Article

A counterfactual approach to bias and effect modification in terms of response types

BMC Medical Research Methodology (Impact Factor: 2.27). 07/2013; 13(1):101. DOI: 10.1186/1471-2288-13-101
Source: PubMed

ABSTRACT

The counterfactual approach provides a clear and coherent framework to think about a variety of important concepts related to causation. Meanwhile, directed acyclic graphs have been used as causal diagrams in epidemiologic research to visually summarize hypothetical relations among variables of interest, providing a clear understanding of underlying causal structures of bias and effect modification. In this study, the authors aim to further clarify the concepts of bias (confounding bias and selection bias) and effect modification in the counterfactual framework.
The authors show how theoretical data frequencies can be described by using unobservable response types both in observational studies and in randomized controlled trials. By using the descriptions of data frequencies, the authors show epidemiologic measures in terms of response types, demonstrating significant distinctions between association measures and effect measures. These descriptions also demonstrate sufficient conditions to estimate effect measures in observational studies. To illustrate the ideas, the authors show how directed acyclic graphs can be extended by integrating response types and observed variables.
This study shows a hitherto unrecognized sufficient condition to estimate effect measures in observational studies by adjusting for confounding bias. The present findings would provide a further understanding of the assumption of conditional exchangeability, clarifying the link between the assumptions for making causal inferences in observational studies and the counterfactual approach. The extension of directed acyclic graphs using response types maintains the integrity of the original directed acyclic graphs, which allows one to understand the underlying causal structure discussed in this study.
The present findings highlight that analytic adjustment for confounders in observational studies has consequences quite different from those of physical control in randomized controlled trials. In particular, the present findings would be of great use when demonstrating the inherent distinctions between observational studies and randomized controlled trials.

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Available from: Toshihide Tsuda, Feb 10, 2014
RES E AR C H A R T I C L E Open Access
A counterfactual approach to bias and effect
modification in terms of response types
Etsuji Suzuki
1*
, Toshiharu Mitsuhashi
1
, Toshihide Tsuda
2
and Eiji Yamamoto
3
Abstract
Background: The counterfactual approach provides a clear and coherent framework to think about a variety of
important concepts related to causation. Meanwhile, directed acyclic graphs have been used as causal diagrams in
epidemiologic research to visually summarize hypothetical relations among variables of interest, providing a clear
understanding of underlying causal structures of bias and effect modification. In this study, the authors aim to
further clarify the concepts of bias (confounding bias and selection bias) and effect modification in the
counterfactual framework.
Methods: The authors show how theoretical data frequencies can be described by using unobservable response
types both in observational studies and in randomized controlled trials. By using the descriptions of data
frequencies, the authors show epidemiologic measures in terms of response types, demonstrating significant
distinctions between association measures and effect measures. These descriptions also demonstrate sufficient
conditions to estimate effect measures in observational studies. To illustrate the ideas, the authors show how
directed acyclic graphs can be extended by integrating response types and observed variables.
Results: This study shows a hitherto unrecognized sufficient condition to estimate effect measures in observational
studies by adjusting for confounding bias. The present findings would provide a further understanding of the
assumption of conditional exchangeability, clarifying the link between the assumptions for making causal inferences
in observational studies and the counterfactual approach. The extension of directed acyclic graphs using response
types maintains the integrity of the original directed acyclic graphs, which allows one to understand the underlying
causal structure discussed in this study.
Conclusions: The present findings highlight that analytic adjustment for confounders in observational studies has
consequences quite different from those of physical control in randomized controlled trials. In particular, the
present findings would be of great use when demonstrating the inherent distinctions between observational
studies and randomized controlled trials.
Keywords: Bias, Causal inference, Counterfactual, Directed acyclic graphs, Effect modification, Exchangeability,
Randomization, Response types
Background
The counterfactual approach provides a clear and coherent
framework t o think about a variety of important concepts
related to causation [1,2]. In particular, the counterfactual
approach to confounding has been widely accessible to epi-
demiologists since the publication of a classic methods
paper by Greenland and Robins [3], and the concept of bias
is now explained in the counterfactual framework [4-12].
(Note that an update of the classic methods paper was re-
cently published [13]). Meanwhile, directed acyclic graphs
(DAGs) have long been used as causal diagrams in epidemi-
ologic research to visually summarize hypothetical relations
among variables of interest [14,15]. DAGs have been used
extensively to determine the variables for which it is neces-
sary to control for confounding bias to estimate causal
effects [14-20]. Besides, Hernán et al. [21] showed that vari-
ous types of selection bias share a common underlying
causal structure, and referred to conditioning on common
* Correspondence: etsuji-s@cc.okayama-u.ac.jp
1
Department of Epidemiology, Graduate School of Medicine, Dentistry and
Pharmaceutical Sciences, Okayama University, 2-5-1 Shikata-cho, Kita-ku,
Okayama 700-8558, Japan
Full list of author information is available at the end of the article
© 2013 Suzuki et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Suzuki et al. BMC Medical Research Methodology 2013, 13:101
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Page 1
effects as selection bias. Furthermore, VanderWeele and
Robins [22] provided a structural classification of effect
modification by using DAGs. Indeed, the different ap-
proaches provide complementary perspectives, and can be
employed together to provide a clearer understanding of
causality [23].
In this study, we aim to further clarify the concepts of
bias (confound ing bias and selection bias) and effect modi-
fication in the counterfactual framework. To achieve this,
we show how theoretical data frequencies can be described
by using unobservable response types both in observational
studies and in randomized controlled trials. These des-
criptions also demonstrate sufficient conditions to estimate
effect measures in observational studies, which would pro-
vide a further understanding of the assumption of con-
ditional exchangeability. To illustrate the ideas, DAGs are
employed, and we show how one can extend the original
DAGs by integrating response types and observed variables.
We deal only with structural (systematic) relations among
the under lying variables of interest, so that an issue of ran-
dom variation does not arise. Throughout this article, we
assume that the consistency condition is met [24-28].
Methods
Definitions and notation
A causal diagram and causal effects
We use a total of 4 binary variables as shown in Figure 1.
We let D denote a binary outcome of interest (1: outcome
occurred, 0: outcome did not occur) and let E denote a
binary cause of interest (1: exposed, 0: unexposed) that is
potentially manipulable. We let C denote a binary common
cause of E and D (1:present,0:absent),whichisalsopo-
tentially manipulable. Typically, C is called a confounder of
the effect of E on D. (Note that we assume that C precedes
E temporally in this study, which is in general not neces-
sary for C to be a confounder. Recently, VanderWeele and
Shpitser [29,30] further discussed the definition of a con-
founder.) As explained later, C can also act as a direct effect
modifier for the causal effect of E on D because C is a dir -
ect cause of D [22]. Finally, we let S denote selection vari-
able (1: selected, 0: not selected), which is a common effect
of E and D.AdjustmentforS yields a spurious association
between E and D, which is call ed selection bias [21]. Alter-
natively, one may assume that S is also directly influenced
by C, as shown by using a dashed arrow in Figure 1. Al-
though the arrow is assumed to be absent throughout this
paper to avoid technical complications, the following dis-
cussion can be readily extended to the situations in which
the dashed arrow is present.
Throughout this paper, we discuss the case where the
causal effect of primary interest is the effect of E on D in
the total population, including the subpopulations of C =0
and C = 1. In the counterfactual framework, this causal
effect is given by comparing P[D
e=0
=1] and P[D
e=1
=1],
where P[D
e
= 1] denotes the proportion of subjects that
would have developed the outcome D = 1 had all subjects
in the total population received exposure value e.Thevar-
iables D
e=0
and D
e=1
are referred to as potential outcomes.
Therefore, when we use risk ratios (RRs) as a measure of
interest, a causal RR in the total population is given by
PD
e¼1
¼ 1½
PD
e¼0
¼ 1½
: ð1Þ
Causal RRs in the total population can be consistently esti-
mated under the assumption of (conditional) exchangeabil-
ity, or, equivalently, no unmeasured confounding (i.e.,
ED
e
for e). Furthermore, in addition to the effect of
E on D in the total population, we also discuss causal effect
of E on D within the subsets of C. In this case, a causal RR
within the subsets is given by
PD
e¼1
¼ 1
j
C ¼ c½
PD
e¼0
¼ 1 C ¼ c
j
½
: ð2Þ
Causal RRs in the subsets of C can be consistently esti-
mated under the assumption of conditional exchangeability,
or, equivalently, no unmeasured confounding given data on
C (i.e., ED
e
|C for e). Note that, when the causal effect of
interest is the effect of E on D either in the total population
C
E
D
S
Figure 1 A causal diagram depicting a hypothetical example.
E, D, C, and S denote exposure, disease, confounder, and selection
variable, respectively. C may act as a direct effect modifier
simultaneously. In this paper, S is assumed not to be influenced
directly by C (i.e., the dashed arrow is absent). The present
discussion, however, can be readily extended to the situations in
which one assumes that the dashed arrow is present.
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or in the subsets of C, intervening on E is of concern, and
one does not consider intervening on C. Indeed, as outlined
by VanderWeele [31], intervening on C would only be of
concern if the joint effect of E and C on D was of interest.
Therefore, under the situation in which C is being consid-
ered as a potential confounder as well as direct effect modi-
fier (Figure 1), intervening on C is not of interest.
When we show how theoretical data frequencies can
be described by using unobservable response types in
observational studies, however, it is of great use to eluci-
date the relations between C and E in the counterfactual
framework. By so doing, we demonstrate sufficient con-
ditions to esti mate effect measures in observation al stud-
ies, which would provide a further understanding of the
assumption of conditional exchangeability.
Response types
First, we elucidate the relations between C and E by hypo-
thetically conceptualizing potential outcomes of E in the
counterfactual framework. We let E
c
(ω) denote the poten-
tial outcomes of E for individual ω if, possibly contrary to
fact, there had been interventions to set C to c. (In the fol-
lowing sections, we explicitly show ω to discuss response
types.) Then, for each individual ω, there would be 2 rele-
vant potential outcomes of E,i.e.,E
1
(ω)andE
0
(ω), which
correspond to exposure status of that individual when C is
present and absent, respectively. As a result, individuals
can be classified into 4 (i.e., 2
2
)differentE response types
as enumerated in Table 1. We let E
T
(ω)denoteE response
type of individual ω. In some cases, the effect of C may be
in the same direction for all individuals in the population.
We say that C has a positive monotonic effect on E if E
c
(ω) is non-decreasing in c for all individuals, i.e., E
1
(ω) E
0
(ω)forω, which excludes E response type 3. Note that
this should be clearly distinguished from the assumptions
of no preventive action or no preventive sequence [32,33].
In a similar manner, we let D
ce
(ω) denote the potential
outcomes of D for individual ω if, possibly contrary to fact,
there had been interventions to set C to c and to set E to
e. For each individual ω, there would thus be 4 possible
potential outcomes D
11
(ω), D
01
(ω), D
10
(ω), and D
00
(ω),
resulting in 16 (i.e., 2
4
) different D response types as enu-
merated in Table 2 [34]. We let D
T
(ω) denote D response
type of individual ω. We say that C and E have positive
monotonic effects on D if D
ce
(ω) is non-decreasing in c
and e for all individuals, i.e., D
ce
(ω) D
c ' e '
(ω)forω
whenever c c 'ande e '. Under this assumption, the in-
dividuals of D response types 3, 5, 7, and 9 through 15 are
excluded; and individuals of D response types 1, 2, 4, 6, 8,
and 16 may remain [32].
Likewise, we let S
ed
(ω) denote the potential outcomes of
S for individual ω if, possibly contrary to fact, there had
been interventions to set E to e and to set D to d.Foreach
individual ω, there would thus be 4 possible potential out-
comes S
11
(ω), S
01
(ω), S
10
(ω), and S
00
(ω), resulting in 16
(i.e., 2
4
)differentS response types as enumerated in Table 3.
We let S
T
(ω)denoteS response type of individual ω.
Finally, we integrate informatio n about the potential out -
comes discussed above to produce 2 types of compound
potential outcomes, which are also called nested counter-
factuals [2]. (Note that compound potential outcomes ha ve
been extensively used in the issues of mediation and direct/
indirect effects [35-38].) First, we combine the potential
outcomes of E and the potential outcomes of D to define
Table 1 Enumeration of 4 response types for exposure E
and corresponding potential outcomes
E type Potential outcomes of E
E
T
(ω) E
c
(ω)
E
1
(ω) E
0
(ω)
111
210
3
a
01
400
a
Under the assumption of positive monotonicity (i.e., E
1
(ω) E
0
(ω) for ω), this
response type is excluded.
Table 2 Enumeration of 16 response types for outcome D
and corresponding potential outcomes
D type Potential outcomes of D
D
T
(ω) D
ce
(ω)
D
11
(ω) D
01
(ω) D
10
(ω) D
00
(ω)
11111
2
b, c
1110
3
a, b, c
1101
41100
5
a, b, c
1011
61010
7
a, b, c
1001
8
b
1000
9
a, b, c
0111
10
a, b, c
0110
11
a
0101
12
a, b
0100
13
a
0011
14
a, b
0010
15
a, b
0001
16 0000
a
Under the assumption of positive monotonicity (i.e., D
ce
(ω) D
c ' e '
(ω) for ω
whenever c c ' and e e '), these response types are excluded.
b
Given no interaction at the individual level on the additive scale between C
and E in the counterfactual framework (i.e., D
11
(ω) D
01
(ω) D
10
(ω)+D
00
(ω)=
0 for ω), these response types are excluded.
c
Given no interaction at the individual level on the multiplicative scale
between C and E in the counterfactual framework (i.e., D
11
(ω)D
00
(ω)=D
01
(ω)
D
10
(ω) for ω), these response types are excluded.
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D
cE
c
0
ωðÞ. In other words, the compound potential out-
comes of D are defined by (i) confounder status (C(ω)=1,
C(ω) = 0) and (ii) potential exposure status following an
intervention on confounder (E
1
(ω), E
0
(ω)). For each individ-
ual ω, there would thus be 4 possible compound potential
outcomes D
1E
1
ωðÞ, D
1E
0
ωðÞ, D
0E
1
ωðÞ,andD
0E
0
ωðÞ.Second,
we combine the potential outcomes of E,thepotentialout-
comes of D, and the potential outcomes of S to define
S
E
c
D
c
0
E
c
00
ωðÞ. Note that the compound potential outcomes
of S are defined by (i) potential exposure status following
an intervention on confounder (E
1
(ω), E
0
(ω)) and (ii) the
compound potential outcomes of D ( D
1E
1
ωðÞ, D
1E
0
ωðÞ,
D
0E
1
ωðÞ,andD
0E
0
ωðÞ). Thus, for each individual ω,there
would be 8 possible compound potential outcomes S
E
1
D
1E
1
ωðÞ, S
E
1
D
1E
0
ωðÞ, S
E
1
D
0E
1
ωðÞ, S
E
1
D
0E
0
ωðÞ, S
E
0
D
1E
1
ωðÞ, S
E
0
D
1E
0
ωðÞ, S
E
0
D
0E
1
ωðÞ,andS
E
0
D
0E
0
ωðÞ.
Combination of 4 E response types, 16 D response
types , and 16 S response types yields 1,024 (i.e., 4 × 16 ×
16) EDS response types. As noted above, under the as-
sumption of positive monotonic effect of C on E, the
number of possible E response types is reduced from 4
to 3. Further, under the assumptions of both positive
monotonic effects of C and E on D and no interaction at
the individual level on the additive scale between C and
E on D, the number of possible D response types is re-
duced from 16 to 4 (see footnote of Table 2). Analogous
argument applies to S response types (see footnote of
Table 3). Consequently, the number of possible EDS re-
sponse types is reduced from 1,024 to 48 (i.e., 3 × 4 × 4).
In Table 4, we show a complete enumeration of these 48
EDS response types. To enhance readability, Table 4
shows only selection status when C = 1 (i.e., S
E
1
D
1E
1
ωðÞ)
and when C = 0 (i.e., S
E
0
D
0E
0
ωðÞ) among S
E
c
D
c
0
E
c
00
ωðÞ. Note
that we made these restrictive assumptions to show the
correspondence between E response types, D response
types , and S response types in Table 4, which would be
of great help to understand the present findings. The fol-
lowing discussion however applies even without these
assumptions. Thus, in the following sections, we use a
total of 1,024 EDS response types, considering general
cases in which these assumptions are not met.
Four hypothetical situations
In Figure 2, we give an overview of 4 hypothetical situations
by using DAGs. Figure 2A describesasituationinwhichre-
searchers conduct an observational study and the informa-
tion about a portion of subjects is unavailable due to loss to
follow-up. Note that the square around S indicates that the
analysis is restricted to those who do not drop out (i.e.,
S = 1). Investigators often encounter this situation in obser-
vational studies. Researchers should be concerned about
both confounding bias and selection bias in this situation.
Subsequently, Figure 2B shows a situation in which re-
searchers can obtain the information about the total
population, including those who dropped out. In this
situation, a possibility of selection bias can be ruled out
since researchers do not condition on S.
In observational studies, researchers usually aim to
eliminate confounding bias by employing some statistical
procedures, e.g., standardization and inverse-probability
weighting method. In other words, they aim to analy-
tically block or remove the path between C and E by
making an adequate adjustment. (Note that outcome
modeling techniques such as disease risk scores focus on
the path between C and D [39].) By contrast, in random-
ized controlled trials, researchers manipulate the value
of E by employing certain interventions; they physically
prevent E from varying in response to variations in C.
Thus, as shown in Figure 2C and D, C would no longer
have effects on E, and the arrow from C to E is erased
or removed [14]. This should be clearly distinguished
from analytic control of C in observational studies.
In the following sections, we demonstrate significant dif-
ferences between these 4 hypothetical situations, by describ-
ing theoretical data frequencies in terms of response types.
Table 3 Enumeration of 16 response types for selection
variable S and corresponding potential outcomes
S type Potential outcomes of S
S
T
(ω) S
ed
(ω)
S
11
(ω) S
01
(ω) S
10
(ω) S
00
(ω)
1 1111
2
b, c
1110
3
a, b, c
1101
4 1100
5
a, b, c
1011
6 1010
7
a, b, c
1001
8
b
1000
9
a, b, c
0111
10
a, b, c
0110
11
a
0101
12
a, b
0100
13
a
0011
14
a, b
0010
15
a, b
0001
16 0000
a
Under the assumption of positive monotonicity (i.e., S
ed
(ω) S
e ' d '
(ω) for ω
whenever e e ' and d d '), these response types are excluded.
b
Given no interaction at the individual level on the additive scale between E
and D in the counterfactual framework (i.e., S
11
(ω) S
01
(ω) S
10
(ω)+S
00
(ω)=0
for ω), these response types are excluded.
c
Given no interaction at the individual level on the multiplicative scale
between E and D in the counte rfactual framework (i.e., S
11
(ω)S
00
(ω)=S
01
(ω)
S
10
(ω) for ω), these response types are excluded.
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Table 4 Enumeration of 48 EDS response types and corresponding potential outcomes
E type D type S type Potential
outcomes of E
Potential outcomes
of D
Compound potential
outcomes of D
Potential outcomes
of S
Selection
status
E
T
(ω) D
T
(ω) S
T
(ω) E
c
(ω) D
ce
(ω) D
cE
c
0
ωðÞ S
ed
(ω) S
E
c
D
cE
c
ωðÞ
E
1
E
0
D
11
D
01
D
10
D
00
D
1E
1
D
1E
0
D
0E
1
D
0E
0
S
11
S
01
S
10
S
00
S
E
1
D
1
E
1
S
E
0
D
0
E
0
1 1 1 1 1 1 1 (1)
a
(1) 1 (1) (1) 1 1 (1) (1) (1) 1 1
1 1 4 1 1 1 1 (1) (1) 1 (1) (1) 1 1 (1) (0) (0) 1 1
1 1 6 1 1 1 1 (1) (1) 1 (1) (1) 1 1 (0) (1) (0) 1 1
1 1 16 1 1 1 1 (1) (1) 1 (1) (1) 1 0 (0) (0) (0) 0 0
1 4 1 1 1 1 1 (0) (0) 1 (1) (1) 1 1 (1) (1) (1) 1 1
1 4 4 1 1 1 1 (0) (0) 1 (1) (1) 1 1 (1) (0) (0) 1 1
1 4 6 1 1 1 1 (0) (0) 1 (1) (1) 1 1 (0) (1) (0) 1 1
1 4 16 1 1 1 1 (0) (0) 1 (1) (1) 1 0 (0) (0) (0) 0 0
1 6 1 1 1 1 0 (1) (0) 1 (1) (0) 0 1 (1) 1 (1) 1 1
1 6 4 1 1 1 0 (1) (0) 1 (1) (0) 0 1 (1) 0 (0) 1 0
1 6 6 1 1 1 0 (1) (0) 1 (1) (0) 0 1 (0) 1 (0) 1 1
1 6 16 1 1 1 0 (1) (0) 1 (1) (0) 0 0 (0) 0 (0) 0 0
1 16 1 1 1 0 0 (0) (0) 0 (0) (0) 0 1 (1) 1 (1) 1 1
1 16 4 1 1 0 0 (0) (0) 0 (0) (0) 0 1 (1) 0 (0) 0 0
1 16 6 1 1 0 0 (0) (0) 0 (0) (0) 0 1 (0) 1 (0) 1 1
1 16 16 1 1 0 0 (0) (0) 0 (0) (0) 0 0 (0) 0 (0) 0 0
2 1 1 1 0 1 (1) (1) 1 1 (1) (1) 1 1 1 (1) (1) 1 1
2 1 4 1 0 1 (1) (1) 1 1 (1) (1) 1 1 1 (0) (0) 1 1
2 1 6 1 0 1 (1) (1) 1 1 (1) (1) 1 1 0 (1) (0) 1 0
2 1 16 1 0 1 (1) (1) 1 1 (1) (1) 1 0 0 (0) (0) 0 0
2 4 1 1 0 1 (1) (0) 0 1 (0) (1) 0 1 (1) (1) 1 1 1
2 4 4 1 0 1 (1) (0) 0 1 (0) (1) 0 1 (1) (0) 0 1 0
2 4 6 1 0 1 (1) (0) 0 1 (0) (1) 0 1 (0) (1) 0 1 0
2 4 16 1 0 1 (1) (0) 0 1 (0) (1) 0 0 (0) (0) 0 0 0
2 6 1 1 0 1 (0) (1) 0 1 (1) (0) 0 1 (1) (1) 1 1 1
2 6 4 1 0 1 (0) (1) 0 1 (1) (0) 0 1 (1) (0) 0 1 0
2 6 6 1 0 1 (0) (1) 0 1 (1) (0) 0 1 (0) (1) 0 1 0
2 6 16 1 0 1 (0) (1) 0 1 (1) (0) 0 0 (0) (0) 0 0 0
2 16 1 1 0 0 (0) (0) 0 0 (0) (0) 0 (1) (1) 1 1 1 1
2 16 4 1 0 0 (0) (0) 0 0 (0) (0) 0 (1) (1) 0 0 0 0
2 16 6 1 0 0 (0) (0) 0 0 (0) (0) 0 (1) (0) 1 0 1 0
2 16 16 1 0 0 (0) (0) 0 0 (0) (0) 0 (0) (0) 0 0 0 0
4 1 1 0 0 (1) (1) 1 1 1 (1) (1) 1 (1) 1 (1) (1) 1 1
4 1 4 0 0 (1) (1) 1 1 1 (1) (1) 1 (1) 1 (0) (0) 1 1
4 1 6 0 0 (1) (1) 1 1 1 (1) (1) 1 (1) 0 (1) (0) 0 0
4 1 16 0 0 (1) (1) 1 1 1 (1) (1) 1 (0) 0 (0) (0) 0 0
4 4 1 0 0 (1) (1) 0 0 0 (0) (0) 0 (1) (1) (1) 1 1 1
4 4 4 0 0 (1) (1) 0 0 0 (0) (0) 0 (1) (1) (0) 0 0 0
4 4 6 0 0 (1) (1) 0 0 0 (0) (0) 0 (1) (0) (1) 0 0 0
4 4 16 0 0 (1) (1) 0 0 0 (0) (0) 0 (0) (0) (0) 0 0 0
4 6 1 0 0 (1) (0) 1 0 1 (1) (0) 0 (1) 1 (1) 1 1 1
4 6 4 0 0 (1) (0) 1 0 1 (1) (0) 0 (1) 1 (0) 0 1 0
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Results
Describing data from observational studies in terms of
response types
As demonstrated above, under the situation described in
Figure 1, individuals can be classified into one of the
maximum of 1,024 EDS response types. Despite its sophis-
tication and usefulness, however, the response type of each
individual is unobservable. Indeed, this is called a funda-
mental problem of causal inference [40]. Nonetheless, we
can show the conceptual link between unobservable re-
sponse types and observed, or observable, data frequencies
in the population. In this respect, the concept of com-
pound potential outcomes is quite useful.
In Figure 3, we describe theoretical data frequencies from
observational studies in terms of the 1,024 possible EDS re-
sponse types. We let EiDjSk denote the EDS response type
of [E
T
= i, D
T
= j, S
T
= k](i =1,,4, j =1,, 16, k =1,,
16), and let P
EiDjSk
denote a p revalence of t he individuals of
EiDjSk response type in the total population. We also let
P
C|EiDjSk
and P
C EiDjSk
j
denote probabilities of being exposed
and unexposed to C among the individuals of EiDjSk
response type, respectively. When no confusion occurs for
a dichotomous variable X,weusethenotationsX and
X in
the terminologies of events of X =1 andX = 0, respectively.
For example, C and
C mean C =1 and C = 0, respectively.
Further, N denotes the number of total population. Then,
data frequencies in each cell in Figure 3 can be described
either as N
P
ijk
P
C|EiDjSk
P
EiDjSk
or N
X
ijk
P
C
j
EiDjSk
P
EiDjSk
.
(Note that the former can be also expressed as NP
C
P
ijk
P
EiDjSk|C
and that the latter can be expressed as NP
C
X
ijk
P
EiDjSk
C
j
,whereP
C
and P
C
denote probabilities of C
and
C in the total population, respectively). It should be
noted that individuals can be classified into 16 cells,
which is equivalent to a ma ximum possible number
of 4 independent random event s (i.e., E, D, C,andS).
TheupperandlowerpartsofFigure3showdata
frequencies among the sub population with C =1 and
C = 0, respectively. Those who are classified into inner
dashed rectangles represent individuals selected for
analyses (i.e., S =1)whilethosewhoarenotclassified
into the rectangles represent non-selected individuals
(i.e., S = 0). In other words, the information about the
individuals outside the rectangles is unavailable to
researchers.
Notably, individuals of the same EDS response types can
be potentially classified into 2 cells. For example, consider
individual ω who is classified as E1D6S4 response type
(see Table 4). This individual is, by definition, exposed to
E = 1 irrespective of the value of C (i.e., E
1
(ω)=E
0
(ω) = 1).
Further, individual ω is expected to experience outcome
D if there had been interventions to set C to 1 (i.e., D
1E
1
ωðÞ¼D
11
ωðÞ¼1 ), whereas this individual is expected
Table 4 Enumeration of 48 EDS response types and corresponding potential outcomes (Continued)
4 6 6 0 0 (1) (0) 1 0 1 (1) (0) 0 (1) 0 (1) 0 0 0
4 6 16 0 0 (1) (0) 1 0 1 (1) (0) 0 (0) 0 (0) 0 0 0
4 16 1 0 0 (0) (0) 0 0 0 (0) (0) 0 (1) (1) (1) 1 1 1
4 16 4 0 0 (0) (0) 0 0 0 (0) (0) 0 (1) (1) (0) 0 0 0
4 16 6 0 0 (0) (0) 0 0 0 (0) (0) 0 (1) (0) (1) 0 0 0
4 16 16 0 0 (0) (0) 0 0 0 (0) (0) 0 (0) (0) (0) 0 0 0
We consider 4 binary variables as follows: exposure E, outcome D, confounder C, and selection variable S (see Figure 1). We show the enumeration under the
assumptions of positive monotonicity of E, D, and S and no interaction at the individual level on the additive scale between C and E on D and E and D on S in the
counterfactual framework.
a
Parentheses indicate that this particular outcome will never be observed.
Selected individuals
Total population
Observational
studies
A.
B.
Randomized
controlled
trials
C.
D.
C
E
D
C
E
D
S
C
E
D
C
E
D
S
Figure 2 Four causal diagrams depicting hypothetical
situations. E, D, C, and S denote exposure, disease, confounder, and
selection variable, respectively. C may act as a direct effect modifier
simultaneously. The square around S in Figure 2A and C indicates
that the analysis is restricted to those who do not drop out (i.e., S =1).
By contrast, 2 diagrams in Figure 2B and D show the situations in
which information about total population is available to researchers.
See text for details.
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Figure 3 Frequencies of individuals with 1,024 possible EDS response types in observational studies. We consider 4 binary variables as
follows: exposure E, outcome D, confounder C, and selection variable S (see Figure 1). As shown in Tables 1, 2 and 3, we consider 4 response
types for E, 16 response types for D, and 16 response types for S. We let EiDjSk denote the EDS response type of [E
T
= i, D
T
= j, S
T
= k], and let
P
EiDjSk
denote a prevalence of the individuals of EiDjSk response type in the total population. We also let P
C|EiDjSk
and P
C EiDjSkj
denote probabilities
of being exposed and unexposed to C among the individuals of EiDjSk response type, respectively. Further, N denotes the number of total
population. Those who are classified into inner dashed rectangles represent individuals selected for analyses (i.e., S = 1) while those who are not
classified into the rectangles represent non-selected individuals (i.e., S = 0). See text for details.
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Page 7
not to experience outcome D if there had been interven-
tions to set C to 0 (i.e., D
0E
0
ωðÞ¼D
01
ωðÞ¼0). Finally,
the information about this individual is, by definition,
available to researchers had there been interventions to set
C to 1 (i.e., S
E
1
D
1E
1
ωðÞ¼S
11
ωðÞ¼1), whereas this individ-
ual is lost to follow-up had there been interventions to set
C to 0 (i.e., S
E
0
D
0E
0
ωðÞ¼S
10
ωðÞ¼0 ). Thus, in observa-
tional studies, individual ω of E1D6S4responsetypecan
be classified into either one of the following 2 cells in
Figure 3; one is E =1, D =1, C =1, and S =1 while the
other is E =1, D =0, C =0, and S =0.Notethatthisde-
pends on the probabilities that C is present or absent in
individual ω (i.e., P
C|E1D6S4
and P
CE1D6S4
j
).
To summarize, Figure 3 shows theoretical data fre-
quencies in an observational study (i.e., Figure 2A and
B). The situation is, however, strikingly different when
we conduct a randomized controlled trial, which will be
demonstrated in the next section.
Describing data from randomized controlled trials in
terms of response types
As noted above, researchers manipulate the value of E in
randomized controlled trials. Since researchers physically
prevent E from varying in response to variations in C,we
do not need to consider E response types when describing
theoretical data frequencies in ideal randomized controlled
trials; rather we focus on D response types and S response
types. In other words, observed exposure status and E re-
sponse types become independent (i.e., E E
T
)whenre-
searchers marginally intervene on E. Thus, theoretical data
frequencies from randomized controlled trials can be de-
scribed in terms of 256 (i.e., 16 × 16) possible DS response
types, in contrast with 1,024 possible EDS response types.
We let P
E
and P
E
denote the probabilities of E and
E in
the total population, respectively. (For simplicity, we de-
scribe the situation of marginal randomization of E.How-
ever, the following discussion can be extended to the
situation of stratified randomization, in which P
E
and P
E
may vary across the strata of C.) Figure 4 shows distribu-
tions of individuals of the 256 possible DS response types in
a randomized controlled trial. Note that data frequencies in
each cell in Figure 4 can be described as NP
E
P
jk
P
C|
DjSk
P
DjSk
, NP
E
X
jk
P
CjDjSk
P
DjSk
, NP
E
X
jk
P
C
j
DjSk
P
DjSk
,orN
P
E
X
jk
P
C jDjSk
P
DjSk
. (Note that these can be also expressed
as NP
E
P
C
P
jk
P
DjSk|C
, NP
E
P
C
X
jk
P
DjSk Cj
, NP
E
P
C
X
jk
P
DjSk
Cj
,andNP
E
P
C
X
jk
P
DjSk
Cj
, respectively.) As in the
case of observational studies (Figure 3), individuals can be
classified into 16 cells in Figure 4.
The theoretical data frequencies in Figure 4 can be
explained as a re-distribution of individuals in Figure 3.
For example, consider individual ω who is classified as
either E1D6S4responsetypeorE2D6S4responsetype.In
observational studies, if the value of C of individual ω is 1,
this individual is classified into an upper-left cell within
the inner dashed rectangle of the upper part of Figure 3,
i.e., E =1,D =1,C =1,andS = 1. Then, when this individual
is forced to be exposed to E in a randomized controlled
trial, this individual would remain in the upper-left cell
within the inner dashed rectangle of the upper part of
Figure 4. Note that neither D response types nor S re-
sponse types of this individual change by the intervention
on E. By contrast, if individual ω is forced to be unexposed
to E, this individual moves to a lower-left cell within the
inner dashed rectangle of the upper part of Figure 4, i.e.,
E =0, D =1,C =1, and S = 1. On the other hand, consider
individual ω who is classified as either E3D6S4 response
type or E4D6S4 response type. In observational studies, if
the value of C of individual ω is 1, this individual is classi-
fied into a lower-left cell within the inner dashed rectangle
of the upper part of Figure 3, i.e., E =0, D =1,C =1,and
S
= 1. Then, in randomized controlled trials, if this individ-
ual is forced to be unexposed to E,thisindividualwould
remain in the lower-left cell within the inner dashed rect-
angle of the upper part of Figure 4. Meanwhile, if this indi-
vidual is forced to be exposed to E,thisindividualmoves
to an upper-left cell within the inner dashed rectangle of
the upper part of Figure 4, i.e., E =1,D =1,C =1,andS =
1. These re-distributions can be summarized as
P
E
N
X
i¼1;2
P
CjEiD6S4
P
EiD6S4
þ N
X
i¼3;4
P
CjEiD6S4
P
EiD6S4
¼ NP
E
X
i¼1;2;3;4
P
C
P
EiD6S4jC
¼ NP
E
P
C
P
D6S 4jC
¼ NP
E
P
C
j
D6S4
P
D6S 4
; ð3Þ
and
P
E
N
X
i¼1;2
P
CjEiD6S4
P
EiD6S4
þ N
X
i¼3;4
P
CjEiD6S4
P
EiD6S4
¼ NP
E
X
i¼1;2;3;4
P
C
P
EiD6S4jC
¼ NP
E
P
C
P
D6S 4jC
¼ NP
E
P
C
j
D6S 4
P
D6S 4
: ð4Þ
Note that the numbers in the parentheses of left-hand
sides of equations 3 and 4 are based on the subpopulation
of C = 1 in observational studies (i.e., upper part of Figure 3),
whereas the right-hand sides of these equations are based
on the subpopulation of C = 1 in randomized controlled
trials (i.e., upper part of Figure 4). In other words, these
equations explain how individuals of subpopulation of
C = 1 are re-distributed as a result of intervention on E.
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Figure 4 Frequencies of individuals with 256 possible DS response types in randomized controlled trials. We consider 4 binary variables
as follows: exposure E, outcome D, confounder C, and selection variable S (see Figure 1). As shown in Tables 2 and 3, we consider 16 response
types for D and 16 response types for S. We let P
E
and P
E
denote the probabilities of E and
E in the total population, respectively. For the
definition of other notations, see Figure 3. Those who are classified into inner dashed rectangles represent individuals selected for analyses
(i.e., S = 1) while those who are not classified into the rectangles represent non-selected individuals (i.e., S = 0). See text for details.
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Analogous discussion applies when the value of C is 0
among the individuals of E1D6S4, E2D6S4, E3D6S4, or
E4D6S4 response types. Note that, in observational stud-
ies, these individuals are classified in either an upper-right
cell (i.e., E =1,D =0,C =0,andS = 0) or a lower-right cell
(i.e., E =0,D =0,C =0,andS = 0) outside the inner dashed
rectangle of the lower part of Figure 3. The re-
distributions of these individuals as a result of intervention
on E can be summarized as
P
E
N
X
i¼1;3
P
C jEiD6S4
P
EiD6S4
þ N
X
i¼2;4
P
C jEiD6S4
P
EiD6S4
!
¼ NP
E
X
i¼1;2;3;4
P
C
P
EiD6S4
j
C
¼ NP
E
P
C
P
D6S 4
j
C
¼ NP
E
P
C
j
D6S 4
P
D6S4
; ð5Þ
and
P
E
N
X
i¼1;3
P
C
j
EiD6S4
P
EiD6S4
þ N
X
i¼2;4
P
C
j
EiD6S4
P
EiD6S4
!
¼ NP
E
X
i¼1;2;3;4
P
C
P
EiD6S4j
C
¼ NP
E
P
C
P
D6S 4j
C
¼ NP
E
P
C jD6S4
P
D6S 4
: ð6Þ
Again, the numbers in the parentheses of left-hand sides
of equations 5 and 6 are based on the subpopulation of
C = 0 in observational studies (i.e., lower part of Figure 3),
whereas the right-hand sides of these equations are based
on the subpopulation of C = 0 in randomized controlled
trials (i.e., lower part of Figure 4). In other words, these
equations explain how individuals of subpopulation of
C = 0 are re-distributed as a result of intervention on E.It
should be noted that these re-distributions do not occur
across the upper and lower parts of Figures 3 and 4 be-
cause C precedes E temporally and the value of C is, by
definition, predetermined before intervention on E. These
discussions also demonstrate that, in Figure 4, individuals
of the same DS response types can be potentially classified
into 4 cells, depending on the probability of being exposed
or unexposed to C (i.e., P
C|DjSk
or P
CDjSkj
) and the prob-
ability of being exposed or unexposed to E (i.e., P
E
or P
E
).
Note that, when the information about the total popula-
tion is available, both marginal and conditional exchange-
ability assumptions are met in Figure 4; the distributions
of DS response types are comparable between the exposed
and unexposed groups. However, when the information
about those who dropped out is not available, exchange-
ability assumptions do not hold, either conditionally or
unconditionally. See (Additional file 1: Appendix 1) for a
discussion of positivity another fundamental assumption
for causal inference [41-43].
Epidemiologic measures in terms of response types
The descriptions of data frequencies in Figures 3 and 4
have a crucial implication, demonstrating significant
distinctions between association measures and effect mea-
sures [9]. In the following sections, we continue to focus our
discussion on RRs, which can be extended to other mea-
sures. Note also that, although epidemiologic measures can
be defined for a variety of target population (e.g., the ex-
posed and the unexposed), the following discussion focuses
on the situation in which target population is the total popu-
lation. Furthermore, we also discuss epidemiologic measures
in the subpopulation defined by C or S.
In observational studies (Figure 2A and B), researchers
can readily calculate associational RRs by referring to the
notation s in Figure 3. In particular, when no information is
available about those who dropped out (Figure 2A), one can
calculate an associational RR
S=1
by using the information
about individuals classified into the inner rectangles in Figure 3.
Then, as shown in (Additional file 2: Table S1), associ-
ational RR
S=1
can be described in terms of a probability
of being exposed or unexposed to C among the individ-
uals of EiDj Sk response type (i.e., P
C|EiDjSk
or P
C EiDjSk
j
)
and a pre valence of the individuals of EiDjSk response
type in the to tal population (i.e., P
EiDjSk
)(equationA1).
Meanwhile, when researchers are capable of gathering
information about those who dropped out (Figure 2B),
the information about individuals of S response types 1
through 16 is available, which yields an a ssociational RR
(equation A4).
By contrast, when researchers obtain data from random-
ized controlled trials (Figure 2C and D), their frequencies
can be described in a different way, as shown in Figure 4.
In these cases, researchers can calculate causal RRs to
infer causality between E and D. When no information is
available about those who dropped out (Figure 2C), one
cancalculateacausalRR
S=1
by using the information
about individuals classified into the inner rectangles in
Figure 4. Then, as shown in (Additional file 2: Table S2),
causal RR
S=1
can be described in terms of a probability of
being exposed or unexposed to C among the individuals
of DjSk response type (i.e., P
C|DjSk
or P
CDjSkj
) and a preva-
lence of the individuals of DjSk response type in the total
population (i.e., P
DjSk
) (equation A7). In ideal randomized
controlled trials without loss to follow-up (Figure 2D), the
information about individuals of S response types 1
through 16 is available, which yields a causal RR
(equation A10). We should note that the causal RR shown
in equation A10 is an alternative notation of the causal
RR shown in equation 1 in terms of response types
(see Additional file 1: Appendix 2).
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Note that, even in ideal (either marginal or stratified)
randomized controlled trials, one may observe a hetero-
geneity between stratum-specific causal RRs, which will
be addr essed in the section entitled Modification of epi-
demiologic measures.
Confounding bias
In this section, we aim to further clarify the concept of
confounding bias in the counterfactual framework, by
describing it in terms of response types.
We show a sufficient condition to estimate effect mea-
sures in observational studies by adjusting for confounding
bias in terms of response types. In this case, we use effect
measures in the total population in ideal randomized con-
trolled trials (i.e., causal RR) as a gold standard. As noted
above, confounding bias is induced by a common cause C
of E and D. Thus, to show a sufficient condition to adjust
for confounding bias, we need to compare association mea-
sures in the total population in observational studies
(Figure 2B) and effect measures in the total population in
randomized controlled trials (Figure 2D) In other words, a
sufficient condition to adjust for confounding bias can be
described as: adjusted associational RR = causal RR. Note
that we here compare 2 distinct types of epidemiologic
measures, which are obtained from distinct study designs.
To adjust for confounding bias in observational studies,
one may calculate a weighted average of stratum-specific
associational RRs, or standardization, expecting to esti-
mate a causal RR. By using stratum-specific associational
RRs (equations A5 and A6), this procedure can be de-
scribed in terms of response types as follows:
Notably, this is not equivalent to a causal RR (equation
A10). In other words, this stratification-based procedure
does not delete the arrow from C to E in Figure 2B,
yielding subtly different measures from effect measures.
When one can assume conditional exchangeability (i.e.,
ED
e
|C for e), the weighted average of stratum-specific
associational RR can be expressed as
X
j¼1;2;3;4;5;6;7;8
P
C
j
Dj
P
Dj
þ
X
j¼1;2;3;4;9;10;11;12
P
C
j
Dj
P
Dj
X
j¼1;2;5;6;9;10;13;14
P
C
j
Dj
P
Dj
þ
X
j¼1;3;5;9;11;13;15
P
C
j
Dj
P
Dj
; ð8Þ
which is equivalent to a causal RR in equation A10 (see
Additional file 1: Appendix 3). Indeed, the condition
ED
e
|C for e is a sufficient condition to estimate effect
measures in observational studies by adjusting for
confounding bias, and the assumption of exchangeability
often gets most of the attention in discussions about
causal inference [44]. Unfortunately, however, the condi-
tion is not guaranteed in observational studies, and expert
knowledge is required. Importantly, the assumption of
conditional exchangeability, i.e., ED
e
|C for e,issubtly
weaker than another sufficient condition to estimate effect
measures in observational studies, i.e., full conditional
exchangeability, or E(D
e=1
, D
e=0
)|C [45]. It may be diffi-
cult, however, to imagine a practical scenario where the
former holds but not the latter [46], and the word ex-
changeability has been sometimes used interchangeably
in the literature. (A combination of full exchangeability
P
C
X
i ¼ 1; 2
j ¼ 1; 2; 3; 4; 5; 6; 7; 8
P
C
j
EiDj
P
EiDj
X
i¼1;2
P
CjEi
P
Ei
0
B
B
B
B
@
1
C
C
C
C
A
þ P
C
X
i ¼ 1; 3
j ¼ 1; 2; 3; 4; 9; 10; 11; 12
P
C
j
EiDj
P
EiDj
X
i¼1;3
P
C jEi
P
Ei
0
B
B
B
B
@
1
C
C
C
C
A
P
C
X
i ¼ 3; 4
j ¼ 1; 2; 5; 6; 9; 10; 13; 14
P
C
j
EiDj
P
EiDj
X
i¼3;4
P
C
j
Ei
P
Ei
0
B
B
B
B
@
1
C
C
C
C
A
þ P
C
X
i ¼ 2; 4
j ¼ 1; 3; 5; 9; 11; 13; 15
P
C
j
EiDj
P
EiDj
X
i¼2;4
P
C jEi
P
Ei
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼
X
i ¼ 1; 2
j ¼ 1; 2; 3; 4; 5; 6; 7; 8
P
C
P
EiDj
j
C
X
i¼1;2
P
EijC
þ
X
i ¼ 1; 3
j ¼ 1; 2; 3; 4; 9; 10; 11; 12
P
C
P
EiDj
j
C
X
i¼1;3
P
Eij
C
0
B
B
B
B
@
1
C
C
C
C
A
X
i ¼ 3; 4
j ¼ 1; 2; 5; 6; 9; 10; 13; 14
P
C
P
EiDj
j
C
X
i¼3;4
P
Ei
j
C
þ
X
i ¼ 2; 4
j ¼ 1; 3; 5; 9; 11; 13; 15
P
C
P
EiDj
j
C
X
i¼2;4
P
Ei
j
C
0
B
B
B
B
@
1
C
C
C
C
A
: ð7Þ
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Page 11
and positivity has been termed strongly ignorable treat-
ment assignment assumption or strong ignorability,
whereas a combination of exchangeability and positivity
has been termed weakly ignorable treatment assignment
assumption or weak ignorability [2,47,48].) By compar-
ing equations 7 and 8, we can show that the conditions
E
T
D
T
|C and E D
T
|C are also sufficient conditions to
estimate effect measures in observational studies by
adjusting for confounding bias (see Additional file 1:
Appendix 3). In Additional file 1: Appendix 4, we show a
proof of the following inclusion relation:
E
T
D
T
CED
T
CE D
e¼1
; D
e¼0
ðÞjCED
e
jC for e
The subtle differences between E
T
D
T
|C and E D
T
|C
are described graphically in the section entitled Extended
causal diagrams integrating response types.Itisworth-
while to mention that the condition E
T
D
T
|C is not
guaranteed in randomized controlled trials.
The above discussion implies that analytic adjustment
for C in observational studies has consequences quite
different from those of physical control in randomized
controlled trials. Even when adequate analytic control of
C may be envisaged in observation al studi es, researchers
cannot estimate effect measures without the assumption
external to data. See Additional file 1: Appendix 5 for a
discussion of recently-int roduced assumptions to com-
pensate for a lack of randomization.
Selection bias
In this section, we aim to further clarify the concept of
selection bia s in the counterfactual framework, by de-
scribing it in terms of response types.
We show sufficient conditions for non-selection bias in
terms of response types. As explained above, selection bias
is induced by conditioning on a common effect of E an d D
(Figure 2A and C). Thus, to show sufficient conditions for
non-selection bias, we need to specify epidemiologic mea-
sures, i.e., association measures or effect measures. With re-
gard to association measures, a sufficient condition for
non-selection bias is described as associational RR
S=1
=
associational RR (see equations A1 and A4). Likewise, a suf-
ficient condition for non-selection bias for effect measures
is described as causal RR
S=1
= causal RR (see equations A7
and A10). It is worthwhile to mention that, when discussing
selection bias, one need to specify a stratum of S [21]. In
most cases, researchers are interested in the presence and
the degree of selection bias among t he subjects who do not
drop out. Thus, we here show sufficient conditions for
non-selection bias in a stratum S = 1. As explained later by
using extended causal diagrams, selection bias results in
violation of E D
T
even when exposure is randomly
assigned.
Modification of epidemiologic measures
For decades, epidemiologists have used the term effect
modification in a broad context , simply referring to
a v ariation in the selected effect mea sure for the
factor under study across levels of another factor [49].
In this respect, a recent paper clarified the distinction
between interaction and effect modification within the
counterfactual framework [31]. It is also well known that
the presence, direction, and size of modification can be
dependent on the choice of measure [50]. Since the term
effect modification is ambiguous, it is now recommended
to specify the measures more precisely, e.g., risk-difference
modification [50]. The above discussion implies that re-
searchers need to distinguish association-measure modi-
fication and effect-measure modification. For example,
when the information about total population is a vailable in
a randomized controlled trial, causal-RR modification is
defined to be present if stratum-specific causal RRs from
each subpopulation varies across the strata of C, i.e., causal
RR
C=1
causal RR
C=0
(see equations A11 and A12).
When stratum-specific causal RRs are (approximately)
homogeneous or uniform across strata, researchers usually
pool the data to calculate a causal RR in the total popula-
tion (i.e., causal RR). In a similar manner, one can define
associational-RR modification (see equations A5 and A6).
Only if it is appropriate to pool the data across the strata
of C, one can validly interpret associational RRs in the
total population.
Notably, the presence of association-measure modifi-
cation does not necessarily imply the presence of effect-
measure modification, and vice versa.
Extended causal diagrams integrating response types
In this section, we attempt to explain the concept of bias
by extending causal diagrams, which integrate response
types and observed variables. Although these causal dia-
grams , or extended DAGs, may appear less intuitive,
they maintain the integrity of the original DAGs and
would be of great use in graphically describing the find-
ings discussed in this study. In particular, by integrating
response types and observed variables, we can readily
understand subtle differences between E
T
D
T
|C and
E D
T
|C, demonstrating sufficient conditions to esti-
mate effect measures in observational studies.
Figure 5 shows the hypo thetical situation described in
Figure 1 by integrating response types of E, D, and S
(i.e., E
T
, D
T
, and S
T
, respectively). First, note that the
only arrows emanating from the response types point to
the corresponding observed variables, i.e., E
T
E, D
T
D, and S
T
S. Then, to describe the underlying correl-
ation between E
T
, D
T
, and S
T
, we use a total of 3 un-
measured common causes, U1, U2, and U3, which are
independent of each other. In other words, these
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unmeasured common causes as a whole represent un-
derlying personal characteristics determining his/her re-
sponse types of E, D, and S. (Note that, unlike U1, U2,
and U3, we assume that C does not determine his/her
response types. In other words, we assume that U1, U2,
and U3 precede response types, while C does not.) It
would be worth to mention that Figure 5 well describes
how the observed variables are determined in response
to the corresponding response types and their measured
parent(s). For example, each individual has 2 potential
outcomes of E: the outcome that would occur if C is
present in that individual, E
1
, and the outcome that
would occur if C is absent in that individual, E
0
. Thus,
we observe E = CE
1
+(1 C)E
0
. In Figure 5, this equation
is illustrated by the only 2 arrows from E
T
and C to E.
In randomized controlled trials, in which E is margin-
ally randomized, researchers physically prevent E from
varying in response to variations in C by intervening on
E. Thus, by applying the rule used in the sta ndard DAG
theory, the 2 arrows pointing to E in Figure 5 are re-
moved. As a result, observed exposure status and E re-
sponse types become independent (i.e., E E
T
), as
shown in Figure 6. In addition, observed exposure status
becomes independent of D response types and S re-
sponse types (i.e., ED
T
and ES
T
, respectively) be-
cause the value of E is, by definition, determined
randomly. Note that ED
T
implies an assumption of
(full) exchangeability. Trivially, observed exposure status
is also independent of D
T
given C, i.e., ED
T
C
j
, thus
implying the assumption of (full) conditional exchange-
ability. Importantly, even when adjusting for C, marginal
randomization of E does not result in independence be-
tween E response types and D response types due to 2
open paths, E
T
U1 D
T
and E
T
U2 D
T
.Ifwe
adjust for U1 and U2, they become independent. Finally,
Figure 6 also clearly shows that selection bias res ults in
violation of ED
T
; when some of the subjects are lost
to follow up, 7 marginally blocked paths between E and
D
T
(i.e., E S D D
T
, E S S
T
U1 D
T
, E
S S
T
U2 D
T
, E S S
T
U3 D
T
, E D
S S
T
U1 D
T
, E D S S
T
U2 D
T
, and
E D S S
T
U3 D
T
) become open because we
condition on the collider S. Indeed, extended DAGs are
of great use to demonstrate that both confounding bias
and selection bias result in lack of (full) exchangeability
of the exposed and unexposed groups.
Meanwhile, when using stratified randomiz ation of E
by C, researchers physically prevent E from varying in
response to variations in E response types, but the prob-
ability of E may vary across the strata of C. Thus , unlike
the marginal randomization of E,ofthe2arrows
pointing to E in Figure 5, only the arrow from E
T
to E is
removed (Figure 7). A s a result, there is an open path
between E and E
T
, i.e., E C U1 E
T
, which can be
blocked by adjusting for C (i.e., EE
T
C
j
). Further, al-
though there is an open path between E and D
T
, i.e.,
E C U1 D
T
, this path can be blocked by
adjusting for C (i.e., ED
T
C
j
), which implies that the
assumption of (full) conditional exchangeability can be
readily met in stratified randomization of E. Finally,
there is an open path between E and S
T
, i.e., E C
U1 S
T
, which can be also blocked by adjusting for C
(i.e., ES
T
C
j
). Note that, like Figure 6, even when
adjusting for C, stratified randomization of E does not
result in independence between E response types and D
responsetypes,andweneedtoadditionallyadjustfor
U1andU2.
Figure 8 shows the situation in observational studies, in
which researchers stratify by C to calculate a weighted aver -
age of stratum-specific association measures. (A square
around C means that we condition on C.) Unlike marginal
or stratified randomization of E, observed exposure status
is determined in response to E response types as well as the
status of C. Therefore, no arrows pointing to E are removed
in Figure 8. Note that, in Figure 8, E and D
T
would be mar-
ginally connected via the foll owing 3 paths, i.e., E E
T
C
E
D
S
E
T
D
T
S
T
U1
U3
U2
Figure 5 An extended causal diagram depicting a
hypothetical example.
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U1 D
T
, E E
T
U2 D
T
,andE C U1 D
T
.
When we condition on C, only the third path can be
blocked, and E and D
T
remain connected via the first 2
paths. Notably, the 3 paths can be theoretically blocked by
conditioning on U1andU2. In other words, a sufficient
condition of ED
T
C
j
is to adjust for U1andU2inobser-
vational studies. Meanwhile, in equation 8, we demonstrate
that the condition E
T
D
T
C
j
is a sufficient condition to es-
timate effect measures in observational studies without loss
to follow-up. Indeed, E
T
and D
T
are not independent given
C in Figure 8, and they are connected via the following 2
paths, conditional on C,i.e.,E
T
U1 D
T
and E
T
U2 D
T
. Note that both paths can be theoretically blocked
by conditioning on U1andU2. To summarize, although
E
T
D
T
C
j
and ED
T
C
j
are sufficient conditions to esti-
mate effect measures in observational studies, neither is
guaranteed in observational studies, and expert knowledge
is required. In particular, although E
T
D
T
C
j
is stronger
than ED
T
C
j
, we need to adjust for U1andU2 to achieve
either condition as shown in Figure 8.
Finally, it is worthwhile to mention that the perspectives
of the extended DAGs are different from those of the twin
network method, which has been developed to deal with
counterfactual values in DAGs [2]. This graphical method
uses two networks, one to represent the actual world and
one to represent the hypothetical world. Thus, this method
is used to represent the causal relations under intervention.
The aim of our extended DAGs is to integrate response
types and observed variables, which is thus applicable
to obser vational studies a s well as randomized con-
trolled trials. As a consequence, we can use the ex-
tended DAGs to describe the sufficient conditions to
infer causality in observational studies in terms of re-
sponse type s.
Discussion
We have clarified the concepts of bias and effect modifica-
tion in the counterfactual framework, by describing theoret-
ical data frequencies from observational studies and
randomized controlled trials in terms of response types. Al-
though these concepts have been extensively explained in
the epidemiologic literature, most of the studies have
discussed them separately. In this article, we have highlighted
the relations between these concepts, by discussing them
simultaneously. The present findings would somehow clarify
the link between the assumptions for making causal
C
E
D
S
E
T
D
T
S
T
U1
U3
U2
Figure 6 An extended causal diagram depicting a hypothetical
situation under marginal randomization.
C
E D
S
E
T
D
T
S
T
U1
U3
U2
Figure 7 An extended causal diagram depicting a hypothetical
situation under stratified randomization.
Suzuki et al. BMC Medical Research Methodology 2013, 13:101 Page 14 of 17
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Page 14
inferences in observational studies and the counterfactual
approach, demonstrating the inherent distinctions between
observationalstudiesandrandomizedcontrolledtrials.The
extension of DAGs using response types maintains the integ-
rity of the original DAGs, which allows one to understand
the underlying causal structure discussed in this study.
We have shown a hitherto unrecognized sufficient condi-
tion E
T
D
T
C
j
to estimate effect measures in observational
studies by adjusting for confounding bias. This condition is
stronger than the assumption of (full) conditional exchange-
ability, and it is not straightforward to discuss technical ad-
vantages of the hitherto unrecognized condition. Such
consideration however would enable one to further under-
stand the conceptual link between unobservable response
types and observed, or observable, data frequencies in the
population. This would also facilitate understanding of the
underlying causal structures of bias and effect modification.
In this article, we use a simple hypothetical situation,
including only 4 binary variables. Thus , it should be
noted that the present study does not encompa ss more
complicated situations , e.g., M-bias [51]. It is however
worthwhile to mention that the condition E
T
D
T
Cj is
applicable even when an exposure and an outcome are
polytomous variables, because our discussion based on
the extended DAGs does not restrict the type of vari-
ables. When considering situations in which there are
some confounders, the present finding would apply by de-
fining and estimating a function of measured confounders
that can be treated as a single confounder. It should be
also noted that we focused only on direct effect modifica-
tion, and thus, the present discussion does not necessarily
apply to other types of effect modification, i.e., indirect ef-
fect modification, effect modification by proxy, and effect
modification by a common cause [22]. Further, this study
does not address the issue of information bias or measure-
ment error. Recent studies have discussed how DAGs can
be used to represent them [52-55], which should be
addressed further in future studies.
Conclusion
As shown in the present study, researchers should
recognize inherent limitations of obser vational studies
in estimating causal effects. It should be emphasized,
however, that the recognition should come in the inter-
pretation of the evidence when trying to draw c onclu-
sions, not in the statement of research goals or s tudy
design and conduct phases [56]. The data from obser-
vational studies yield mea sures of association and those
whoexaminethedatashouldstrivetoimposeamean-
ing based on their expert knowledge on each occasion,
which would improve causal interpretations.
Additional files
Additional file 1: Appendices 1 to 5.
Additional file 2: Tables S1 and S2. Risk ratios in terms of response
types.
Abbreviations
DAG: Directed acyclic graph; RR: Risk ratio.
Competing interests
The authors declare that they have no competing interests.
Authors contributions
ES conceptualized the authors' views and drafted the manuscript. TM, TT,
and EY critically revised the manuscript for intellectual content. EY supervised
the study. All authors read and approved the final manuscript.
Authors information
ES is Assistant Professor of Epidemiology at Okayama University. His primary
research interest concerns improving causal interpretations of observational
studies. TM was a Research Fellow of Epidemiology when this study was
conducted. He is currently working as Assistant Professor in Center for
Innovative Clinical Medicine at Okayama University Hospital. TT, as a
Professor of Environmental Epidemiology, has evaluated a variety of health
effects of environmental factors to advance the publics health. EY, as a
Professor of Statistics, is interested in contributing to the advancement of
statistical theories necessary for causal inference.
C
E
D
S
E
T
D
T
S
T
U1
U3
U2
Figure 8 An extended causal diagram depicting a hypothetical
situation in observational studies.
Suzuki et al. BMC Medical Research Methodology 2013, 13:101 Page 15 of 17
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Page 15
Author details
1
Department of Epidemiology, Graduate School of Medicine, Dentistry and
Pharmaceutical Sciences, Okayama University, 2-5-1 Shikata-cho, Kita-ku,
Okayama 700-8558, Japan.
2
Department of Human Ecology, Graduate School
of Environmental and Life Science, Okayama University, 3-1-1 Tsushima-naka,
Kita-ku, Okayama 700-8530, Japan.
3
Department of Information Science,
Faculty of Informatics, Okayama University of Science, 11 Ridai-cho, Kita-ku,
Okayama 700-0005, Japan.
Received: 17 October 2012 Accepted: 15 July 2013
Published: 31 July 2013
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doi:10.1186/1471-2288-13-101
Cite this article as: Suzuki et al.: A counterfactual approach to bias and
effect modification in terms of response types. BMC Medical Research
Methodology 2013 13:101.
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