can be
achieved simultaneously. This leads to an achievable lower bound for PC:

$\mbox{PC}\geq L=\sum_{s}l(s)\times\Pr(S=s\mid X=1,Y=1).$ (33)

We can express

$\displaystyle L$ $\displaystyle=$ $\displaystyle\frac{1}{\Pr(Y=1\mid X=1)}$
$\displaystyle{}\times\sum_{s}\max\left\\{0,\Pr(Y=1\mid X=1,S=s)-\Pr(Y=1\mid
X=0,S=s)\right\\}\times\Pr(S=s\mid X=1).$ (34)

Similarly we obtain upper bound

$\displaystyle U$ $\displaystyle=$ $\displaystyle 1-\frac{1}{\Pr(R=1\mid
E=1)}$ $\displaystyle{}\times\sum_{s}\max\\{0,\Pr(Y=1\mid X=1,S=s)-\Pr(Y=0\mid
X=0,S=s)\\}\times\Pr(S=s\mid X=1).$ (35)

We can not compare these bounds directly with those of (30), since when we
don’t take account of $S$ the relation between $X$ and $Y$ is generally non-
ignorable: $\Pr(Y=y\mid X\leftarrow x)\neq\Pr(Y=y\mid X=x)$.

####  15.3 Non-ignorability

Tian & Pearl (2000) analysed this non-ignorable case where we don’t observe
$S$, either for Juanita or in the external data (still considered exchangeable
with Juanita). We now need both observational and experimental data on $X$ and
$Y$. Tian & Pearl (2000) develop the following lower bound for
$\mbox{PC}=\Pr\\{Y(0)=0\mid X=1,Y(1)=1\\}$:

$L^{\prime}=\max\left\\{0,\frac{\Pr(Y=1)-\Pr(Y=1\mid X\leftarrow
0)}{\Pr(X=1,Y=1)}\right\\}.$ (36)

Dawid & Musio (2021) show that this can also be derived as a special case of
our expression (34), if we substitute for $S$ the binary variable $D=$
“desired exposure” (Corradi & Musio, 2020). $D$ will be identical with $X$ in
an observational context, but need not be so in an experimental setting, where
$D$ may not be observable.

In our case, with access to information on $S$, we could compute $\Pr(Y=1\mid
X\leftarrow 0)$ by the “back-door formula”:

$\Pr(Y=1\mid X\leftarrow 0)=\sum_{s}\Pr(Y=1\mid X=0,S=s)\times\Pr(S=s),$ (37)

and thus compute $L^{\prime}$ of (36). It can be shown (Dawid & Musio, 2021)
that $L^{\prime}\leq L$ of (34), with equality if and only if all the
conditional risk ratios

$\frac{\Pr(Y=1\mid X=1,S=s)}{\Pr(Y=1\mid X=0,S=s)}\quad\quad(s\in S)$

lie on the same side of 1: knowing, and using, the information about $S$ is at
least as good as ignoring it. Similarly we can show that the upper bound $U$
of (35) does not exceed the upper bound $U^{\prime}$ derived by Tian & Pearl
(2000),

$U^{\prime}=\min\left\\{1,\frac{\Pr(Y=0\mid X\leftarrow
0)-\Pr(X=0,Y=0)}{\Pr(X=1,Y=1)}\right\\},$

with equality if and only if all the ratios

$\frac{\Pr(Y=1\mid X=1,S=s)}{\Pr(Y=0\mid X=0,S=s)}\quad\quad(s\in S)$

lie on the same side of 1.

####  15.4 Mediators

We now consider the case that a third variable $M$ acts a complete mediator in
the causal pathway $X\rightarrow M\rightarrow Y$ between the exposure $X$ and
the response $Y$. Again we restrict to the case that all variables are binary.
We introduce the potential value $M(x)$ of $M$ for $X=x$, and $Y(m)$, the
potential value of $Y$ for $M=m$, and define
$\mbox{$\mathbf{M}$}=(M(0),M(1))$, $\mbox{$\mathbf{Y}$}=(Y(0),Y(1))$. We
observe $X$, $M=M(X)$ and $Y=Y(M)$. We assume the exchangeability and the
ignorability conditions, the latter expressed as mutual independence between
$X$, $\mathbf{M}$ and $\mathbf{Y}$. This implies the observational conditional
independence

$Y\,\mbox{$\perp\\!\\!\\!\perp$}\,X\mid M$ (38)

which is a testable implication of our assumptions. We assume we have data
supplying values for $\Pr(M=m\mid X=x)$ and $\Pr(Y=y\mid M=m)$, and compute,
by (38),

$\Pr(Y=y\mid X=x)=\sum_{m}\Pr(Y=y\mid M=m)\Pr(M=m\mid X=x),$ (39)

For the case that $M$ is observed in the experimental data, but not for
Juanita, Dawid et al. (2016) showed that this additional information does not
change the lower bound $l$ on PC in (30), but does lower the upper bound, $u$.
Dawid & Musio (2021) extend this analysis to cases with additional covariates,
while Dawid et al. (2019) deal with the case that we have a complete mediation
sequence $X=M_{0}\rightarrow M_{1}\rightarrow\cdots\rightarrow
M_{n-1}\rightarrow M_{n}=Y$, and know the probabilistic structure of each link
in the chain; with all, some, or none of the $M$s being observed for Juanita.

###  16 Further CoE problems

We have only dealt here with the case of a single putative cause,
understanding causation in terms of the “but for” criterion. There are many
more complex problems that can not be handled in this way: in particular, the
whole field of legal causation has to handle a wide variety of problems
involving multiple competing causes and other concepts of causality (Hart &
Honore, 1985, Goldberg, 2011). While there have been some interesting
statistical treatments of specific problems, e.g., Cox (I984), it seems fair
to say that general philosophical understandings of causality in such problems
have not reached maturity. To the extent that problems are modelled in formal
terms, this often involves a purely deterministic understanding of causality,
which is not easily translated into a stochastic framework. Halpern (2016)
makes an interesting attempt to pin down the concept of “the actual cause”
using the SCM framework, but admits that he is unable to reach a fully
satisfying conclusion.

There is clearly much ground remaining to be covered in understanding “causes
of effects”, but it is perhaps premature to attempt more detailed statistical
treatment before clearer general principles have emerged.

## CONCLUSION

We have presented a thorough account of a number of ways in which the
statistical problems of effects of causes (EoC) and of causes of effects (CoE)
have been formulated. Although most treatments of statistical causality use
essentially identical tools to address both these problem areas, we consider
that this is inappropriate. Popular formalisms, such as potential outcomes and
structural causal models, involve deterministic relations, and allow formal
statements concerning two or more parallel worlds simultaneously. While
something of this nature appears unavoidable for CoE considerations, it is
unnecessary for EoC analyses, which can proceed using stochastic models and
statistical decision theory. Furthermore, the use of an inappropriate formal
framework brings with it the danger that we treat any mathematically well-
formed formula (such as those describing “individual causal effect” and “local
average treatment effect”) as meaningful, when it may not be.

### DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

### ACKNOWLEDGMENTS

The second author was partially supported by the project STAGE of Fondazione
di Sardegna.

### References

  * Angrist et al. (1996) Angrist J, Imbens G, Rubin DB. 1996. Identification of causal effects using instrumental variables. Journal of the American Statistical Association 91:444–455
  * Balke & Pearl (1997) Balke AA, Pearl J. 1997. Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association 92:1172–6
  * Beyea & Greenland (1999) Beyea J, Greenland S. 1999. The importance of specifying the underlying biologic model in estimating the probability of causation. Health Physics 76:269–274
  * Bowden & Turkington (1984) Bowden RJ, Turkington DA. 1984. Instrumental Variables. Cambridge University Press
  * Bühlmann (2020) Bühlmann P. 2020. Invariance, causality and robustness (with Discussion). Statistical Science 35:404–436
  * Case Report (2015) Case Report. 2015. Hempstead v. Pfizer, Inc. (In re Lipitor (Atorvastatin Calcium) Mktg., Sales Practices & Prods. Liab. Litig. 150 F. Supp. 3d 644 (D.S.C. 2015)
  * Constantinou & Dawid (2017) Constantinou P, Dawid AP. 2017. Extended conditional independence and applications in causal inference. Annals of Statistics 45:2618–2653
  * Corradi & Musio (2020) Corradi F, Musio M. 2020. Causes of effects via a Bayesian model selection procedure. Journal of the Royal Statistical Society, Series A 183:1777–1792
  * Cowell et al. (1999) Cowell RG, Dawid AP, Lauritzen SL, Spiegelhalter DJ. 1999. Probabilistic Networks and Expert Systems. New York: Springer
  * Cox (I984) Cox LA. I984. Probability of causation and the attributable proportion of risk. Risk Analysis 4:221–230
  * Dawid (1976) Dawid AP. 1976. Properties of diagnostic data distributions. Biometrics 32:647–658
  * Dawid (1979) Dawid AP. 1979. Conditional independence in statistical theory (with Discussion). Journal of the Royal Statistical Society, Series B 41:1–31
  * Dawid (2000) Dawid AP. 2000. Causal inference without counterfactuals (with Discussion). Journal of the American Statistical Association 95:407–448
  * Dawid (2003) Dawid AP. 2003. Causal inference using influence diagrams: The problem of partial compliance (with Discussion), In Highly Structured Stochastic Systems, eds. PJ Green, NL Hjort, S Richardson, pp. 45–81, Oxford University Press
  * Dawid (2010) Dawid AP. 2010. Beware of the DAG!, In Proceedings of the NIPS 2008 Workshop on Causality, eds. I Guyon, D Janzing, B Schölkopf, vol. 6 of Journal of Machine Learning Research Workshop and Conference Proceedings, pp. 59–86. http://tinyurl.com/33va7tm
  * Dawid (2015) Dawid AP. 2015. Statistical causality from a decision-theoretic perspective. Annual Review of Statistics and its Application 2:273--303
  * Dawid (2017) Dawid AP. 2017. On individual risk. Synthese 194:3445--3474
  * Dawid (2021a) Dawid AP. 2021a. Decision-theoretic foundations for statistical causality (with Discussion). Journal of Causal Inference (in Press) arXiv:2004.12493
  * Dawid (2021b) Dawid AP. 2021b. The tale wags the DAG. In Probabilistic and Causal Inference: The Works of Judea Pearl, eds. R Dechter, H Geffner, J Halpern. Association for Computing Machinery. (In Press).
  * Dawid & Didelez (2012) Dawid AP, Didelez V. 2012. ‘‘Imagine a can opener’’ --- The magic of principal stratum analysis. International Journal of Biostatistics 8\. DOI: 10.1515/1557-4679.1391
  * Dawid et al. (2014) Dawid AP, Faigman DL, Fienberg SE. 2014. Fitting science into legal contexts: Assessing effects of causes or causes of effects? (with Discussion and authors’ rejoinder). Sociological Methods and Research 43:359--421
  * Dawid et al. (2015) Dawid AP, Faigman DL, Fienberg SE. 2015. On the causes of effects: Response to Pearl. Sociological Methods and Research 44:165--174
  * Dawid et al. (2019) Dawid AP, Humphreys M, Musio M. 2019. Bounding causes of effects with mediators. arXiv:1907.00399
  * Dawid et al. (2016) Dawid AP, Murtas R, Musio M. 2016. Bounding the probability of causation in mediation analysis, In Topics on Methodological and Applied Statistical Inference, eds. TD Battista, E Moreno, W Racugno, pp. 75--84, Springer
  * Dawid & Musio (2021) Dawid AP, Musio M. 2021. What can group level data tell us about individual causality? In Statistics in the Public Interest, eds. A Carriquiry, W Eddy, J Tanur. Springer. In Press
  * Dawid et al. (2017) Dawid AP, Musio M, Murtas R. 2017. The probability of causation. Law, Probability and Risk 16:163--179
  * Faigman et al. (2014) Faigman DL, Monahan J, Slobogin C. 2014. Group to individual (G2i) inference in scientific expert testimony. University of Chicago Law Review 81:417--480
  * Goldberg (2011) Goldberg R, ed. 2011. Perspectives on Causation, Oxford. Hart Publishing
  * Greenland (1999) Greenland S. 1999. Relation of probability of causation to relative risk and doubling dose: A methodologic error that has become a social problem. American Journal of Public Health 89:1166--1169
  * Halpern (2016) Halpern JY. 2016. Actual Causality. Cambridge, Massachusetts: MIT Press
  * Hart & Honore (1985) Hart HLA, Honore AM. 1985. Causation in the Law. Oxford: Clarendon Press
  * Hausman (1998) Hausman D. 1998. Causal Asymmetries. Cambridge: Cambridge University Press
  * Holland (1986) Holland PW. 1986. Statistics and causal inference (with Discussion). Journal of the American Statistical Association 81:945--970.
  * Holland (1988) Holland PW. 1988. Causal inference, path analysis, and recursive structural equations models. Sociological Methodology 18:449--484
  * Imbens & Angrist (1994) Imbens GW, Angrist J. 1994. Identification and estimation of local average treatment effects. Econometrica 62:467--476
  * Katan (1986) Katan MB. 1986. Apolipoprotein E isoforms, serum cholesterol, and cancer. The Lancet 327:507--508
  * Kuroki & Cai (2011) Kuroki M, Cai Z. 2011. Statistical analysis of ‘probabilities of causation’ using co-variate information. Scandinavian Journal of Statistics 38:564--577
  * Lauritzen et al. (1990) Lauritzen SL, Dawid AP, Larsen BN, Leimer HG. 1990. Independence properties of directed Markov fields. Networks 20:491--505
  * Lewis (1973) Lewis DK. 1973. Counterfactuals. Oxford: Blackwell
  * Mackie (1980) Mackie JL. 1980. The Cement of the Universe: A Study of Causation. Oxford University Press
  * Mill (1843) Mill JS. 1843. A system of logic, ratiocinative and inductive: Being a connected view of the principles of evidence, and methods of scientific investigation. London: John W. Harper
  * Pearl (2009) Pearl J. 2009. Causality: Models, Reasoning and Inference. Cambridge: Cambridge University Press, Second ed.
  * Pearl (2015) Pearl J. 2015. Causes of effects and effects of causes. Sociological Methods and Research 44:149--164
  * Pearl & Mackenzie (2018) Pearl J, Mackenzie D. 2018. The Book of Why. New York: Basic Books
  * Price (1991) Price H. 1991. Agency and probabilistic causality. British Journal for the Philosophy of Science 42:157--176
  * Reichenbach (1956) Reichenbach H. 1956. The Direction of Time. Berkeley: University of Los Angeles Press
  * Robins & Greenland (1989) Robins JM, Greenland S. 1989. The probability of causation under a stochastic model for individual risk. Biometrics 45:1125--1138
  * Rubin (1974) Rubin DB. 1974. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology 66:688--701
  * Tian & Pearl (2000) Tian J, Pearl J. 2000. Probabilities of causation: Bounds and identification. Annals of Mathematics and Artificial Intelligence 28:287--313
  * Verma & Pearl (1990) Verma T, Pearl J. 1990. Causal networks: Semantics and expressiveness, In Uncertainty in Artificial Intelligence 4, eds. RD Shachter, TS Levitt, LN Kanal, JF Lemmer, pp. 69--76, Amsterdam: North-Holland
  * Woodward (2003) Woodward J. 2003. Making Things Happen: A Theory of Causal Explanation. Oxford: Oxford University Press
  * Woodward (2016) Woodward J. 2016. Causation and manipulability. The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.).
https://plato.stanford.edu/entries/causation-mani/

  * Wright (1921) Wright SS. 1921. Correlation and causation. Journal of Agricultural Research 20:557–--585