American Journal of Epidemiology
Copyright O 1997 by The Johns Hopkins University School of Hygiene and Pubfic Health
All rights reserved
Vol 145, No 7
Printed In U S.A
Biologic Synergism and Parallelism
John Darroch
In epidemiologic studies of two binary exposure factors, much attention has been given to the concept of
synergism of the factors. The leading dictionary of epidemiology offers two definitions of synergism, one of
which this author labels statistical and the other biologic. The epidemiologic literature has been largely
concerned with statistical synergism, which is typically measured using additive or multiplicative interaction.
This paper focuses on biologic synergism, on the related concept of biologic parallelism, and on the question
of how much information can be gleaned about population amounts of biologic synergism and parallelism—
information which is of vital interest to epidemiologists. A fundamental identity equates the difference between
the amounts of biologic synergism and parallelism to the additive interaction. Two biologic models, the
multistage model and the no-hrt or immunity model, enhance the interpretation of multiplicative interaction as
a measure of statistical synergism, but it is pointed out here that, unfortunately, both models incorporate the
strong assumption that there is no parallelism. A third model, the single-hit or vulnerability model, makes the
even stronger assumption that there is no biologic synergism and consequently that the additive interaction
is equal to minus the amount of parallelism A consequence of this fact is that a link which has been perceived
in the literature to exist between the single-hit model and the additive interaction is false. Am J Epidemiol 1997;
145:661-8.
additive interaction, antagonism; multiplicative interaction; multistage model; no-hit model; parallelism,
single-hit model; synergism
BIOLOGIC SYNERGISM
tical interaction or by multiplicative statistical interaction. These publications may therefore be considered
to relate to the dictionary's first definition because,
when "effect" is interpreted at the population level as
risk difference (attributable risk) or risk ratio (relative
risk), synergism is equated to additive or multiplicative interaction. These notions of "statistical synergism" will be revisited in this paper, but the principal
focus will be on "biologic synergism" as given in the
dictionary's second definition.
According to this definition, which has been discussed by Miettinen (2), Rothman (3, 4), Koopman
(5), Hamilton (6), Greenland and Poole (7), and
Darroch and Borkent (8), synergism of two factors A
and B is a combination of biologic properties possessed by a single individual in relation to some disease D. Three of the properties are those of getting D
if exposed to both A and B and, other things being
equal, of not getting D if exposed to either A or B
alone. There is also the implied fourth property of not
getting D if exposed to neither of the factors, provided
that we make the assumption, a very reasonable one in
most cases of practical interest, that the two factors are
"causal." For a given disease D (lung cancer, for
example), a risk factor is causal (smoking or asbestos
exposure, for example) if an individual who does not
The definition of synergism in epidemiology is
somewhat controversial. In A Dictionary of Epidemiology, Last (1) offers two definitions, the first a common dictionary definition and the second a more specific definition encountered in bioassay:
SYNERGISM, SYNERGY.
1. A situation in which the combined effect of two or
more factors is greater than the sum of their solitary
effects.
2. Two factors act synergistically if there are persons
who will get the disease when exposed to both factors
but not when exposed to either alone. ANTAGONISM, the opposite of synergism, exists if there are
persons who will get the disease when exposed to one
of the factors alone, but not when exposed to both.
Note that under these definitions two factors may act
synergistically in some persons and antagonistically
in others.
In the vast majority of epidemiologic publications on
synergism, synergism is measured by additive statisReceived for publication January 22, 1996, and in final form
October 4, 1996.
From the Department of Mathematics and Statistics, Faculty of
Science and Engineering, Hinders University of South Australia,
GPO Box 2100, Adelaide 5001 SA, Australia. (Reprint requests to
Dr. John Darroch at this address).
661
662
Darroch
or would not get D in its presence would, other things
being equal, also not get D in its absence. For an
individual having the above four properties, we shall
say that the action of the two risk factors exhibits
biologic synergisra. If either factor is "preventive"
(rural living, for example) instead of causal, it can be
replaced by its causal opposite, whose absence/presence corresponds to the original factor's presence/
absence. Little will be said here about factors which
are "directionless," that is, neither causal nor preventive (being male, for example).
The qualification "other things being equal" is important in the above definitions and can be given
substance using Rothman's (3) notion of "sufficient
cause." The sufficient cause of D, for a given individual, is the minimal set of conditions and events that
produced D—minimal in the sense that none of the
conditions and events were superfluous. As noted by
Koopman (5), the component causes of a sufficient
cause for a given individual comprise both attributes
of the individual and attributes of the individual's
environment. Koopman suggested that events occurring along a pathogenic pathway might also be considered component causes, but this author prefers to
think of them as effects of the sufficient cause. As
Rotnman pointed out (4, p. 16), the sufficient cause
paradigm permits us to speak deterministically about
the influence of a factor on an individual, all of the
chance influences in the individual's environment having been absorbed into the constellation of component
causes, most of which are hidden from us.
Biologic synergism, called "epidemiologic interaction" by Rotnman (4, p. 313), is present in an individual whose sufficient cause contains both A and B as
component causes; if A or B or both A and B are
removed from the individual's environment, all of the
other component causes remaining the same, D is not
incurred. If an individual exposed to A and B does get
D, we of course cannot know whether A and B acted
synergistically. The most that we can hope to know is
the population fraction of such individuals, that is,
individuals whose sufficient cause contained A and B.
One of the contributions of this paper is the delineation
of the extent to which this fraction, the proportional
size of die "synergistic class," can be measured.
Two risk factors A and B generate four subpopulations according to whether each factor is absent or
present. A given individual necessarily belongs to only
one of these subpopulations, so that, of the four properties constituting biologic synergism, only one is observable, as mentioned above. The other three properties, concealed widiin the biology of the individual,
depend upon our imagining what would have happened regarding D had the individual belonged to each
of the other three subpopulations. For simplicity, we
shall focus our attention on the individuals in one of
the four subpopulations, and we will choose the subpopulation exposed to both factors; this group will be
referred to as "the population" in the ensuing discussion. Within this population, the class of individuals
who get D but would not have done so under any of
the three milder exposure conditions is the class evincing biologic synergism.
OTHER CLASSES
The synergistic class is just one of several classes. If
both factors are directionless, there are 16 classes in
total, because each individual either would or would
not get D under each of the four exposure conditions.
Greenland and Poole (7) listed and named all 16
classes and put them into categories according to invariance considerations which arise when the factor
levels, absence and presence, are recoded as presence
and absence. Earlier, Miettinen (2), in a discussion of
causal and preventive factors, synergism, antagonism,
independence, and interdependence, had listed and
named most of the 16 classes. Hamilton (6) observed
that when assuming as we do here that both factors are
causal, the number of classes is reduced to six. In
seeking to measure the sizes of these classes, he complicated the task by trying to distinguish and measure
separately "real" and "spurious" synergistic action.
Darroch and Borkent (8) commented on this distinction and, more importantly, provided a mathematical
discussion of the six classes. The proofs of results
given here either were presented in that paper (8) or
can be constructed within its framework.
THE SIX CLASSES AND THEIR SIZES
As stated above, when both factors are causal, there
are six classes of individuals in the population (that is,
the subpopulation exposed to both A and B). Two of
these classes are of only minor interest. These are the
"all" class of individuals who would get D under all
four exposure conditions, and the "none" class who
would get D under none of the conditions.
The remaining four classes are the ones of real
interest. Besides the synergistic class, there is the
"A-only" class of individuals who get D if and only if
exposed to A, regardless of the presence or absence of
B; for each of these individuals, A, but not B, is a
component cause of his or her sufficient cause. Likewise, there is the 5-only class. Finally, there is the
class containing individuals for whom the risk factors
act "in parallel." These individuals would get D if they
were exposed to just one of A and B, but would not if
they were exposed to neither A nor B; since they are in
Am J Epidemiol
Vol. 145, No. 7, 1997
Biologic Synergism and Parallelism
fact exposed to both A and B, they do get D because of
the assumption that the factors are causal. Parallelism
does not fit naturally into the sufficient cause paradigm, but it can be accommodated within a stretched
paradigm by imagining that there is one component of
the sufficient cause that can take any one of the three
alternative values A, B, and AB. The four classes—
synergistic, A-only, fi-only, and parallel—thus contain
individuals whose sufficient causes include, as components, both A and B in the first case, A in the second,
B in the third, and "A or B" in the fourth.
Nothing can be inferred about the sizes of the six
classes without assuming that the proportions of individuals in the subpopulation exposed to both factors
who would have gotten D under each of the three
milder exposure conditions are equal to the actual
proportions in the subpopulations so exposed. This
"comparability assumption" will be made in this paper.
Let the risks (proportions of individuals getting D)
for the four subpopulations exposed to neither A nor B,
to A alone, to B alone, and to both A and B be denoted
R, R(A), R(B), and R(AB), respectively. The proportional sizes of the six classes sum to 1 and therefore
constitute five unknown quantities—one more, unfortunately, than the number of the known quantities R,
R(A), R(B), and R(AB). Now the proportional sizes of
the "all" class and the "none" class are easily seen to
be R and 1 - R(AB), and it is the other four class sizes,
denoted |synergism|, |parallelism|, |A-only|, and \Bonly|, which cannot in general be fully determined.
The first thing to note about them, given the causal
factor assumption and the comparability assumption,
is that their sum is the risk difference R(AB)—R. This is
because the four classes together contain all of the
individuals who do get D but would not have done so
under the null exposure.
Certain pairs of the four sizes sum to other risk
differences, as indicated in table 1. For example, the
sum of the sizes |synergism| and |A-only| is the risk
difference R(AB)-R(B), while the sum of the sizes
|A-only| and |parallelism| is the risk difference
R(A)-R.
TABLE 1. Two-by-two table of the four unknown class sizes
and their known marginal totals
Isynergisrrd
IB-onlyi
l/4-onlyi
Iparallellsrrt
R(AB) - R{B)
R{B) - R
Am J Epidemiol
Vol 145, No. 7, 1997
R{AB) - R{A)
R(A) - R
R(AB)-R
663
An important result, reported by Darroch and
Borkent (8) and easily deduced from table 1, is that
|synergism| — |parallelism| = R(AB) — R(A)
- R(B)
R.
The significance of this equation is that it provides a
new interpretation of additive interaction as the difference between the population amounts of biologic synergism and biologic parallelism. It contrasts with the
traditional interpretation of additive interaction as one
of the measures of statistical synergism, measuring the
excess of the combined effect over the sum of the
solitary effects. The equation shows that, if the additive interaction is positive, so is |synergism|. It also
shows that biologic parallelism, in contributing negatively to additive interaction, can be thought of as an
opposite of biologic synergism. It has been traditional
to think of the negative influence on the additive
interaction as arising from antagonism. However, biologic antagonism, described in the dictionary's second definition as being present in individuals "who
will get the disease when exposed to one of the factors
alone, but not when exposed to both," has been ruled
out of existence in this paper by our assumption that
the risk factors are causal. Without this assumption
there would be several classes that could be given the
antagonism label, and their sizes would indeed contribute negatively to the additive interaction. The formula for their contributions is implicit in the paper by
Greenland and Poole (7), where the additive interaction is displayed as a function of the sizes of the 16
classes which exist when the factors are directionless.
Rothman (4, p. 314) omitted the existence of parallelism and was led to equate |synergism| with the
additive interaction, whereas |synergism| is, in fact,
greater than or equal to it. Koopman had earlier stated
that "The interaction contrast of disease rates . . . is
shown always to be zero or slightly negative when the
assumptions of no interaction in the sufficient component discrete causes model hold" (5, p. 716). In the
terminology of this paper, Koopman said that the
additive interaction is shown to be zero or slightly
negative when |synergism| is zero. This statement is
incorrect in its use of the adjective "slightly" because,
when |synergism| is zero, the additive interaction is
equal to — |parallelism|, and no restriction exists on the
amount of parallelism that can be present.
Table 1 encapsulates all of the information that can
be gleaned about the four unknown sizes from the four
risks R, R(A), R(B), and R(AB) and only provides three
equations for the four unknowns. It is impossible to
find a fourth equation. This is because it is impossible
(8) to unearth any information about the population
664
Darroch
proportion of individuals who would get D if exposed
to A alone and also would get D if exposed to B alone,
and it means that, in general, it is impossible to calculate the four sizes exactly from the four risks.
However, the four class sizes are fully determined
by the four risks in some extreme circumstances, one
kind of which will be mentioned below in comments
on biologic models for disease causation. The other
kind of extreme circumstance occurs when two of the
risks are equal. If, for example, R(B)=R, then we
immediately know that both |fi-only| and |parallelism|
are zero, that |synergism| is R(AB)—R(A), and that
|A-only| is R(A)—R. In the even more extreme case of
two equalities among the four risks, only one of
the four class sizes is nonzero. If R(A) = R(B) = R,
the only nonzero size is |synergism|; if R(AB) =
R(A) = R(B), it is |parallelism|; if R(AB) = R(A) and
R(B) = R, it is |A-onlv|; and if R(AB) = R(B) and
R(A) = R, it is |B-only|.
Before specifying what can be determined about the
class sizes in the general case, let us first consider an
example. Hertz-Picciotto et al. (9), in a paper on the
synergism of arsenic and smoking in the causation of
lung cancer, reported the results of several studies,
among them a study of retired smelter workers from
Tacoma, Washington. The factors are A = smoking
and B = arsenic exposure, while D = lung cancer. The
value of the risk R in the unexposed subpopulation
was not reported in the article (9), but, relative to it,
the other risks were R(A) = 7.2/?, R(B) = 5.1R, and
R(AB) = 20.7/?, so that
|synergism| — |parallelism| = 9.4/?.
The size table is shown in table 2, and we see that
|5-only|, the amount of arsenic-only action, must lie
between 0 and 4.1/?, from which it follows that |synergism| must be confined to the range from 9.4/? to
13.5/?. Thus, 45 percent (9.4/20.7) to 65 percent (13.5/
20.7) of cases among workers exposed to both smoking and arsenic were attributable to the synergistic
action of these two factors. This is useful information
with which we must be satisfied in the face of the
impossibility of knowing |synergism| exactly. Furthermore, |parallelism| is between 0 and 4.1/? and the
TABLE 2.
Size table: first example*
amount |A-only| of smoking-only action is between
2.1/? and 6.2/?.
In this example, the dominant class is the synergistic
one, because R(AB) strongly exceeds both R(A) and
/?(/?). The result would be very different if, for instance, R(A) and/?(B) were equal to 17.2/? and 15.1/?
instead of 7.2/? and 5.1/?, with R(AB) remaining equal
to 20.7/?. The size table is shown in table 3. Now
|synergism| — |parallelism| = — 10.6R
and |parallelism| is dominant, lying between 10.6/? and
14.1/?.
In other cases, nothing precise can be deduced from
the size table. Suppose that R(A) = 5R, R(B) = 6R,
and R(AB) = 10/?, so that the risk differences R(A) R, R(B) - R, R(AB) - R(A), and R(AB) - R(B) are all
roughly equal, and
|synergism| — |parallelism| = 0.
The size table is shown in table 4. All that can be said
here regarding |synergism| is that it lies somewhere
between 0 and 4/?, with correspondingly not very
useful statements for the other sizes.
The inequalities for the sizes flowing from the general size table are:
max[/?(A5) - R(A) - R(B) + R, 0]
< |synergism| < min[R(AB) - R(B),
R(AB) max[0,-/?(A8) + R(A) + R(B) - R]
< |parallelism| < min[/?(A) - /?, R(B) max[0, R(A) - R(B)]
< |i4-only| =£ mh\[R(A) - R, R(AB) max[/?(B) - R(A), 0]
< |B-only| < min[/?(fi) - /?, R(AB) The left (right) inequalities for the first two sizes are
equivalent to each other and to the right (left) inequalities for the last two sizes.
TABLE 3.
Size table: second example*
Isynerglsmi
IB-onlyl
13.5R
bynerglsrrd
IB-onlyl
3.5R
M-onlyt
(parallelism)
B2R
M-onlyt
Iparallellsrrt
16.2R
15.6fl
4.1 R
19.7/?
5.6R
UAR
19.7R
• Here R{A) = 7.2R, R{B) = 5.1 R, and R(AB) = 20.7R
• Here R(A) » 17.2H, R(B) = 15.1 R. and R(AB) = 2O.7R
Am J Epidemiol
Vol. 145, No. 7, 1997
Biologic Synergism and Parallelism
TABLE 4.
Size table: third example*
feynergisrrd
IS-onlyi
M-onlyl
[parallelism!
4/?
1
Hero R(A) = 5R, R{B) = 6fl, and fl(/AB) = 10R
STATISTICAL SYNERGISM
The Dictionary of Epidemiology's (1) first definition of synergism is not one definition but an infinite
collection of definitions which result from the infinity
of ways of defining the "effect" of a factor. One
member of this infinite collection is biologic synergism, obtained by defining effect at the individual
level as the all-or-nothing property of getting or not
getting D; in order to exceed the sum of the solitary
effects, the combined effect of A and B must be D and
the solitary effects of A and of B must be not-D, and
this exactly describes biologic synergism.
It is more usual to define effect at the population
level and so produce a definition of "statistical synergism." When effect is risk difference, the combined
effect of A and B is R(AB)—R, and the excess of this
over the sum of the solitary effects of A and of B is
[R(AB) - R] - [R(A) - R\ - [R(B) - R]
= R(AB) - R(A) - R(B) + R.
Statistical synergism is thus equated to the additive
interaction; and if, in a given application, the interaction turns out to be negative, so that the combined
effect is less than the sum of the solitary effects, it has
been customary to then say that there is (statistical)
antagonism.
When effect is measured by risk ratio or, to precisely fit the definition 1 reference to the sum of
effects, by log(risk ratio), the excess of the combined
effect over the sum of the solitary effects is equal to
\og[R(AB)/R] - \og[R(A)/R] - \og[R{B)/R]
= log[R(AB)RJR(A)R(B)l
which is the logarithm of the multiplicative interaction
of A and B, so that statistical synergism and antagonism are measured by multiplicative interaction.
Breslow and Storer (10) considered a "power-law"
family of definitions of effect, giving rise to a powerlaw family of definitions of statistical synergism, two
members of which are additive interaction and multiplicative interaction.
The ambiguity in definition 1, which is manifested
in the infinite choice of definitions of statistical synAm J Epidemiol
Vol. 145, No. 7, 1997
665
ergism, greatly weakens the usefulness of any one of
these definitions. Many authors, too numerous to list
here, have confined their attention to a choice between
additive and multiplicative interaction. Thompson
(11), in his review of the literature, describes some of
the difficulties presented by this choice. One difficulty
lies in the notorious fact that statistical synergism
according to the additive definition can coexist with
statistical antagonism according to the multiplicative
definition. This phenomenon occurs in the smokingarsenic-lung cancer data discussed above: the additive
interaction is 9AR, while the multiplicative interaction
is 0.564.
By contrast with the ambiguous definition 1, definition 2 provides us with the unambiguous concept
that we have called biologic synergism. Parallelism is
also unambiguously defined, and the interpretation of
additive interaction as the amount by which biologic
synergism exceeds parallelism confers a special status
on additive interaction, setting it apart from the other
members of the infinite family of definitions of statistical synergism.
BIOLOGIC MODELS FOR DISEASE CAUSATION
A number of biologic models for disease causation
have been advanced in the literature. Two of them, the
multistage model and the no-hit model or immunity
model, are linked to multiplicative interaction, while
the single-hit or vulnerability model has been linked
(falsely, as we shall explain) to additive interaction.
Siemiatycki and Thomas (12) and Thompson (11)
reviewed these models and emphasized, with many
examples, what might be called the "fallacy of the
inverse implication." For example, when combined
with an independence property, the multistage model
implies no multiplicative interaction. However, we
cannot conclude from the absence of multiplicative
interaction that the multistage model and the independence property are true, tempting as it is to do draw
these conclusions; there are other biologic models
which also imply no multiplicative interaction, and
any one of them, or none of them, could be true.
The usual formulations of these biologic models are
probabilistic: the individual makes stochastic transitions from stage to stage in the case of the multistage
model, receives random "health hits" in the no-hit
model, and receives random "disease hits" in the
single-hit model. Here we provide formulations in a
deterministic framework which accords with the
sufficient-cause paradigm, and then we examine the
models in the light of the six-class structure.
666
Darroch
The vulnerability model
The single-hit model was named by Walter and
Holford (13) and has been proposed, in one form or
another, by many authors, including Plackett and
Hewlett (14), Finney (15), Valeriote and Lin (16),
Rothman (17), Hogan et al. (18), Miettinen (2), and
Weinberg (19). Thompson (11) provided a recent description of the model. Darroch and Borkent (8) formulated it deterministically as the "vulnerability model," as follows. An individual gets D if and only if he
or she possesses at least one type of vulnerability to D.
There are three kinds of vulnerability. One manifests
itself when A is present and one when B is present, and
the other is a blanket vulnerability which manifests
itself universally. An individual having no vulnerabilities is in the "none" class; someone having only the
first type of vulnerability is in the A-only class; someone having only the first two types of vulnerability is
in the parallel class; and someone having the third type
of vulnerability is in the "all" class. The vulnerability
model implies that the synergistic class is empty and
that the additive interaction therefore equals — |parallelism| and is less than or equal to zero. It follows that,
when the additive interaction is positive, the vulnerability model cannot hold.
We have interpreted statistical synergism in accordance with Last's first definition (1) as a positive
difference between the combined effects of A and B
and the sum of their solitary effects, and statistical
antagonism as a negative difference. In the literature
on the vulnerability model, alias the single-hit model,
there is an alternative concept of statistical synergism
and antagonism as positive and negative dependence
in the distribution of the A and B vulnerabilities.
Moreover, there is a perception in this literature that
this dependence can be measured by the additive interaction. It is apparent from the previous paragraph
that this perception must be incorrect, and we need to
examine the false argument which led to it. What is
true is that positive or negative statistical dependence
in the distribution of the A and B vulnerabilities in that
part of the population not having the blanket vulnerability is equivalent to the positiveness or negativeness
of the quantity
- R(AB)]\_l - R] It is the next step in the argument that is the false one.
It reasons that, if the four risks are small, which they
invariably are, then their products can be neglected
and the positiveness or negativeness of the above
quantity is the same as the negativeness or positiveness (note the sign reversal, which is sometimes
missed) of the additive interaction
R(AB) - R(A) - R(B) + R.
While this reasoning would be valid in general, it is
not so under model conditions which demand that the
additive interaction is negative.
The immunity model
The "no-hit model," proposed by Walter and
Holford (13), supposes that individuals who become
diseased are those who receive no random hit, while
individuals who receive at least one hit remain
healthy. Darroch and Borkent (8) reformulated it as
the "immunity model." It is the mirror image of the
vulnerability model. It is assumed that an individual
does not get D if and only if he or she possesses at
least one type of immunity to D, there being three
kinds of immunity. One manifests itself when A is
absent and one when B is absent, and the other is a
blanket immunity which manifests itself universally.
(Thompson (11) describes this model in terms of two
preventive factors whose presence corresponds to the
absence of our causal factors A and B.) Thus, an
individual with no immunity is in the "all" class, a
person with just the first type of immunity is in the
A-only class, one with just the first two types of
immunity is in the synergistic class, and one with the
third type of immunity is in the "none" class. The
immunity model implies that there is no parallelism
and therefore that the additive interaction is positive.
A negative additive interaction immediately invalidates the immunity model.
The immunity model is linked to multiplicative interaction in the following way. There is positive or
negative multiplicative interaction depending on
whether the distribution of the A and B immunities is
positively or negatively dependent in that part of the
population comprising the individuals who do not
have the blanket immunity and who therefore do get D
as they are exposed to both A and B.
The multistage model
The multistage model of carcinogenesis was proposed by Armitage and Doll (20) and developed by
Moolgavkar (21), among others. In the two-stage version, it is assumed that D is the second stage of a
disease which is only reached via a first stage D', and
that the factors A and B each influence only one stage.
The model can be given a deterministic formulation
appropriate to the sufficient cause paradigm as follows. Let D" denote that property of the biology of an
individual which ensures that, if the individual gets D',
he or she will also get D. Factors A and B are assumed
to be causal with respect to both D' and D". IndividAm J Epidemiol
Vol. 145, No. 7, 1997
Biologic Synergism and Parallelism
uals without D" do not get D, but they may get D'\ and
individuals who do not get D' may nevertheless have
the property D". The model is completed with the
assumption that D' is insensitive to B and D" is insensitive to A. (Alternatively, it could be assumed that D'
is insensitive to A and D" is insensitive to B.) Saying
that D' is "insensitive" to B means that the question of
whether an individual gets D' is independent of the
level (absence or presence) of B and depends only on
the level of A. It thus means that, as far as the getting
of D' is concerned, there are only three classes of
individual: "none," A-only, and "all." Likewise, as far
as the possession of the property D" is concerned,
there are only three classes: "none," fl-only, and "all."
In the complete two-stage model for the occurrence
of D=(D', IT), there are just five classes of individuals. The parallel class is empty, from which it follows, as for the immunity model, that the additive
interaction must be non-negative. If the additive interaction is negative, we know that the two-stage model
cannot hold.
The statistical interest shown in the literature towards the two-stage model is centered on a link between it and multiplicative interaction. Let a prime
denote the property of getting D' when A is absent and
a double prime the property of having D" when B is
absent. Then the multiplicative interaction is positive
or negative depending on whether the distribution of
prime and double prime is positively dependent or
negatively dependent in that part of the population
which does get D=(D', IT). Thus, within the framework of the two-stage model, statistical synergism, as
measured by multiplicative interaction, can be interpreted as meaning that prime and double prime occur
together in the same individual more often than independently, and statistical antagonism as meaning that
they occur less often. If prime and double prime are
distributed independently, then there is no multiplicative interaction and A and B may be said to "act
multiplicatively." It is important to note that this multiplicative action is a population property and says
nothing about any single individual; with regard to
prime and double prime, an individual simply has
none, one, or both.
These results for a two-stage disease immediately
extend to cover a multistage disease, some stages of
which are insensitive to A while the others are insensitive to B.
DISCUSSION
Much attention been directed in the literature
towards determining whether a no-interaction constraint can be placed on the four risks R, R(A), R(,B),
Am J Epidemiol
Vol. 145, No. 7, 1997
667
and R(AB), thus effectively reducing the four parameters to three. The emphasis in this paper has been in
the opposite direction of not seeking to constrain the
four parameters but, instead, saying something about
the five underlying parameters which determine the
proportional sizes of the six classes into which a
population exposed to two causal factors can be partitioned. The sizes of the "none" and "all" classes are
fully determined by the four risks, but the sizes of the
other four classes—the synergistic, parallel, A-only,
and S-only classes—are determined only to a partial
degree, which varies from context to context. The
incompleteness of our knowledge of the four sizes
does not imply any shortcoming in the theory, but
exists simply because Nature only partially reveals the
sizes to us. In addition to this fundamental uncertainty
about |synergism|, |parallelism|, |A-only|, and |B-only{,
there is the usual inferential uncertainty which arises
from estimating the population risks R(AB), R(A),
R(B), and R from sample information. We have, for
simplicity, ignored inferential uncertainty in this paper
and spoken of the four risks as if they were known,
unlike Walker (22), who did provide some formulae
for estimation variances.
The notions of "independent action" and "interdependent action" are featured in the literature, but have
not been required in this paper. It would in fact be
difficult to know which adjective, independent or interdependent, to apply to some of the six types of
action represented by the six classes.
Implicit to this paper is the belief that epidemiologists would like to know the sizes of the six classes. If
the synergistic class were found to be dominant, the
public health concern would be a reduction of exposure to either A or B. This raises the issue of which
reduction should be given priority, as noted recently
by Saracci and Boffetta (23); information on the sizes
of the A-only and fi-only classes, particularly the fact,
not mentioned previously, that
|A-only| - |fl-only| = R(A) -
R(B),
should help to resolve this issue. If it happened that
parallelism were dominant, the public health concern
would be to reduce exposure to both A and B.
In seeking to understand why there is biologic synergism, the epidemiologist postulates pathogenic pathways to D which require exposure to both A and B. In
the example from Hertz-Picciotto et al. (9), we found
that 45-65 percent of the cases exhibited biologic
synergism. These authors suggested several possible
pathways to lung cancer which require exposure to
both smoking and arsenic. One of these suggestions
was that "cigarette smoking impedes tracheobronchial
clearance, which could prolong the exposure of bron-
668
Darroch
chial epithelium to arsenic-containing participates and
magnify the dose to the target tissue"; another that
"arsenic deposited in the lung may enhance proliferation of cells already subjected to the initiating and
promoting activities of other carcinogens in mainstream tobacco smoke" (9, p. 29).
The single-hit, no-hit, and multistage models were
designed to assist the study of how two exposure
factors affect the incidence of a disease. Their status in
this role has been reduced by the increased awareness
that, while it is true that these biologic models imply
certain interaction properties, it is false to inversely
conclude that the biologic model is valid when the
interaction property holds. The status of these models
is reduced further by the facts reported here—namely,
that the first model incorporates the restriction that
there is no synergism and the second and third models
the restriction that there is no parallelism. While it is
impossible to infer the truth, in a given study, of any
one of these models, with or without an assumption of
statistical independence, falsity can certainly be inferred if the additive interaction has the wrong sign.
The single-hit model must be false if the additive
interaction is positive, and the other two must be false
if the additive interaction is negative. Finally, an investigator who is bold enough to assume a priori the
validity of any of these models knows that either
|synergism| or |parallelism| is zero, and hence knows
the sizes of the other five classes.
REFERENCES
1. Last JM.ed. A dictionary of epidemiology. 3rd ed New York,
NY: Oxford University Press, 1995:164-5.
2 Miettinen OS. Causal and preventive interdependence: elementary principles. Scand J Work Environ Health 1982,8:
159-68.
3 Rothman KJ. Causes. Am J Epidemiol 1976;104587-92
4. Rothman KJ. Modern epidemiology. Boston, MA. Little,
Brown and Company, 1986.
5. Koopman JS. Interaction between discrete causes. Am J Epidemiol 1981;113:716-24.
6. Hamilton MA. Choosing the parameter for a 2 X 2 table or a
2 X 2 X 2 table analysis. Am J Epidemiol 1979;109:362-75.
7. Greenland S, Poole C. Invariants and norunvariants in the
concept of interdependent effects. Scand J Work Environ
Health 1988;14:125-9.
8. Darroch JN, Borkent M. Synergism, attributable risk and
interaction for two binary exposure factors. Biometnka 1994;
81:259-70.
9. Hertz-Picciotto L Smith AH, Holtzman D, et al Synergism
between occupational arsenic exposure and smoking in the
induction of lung cancer Epidemiology 1992,3.23-31.
10. Breslow NE, Storer BE General relative risk functions for
case-control studies. Am J Epidemiol 1985;122149-61.
11. Thompson WD. Effect modification and the limits of biological inference from epidemiologic data. J Clin Epidemiol
1991;44.221-32.
12. Siemiatycki J, Thomas DC. Biological models and statistical
interactions' an example from multistage carcinogenesis. Int J
Epidemiol 1981;10:383-7.
13. Walter SD, Holford TR. Additive, multiplicative, and other
models for disease risks. Am J Epidemiol 1978;108:341-6.
14. Plackett RL, Hewlett PS. A comparison of two approaches to
the construction of models for quanta! responses to mixtures
of drugs. Biometrics 1967;23:27-44.
15. Finney DJ. Probit analysis. Cambridge, England. Cambridge
University Press, 1971.
16. Valeriote F, Lin H. Synergistic interaction of anticancer
agents: a cellular perspective. Cancer Chemother Rep 1975;
59:895-900.
17. Rothman KJ. The estimation of synergy or antagonism. Am J
Epidemiol 1976;103:506-ll.
18. Hogan MD, Kupper LL, Most BM, et al. Alternatives to
Rothman's approach for assessing synergism (or antagonism)
in cohort studies. Am J Epidemiol 1978,108:60-7.
19 Weinberg CR. Applicability of the simple independent action
model to epidemiologic studies involving two factors and a
dichotomous outcome. Am J Epidemiol 1986,123:162-73.
20. Armitage P, Doll R. The age distribution of cancer and a
multistage theory of carcinogenesis. Br J Cancer 1954,8.1-12.
21. Moolgavkar SH. The multistage theory of carcinogenesis and
the age distribution of cancer in man. J Natl Cancer Inst
1978;61 49-52.
22. Walker AM The proportion of disease attributable to the
combined effect of two factors. Int J Epidemiol 1981,10:81-5.
23. Saracci R, Boffetta P. Interactions of tobacco smoking with
other causes of lung cancer In: Samet JM, ed. Epidemiology
of lung cancer New York, NY- Marcel Dekker, Inc, 1994:
465-93.
Am J Epidemiol
Vol. 145, No. 7, 1997