American Journal of Epidemiology
ª The Author 2011. Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of
Public Health. All rights reserved. For permissions, please e-mail: [email protected].
Vol. 173, No. 10
DOI: 10.1093/aje/kwr009
Advance Access publication:
April 13, 2011
Practice of Epidemiology
Remarks on Antagonism
Tyler J. VanderWeele* and Mirjam J. Knol
* Correspondence to Dr. Tyler J. VanderWeele, Departments of Epidemiology and Biostatistics, Harvard School of Public Health,
677 Huntington Avenue, Boston, MA 02115 (e-mail: [email protected]).
Initially submitted August 18, 2010; accepted for publication January 5, 2011.
Different forms of antagonism are classified in terms of response types and are related to the sufficient-cause
framework. These forms of antagonism include ‘‘synergy under recoding of an exposure,’’ ‘‘synergism under
recoding of the outcome,’’ and so-called ‘‘competing response types,’’ with synergism itself conceived of as causal
co-action within the sufficient-cause framework. In this paper, the authors show that subadditivity necessarily
implies at least one of these 3 forms of antagonism. Empirical conditions for specific forms of antagonism are
given for settings in which monotonicity assumptions are and are not considered plausible. The implications of
subadditivity and superadditivity for causal co-action for either an outcome or its absence are characterized under
various assumptions about monotonicity. A simple computational procedure is described for assessing whether
any specific form of causal co-action can be detected for either an outcome or its absence for both cohort and casecontrol data. The results in this paper are illustrated by application to examples drawn from the existing literature on
gene-gene and gene-environment interactions.
causality; dichotomous response; interaction; minimal sufficiency; synergism
Abbreviations: CI, confidence interval; RERI, relative excess risk due to interaction; RR, risk ratio.
In this paper, we offer a series of remarks concerning the
notion of antagonism as it relates to epidemiologic research.
We draw upon prior literature relating to the sufficient-cause
framework and response types (1–7) in order to develop
a comprehensive classification for antagonism and to derive
a series of conditions for empirically detecting different
forms of antagonism.
Our remarks here are restricted to 2 dichotomous exposures and dichotomous outcomes. The term ‘‘antagonism,’’
like numerous other epidemiologic terms, is ambiguous
and is subject to different uses by different researchers. Here
we seek to relate possible uses of ‘‘antagonism’’ to causal
co-action within the sufficient-cause framework (4, 5, 8). We
essentially use ‘‘antagonism’’ to refer to either 1) synergism
under exposure recoding, 2) synergism under outcome recoding, or 3) the outcome occurring if either exposure is
present (‘‘competing antagonism’’). Here we take these 3
forms as the definition of ‘‘antagonism’’ (remarks 1 and 3
below) and then consider the implications. These 3 forms are
exemplified by Table 1, Table 2, and Table 3, respectively.
SUFFICIENT CAUSES AND RESPONSE TYPES
Consider a setting in which there are 2 binary causes, X1
and X2, for the occurrence of a binary outcome D by a specific time. We use X1 and X2 to denote the absence of X1 and
X2, respectively, so that X1 ¼ 1 if X1 ¼ 0 and X1 ¼ 0 if X1 ¼
1, and likewise for X2. We will use Xi and (Xi ¼ 1) and also
X1 and Xi ¼ 0 interchangeably. For each individual, let Dx1 x2
denote the potential outcome (or ‘‘counterfactual outcome’’) if we had set the person’s exposures X1 to x1 and
X2 to x2. Each individual has 4 potential outcomes: D11, D10,
D01, and D00, each either 0 or 1. People can be classified into
one of 16 different types, as shown in Table 4 (1); for example, response type 8 indicates a person for whom D11 ¼ 1
and D10 ¼ D01 ¼ D00 ¼ 0. For simplicity, we will assume
that the associations between the 2 exposures of interest and
the outcome reflect the actual causal effects of the exposures
on the outcome (i.e., there is no confounding, selection bias,
or measurement error). If the effects of the exposures on
the outcome are unconfounded, conditional on some set of
1140
Am J Epidemiol. 2011;173(10):1140–1147
Remarks on Antagonism
Table 1. Example of ‘‘Exposure-Based Antagonism’’ (Class I)
Using Data From the Article by Stern et al. (20), With Odds Ratios for
Gene-Gene Interaction
X2 5 0
No. of
Cases
No. of
Controls
Table 3. Example of ‘‘Competing Antagonism’’ (Class III) Using
Data From the Article by Stern et al. (22), With Odds Ratios for GeneGene Interaction
X2 5 1
OR
No. of
Cases
No. of
Controls
1141
X2 5 0
No. of
Cases
OR
No. of
Controls
X2 5 1
No. of
Cases
OR
No. of
Controls
OR
X1 ¼ 0
29
68
1.0
9
8
2.6
X1 ¼ 0
6
18
1.0
83
76
3.3
X1 ¼ 1
171
111
3.6
21
23
2.1
X1 ¼ 1
20
19
3.2
123
93
4.0
Abbreviation: OR, odds ratio.
Abbreviation: OR, odds ratio.
covariates C, then our conclusions hold within strata of the
covariates C.
We will make reference to ‘‘monotonicity’’ assumptions,
meaning that for all persons in a population, the exposure
affects the outcome in the same direction (e.g., it is neutral
or harmful for all members of the population). We will
say that X1 has a positive monotonic effect on D if Dx1 x2 is
nondecreasing in x1, ruling out persons of response types 3,
7, 9–12, and 15. We will likewise say that X1 has a negative
monotonic effect on D (equivalently, X1 has a positive
monotonic effect on D) if Dx1 x2 is nonincreasing in x1; likewise for X2.
If we consider the exposures X1 and X2 and outcome D,
then, under the sufficient-cause framework (8), there are 9
different possible sufficient causes for D: A1, A2X1, A3 X1 ,
A4X2, A5 X2 , A6X1X2, A7 X1 X2 , A8 X1 X2 , and A9 X1 X2 , where
each sufficient cause includes a ‘‘background cause’’ Ai, the
other factors beyond possibly X1 and X2 required to complete a given sufficient cause. We say that there is synergism
between X1 and X2 if there is a sufficient cause A6X1X2; we
say that there is causal co-action if there is at least 1 sufficient cause of the form A6X1X2, A7 X1 X2 , A8 X1 X2 , or
5
A9 X1 X2 . The early literature on synergism and antagonism
assumed that the background causes of each sufficient cause
were independent of one another (9–12); such assumptions
are often unrealistic, are difficult to assess in practice, and
are not generally necessary (3, 13), and we will not be
making such assumptions here.
Several authors have considered the relation between response types and the sufficient-cause framework (1–5, 14,
15). In this paper, we will be drawing upon a result given by
VanderWeele and Robins (3, 4)—namely, that if there were
a person for whom D11 ¼ 1 but D10 ¼ D01 ¼ 0, then there
must be a sufficient cause A6X1X2. We use this result, and
occasionally others (3–7), to give remarks on antagonism.
Remarks 4–7 specifically consider empirical conditions that
Table 2. Example of ‘‘Outcome-Based Antagonism’’ (Class II)
Using Data From the Article by Xu et al. (21), With Odds Ratios for
Gene-Environment Interaction
X2 5 0
No. of
Cases
No. of
Controls
X2 5 1
OR
No. of
Cases
No. of
Controls
OR
can be used for testing: remark 4 for specific forms of antagonism, remark 5 for any form of causal co-action for D
and remarks 6 and 7 for particular forms of causal
or D,
co-action specifically for D (remark 6) or D (remark 7).
REMARKS ON ANTAGONISM
Remark 1. Within the sufficient-cause framework, ‘‘antagonism’’ might be understood as ‘‘synergism under recoding,’’
either a recoding of one of the exposures or a recoding of the
outcome.
We might first consider antagonism as synergism under
the recoding of an exposure, so that antagonism is present
if there is a sufficient cause with X1 X2 or, similarly, if there
were a sufficient cause with X1 X2 . Note that even if we
restrict our attention to instances of ‘‘synergism under recoding of one of the exposures,’’ the phrase may refer to
a sufficient cause with X1 X2 or a sufficient cause with X1 X2 ;
it would thus be important to clarify which of the 2 exposures is being recoded, that is, whether a sufficient cause
with X1 X2 or a sufficient cause with X1 X2 is in view.
Table 4. Response Patterns to 4 Possible Exposure Combinations
According to 4 Possible Potential Outcomes
Potential Outcome
Response Type
D11
D01
D10
D00
1
1
1
1
1
2
1
1
1
0
3
1
1
0
1
4
1
1
0
0
5
1
0
1
1
6
1
0
1
0
7
1
0
0
1
8
1
0
0
0
9
0
1
1
1
10
0
1
1
0
11
0
1
0
1
12
0
1
0
0
13
0
0
1
1
0
0
1
0
X1 ¼ 0
655
629
1.0
291
272
1.0
14
X1 ¼ 1
50
61
0.8
15
42
0.3
15
0
0
0
1
16
0
0
0
0
Abbreviation: OR, odds ratio.
Am J Epidemiol. 2011;173(10):1140–1147
1142 VanderWeele and Knol
Alternatively, we might conceive of antagonism as synergism under the recoding of the outcome; that is, antagonism
As we will see, this is
is present if there is synergism for D.
not equivalent to ‘‘synergism under recoding of one of the
exposures’’; one may hold without the other.
Remark 2. Response types 10, 12, and 14 necessarily
imply antagonism understood as synergism under the recoding of one of the 2 exposures (class I antagonism, ‘‘exposurebased antagonism’’). Response types 9 and 10 necessarily
imply antagonism understood as synergism under the recoding of the outcome (class II antagonism, ‘‘outcome-based
antagonism’’).
If there is a person of response type 14, then, using the
result of VanderWeele and Robins (3, 4), there must be
a sufficient cause with X1 X2 . Likewise, if there is a person
of response type 12, there must be a sufficient cause with
X1 X2 . If there is a person of response type 10, there must be
a sufficient cause with X1 X2 and another sufficient cause
with X1 X2 . All of these settings are instances of synergism
under the recoding of the exposure, ‘‘exposure-based
antagonism’’; this will constitute our first class of
antagonism.
Suppose now that there were a person of response type 9
or 10. Again using the result of VanderWeele and Robins
This
(4), there must be a sufficient cause X1X2 for D.
would imply synergism under the recoding of the outcome
(‘‘outcome-based antagonism’’); this will constitute our
second class of antagonism. As discussed below, causal
co-action is not invariant to the recoding of the outcome.
Note that response type 10 is an instance of both exposurebased antagonism and outcome-based antagonism. Below,
we will consider how we can empirically test for these
different forms of antagonism.
Remark 3. We might understand ‘‘antagonism’’ to also
include response type 2, that is, persons for whom the outcome occurs if one or the other of the exposures is present but
not if both are absent, so that, when both are present, the
exposures effectively compete to cause the outcome (class III
antagonism, ‘‘competing antagonism’’). If we include such
persons as manifesting antagonism, then the antagonistic response types are equivalent to the class of ‘‘subadditive’’
types.
Response type 2 is sometimes referred to as a ‘‘competing’’ or ‘‘antagonistic’’ (1, 5) response type, because if both
exposures are present, they will effectively compete to cause
the outcome (1, 5). If response type 2 is taken as an antagonistic type, this will be our third class of antagonism
(‘‘competing antagonism’’).
If we consider all 3 classes of antagonism together, this
gives us response types 2, 9, 10, 12, and 14. Rothman et al.
(5) note that for each of these response types, it is the case
that D11 D10 D01 þ D00 < 0 or, equivalently, (D11 D00)
< (D10 D00) þ (D01 D00)—that is, the effect of both
exposures together is less than the sum of the effects of each
considered separately. These are the only response types for
which this is true (5). Rothman et al. (5) thus refer to these
persons as ‘‘subadditive’’ types; terminology concerning
‘‘subadditive types’’ could be used instead without making
linguistic commitments as to what is and is not to be included under the category of ‘‘antagonism.’’ Antagonism of
class I, class II, or class III is equivalent to the class of
‘‘subadditive’’ types. As discussed below, Table 1 is an example of exposure-based antagonism (class I), Table 2 an
example of outcome-based antagonism (class II), and Table
3 an example of competing antagonism (class III).
Remark 4. It is possible to empirically test for each form
of antagonism or for any form of antagonism, both with and
without monotonicity assumptions.
Let px1 x2 ¼ Pr D ¼ 1j X1 ¼ x1 ; X2 ¼ x2 , that is, let p11,
p10, p01, and p00 denote the overall levels of risk in the
population for the outcome under each possible exposure
combination. Using the results of VanderWeele and Robins
(3, 4) (recoding the exposure X2), there must be a sufficient
cause for D with X1 X2 if
p10 p11 p00 > 0:
ð1Þ
If condition 1 is satisfied, there must be a person of response
type 10 or 14 present, and thus there must be a sufficient
cause for D with X1 X2 . If X1 and X2 have positive monotonic
effects on the outcome, then assessing the weaker condition
p10 p11 p00 þ p01 > 0
ð2Þ
will suffice for this conclusion (3). Likewise, there must be
persons of response type 10 or 12 and consequently a sufficient cause for D with X1 X2 if
p01 p11 p00 > 0:
ð3Þ
If X1 and X2 have positive monotonic effects on D, then
p01 p11 p00 þ p10 > 0
ð4Þ
would suffice. Conditions 1–4 thus constitute empirical conditions for detecting antagonism of class I, ‘‘exposure-based
antagonism.’’
We can also empirically assess antagonism of class II,
‘‘outcome-based antagonism.’’ Again using the results of
VanderWeele and Robins (3, 4), we could conclude that
there must be a person of response type 9 or 10 and consequently a sufficient cause for D with X1X2 if
ð1 p11 Þ ð1 p10 Þ ð1 p01 Þ > 0:
ð5Þ
If X1 and X2 have negative monotonic effects on D (i.e.,
then
positive monotonic effects on D),
ð1 p11 Þ ð1 p10 Þ ð1 p01 Þ þ ð1 p00 Þ > 0
ð6Þ
would suffice.
VanderWeele (6, 7) showed that one can in fact empirically test for the presence of specific response types, both
with and without monotonicity assumptions (see Appendix
for further discussion). We can thus use these results, under
appropriate recoding, to empirically assess not only class I
and class II antagonism but also specific antagonistic
Am J Epidemiol. 2011;173(10):1140–1147
Remarks on Antagonism
Table 5. Empirical Conditions for Different Forms of Antagonism
Form of
Antagonism and
Monotonicity
Assumption
No assumption
1 and X2
X
with positive
monotonicity
Table 6. Conclusions About the Presence of Response Types
Under Various Monotonicity Assumptions and Superadditivity or
Subadditivity
Empirical Test
Assumption
2 for D
Class I: X1 X
No assumption
2
X1 and X
with positive
monotonicity
1 X2 for D
Class I: X
1143
p10 p11 p00 > 0
p10 p11 p00 þ p01 > 0
Superadditivity
Subadditivity
No monotonicity
assumption
3, 5, 7, 8, or 15
2, 9, 10, 12, or 14
X1 positive monotonic
5 or 8
2 or 14
X1 negative monotonic
3 or 15
9 or 12
X2 positive monotonic
3 or 8
2 or 12
X2 negative monotonic
5 or 15
9 or 14
p01 p11 p00 > 0
X1 positive, X2 positive
8
2
p01 p11 p00 þ p10 > 0
X1 positive, X2 negative
5
14
X1 negative, X2 positive
3
12
X1 negative, X2 negative
15
9
Class II: X1 X2 for D
No assumption
ð1 p11 Þ ð1 p10 Þ ð1 p01 Þ > 0
X1 and X2
with negative
monotonicity
for D
ð1 p11 Þ ð1 p10 Þ ð1 p01 Þ þ ð1 p00 Þ > 0
Class III: competing
No assumption
ð1 p00 Þ ð1 p10 Þ ð1 p01 Þ ð1 p11 Þ > 0
Either X1 or X2
with positive
monotonicity
ð1 p00 Þ ð1 p10 Þ ð1 p01 Þ > 0
Both X1 and X2
with positive
monotonicity
p11 p10 p01 þ p00 < 0
response types. One could conclude that a person of response type 14 were present if
p10 p11 p00 p01 > 0:
ð7Þ
If one of X1 or X2 had a positive monotonic effect on D, then
condition 1 would suffice for this conclusion; if both did,
then condition 2 would suffice. One could conclude that
a person of response type 12 were present if
p01 p11 p00 p10 > 0:
ð8Þ
If either X1 or X2 has a positive monotonic effect on D, then
condition 3 suffices; if both do, then condition 4 suffices. One
could conclude that a person of response type 9 were present if
ð1 p11 Þ ð1 p10 Þ ð1 p01 Þ ð1 p00 Þ > 0:
ð9Þ
If either X1 or X2 has a negative monotonic effect on D, then
condition 5 suffices; if both do, then condition 6 suffices.
Finally, one can similarly empirically assess the third
class of antagonism, what we called ‘‘competing antagonism’’ constituted by response type 2. One could conclude
that a person of response type 2 were present if
ð1 p00 Þ ð1 p10 Þ ð1 p01 Þ ð1 p11 Þ > 0: ð10Þ
If either X1 or X2 has a positive monotonic effect on D, then
ð1 p00 Þ ð1 p10 Þ ð1 p01 Þ > 0
p11 p10 p01 þ p00 < 0;
ð12Þ
that is, it would suffice to test for ‘‘subadditivity.’’ If condition
12 is satisfied, there must be a person of one of the antagonistic response types—that is, one of types 2, 9, 10, 12, and
14. If condition 12 is satisfied and one or both of X1 or X2
have positive or negative monotonic effects on D, then more
specific conclusions can be made about which antagonistic
response types must be present, as described in Table 6.
By dividing any of the inequalities above by p00, one obtains
conditions on relative risks rather than risks. Such tests could
sometimes also be applied to case-control data under incidence
density sampling or in situations where the outcome was rare
enough that odds ratios approximated risk ratios. Statistical
inference can be conducted with t-test-like test statistics if risks
are used (3, 17) or with other methods for confidence intervals
if risk ratios or odds ratios are used (18, 19). Additional remarks about such tests are made below, and remarks about
modeling are made in the Conclusion section.
Remark 5. What is ordinarily described as ‘‘superadditivity’’ or ‘‘subadditivity’’ necessarily implies some form of causal
co-action for either the presence or the absence of the outcome.
Thus, nonadditivity implies that some form of causal co-action
must be present either for the outcome or for its absence.
ð11Þ
will suffice for this conclusion; if both X1 and X2 have a
positive monotonic effect on D, then (1 – p00) (1 – p10) Am J Epidemiol. 2011;173(10):1140–1147
(1 – p01) þ (1 – p11) > 0 will suffice. This can be rewritten as
p11 p10 p01 þ p00 < 0. The empirical conditions for
each of the 3 forms of antagonism (rather than specific response types) are summarized in Table 5. In each case, the
contrast itself also serves as a lower bound for the prevalence of persons with that form of antagonism (16).
If we are simply interested in testing for any form of antagonism (class I, II, or III)—that is, the presence of any of
the response types 2, 9, 10, 12, and 14—then, from remark 3
above and the discussion in the paper by Rothman et al. (5),
we have that, without any assumptions about monotonicity,
it would suffice to test
Typically, risks are said to be superadditive or to manifest
positive effect-measure modification on the risk difference
scale if
1144 VanderWeele and Knol
p11 p10 p01 þ p00 > 0
and are said to be subadditive (negative effect-measure
modification on the risk difference scale) if
p11 p10 p01 þ p00 < 0:
If risks are subadditive, there must be a person of response
type 2, 9, 10, 12, or 14 (i.e., of one of the antagonistic types),
and thus some form of causal co-action, either for D or for
By similar arguments, if risks are superadditive, there
D.
must be a person of response type 3, 5, 7, 8, or 15 (5) and
thus some form of causal co-action, either for D or for D.
It thus follows that nonadditivity, that is,
p11 p10 p01 þ p00 6¼ 0;
must imply one of the response types 2, 3, 5, 7–10, 12, 14, or
Greenland and
15 and causal co-action for either D or D.
Poole (1) call these 10 types instances of ‘‘causal interdependence.’’ This notion of ‘‘causal interdependence’’ (1) is
somewhat different from the notion of ‘‘definite interdependence,’’ introduced by VanderWeele and Robins (3), which
is constituted by 6 types: 7, 8, 10, 12, 14, and 15. ‘‘Causal
interdependence’’ allows us to know that there is causal co ‘‘definite interdependence’’ allows
action for either D or D;
us to know the specific form of causal co-action for D.
Remark 6. For the presence of a particular outcome, a
single simple calculation, even in case-control studies, will
make clear whether any specific form of causal co-action
between the presence or absence of 2 exposures can be
detected from the data.
Subadditivity and superadditivity allow us to conclude that
some form of causal co-action is present but generally do not
allow us to determine what specific form is present. Here we
will consider a simple calculation that allows one to determine
whether any specific form of causal co-action may be present.
This simple calculation, along with the one in the following
remark, will essentially automate the process of choosing the
relevant empirical contrast from remark 4 and for synergism
(3–7). We will consider 2 cases, one in which monotonicity
assumptions may be plausible and a second in which no
monotonicity assumptions are made; we will also consider
data on both risks and risk ratios (or odds ratios that may have
been obtained from case-control data with a rare outcome or
using incidence density sampling).
If data are available on the risks, p11, p10, p01, and p00, and it
is thought that monotonicity assumptions may be plausible,
select A as one of X1 or 1 X1 and B as one of X2 or 1 X2, so
that pAB is the largest of the 4 risks, that is, the largest of pAB,
p(1 A)B, pA(1 B), and p(1 A)(1 B) (equivalently of p11, p10,
p01, and p00). We use A and B here, rather than X1 and X2, to
notationally distinguish between the coding chosen by the
investigator (i.e., A and B) and what might be the more natural
coding (i.e., X1 and X2). It is shown in the Appendix that if
pAB p(1 A)B pA(1 B) þ p(1 A)(1 B) > 0 and if A and B
have positive monotonic effects on D, then there is specifically
causal co-action for D between A and B. For example, if A had
been selected as X1 and B had been selected as 1 X2, we
would conclude that there was specifically a sufficient cause
for D with X1 X2 . It is also shown in the Appendix that if pAB p(1 A)B pA(1 B) þ p(1 A)(1 B) 0, no specific form of
causal co-action for D can be detected from the data simply
from the probabilities p11, p10, p01, and p00. Whether A and
B have positive monotonic effects on D must be evaluated
on subject matter grounds, although the true probabilities
must at least satisfy pAB maximum (p(1 A)B, pA(1 B))
and p(1 A)(1 B) minimum (p(1 A)B, pA(1 B)). Without
assumptions on monotonicity, again select A and B as above.
If pAB p(1 A)B pA(1 B) > 0, there is specifically causal
co-action between A and B; if pAB p(1 A)B pA(1 B) 0, no specific form of causal co-action for D can be detected
from the data without monotonicity assumptions.
If data on risk ratios (RRs) are instead available, RR11,
RR10, RR01, and RR00, where RRij ¼ pij/p00 and RR00 by
default is 1 (or if data are available on odds ratios from
a case-control study with a rare outcome so that these approximate risk ratios), then select A as one of X1 or 1 X1
and B as one of X2 or 1 X2, so that RRAB is the largest (in
magnitude on an absolute scale, i.e., most above 0) of the 4
risk ratios. The approach of the previous paragraph applies,
replacing the probabilities with risk ratios.
Note that the above calculations presuppose that it is
known which risk is the largest; this would be the case if
both exposures were assumed to have monotonic effects, but
it would be an additional assumption otherwise. A similar
point holds with remark 7 below. Future work could consider
settings in which the largest probability is assumed to be
unknown and could derive statistical properties of a 2-stage
test procedure, first testing for the largest risk and then testing the relevant contrast. Simulations indicate that a naive 2stage testing procedure could be conservative or anticonservative depending on the true parameter values.
Remark 7. If we wish to detect whether any specific form of
causal co-action for the absence of an outcome can be detected
from the data, a different simple calculation can be employed.
VanderWeele and Robins (3) noted that whether or not
some form of causal co-action was necessarily implied by
a response type was invariant to the recoding of the exposures but not invariant to recoding of the outcome. That is to
say, we might have causal co-action for the presence of the
outcome but not for the absence of the outcome or vice
versa. The example given by VanderWeele and Robins (3)
was one in which X1 and X2 denote 2 genetic factors such
that the person will have the outcome D if and only if both
are present; thus, there is causal co-action between X1 and
X2 for D, as X1X2 is a sufficient cause for D. If instead we
then either
consider the absence of the outcome, that is, D,
the absence of the first factor (i.e., X1 ) or the absence of the
2 ) will suffice for the absence of the
second factor (i.e., X
the sufficient causes for D could then
outcome (i.e., for D);
2 , and there is no causal co-action of
1 and X
be considered X
any form between X1 and X2 for D.
In some cases, we might want to determine whether we
can empirically detect any specific form of causal co-action
Because causal
for the absence of the outcome, that is, for D.
Am J Epidemiol. 2011;173(10):1140–1147
Remarks on Antagonism
co-action is not invariant to recoding of the outcome, we
need to use different simple calculations than those given in
the previous remark. If data are available on the risks for D,
p11, p10, p01, and p00, and it is thought that monotonicity
assumptions may be plausible, select A and B so that pAB is
the smallest of the 4 risks for D. If p(1 A)B þ pA(1 B) >
pAB þ p(1 A)(1 B) and if A and B have negative monotonic
then
effects on D (i.e., positive monotonic effects on D),
there is specifically causal co-action between A and B for
if p(1 A)B þ pA(1 B) pAB þ p(1 A)(1 B), then no
D;
specific form of causal co-action for D can be detected
from the data simply from the risks. Without assumptions on
monotonicity, again select A and B as above. If p(1 A)B þ
pA(1 B) > 1 þ pAB, then there is specifically causal co-action
if p(1 A)B þ pA(1 B) > 1 þ pAB, then
between A and B for D;
no specific form of causal co-action for D can be detected from
the data without monotonicity assumptions.
If data are instead available on risk ratios (RR11, RR10,
RR01, and RR00) for D (or if data are available on odds ratios
from a case-control study with a rare outcome), select A and B
so that RRAB is the smallest of the 4 risk ratios. If RR(1 A)B þ
RRA(1 B) > RRAB þ RR(1 A)(1 B) and if A and B have
negative monotonic effects on D (i.e., positive monotonic
then there is specifically causal co-action
effects on D),
if RR(1 A)B þ RRA(1 B) between A and B for D;
RRAB þ RR(1 A)(1 B), then no specific form of causal
co-action for D can be detected from the data simply from
the risk ratios. Without assumptions on monotonicity, again
select A and B as above. If
1
;
RRð1AÞB þ RRAð1BÞ > RRAB þ
pð1AÞð1BÞ
then there is specifically causal co-action between A and
Note that some information on the prevalence
B for D.
p(1 A)(1 B) is needed to evaluate this inequality, or alternatively one could consider the values of p(1 A)(1 B) for
which the inequality is satisfied. If
1
;
RRð1AÞB þ RRAð1BÞ RRAB þ
pð1AÞð1BÞ
then no specific form of causal co-action for D can be detected from the data simply from the risk ratios.
EXAMPLES
We present 3 examples illustrating the 3 forms of antagonism drawn from the existing literature on gene-gene and
gene-environment interactions. More detailed calculations
for these examples using remarks 6 and 7 are presented in
the Web Appendix, which is posted on the Journal’s Web
site (http://aje.oxfordjournals.org/). All confidence intervals
are obtained via the methods described by Richardson and
Kaufman (19) for the relative excess risk due to interaction
(RERI), RERI ¼ RR11 RR10 RR01 þ 1, under appropriate recoding; when a contrast such as RR11 RR10 RR01 is in view, a confidence interval can be obtained simply
by subtracting 1 from both limits of the RERI confidence
interval. The examples were chosen for illustrative purposes
and do not constitute formal tests for hypotheses concerning
Am J Epidemiol. 2011;173(10):1140–1147
1145
antagonism; we assume unconfoundedness in all examples,
which may not be realistic here (see also the comment on
testing at the end of remark 6).
Consider first data presented by Stern et al. (20) (summarized in Table 1), where X1 denotes ever smoking, X2
denotes the Gln/Gln genotype versus the Lys/Lys or Lys/
Gln genotype for XPD codon 751, and D denotes bladder
cancer. If we follow the procedure in remark 6 to detect
a specific form of causal co-action for D, we select RR10
as the highest relative risk. Monotonicity is empirically
violated in Table 1, but if we assess the contrast for causal
co-action without monotonicity, we have RR10 RR11 RR00 ¼ 3.6 2.1 1.0 ¼ 0.5 (95% confidence interval
(CI): 1.2, 2.1). By means of the results in remark 6, the
point estimate of 0.5 would suggest evidence for causal coaction for D between X1 and X 2 (antagonism of class I,
exposure-based antagonism) without any assumptions about
monotonicity; this could also be seen from remark 4, condition 1; note, however, that the confidence interval here comfortably includes 0. Table 1 is representative of data
suggesting antagonism of form I, exposure-based antagonism.
Now consider data presented by Xu et al. (21), considering
possible interactive effects between CYP19A1 rs1870050
polymorphisms and the consumption of polyphenol-rich
foods and beverages on endometrial cancer. Specifically, let
X1 denote high versus low tea consumption and let X2 denote
the CC genotype versus the AA/AC genotype. Odds ratios
(which approximate risk ratios) obtained from the numbers
of cases and controls given in the paper by Xu et al. (21) are
shown in Table 2. If we follow the procedure in remark 7 to
then we have
detect a specific form of causal co-action for D,
that RR11 is the lowest relative risk, and RR01 þ RR10 RR00 RR11 ¼ 0.5 (95% CI: 0.0, 0.9). By the results described in remark 7, if X1 and X2 have negative monotonic
effects on D (i.e., are preventive or neutral for all persons),
then there is causal co-action between X1 and X2 for D (antagonism of class II, outcome-based antagonism); this could
also be seen from remark 4, condition 6. Said another way,
high tea consumption and the CC genotype interact synergistically to prevent D. Table 2 is representative of data suggesting antagonism of class II, outcome-based antagonism.
Finally, consider the data (22) summarized in Table 3,
where X1 denotes the Arg/Arg genotype for XRCC1 codon
194 and X2 denotes the presence of methionine variants at
XRCC3 codon 241 and D denotes bladder cancer. If we
follow the procedure in remark 6, the condition for a specific
form of causal co-action for D is not satisfied. If we follow
the procedure in remark 7 to detect a specific form of causal
then RR00 is the lowest relative risk and
co-action for D,
RR10 þ RR01 RR11 RR00 ¼ 3.2 þ 3.3 4.0 1.0 ¼ 1.5
(95% CI: 1.9, 4.8). If 1 X1 and 1 X2 have negative
monotonic effects on D (i.e., X1 and X2 have positive monotonic effects on D), then, by remark 7, the point estimate of
1 and
1.5 gives some evidence of causal co-action between X
2 for D,
that is, under monotonicity, evidence for persons
X
of response type 2 (antagonism of class III, competing antagonism); this could also be seen from remark 4, condition
12. Note, however, that the confidence interval here includes
0. Table 3 is representative of data suggesting antagonism of
class III, competing antagonism.
1146 VanderWeele and Knol
CONCLUSION
In this paper, we have presented a classification of forms
of antagonism. We have considered what we called class I
antagonism or ‘‘exposure-based antagonism,’’ equivalent to
synergism under the recoding of one of the exposures; we
have considered what we called class II antagonism or ‘‘outcome-based antagonism,’’ equivalent to synergism under the
recoding of the outcome; and we have also considered what
we called class III antagonism or ‘‘competing antagonism,’’
which arises when either of 2 exposures suffices for the
outcome and thus the exposures ‘‘compete’’ in producing
the outcome. The 3 classes taken together are equivalent to
the class of ‘‘subadditive’’ response types. We have shown
how to empirically assess each class of antagonism. Moreover, we have considered empirical conditions with which to
detect any form of causal co-action either for the presence
of a particular outcome or for the absence of a particular
outcome. We hope that our remarks will be useful in clarifying the conceptualization of antagonism and for testing for
it in actual practice.
A few additional comments merit attention. First, we have
considered here the setting in which the effects of the 2
exposures on the outcome are unconfounded. We have assumed that if unconfoundedness only holds conditional on
the covariates, then the tests we have described are conducted within strata of the confounding variables. This is
reasonable if there are a few categorical covariates that serve
as confounders; however, in settings in which there many
covariates or in which some of the covariates are continuous,
it will no longer be possible to test within each stratum of the
covariates; modeling will be needed instead. Various modeling approaches for interactions and synergism within the
sufficient-cause framework have been described elsewhere
(16, 23, 24), and such approaches could be applied to the
tests for antagonism described above as well. Second, we
have assumed no confounding and no measurement error;
future work could consider the sensitivity of conclusions to
such assumptions. Third, we have restricted our setting to
that in which there are 2 binary exposures; in future work,
researchers could consider what might be meant by antagonism in settings with more than 2 exposures (4) or when the
exposures of interest are categorical or ordinal (25). The
sufficient-cause framework is not, however, generally applicable to continuous exposures; some progress is possible
with time-to-event outcomes (26). Finally, we have assumed
a deterministic framework, but similar remarks would also
hold in a framework that allowed for stochastic counterfactuals and sufficient causes (27).
ACKNOWLEDGMENTS
Author affiliations: Departments of Epidemiology and
Biostatistics, Harvard School of Public Health, Boston,
Massachusetts (Tyler J. VanderWeele); and Julius Center
for Health Sciences and Primary Care, University Medical
Center Utrecht, Utrecht, the Netherlands (Mirjam J. Knol).
Dr. Tyler J. VanderWeele was supported by National
Institutes of Health grant ES017876.
The authors thank Drs. Sander Greenland and Charles
Poole for helpful comments on this paper.
Conflict of interest: none declared.
REFERENCES
1. Greenland S, Poole C. Invariants and noninvariants in the
concept of interdependent effects. Scand J Work Environ
Health. 1988;14(2):125–129.
2. Greenland S, Brumback B. An overview of relations among causal
modelling methods. Int J Epidemiol. 2002;31(5):1030–1037.
3. VanderWeele TJ, Robins JM. The identification of synergism
in the sufficient-component-cause framework. Epidemiology.
2007;18(3):329–339.
4. VanderWeele TJ, Robins JM. Empirical and counterfactual
conditions for sufficient cause interactions. Biometrika.
2008;95(1):49–61.
5. Greenland S, Lash TL, Rothman KJ. Concepts of interaction. In:
Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 3rd
ed. Philadelphia, PA: Lippincott Williams & Wilkins; 2008:71–83.
6. VanderWeele TJ. Empirical tests for compositional epistasis [letter]. Nat Rev Genet. 2010;11(2):166. (doi: 10.1038/nrg2579-c1).
7. VanderWeele TJ. Epistatic interactions. Stat Appl Genet Mol
Biol. 2010;9((1):Article 1. (doi: 10.2202/1544-6115.1517).
8. Rothman KJ. Causes. Am J Epidemiol. 1976;104(6):587–592.
9. Rothman KJ. Synergy and antagonism in cause-effect
relationships. Am J Epidemiol. 1974;99(6):385–388.
10. Rothman KJ. The estimation of synergy or antagonism. Am J
Epidemiol. 1976;103(5):506–511.
11. Rothman KJ. Estimation versus detection in the assessment
of synergy. Am J Epidemiol. 1978;108(1):9–11.
12. Hogan MD, Kupper LL, Most BM, et al. Alternatives to
Rothman’s approach for assessing synergism (or antagonism)
in cohort studies. Am J Epidemiol. 1978;108(1):60–67.
13. VanderWeele TJ, Robins JM. Directed acyclic graphs,
sufficient causes, and the properties of conditioning on
a common effect. Am J Epidemiol. 2007;166(9):1096–1104.
14. Flanders WD. On the relationship of sufficient component
cause models with potential outcome (counterfactual) models.
Eur J Epidemiol. 2006;21(12):847–853.
15. VanderWeele TJ, Hernán MA. From counterfactuals to
sufficient component causes and vice versa. Eur J Epidemiol.
2006;21(12):855–858.
16. VanderWeele TJ, Vansteelandt S, Robins JM. Marginal
structural models for sufficient cause interactions. Am J
Epidemiol. 2010;171(4):506–514.
17. Greenland S. Tests for interaction in epidemiologic studies:
a review and a study of power. Stat Med. 1983;2(2):243–251.
18. Hosmer DW, Lemeshow S. Confidence interval estimation
of interaction. Epidemiology. 1992;3(5):452–456.
19. Richardson DB, Kaufman JS. Estimation of the relative excess
risk due to interaction and associated confidence bounds. Am J
Epidemiol. 2009;169(6):756–760.
20. Stern MC, Johnson LR, Bell DA, et al. XPD codon 751 polymorphism, metabolism genes, smoking, and bladder cancer risk.
Cancer Epidemiol Biomarkers Prev. 2002;11(10):1004–1011.
21. Xu WH, Dai Q, Xiang YB, et al. Interaction of soy food
and tea consumption with CYP19A1 genetic polymorphisms
in the development of endometrial cancer. Am J Epidemiol.
2007;166(12):1420–1430.
22. Stern MC, Umbach DM, Lunn RM, et al. DNA repair gene
XRCC3 codon 241 polymorphism, its interaction with
smoking and XRCC1 polymorphisms, and bladder cancer risk.
Cancer Epidemiol Biomarkers Prev. 2002;11(9):939–943.
Am J Epidemiol. 2011;173(10):1140–1147
Remarks on Antagonism
23. Vansteelandt S, VanderWeele TJ, Tchetgen E, et al. Multiply
robust inference for statistical interactions. J Am Stat Assoc.
2008;103(484):1693–1704.
24. VanderWeele TJ. Sufficient cause interactions and statistical
interactions. Epidemiology. 2009;20(1):6–13.
25. VanderWeele TJ. Sufficient cause interactions for categorical and
ordinal exposures with three levels. Biometrika. 2010;97:647–659.
26. VanderWeele TJ. Causal interactions in the proportional
hazards model. Epidemiology. In press.
27. VanderWeele TJ, Robins JM. Stochastic counterfactuals and
sufficient causes. Stat Sin. In press.
28. Cordell HJ. Detecting gene-gene interactions that underlie
human diseases. Nat Rev Genet. 2009;10(6):392–404.
29. Phillips PC. Epistasis—the essential role of gene interactions
in the structure and evolution of genetic systems. Nat Rev
Genet. 2008;9(11):855–867.
APPENDIX
Empirical conditions for specific response types
In the text, we give empirical conditions for antagonistic
response types 2, 12, and 14. Through appropriate recoding
of the exposure and/or the outcome, the results presented by
VanderWeele (6, 7) can also be used to empirically test for
response types 3, 5, 8, 9, and 15. Specifically, VanderWeele
(6, 7) showed that one could test for persons of type 8 (i.e.,
persons for whom D11 ¼ 1 but D10 ¼ D01 ¼ D00 ¼ 0) by
testing p11 p10 p01 p00 > 0. Note that this contrasts
with the usual condition for positive effect-measure modification on the risk difference scale, because we subtract rather
than add back in the probability p00. This test did not require
any monotonicity assumptions or assumptions about the absence of other response types. When X1 and X2 are 2 genetic
factors, the presence of such response types for whom D11 ¼
1 but D10 ¼ D01 ¼ D00 ¼ 0 is sometimes referred to as an
instance of ‘‘compositional epistasis’’ (28, 29) because, for
those persons, the effect of one of the genetic factors is completely masked if the other genetic factor is absent. Other
conditions for this specific response type can also be given
if monotonicity assumptions can be made (6, 7). By appropriately recoding the exposure and/or the outcome of interest,
the results of VanderWeele (6, 7) give empirical tests for any
of response types 2, 3, 5, 8, 9, 12, 14, and 15, both with and
without monotonicity assumptions. Thus, each of these response types can, in some instances, be empirically detected
without making any assumptions about monotonicity or
about the absence of other response types, contrary to the
claim made by Rothman et al. (5, p. 79). Note that these
8 response types are those for which either exactly 1 or exactly 3 of the potential outcomes in Table 4 take the value 1.
Proof of results in remark 6
If pAB p(1A)B pA(1B) þ p(1A)(1B) > 0 and if A
and B have positive monotonic effects on D, then the results
of VanderWeele and Robins (3) imply that there is causal coaction between A and B; if pAB p(1A)B pA(1B) þ
p(1A)(1B) < 0, then we cannot draw conclusions about
causal co-action between A and B for D. Because the ordering of the levels was chosen so that pAB is the largest of pAB,
Am J Epidemiol. 2011;173(10):1140–1147
1147
p(1A)B, pA(1B), and p(1A)(1B), only positive monotonic
effects of A and B on D are possible, so no other form of
causal co-action for D can be detected using monotonicity
assumptions. Without monotonicity, if pAB p(1 A)B pA(1 B) > 0, then the results of VanderWeele and Robins
(3) imply that there is causal co-action between A and B for
D. If pAB p(1 A)B pA(1 B) > 0 holds, then p(1 A)(1 B) p(1 A)B pA(1 B) > 0 might also hold, in which case
one could also empirically detect causal co-action between
(1 A) and (1 B) for D. If pAB p(1 A)B pA(1 B) 0,
then we cannot draw conclusions about causal co-action
between A and B for D. Because the ordering of the levels
was chosen so that pAB is the largest of pAB, p(1 A)B,
pA(1 B), and p(1 A)(1 B), we cannot have p(1 A)(1 B) p(1 A)B pA(1 B) > 0, p(1 A)B pAB p(1 A)(1 B) > 0,
or pA(1 B) pAB p(1 A)(1 B) > 0, so we cannot detect
any specific form of causal co-action for D without monotonicity. The results for case-control studies follow by dividing the inequalities by p(1 A)(1 B) in the argument above.
Proof of results in remark 7
The arguments for the results concerning causal co-action
for D using risks are analogous to those for risks in remark 6,
under recoding of the outcome. For case-control studies, we
would have causal co-action between A and B for D when
both factors have negative monotonic effects on D (positive
if (1 pAB) (1 p(1 A)B) monotonic effects on D)
(1 pA(1 B)) þ (1 p(1 A)(1 B)) > 0, that is, if RR(1 A)B þ
RRA(1 B) > RRAB þ RR(1 A)(1 B). If RR(1 A)B þ
RRA(1 B) RRAB þ RR(1 A)(1 B), we cannot draw
conclusions about causal co-action between between A
Because the ordering of the levels was chosen
and B for D.
so that pAB is the smallest of pAB, p(1A)B, pA(1B), and
p(1A)(1B), only negative monotonic effects of A and B on
D are possible, so no other form of causal co-action can be
detected using monotonicity assumptions. Without monotonicity, we would have causal co-action between A and B
for D if (1 pAB) (1 p(1 A)B) (1 pA(1 B)) > 0,
that is, if p(1 A)B þ pA(1 B) > pAB þ 1, which, dividing by
pð1AÞð1BÞ , is
RRð1AÞB þ RRAð1BÞ > RRAB þ
1
pð1AÞð1BÞ
:
If
RRð1AÞB þ RRAð1BÞ RRAB þ
1
pð1AÞð1BÞ
;
then (1 pAB) (1 p(1 A)B) (1 pA(1 B)) 0,
and we cannot draw a conclusion about causal co-action
Because the ordering of the
between A and B for D.
levels was chosen so that pAB is the smallest of pAB,
p(1A)B, pA(1B), and p(1A)(1B), we cannot have (1 p(1 A)(1 B)) (1 p(1 A)B) (1 pA(1 B)) > 0,
(1 p(1 A)B) (1 p(1 A)(1 B)) (1 pAB) > 0, or
(1 pA(1 B)) (1 p(1 A)(1 B)) (1 pAB) > 0 and
thus cannot detect any specific form of causal co-action for
D without monotonicity.