Attributable Fractions for Sufficient Cause Interactions

Abstract

A number of results concerning attributable fractions for sufficient cause interactions are given. Results are given both for etiologic fractions (i.e. the proportion of the disease due to a particular sufficient cause) and for excess fractions (i.e. the proportion of disease that could be eliminated by removing a particular sufficient cause). Results are given both with and without assumptions of monotonicity. Under monotonicity assumptions, exact formulas can be given for the excess fraction. When etiologic fractions are of interest or when monotonicity assumptions do not hold for excess fractions then only lower bounds can be given. The interpretation of the results in this paper and in a proposal by Hoffmann et al. (2006) are discussed and compared. A method is described to estimate the lower bounds on attributable fractions using marginal structural models. Identification is discussed in settings in which time-dependent confounding may be present.

Introduction

A number of recent papers consider tests for sufficient cause interactions (VanderWeele and Robins, 2007a, 2007b, 2008, 2009; Vansteelandt et al., 2008; VanderWeele, 2009; VanderWeele et al., 2010a, 2010b; VanderWeele and Richardson, 2010). In the case of two binary exposures, these sufficient cause interactions indicate the presence of individuals for whom the outcome would occur if both of two exposures were present but for whom the outcome would not occur if only one of the two exposures were present. Within the context of the sufficient cause framework, such individuals signal the presence of synergism between two exposures as conceived by Rothman (1976). In considering the public health impact of exposures, it is often of interest to know what proportion of the disease or outcome is in some sense due to the exposures under study or what proportion could be eliminated if the exposure under study were removed (Miettinen, 1974; Greenland and Robins, 1988). These questions also arise in a natural way within the sufficient cause framework. Rothman conceived of causation as a series of distinct mechanisms or “sufficient causes” each of which would be sufficient to bring about the outcome. We might then be interested in the proportion of disease that is due to, or could be eliminated by removing, a particular mechanism or sufficient cause (Hoffmann et al., 2006).

In this paper we consider several issues concerning inference for attributable fractions for sufficient cause interactions. We discuss a recent proposal by Hoffmann et al. (2006) to calculate the attributable fraction for a particular sufficient cause. We argue that the approach proposed by Hoffmann et al. implicitly requires monotonicity assumptions that are not acknowledged in the paper. We furthermore show that even within the context of monotonicity, the approach described by Hoffmann et al. gives excess attributable fractions rather than etiologic attributable fractions (Greenland and Robins, 1988; Robins and Greenland, 1989). We give a number of results that extend the approach of Hoffmann et al. First, we discuss how one can obtain lower bounds for the etiologic fraction under the assumption of monotonicity. Second, we consider inference for the excess and etiologic fraction when the assumption of monotonicity cannot be made; we also consider the invariance of these results to the sufficient cause representation. Third, we discuss how a marginal structural model approach to inference for sufficient cause interactions (VanderWeele et al., 2010a) can also be used to draw inferences concerning attributable fractions for sufficient cause interactions. Finally, we consider issues concerning time-dependent confounding in the context of inference for attributable fractions. We draw on recent identification results concerning the effect of treatment on the treated (Shpitser and Pearl, 2009) to show that in general the expressions for the lower bounds for etiologic fractions will not be identified when time-dependent confounding is present. However, in the case when the exposures of interest are binary we give an identification result which shows that the expressions for these lower bounds are identified in some settings.

Sufficient Cause Framework

We will let D denote a binary outcome of interest and let X₁, X₂, . . . , X_k denote binary causes of interest. For some cause X_i we will let X̅_i denote the complement of X_i i.e. that the cause X_i is absent. When a particular outcome is in view, Rothman (1976) conceived of the relationship between cause and effect as a collection of causal mechanisms for D. Each causal mechanism would itself be sufficient for the outcome and thus Rothman referred to these causal mechanisms as “sufficient causes.” Each mechanism might require some combination of the causes of interest, X₁, X₂, . . . , X_k, either their presence or absence to operate. Each mechanism might also require some additional background factors, other than X₁, X₂, . . ., X_k, in order to operate. For the ith mechanism we will denote the presence of these additional background factors as A_i = 1 with A_i = 0 otherwise. Within Rothman’s description, each mechanism or “sufficient cause” would consists of a minimal set of conditions or “component causes” such that when all the component causes for a particular mechanism were present the mechanism would operate and the outcome would inevitably occur. Within every sufficient cause, every component of that sufficient cause would be necessary for the corresponding mechanism to operate. Synergism would be said to be present between X_i and X_j if there were a sufficient cause which required both X_i and X_j to operate. Further introductory discussion of the sufficient cause model with only two exposures can be found elsewhere (Greenland and Brumback, 2002; VanderWeele and Robins, 2007).

More formally, consider the case of two causes of interest, X₁ and X₂. We will let D_x1x2 (ω) denote the counterfactual value of D for individual ω if, possibly contrary to fact, we had set X₁ = x₁ and X₂ = x₂. Let ∨ denote the disjunctive “or” operator defined by X ∨ Y = X + Y – XY so that X ∨ Y = 1 if X = 1 or Y = 1 or both but X ∨ Y = 0 if X = Y = 0. VanderWeele and Robins (2008) defined a sufficient cause representation to be any set of variables {A_i(ω)}_{i=0, . . ., 8} that are functions of {D_x1x2 (ω)}_{x1,x2∈{0,1}} such that

$$\begin{array}{ll}{D}_{x_1{x}_2}=& A_0\vee A_1{x}_1\vee A_2(1-x_1)\vee A_3{x}_2\vee A_4(1-x_2)\vee A_5{x}_1{x}_2\\ & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee A_6(1-x_1)x_2\vee A_7{x}_1(1-x_2)\vee A_8(1-x_1)(1-x_2)\end{array}$$ (1)

A sufficient cause interaction was said to be present between X₁ and X₂ if for every sufficient cause representation there exists an ω such that A₅(ω) ≠ 0. A sufficient cause interaction implies synergism in the sense of Rothman (1976). Sufficient cause interactions and synergism between X₁ and X̅₂, or between X̅₁ and X₂, or X̅₁ and X̅₂ can be defined similarly. The results below are stated in terms of sufficient cause interactions between X₁ and X₂ as the other cases are easily covered by recoding the exposures.

VanderWeele and Robins (2007a, 2008) derived an empirical condition for testing for synergism. Testing for synergism requires that control be made for confounding. Let $A\coprod B|C$ denote that A is conditionally independent of B given C. We will say that the effects of X₁ and X₂ on D are unconfounded given C if $D_{x_1{x}_2}\coprod \{X_1,\hspace{0.17em}{X}_2\}|C$. VanderWeele and Robins (2007a, 2008) showed that if the effects of X₁ and X₂ on D are unconfounded given a set of variables C and if we let p_x1x2c = P(D = 1|x₁, x₂, c) then if for some c,

$$p_{11c}-p_{10c}-p_{01c}>0$$ (2)

then a sufficient cause interaction between X₁ and X₂ must be present. VanderWeele and Robins (2008) furthermore showed that a sufficient cause interaction was equivalent to an individual for whom

$$D_{11}(\omega )-D_{10}(\omega )-D_{01}(\omega )>0$$ (3)

which itself is equivalent to an individual for whom D₁₁(ω) = 1 and D₁₀(ω) = D₀₁(ω) = 0. Condition (3) is more general than condition (2) in that if (2) is satisfied then (3) must be satisfied for some individual ω but (3) may be satisfied for some individual ω without (2) being satisfied. Essentially condition (2) implies condition (3) and condition (3) implies a sufficient cause interaction and thus the presence of synergism between X₁ and X₂. However, condition (2), unlike condition (3), can be tested using data.

In some cases the effects of the cause X_i may be in the same direction for all individuals. We will say that X₁ and X₂ have positive monotonic effects on D if D_x1x2 (ω) is non-decreasing in x₁ and x₂ for all individuals ω. When X₁ and X₂ have positive monotonic effects on D then a condition weaker than (2) can be used to test for sufficient cause interactions. In particular, if the effects of X₁ and X₂ on D are unconfounded given C then a sufficient cause interaction must be present if the following condition holds (VanderWeele and Robins, 2007a, 2008; Rothman et al., 2008):

$$p_{11c}-p_{10c}-p_{01c}+p_{00c}>0.$$

It can furthermore be shown that, under the monotonicity assumption, a sufficient cause interaction is present if there is an individual ω for whom

$$D_{11}(\omega )-D_{10}(\omega )-D_{01}(\omega )+D_{00}(\omega )>0$$

(VanderWeele and Robins, 2008). Empirical and counterfactual conditions are also available for testing for 3-way or k-way interactions between binary exposures (VanderWeele and Robins, 2008; VanderWeele and Richardson, 2010).

VanderWeele et al. (2010a) also noted that the expressions $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁] and $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁+D₀₀] constitute lower bounds on the prevalence of sufficient cause interactions (i.e. the proportion of individuals for whom D₁₁(ω) = 1 and D₁₀(ω) = D₀₁(ω) = 0) without and with the monotonicity assumption respectively. More generally, for some set of variables Q, $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁|Q = q] and $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ + D₀₀|Q = q] constitute lower bounds on the prevalence of sufficient cause interactions within strata Q = q and thus

$${\sum }_q\text{max}(0,\hspace{0.17em}\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|q])P(q)$$

and

$${\sum }_q\text{max}(0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|q])P(q)$$

will constitute lower bounds on the population prevalence of sufficient cause interactions without and with the monotonicity assumption respectively. In general these lower bounds will be larger (i.e. further from 0) than the crude bounds given by $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁] or $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ + D₀₀].

Attributable Fractions for Sufficient Cause Interactions

In many settings it is of interest to consider the proportion of the disease or outcome that is in some sense due to the exposure under study or could be eliminated if the exposure under study were removed (Miettinen, 1974; Robins and Greenland, 1988). For a single binary exposure the “attributable fraction” or “population attributable fraction” is sometimes defined as

$$PAF(X)=\frac{P(D)-P(D_{x=0})}{P(D)}.$$ (4)

Greenland and Robins (1988) note that there is ambiguity in the expression “attributable fraction” and draw a distinction between an excess attributable fraction and an etiologic attributable fraction. Excess fractions are the proportion of the disease that could be eliminated by eliminating the exposure; etiologic fractions are the proportion of the disease due to the exposure. Because of the possibility of competing risks, the two quantities (the excess fraction and etiologic fraction) need not coincide. For example, it may be the case that for some individual the occurrence of the outcome is in fact due to the presence of the exposure under study but, for that individual, if the exposure were eliminated the outcome would still occur through some other mechanism. For such an individual the occurrence of the outcome is in fact due to the exposure but the outcome would not be eliminated by eliminating the exposure. Such an individual would be included in the numerator of the etiologic fraction but not for the excess fraction. The quantity given in (4) is the excess fraction. In the absence of further biological knowledge, the etiologic fraction is not identified (Greenland and Robins, 1988). Miettinen (1974) noted that if the effect of X on D was unconfounded given C then PAF(X) =

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}P(X=1|D)\frac{{\sum }_c\{P(D|X=1,c)-P(D|X=0,c)\}P(c|X=1)}{{\sum }_{c}P(D|X=1,c)P(c|X=1)}\\ =P(X=1|D)\frac{SMR-1}{SMR}\text{where}\hspace{0.17em}SMR=\frac{{\sum }_{c}P(D|X=1,c)P(c|X=1)}{{\sum }_{c}P(D|X=0,c)P(c|X=1)}.\end{array}$$ (5)

In a recent paper, Hoffmann et al. (2006) proposed a method to estimate a quantity they define as the PDC which they conceived of as “the proportion of disease due to a class of sufficient causes”; however, as pointed out below, the method they describe in fact gives the excess fraction, not the etiologic fraction (i.e. the fraction that could be eliminated not the fraction due to the exposure). Hoffmann et al. consider exposures X₁, . . . , X_k and implicitly assume that none of the sufficient causes involve any of the complements of X₁, . . . , X_k so that X₁, . . . , X_k have positive monotonic effects on D. For example, for two exposures X₁ and X₂ they assume that of the sufficient causes A₀, A₁X₁, A₂X̅₁, A₃X₂, A₄X̅₂, A₅X₁X₂, A₆X̅₁X₂, A₇X₁X̅₂, A₈X̄₁X̄₂, only A₀, A₁X₁, A₃X₂, A₅X₁X₂ are present (i.e. A₂ = A₄ = A₆ = A₇ = A₈ = 0). This assumption of monotonicity is not explicitly stated by Hoffmann et al. but is required in their derivations.

Hoffmann et al. then propose using the formula in (5) from Miettinen (1974) to estimate population attributable fraction (PAF) for a specific exposure or a group of exposures. In particular let X₍₁₎, . . . , X_(m) be some subset of X₁, . . . , X_k and define PAF(X₍₁₎, . . . , X_(m)) as the proportion of diseased subjects who would not develop the disease if the exposures X₍₁₎, . . . , X_(m) were eliminated. For I ⊆ {1, . . . , k} let S_I denote the sufficient cause which requires X_i for all i ∈ I; for example, with two exposures, S₀ would be the sufficient cause A₀, S₁ would be the sufficient cause A₁X₁, S₂ would be the sufficient cause A₃X₂, and S₁₂ would be the sufficient cause A₅X₁X₂. Let PDC(S_I) denote the proportion of the diseased subjects who would not develop the disease if the effects of the sufficient cause S_I could be eliminated. Hoffmann et al. argue in their first Appendix that:

$$\begin{array}{l}PDC(S_0)=1-PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)\hfill \\ PDC(S_i)=PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)-PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_{i-1},X_{i+1},\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)\hfill \end{array}$$

and PDC(S_{(1), . . . , (m)}) is given recursively by

$$\begin{array}{ll}PDC(S_{(1),\hspace{0.17em}\dots ,\hspace{0.17em}(m)})\hspace{0.17em}=& PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)-PAF(X_{(m+1)},\hspace{0.17em}\dots ,\hspace{0.17em}{X}_{(k)})\\ & -{\sum }_{I\subset \{(1),\hspace{0.17em}\dots ,\hspace{0.17em}(m)\}}PDC(S_I).\end{array}$$

Thus for two exposures, X₁ and X₂, one would obtain PDC(S₁₂)

$$\begin{array}{l}=PAF(X_1,X_2)-0-{\sum }_{I\subset \{1,2\}}PDC(S_I)\hfill \\ =PAF(X_1,X_2)-0-\{PDC(S_1)+PDC(S_2)\hfill \\ =PAF(X_1,X_2)-\{PAF(X_1,X_2)-PAF(X_1)\}-\{PAF(X_1,X_2)-PAF(X_2)\}\hfill \\ =PAF(X_1)+PAF(X_2)-PAF(X_1,X_2)\hfill \end{array}$$

This final expression could be estimated using (5). By definition of PAF, this final quantity is also equal to

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{P(D)-P(D_{x_1=0})}{P(D)}+\frac{P(D)-P(D_{x_2=0})}{P(D)}-\frac{P(D)-P(D_{00})}{P(D)}\\ =\frac{P(D)-P(D_{x_1=0})-P(D_{x_2=0})+P(D_{00})}{P(D)}.\end{array}$$

Hoffmann et al. propose that the quantity PDC(S_I) be interpreted as “the proportion of disease due to sufficient cause S_I.” From the discussion above, it can be seen that the quantity should be interpreted as “the proportion of disease that would be eliminated by preventing the sufficient cause S_I from operating” i.e. as an excess fraction not as an etiologic fraction. Essentially because the formulas they use for the PAF(X₍₁₎, . . . , X_(m)) correspond to the proportion of diseased subjects who would not develop the disease if the exposures X₍₁₎, . . . , X_(m) were eliminated, the quantity PDC(S_I), calculated by using PAF (X₍₁₎, . . . , X_(m)), corresponds to the proportion of diseased subjects who would not develop the disease if the sufficient cause S_I were eliminated. Note that Hoffmann and Flanders (2006) make a somewhat different clarification concerning the method described by Hoffmann et al. (2006) in that in the application of Hoffmann et al., it was not PDC(S_{(1), . . . , (m)}) that was estimated but rather a different quantity PDC(S_E), where E = (X₁ = 1, . . . , X_m = 1, X_m+1 = 0, . . . , X_k = 0), for which PDC(S_E) is interpreted as the proportion of disease that would be eliminated if all individuals with E had their all exposures X_i set to 0. At the end of their paper Hoffmann et al. note the distinction between excess fraction and etiologic fractions but continue to refer to the quantity PDC as the “the proportion of disease due to a particular sufficient cause [or class of sufficient causes].” The language and interpretation should be modified. The quantities discussed by Hoffmann et al. (2006) correspond to what Greenland and Robins define as the “excess fraction.”

One further point concerning attributable fractions for sufficient causes, not considered by Hoffmann et al. (2006), merits attention. It has been noted (Greenland and Brumback, 2002; VanderWeele and Robins, 2007a) that there will in general be multiple ways to represent the outcome in terms of sufficient causes. For example, as noted above, any set of variables {A_i(ω)}_{i=0, . . . ,8} that are functions of {D_x1x2 (ω)}_{x1, x2∈{0,1}} that satisfy (1) constitutes a sufficient cause representation of the potential outcomes, and more than one representation is in general possible. It is important to know then whether the approach of Hoffmann et al. described above is invariant to the sufficient cause representation. Hoffmann et al. (2006) obtain the formula for PDC(S_{(1), . . . , (m)}) by arguing that the difference between PAF(X₁, . . . , X_k) and PAF (X_(m+1), . . . , X_(k)) must constitute occurrences of the outcome that arise from not having eliminated sufficient causes containing one or more of X₍₁₎, . . . , X_(m) but none of X_(m+1), . . . , X_(k) and from this it follows that PAF(X₁, . . . , X_k) – PAF(X_(m+1), . . . , X_(k)) = ∑_{I⊆{(1), . . . , (m)}} PDC(S_I). Removing PDC(S_{(1), . . . , (m)}) from the sum ∑_{I⊆{(1), . . . , (m)}} PDC(S_I) and rearranging gives the formula for PDC(S_{(1), . . . , (m)}). The logic applies irrespective of the sufficient cause representation and thus the formula given by Hoffmann et al. (2006) is invariant to the sufficient cause representation. The results of Hoffmann et al. (2006) along with the discussion above establishes the following Theorem.

Theorem 1. If the effects of X₁, . . . , X_k on D are monotonic then the formulas for the attributable fractions for sufficient causes given by Hoffmann et al. (2006), namely,

$$\begin{array}{l}PDC(S_0)=1-PAF(X_1,\dots X_k)\hfill \\ PDC(S_i)=PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)-PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_{i-1},X_{i+1},\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)\hfill \end{array}$$

and PDC(S_{(1), . . . , (m)}) given recursively by

$$\begin{array}{ll}PDC(S_{(1),\hspace{0.17em}\dots ,\hspace{0.17em}(m)})\hspace{0.17em}=& PAF(X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k)-PAF(X_{(m+1),\hspace{0.17em}\dots ,\hspace{0.17em}}{X}_{(k)})\\ & -{\sum }_{I\subset \{(1),\hspace{0.17em}\dots ,\hspace{0.17em}(m)\}}PDC(S_I)\end{array}$$

give the excess fraction for a sufficient cause S_{(1), . . . , (m)} irrespective of the sufficient cause representation.

As noted above, the quantity PDC(S₁₂) given by Hoffmann et al. for the sufficient cause, A₅X₁X₂, for example, corresponds to the “excess fraction” for the A₅X₁X₂ sufficient cause. Under the monotonicity assumption, the formula for the excess fraction gives a lower bound on the etiologic fraction (Greenland and Robins, 1988). This is because “the proportion of disease that would be eliminated by blocking some sufficient cause S from operating” provides a lower bound on the “the proportion of disease due to the sufficient cause S.” Using theory for sufficient cause interactions we can draw further inferences concerning the proportion of disease that are in fact “due to” particular sufficient causes such as A₅X₁X₂ i.e. concerning etiologic fractions. As is the case with Hoffmann et al., we will assume, at least initially, that the effects of X₁ and X₂ on D are monotonic. The presence of a sufficient cause interaction indicates, in the terminology of Greenland and Robins, that there is a non-zero “etiologic fraction.” Clearly the outcome will be due to the sufficient cause A₅X₁X₂ only if X₁ = 1, X₂ = 1. Let Q be some subset of C. From the above discussion a lower bound on the prevalence of sufficient cause interactions amongst the group with X₁ = 1, X₂ = 1 with Q = q is given by

$$\text{max}(0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|X_1=1,X_2=1,q]).$$ (6)

For the group with X₁ = 1, X₂ = 1 and Q = q, the quantity in (6) will be a lower bound on the prevalence of sufficient cause interactions; for individuals ω with X₁(ω) = 1, X₂(ω) = 1, Q = q and with a sufficient cause interaction present, we will have that D₁₁(ω) = 1, D₁₀(ω) = D₀₁(ω) = 0 and D = 1. Thus for these individuals the outcome must be due to the sufficient cause A₅X₁X₂ since neither of the sufficient causes A₀ or A₁X₁ can be the cause of the outcome since D₁₀(ω) = 0, the sufficient cause A₃X₂ cannot be the cause of the outcome because D₀₁(ω) = 0, and none of the sufficient causes A₂X̅₁, A₄X̅₂, A₆X̅₁X₂, A₇X₁X̅₂, A₈X̅₁X̅₂ can be the cause of the outcome because X₁ = 1, X₂ = 1. Note that under an assumption of “sufficient cause monotonicity” the sufficient causes cannot be the cause of the outcome because A₂X̅₁, A₄X̅₂, A₆X̅₁X₂, A₇X₁X̅₂, A₈X̅₁X̅₂ are eliminated by definition but even under the weaker counterfactual monotonicity that D_x1x2 (ω) is non-decreasing in x₁ and x₂ for all ω, clearly none of A₂X̅₁, A₄X̅₂, A₆X̅₁X₂, A₇X₁X̅₂, A₈X̅₁X̅₂ can be the cause of D = 1 when X₁(ω) = X₂(ω) = 1. The above argument holds irrespective of the sufficient cause representation. Now, for stratum, Q = q, the set of individuals for whom X₁ = 1, X₂ = 1 and for whom the outcome is due to sufficient cause A₅X₁X₂ must be a subset of the set of individuals for whom the outcome is due to sufficient cause A₅X₁X₂. Thus a lower bound for the proportion of individuals in stratum Q = q for whom the outcome is due to sufficient cause A₅X₁X₂ is given by:

$$\begin{array}{c}\text{max}(0,\mathbb{\text{E}}[1(D_{11}-D_{10}-D_{01}+D_{00}>1\hspace{0.17em}\text{and}\hspace{0.17em}{X}_1=X_2=1)|q])\\ \ge \hspace{0.17em}\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|X_1=X_2=1,Q=q]P(X_1=X_2=1|q)\}.\end{array}$$

From this it follows that a lower bound on the proportion of disease due to sufficient cause A₅X₁X₂ in the population is given by

$$\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|X_1=1,X_2=1,q])P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}.$$

If the effects of X₁ and X₂ on D are unconfounded given C then the quantity $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ + D₀₀|X₁ = 1, X₂ = 1, q] can be estimated by

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sum }_c\{\mathbb{\text{E}}[D|X_1=1,X_2=1,c]-\mathbb{\text{E}}[D|X_1=1,X_2=0,c]\hfill \\ -\mathbb{\text{E}}[D|X_1=0,X_2=1,c]+\mathbb{\text{E}}[D|X_1=0,X_2=0,c]\}P(c|X_1=1,X_2=1,q).\hfill \end{array}$$

Because sufficient cause interactions concern statements about all possible sufficient cause representations, the argument above holds irrespective of the sufficient cause representation. We have thus established the following theorem.

Theorem 2. If the effects of X₁ and X₂ on D are monotonic then a lower bound on the etiologic fraction for the sufficient cause A₅X₁X₂ is given by

$$\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|X_1=1,X_2=1,q])P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}$$ (7)

irrespective of the sufficient cause representation. If the effects of X₁ and X₂ on D are unconfounded given C then the quantity $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ + D₀₀|X₁ = 1, X₂ = 1, q] can be estimated by

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sum }_c\{\mathbb{\text{E}}[D|X_1=1,X_2=1,c]-\mathbb{\text{E}}[D|X_1=1,X_2=0,c]\hfill \\ -\mathbb{\text{E}}[D|X_1=0,X_2=1,c]+\mathbb{\text{E}}[D|X_1=0,X_2=0,c]\}P(c|X_1=1,X_2=1,q).\hfill \end{array}$$

Theorem 2 is stated for two binary exposures. However, as discussed in Appendix 1, the result generalizes to lower bounds for etiologic fraction for sufficient causes with k factors.

The results of Hoffmann et al. (2006) and the theorems given above required that the effects of all of X₁, . . . , X_k on D be monotonic. Using theory for sufficient cause interactions we can, however, derive lower bounds for the etiologic fraction without the monotonicity assumption. Without monotonicity, by arguments similar to those above,

$$\begin{array}{l}\text{max}(0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|X_1=1,X_2=1,q]).\\ \end{array}$$

will be a lower bound on the prevalence of sufficient cause interactions amongst the group with X₁ = 1, X₂ = 1 and Q = q. For individuals ω with X₁(ω) = 1, X₂(ω) = 1, with a sufficient cause interaction present, we will have that D₁₁(ω) = 1, D₁₀(ω) = D₀₁(ω) = 0 and D = 1. Thus for these individuals the outcome must be due to the sufficient cause A₅X₁X₂ since, irrespective of the sufficient cause representation, neither of the sufficient causes A₀ or A₁X₁ can be the cause of the outcome because D₁₀(ω) = 0, and the sufficient cause A₃X₂ cannot be the cause of the outcome because D₀₁(ω) = 0, and none of the sufficient causes A₂X̅₁, A₄X̅₂, A₆X̅₁X₂, A₇X₁X̅₂, A₈X̅₁X̅₂ can be the cause of the outcome because X₁ = 1, X₂ = 1. Thus,

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\text{max}(0,\mathbb{\text{E}}[1(D_{11}-D_{10}-D_{01}>1\hspace{0.17em}\text{and}\hspace{0.17em}{X}_1=X_2=1)|q])\hfill \\ \ge \hspace{0.17em}\hspace{0.17em}\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|X_1=X_2=1,q]P(X_1=X_2=1|q)\}.\hfill \end{array}$$

will be a lower bound on the proportion of individuals in stratum Q = q for whom the outcome is due to sufficient cause A₅X₁X₂ and with X₁ = 1, X₂ = 1 and thus also a lower bound on the proportion of individuals in stratum Q = q for whom the outcome is due to the sufficient cause A₅X₁X₂. From this it follows that a lower bound on the proportion of disease due to sufficient cause A₅X₁X₂ in the population is given by

$$\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|X_1=1,X_2=1,q]P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}.$$

This establishes the following result.

Theorem 3. Without the assumption of monotonicity, a lower bound on the etiologic fraction for the sufficient cause A₅X₁X₂ is given by

$$\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|X_1=1,X_2=1,q]P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}$$ (8)

irrespective of the sufficient cause representation. If the effects of X₁ and X₂ on D are unconfounded given C then the quantity $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁|X₁ = 1, X₂ = 1, q] can be estimated by

$$\begin{array}{l}{\sum }_c\{\mathbb{\text{E}}[D|X_1=1,X_2=1,c]-\mathbb{\text{E}}[D|X_1=1,X_2=0,c]\hfill \\ -\mathbb{\text{E}}[D|X_1=0,X_2=1,c]\}P(c|X_1=1,X_2=1,q).\hfill \end{array}$$

Theorem 3 is stated for two binary exposures but generalizes to lower bounds for etiologic fractions for sufficient causes with k factors. See Appendix 1 for further discussion.

We have seen above that Theorem 1 gives a method for calculating excess fractions for sufficient causes under the assumption of monotonicity. Theorem 2 gives a method for obtaining a lower bound on etiologic fraction for sufficient causes under the assumption of monotonicity. Theorem 3 gives a method for obtaining a lower bound on etiologic fraction for sufficient causes without the assumption of monotonicity. A question thus still remains about how to obtain a lower bound for excess fractions (rather than etiologic fractions) for sufficient causes when the monotonicity assumption does not hold. When monotonicity cannot be assumed, excess fractions for the A₅X₁X₂ sufficient cause are not identified, but a lower bound for the excess fraction can be derived. Theorem 4 give a lower bound for the excess fraction in cases in which neither X₁ nor X₂ can be assumed to have a monotonic effect on D and also in cases in which just one of X₁ or X₂ have a monotonic effect on D. The proof of Theorem 4 is given in Appendix 2.

Theorem 4. Without the assumption of monotonicity, a lower bound on the excess fraction for the sufficient cause A₅X₁X₂ is given by

$$\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}-D_{00}|X_1=1,X_2=1,q]P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}$$ (9)

irrespective of the sufficient cause representation. If one of X₁ or X₂ have a monotonic effect on D then a lower bound on the excess fraction for the sufficient cause A₅X₁X₂ is given by expression (8) above. If the effects of X₁ and X₂ on D are unconfounded given C then the quantity $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ – D₀₀|X₁ = 1, X₂ = 1, q] can be estimated by

$$\begin{array}{l}{\sum }_c\{\mathbb{\text{E}}[D|X_1=1,X_2=1,c]-\mathbb{\text{E}}[D|X_1=1,X_2=0,c]\hfill \\ -\mathbb{\text{E}}[D|X_1=0,X_2=1,c]-\mathbb{\text{E}}[D|X_1=0,X_2=0,c]\}P(c|X_1=1,X_2=1,q).\hfill \end{array}$$

The expression D₁₁ – D₁₀ – D₀₁ – D₀₀ is related to what might be referred a “singular” interaction, defined as the presence of an individual ω for whom D₁₁(ω) = 1 but D₁₀(ω) = D₀₁(ω) = D₀₀(ω) = 0 (VanderWeele and Richardson, 2010); in the context of two genetic factors such interactions are referred to as instances of “compositional epistasis” (Cordell, 2009; VanderWeele, 2010a, 2010b). If either X₁ or X₂ have a monotonic effect on D then a singular interaction and a sufficient cause interaction are equivalent. When neither X₁ nor X₂ have a monotonic effect on D, the condition for a singular interaction is stronger than that for a sufficient cause interaction.

Marginal Structural Models for Bounds on Attributable Fractions for Sufficient Cause Interactions

The expressions for attributable fractions given in Theorems 1–3 require making adjustment for a set of confounding factors C. When the set C consists of a small number of categorical variables the quantities in Theorems 1–3 could be estimated by stratifying on C. However when C contains many variables or some continuous variables such stratification will not in general be possible. Logistic regression could be employed to estimate quantities such as $\mathbb{\text{E}}$[D|X₁ = 1, X₂ = 1, c]. However, as discussed in VanderWeele et al. (2010a), within the sufficient cause framework, such an approach may be undesirable because regression models for $\mathbb{\text{E}}$[D|X₁ = 1, X₂ = 1, c] impose multiplicative relationships between the confounding variables C and the unknown background causes A₀, A₁, . . . , A₈. Imposing such restriction is in general undesirable because in most cases the background causes A₀, A₁, . . . , A₈ will be unknown and it will thus not be entirely clear what substantively is being assumed. To overcome this issue, VanderWeele et al. (2010a) proposed the use of marginal structural models (Robins, 1999; Robins et al., 2000) to draw inferences concerning sufficient cause interactions. In this section we discuss how this approach can be extended to draw inference concerning attributable fractions for sufficient cause interactions.

To apply Theorems 2 and 3, instead of specifying regression models for $\mathbb{\text{E}}$[D|X₁ = 1, X₂ = 1, c ] we could alternatively specify a marginal structural model for $\mathbb{\text{E}}$[D_x1x2|X₁ = 1, X₂ = 1, q]. For instance, if Q were binary, one could specify a saturated marginal structural model of the form

$$\begin{array}{ll}\mathbb{\text{E}}[D_{x_1{x}_2}|X_1=1,X_2=1,Q=q]=& {\alpha }_0+{\alpha }_1{x}_1+{\alpha }_2{x}_2+{\alpha }_3{x}_1{x}_2.\\ & +{\alpha }_{4}q+{\alpha }_{5}q{x}_1+{\alpha }_{6}q{x}_2+{\alpha }_{7}q{x}_1{x}_2.\end{array}$$ (10)

Under the assumption of unconfoundedness conditional on C (i.e. $D_{x_1{x}_2}\coprod \{X_1,\hspace{0.17em}{X}_2\}|C$) marginal structural models can be fit using inverse probability of treatment weighting (IPTW, Robins, 1999; Robins et al., 2000). This IPTW technique has become quite routine in fitting marginal structural model not conditional on the exposures such as

$$\begin{array}{ll}\mathbb{\text{E}}[D_{x_1{x}_2}|Q=q]=& {\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2.\\ & +{\beta }_{4}q+{\beta }_{5}q{x}_1+{\beta }_{6}q{x}_2+{\beta }_{7}q{x}_1{x}_2.\end{array}$$ (11)

For example, if it were the case that $D_{x_1{x}_2}\coprod \{X_1,\hspace{0.17em}{X}_2\}|C$ then consistent estimators for (β₀, β₁, β₂, β₃, β₄, β₅, β₆, β₇) in model (11) can be obtained by fitting a Bernoulli regression with identity link of D on 1, X₁, X₂, X₁X₂, Q, QX₁, QX₂, QX₁X₂ with each subject ω weighted by the inverse probability of treatment weights of

$$w_{\omega }=\frac{1}{P(X_1=x_{1\omega ,}{X}_2=x_{2\omega }|C=c_{\omega })}.$$

where x_1ω, x_2ω and c_ω denote the values of X₁, X₂ and C respectively for individual ω (Robins, 1999). With two exposures, the weights w_ω might in practice be obtained by $w_{\omega }=w_{\omega }^1*w_{\omega }^2$ where

$$\begin{array}{c}{w}_{\omega }^1=\frac{1}{P(X_1=x_{1\omega }|C=c_{\omega })}\\ w_{\omega }^2=\frac{1}{P(X_2=x_{2\omega }|C=c_{\omega },X_1=x_{1\omega })}\end{array}$$

and where models for P(X₁ = x₁|C = c) and P (X₂ = x₂|C = c, X₁ = x₁) may, for example, be fit using logistic regression. When the marginal structural model is not saturated (for example, if q were continuous) then so called stabilized weights, $w_{\omega }^s=w_{\omega }^{1s}*w_{\omega }^{2s}$ where

$$\begin{array}{c}{w}_{\omega }^{1s}=\frac{P(X_1=x_{1\omega }|Q=q_{\omega })}{P(X_1=x_{1\omega }|C=c_{\omega })}\\ w_{\omega }^{2s}=\frac{P(X_2=x_{2\omega }|Q=q_{\omega },X_1=x_{1\omega })}{P(X_2=x_{2\omega }|C=c_{\omega },X_1=x_{1\omega })}\end{array}$$

and where q_ω denotes the value of Q for individual ω, may give smaller variance for the estimates of (β₀, β₁, β₂, β₃, β₄, β₅, β₆, β₇) (Robins, 1999). When a marginal structural model conditional on the exposures such as (10) rather than (11) is under consideration, a modified set of weights is needed (Sato and Matsuyama, 2003). In such cases, consistent estimators for (α₀, α₁, α₂, α₃, α₄, α₅, α₆, α₇) in model (10) can be obtained by fitting a Bernoulli regression with identity link of D on 1, X₁, X₂, X₁X₂, Q, QX₁, QX₂, QX₁X₂ with each subject ω weighted by weights of

$$w_{\omega }^t=\frac{P(X_1=1,X_2=1|C=c_{\omega })}{P(X_1=x_{1\omega },X_2=x_{2\omega }|C=c_{\omega })}.$$

where the weights $w_{\omega }^t$ might in practice be obtained by $w_{\omega }^t=w_{\omega }^{1t}*w_{\omega }^{2t}$ where

$$\begin{array}{c}{w}_{\omega }^{1t}=\frac{P(X_1=1|C=c_{\omega })}{P(X_1=x_{1\omega }|C=c_{\omega })}\\ w_{\omega }^{2t}=\frac{P(X_2=1|C=c_{\omega },X_1=1)}{P(X_2=x_{2\omega }|C=c_{\omega },X_1=x_{1\omega })}\end{array}$$

and where once again models for P(X₁ = x₁|C = c) and P (X₂ = x₂|C = c, X₁ = x₁) might be fit using logistic regression. Once estimates for (α₀, α₁, α₂, α₃, α₄, α₅, α₆, α₇) are obtained, the quantities (7)–(9) for lower bounds for etiologic and excess fraction fractions in Theorems 2–4, could be estimated by:

$$\frac{{\sum }_q\text{max}\{0,({\alpha }_3+{\alpha }_{7}q)P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}$$

and

$$\begin{array}{l}\frac{{\sum }_q\text{max}\{0,({\alpha }_3+{\alpha }_{7}q-{\alpha }_0-{\alpha }_{4}q)P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}\\ \end{array}$$

and

$$\frac{{\sum }_q\text{max}\{0,({\alpha }_3+{\alpha }_{7}q-2{\alpha }_0-2{\alpha }_{4}q)P(X_1=1,X_2=1|q)\}P(q)}{P(D=1)}.$$

respectively. A similar approach could be employed when sufficient causes with k factors are considered.

Time-Dependent Confounding in Inference for Etiologic Fractions for Sufficient Cause Interactions

The preceding discussion assumed that the effects of X₁ and X₂ on D were unconfounded given C in the sense of $D_{x_1{x}_2}\coprod \{X_1,\hspace{0.17em}{X}_2\}|C$. In some settings there may be an effect, L, of the first exposure, X₁, that in turn causes both the second exposure, X₂, and the outcome, D. Such an example is given in Figure 1.

Figure 1.

Example of time-dependent confounding.

In such cases, the unconfoundedness assumption, $D_{x_1{x}_2}\coprod \{X_1,\hspace{0.17em}{X}_2\}|C$, will not in general hold. It may, however, still be the case that a sequential ignorability or unconfoundedness assumptions holds that

$$D_{x_1{x}_2}\coprod X_1|C\hspace{0.17em}\text{and}\hspace{0.17em}{D}_{x_1{x}_2}\coprod X_2\left|\right\{C,X_1,L\}.$$ (12)

Assumption (12) might be intuitively interpreted as that the effect of X₁ on D is unconfounded given C and the effect of X₂ on D is unconfounded given {C, X₁, L}. Robins (1986, 1987) showed that under assumption (12), expected counterfactual outcomes of the form $\mathbb{\text{E}}$[D_x1x2 |q] with Q ⊆ C are identified. Robins (1986, 1987) furthermore conjectured that under assumption (12), expected counterfactual outcomes of the form $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q] would not be identified if x₁ = 0 i.e. the expected counterfactual outcomes $\mathbb{\text{E}}$[D₀₀|X₁ = 1, X₂ = 1, q] and $\mathbb{\text{E}}$[D₀₁|X₁ = 1, X₂ = 1, q] would not be identified under (12). In the discussion in the previous two sections it was the expected counterfactual outcomes of the form, $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q], i.e. of the form not identified under (12), that were used to give lower bounds on etiologic fractions for sufficient causes.

Shpitser and Pearl (2009) recently confirmed the conjecture of Robins (1986, 1987) by providing a set of identification rules on causal directed acyclic graphs (Pearl, 1995) for general expressions involving “the effect of treatment on the treated” including rules for expected counterfactual outcomes of the form $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q]. It follows from Theorem 3 of Shpitser and Pearl (2009) that if Figure 1 constitutes a causal directed acyclic graph then without further restrictions on X₁, X₂, D and without further assumptions, the expected counterfactual outcome $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q] is unidentified for some x₁, x₂. In this case, $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q] is not identified for x₁ = 0. A proof is given in Appendix 2. The assumption that Figure 1 is a causal directed acyclic graph is a stronger assumption than that (12) holds (Robins, 2003) and thus $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, q] is not in general identified for x₁ = 0 under (12).

Although $\mathbb{\text{E}}$[D₀₀|X₁ = 1, X₂ = 1, q] and $\mathbb{\text{E}}$[D₀₁|X₁ = 1, X₂ = 1, q] are not in general identified, we will show that when X₁, X₂, D are all binary some additional progress can be made in identifying the quantity in (7) used to obtain a lower bound for the etiologic fraction under monotonicity. The following theorem states the result formally. The proof is given in Appendix 2.

Theorem 5. Suppose that X₁, X₂ and D are binary, that $D_{x_1{x}_2}\coprod X_1|C$ and $D_{x_1{x}_2}\coprod X_2|\{C,\hspace{0.17em}{X}_1,\hspace{0.17em}L\}$ and furthermore that

$$\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,X_2=0,c,l]=\mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,X_2=0,c,l]$$ (13)

then expression (7) for the lower bound for the etiologic fraction under monotonicity, namely,

$$\frac{1}{P(D=1)}{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}+D_{00}|X_1=1,X_2=1,q]P(X_1=1,X_2=1|q)\}P(q),$$

is identified because

$$\mathbb{\text{E}}[D_{11}|X_1=1,X_2=1,q]=\mathbb{\text{E}}[D|X_1=1,X_2=1,q]$$

$$\begin{array}{l}\mathbb{\text{E}}[D_{10}|X_1=1,X_2=1,q]=\frac{1}{P(X_2=1|X_1=1,q)}\{\sum _{c,l}\mathbb{\text{E}}[D|X_1=1,X_2=0,c,l]P(l|X_1=1,c)P(c|X_1=1,q)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\mathbb{\text{E}}[D|X_1=1,X_2=0,Q=q]P(X_2=0|X_1=1,q)\}\end{array}$$

$$\begin{array}{l}\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,X_2=1,q]=\\ \frac{1}{P(X_2=1|X_1=1,q)}\{\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,X_2=1,c,l]P(l|X_1=0,c)P(c|X_1=1,q)\\ -\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,X_2=0,c,l]P(l|X_1=0,c)P(c|X_1=1,q)\}\end{array}$$

$$\begin{array}{l}-\frac{P(X_2=0|X_1=1,q)}{P(X_2=1|X_1=1,q)}\{\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,X_2=1,c,l]P(l|X_1=1,X_2=0,c)P(c|X_1=1,X_2=0,q)\\ -\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,X_2=0,c,l]P(l|X_1=1,X_2=0,c)P(c|X_1=1,X_2=0,q)\}\end{array}$$

An intuitive interpretation of assumption (13) can be given as follows. Assumption (13) will hold on the causal directed acyclic graph in Figure 1 if there is no interaction on the additive scale between the effects of L and X₂ on D. A formal statement and proof of this assertion is given in Appendix 2. Assumptions such as (13) are useful also in the identification of natural direct and indirect effects. See Hafeman and VanderWeele (2010) and Robins et al. (2010) for further discussion.

Discussion

In this paper we have considered a number of results for attributable fractions for sufficient cause interactions. We have extended the distinction between “excess fractions” and “etiologic fraction” of Greenland and Robins (1988) from the setting of a single exposure to that of sufficient causes. Under assumptions of unconfoundedness we have discussed how to calculate excess fractions and how to obtain lower bounds for etiologic fractions, both with and without monotonicity and also how to obtain a lower bound for excess fractions without the assumption of monotonicity. The results here extend those of Hoffmann et al. in considering etiologic fractions in addition to excess fractions and in considering settings in which monotonicity assumptions do not hold. We have discussed a procedure using marginal structural models to obtain lower bounds for the etiologic fraction both with and without monotonicity and for the excess fraction without monotonicity; within the sufficient cause framework, this approach based on marginal structural models is preferable to a regression approach because it does not impose restrictions on the relationship between the confounding variables and the potentially unknown background causes. Finally we have discussed issues of identifiability of attributable fractions in settings with time-dependent confounding.

The present paper has been concerned principally with attributable fractions related to sufficient cause interactions. In particular, in the paper and the appendix we have considered excess and etiologic fractions for the sufficient cause with k factors in settings in which there are a total of k factors of interest. Future work could consider excess and etiologic fractions, with and without monotonicity, for sufficient sufficient causes with r < k factors in settings in which there are k factors of interest.

As discussed above, under monotonicity, a lower bound for the excess fraction also constitutes a lower bound for the etiologic fraction; thus the maximum of the excess fraction given in Theorem 1 and the lower bound on the etiologic fraction given in Theorem 2 still constitutes a lower bound on the etiologic fraction. Future work could thus consider whether it is in some sense possible to obtain sharp lower bounds on etiologic fractions and such work could also consider optimal choice of Q in the bounds for the etiologic fraction given in Theorems 2–4. As noted in Appendix 1, additional subtleties arise under the monotonicity assumption when etiologic fraction for sufficient causes with more than two factors are considered.

Appendix 1. Attributable fractions for n-way sufficient cause interactions

For k-way interactions, VanderWeele and Richardson (2010) showed that if there were an individual for whom

$$\begin{array}{l}{D}_{x_1=1,\dots ,x_k=1}(\omega )-D_{x_1=0,x_2=1,\dots ,x_k=1}(\omega )-D_{x_1=1,x_2=0,x_3=1,\dots ,x_k=1}(\omega )\\ -\dots -D_{x_1=1,\dots ,x_{n-1}=1,x_k=0}(\omega )>0\end{array}$$

then there was an k-way sufficient cause interaction between X₁, . . . , X_k. For any set of covariates Q,

$$\mathbb{\text{E}}[D_{x_1=1,\dots ,x_k=1}-D_{x_1=0,x_2=1,\dots ,x_k=1}-D_{x_1=1,x_2=0,x_3=1,\dots ,x_k=1}-\dots -D_{x_1=1,\dots ,x_{n-1}=1,x_k=0}|q\hspace{0.17em}]$$ (A1)

constitutes a lower bound on the prevalence of sufficient cause interactions within stratum Q = q. If $D_{x_1\hspace{0.17em}\dots x_k}\coprod \{X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k\}|C$ (i.e. if the effects of {X₁, . . . , X_k} on D are unconfounded given C) and if Q ⊆ C, then the quantity in (A1) can be estimated by using

$$\mathbb{\text{E}}[D_{x_1\dots x_k}|q]={\sum }_c\mathbb{\text{E}}[D|x_1,\dots ,x_k|c]P(c|q).$$

The approach to obtaining lower bounds on etiologic fractions for sufficient causes described in the main text can still be applied to sufficient causes with k factors. Using the same logic as for Theorems 2 and 3, we have that a lower bound for the etiologic fraction for the sufficient cause with X₁, . . . , X_k will be given by

$$\begin{array}{l}\frac{1}{P(D=1)}\{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{x_1=1,\dots ,x_k=1}-D_{x_1=0,x_2=1,\dots ,x_k=1}\\ \hspace{1em}-\dots -D_{x_1=1,\dots ,x_{n-1}=1,x_k=0}|X_1=1,\dots ,X_k=1,q]P(X_1=1,\dots ,X_k=1|q)\}p(q)\}\\ \end{array}$$

where, provided $D_{x_1\hspace{0.17em}\dots x_k}\coprod \{X_1,\hspace{0.17em}\dots ,\hspace{0.17em}{X}_k\}|C$, the expression $\mathbb{\text{E}}$[D_{x1. . . xk}|X₁ = 1, . . . , X_k = 1, q] can be estimated by

$${\sum }_c\mathbb{\text{E}}[D|x_1,\dots ,x_k,c]P(c|X_1=1,\dots ,X_k=1,q).$$

VanderWeele and Robins (2008) and VanderWeele and Richardson (2010) discussed conditions for 3-way and n-way sufficient cause interaction respectively under assumptions of monotonicity. For example, for a 3-way sufficient cause interaction, VanderWeele and Robins (2008) noted that if D_x1x2x3 were non-decreasing in x₁, x₂ and x₃ then if any of the following three expressions are positive this suffices to conclude the presence of a 3-way sufficient cause interaction:

$$\begin{array}{l}\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{010}|q]\hfill \\ \mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{001}|q]\hfill \\ \mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{010}+D_{001}|q].\hfill \end{array}$$

Each of these three quantities also constitutes a lower bound on the prevalence of 3-way sufficient cause interactions within stratum Q = q. Once again, using the logic of Theorems 2 and 3, all three of the following quantities constitute lower bounds on the etiologic fraction for the sufficient cause with X₁, X₂ and X₃:

$$\begin{array}{l}\frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{010}|X_1=1,X_2=1,X_3=1,q]P(X_1=1,X_2=1,X_3=1|q)\}P(q)}{P(D=1)}\hfill \\ \frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{001}|X_1=1,X_2=1,X_3=1,q]P(X_1=1,X_2=1,X_3=1|q)\}P(q)}{P(D=1)}\hfill \\ \frac{{\sum }_q\text{max}\{0,\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{010}+D_{001}|X_1=1,X_2=1,X_3=1,q]P(X_1=1,X_2=1,X_3=1|q)\}P(q)}{P(D=1)}.\hfill \end{array}$$

As before, if $D_{x_1{x}_2{x}_3}\coprod \{X_1,\hspace{0.17em}{X}_2,\hspace{0.17em}{X}_3\}|C$ then $\mathbb{\text{E}}$[D_x1x2x3 |X₁ = 1, X₂ = 1, X₃ = 1, q] can be estimated by

$${\sum }_c\mathbb{\text{E}}[D|x_1,x_2,x_3,c]P(c|X_1=1,X_2=1,X_3=1,q).$$

Since each of the three quantities above constitutes a lower bound on the etiologic fraction for the sufficient cause with X₁, X₂ and X₃ it also follows that each maximum of these three expressions constitutes a lower bound on the etiologic fraction for the sufficient cause with X₁, X₂ and X₃. Thus it is in fact also the case that $\frac{1}{P(D=1)}{\sum }_q\text{max}\{0,\hspace{0.17em}\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{010}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1,\hspace{0.17em}q]P(X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1|q)$,

$$\begin{array}{l}\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{100}+D_{001}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1,\hspace{0.17em}q]P(X_1=\\ 1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1|q),\end{array}$$

$$\begin{array}{l}\mathbb{\text{E}}[D_{111}-D_{011}-D_{101}-D_{110}+D_{010}+D_{001}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1,\hspace{0.17em}q]P(X_1=\\ 1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}{X}_3=1|q)\}P(q)\end{array}$$

also constitutes a lower bound for the etiologic fraction for the sufficient cause with X₁, X₂ and X₃. Whether bounds sharper than this can be obtained is an open question.

Appendix 2. Proofs

Proof of Theorem 4

For any sufficient cause representation satisfying 1, we have that P(D) =

$$\begin{array}{l}P[\{A_0\vee A_1{X}_1\vee A_2(1-X_1)\vee A_3{X}_2\vee A_4(1-X_2)\vee A_5{X}_1{X}_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee A_6(1-X_1)X_2\vee A_7{X}_1(1-X_2)\vee A_8(1-X_1)(1-X_2)\}=1].\end{array}$$ (A2)

If one were able to eliminate the A₅X₁X₂ sufficient cause then the probability of the outcome would be

$$\begin{array}{l}P[\{A_0\vee A_1{X}_1\vee A_2(1-X_1)\vee A_3{X}_2\vee A_4(1-X_2)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee A_6(1-X_1)X_2\vee A_7{X}_1(1-X_2)\vee A_8(1-X_1)(1-X_2)\}=1].\end{array}$$ (A3)

Because the event in (A3) is a subset of the event in (A2) the difference between these two expressions is given by

$$\begin{array}{ll}& P[\{A_5{X}_1{X}_2=1\}\cap \{A_0\vee A_1{X}_1\vee A_2(1-X_1)\vee A_3{X}_2\vee A_4(1-X_2)\\ & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee A_6(1-X_1)X_2\vee A_7{X}_1(1-X_2)\vee A_8(1-X_1)(1-X_2)=0\}]\\ =& P[(X_1{X}_2=1)\cap (A_5=1)\cap (A_0=A_1=A_3=0)].\end{array}$$ (A4)

This final expression in (A4) is an exact formula for the excess fraction for the sufficient cause A₅X₁X₂ but it is not in general identified because A₀, A₁, A₃, A₅ are latent. Clearly,

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}P[(X_1{X}_2=1)\cap (A_5=1)\cap (A_0=A_1=A_3=0)]\\ ={\sum }_q\text{max}\{P[(X_1{X}_2=1)\cap (A_5=1)\cap (A_0=A_1=A_3=0)|q],\hspace{0.17em}0\}P(q)\end{array}$$

since the probability is always non-negative. For any individual ω such that D₁₁(ω) = 1 but D₁₀(ω) = D₀₁(ω) = D₀₀(ω) = 0 we must have by (1) that A₀(ω) = A₁(ω) = A₃(ω) = 0 since D₁₀(ω) = D₀₁(ω) = D₀₀(ω) = 0 and thus that A₅(ω) = 1 since D₁₁(ω) = 1. From this it follows that

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sum }_q\text{max}\{P[(X_1{X}_2=1)\cap (A_5=1)\cap (A_0=A_1=A_3=0)|q],\hspace{0.17em}0\}P(q)\\ \ge {\sum }_q\text{max}\{P[(X_1{X}_2=1)\cap (D_{11}=1)\cap (D_{10}=D_{01}=D_{00}=0)|q],\hspace{0.17em}0\}P(q)\\ ={\sum }_q\text{max}\{P[(D_{11}=1)\cap (D_{10}=D_{01}=D_{00}=0)|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\times \hspace{0.17em}P(X_1=1,\hspace{0.17em}{X}_2=1|q),\hspace{0.17em}0\}P(q)\\ \ge \hspace{0.17em}{\sum }_q\text{max}\{0,\hspace{0.17em}\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}-D_{00}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]P(X_1=1,\hspace{0.17em}{X}_2=1|q)\}P(q)\end{array}$$

From this it follows that the excess fraction for the sufficient cause A₅X₁X₂ is greater than

$$\frac{{\sum }_q\text{max}\{0,\hspace{0.17em}\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}-D_{00}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]P(X_1=1,\hspace{0.17em}{X}_2=1|q)\}P(q)}{P(D=1)}$$

and this establishes (9). If one of X₁ or X₂ has a monotonic effect on D then for any individual ω such that D₁₁(ω) = 1 but D₁₀(ω) = D₀₁(ω) = 0 we also have D₀₀(ω) = 0 and thus must have by (1) that A₀(ω) = A₁(ω) = A₃(ω) = 0 and A₅(ω) = 1. Thus

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sum }_q\text{max}\{P[(X_1{X}_2=1)\cap (A_5=1)\cap (A_0=A_1=A_3=0)|q],\hspace{0.17em}0\}P(q)\\ \ge \hspace{0.17em}{\sum }_q\text{max}\{P[(X_1{X}_2=1)\cap (D_{11}=1)\cap (D_{10}=D_{01}=0)|q],\hspace{0.17em}0\}P(q)\\ =\hspace{0.17em}{\sum }_q\text{max}\{P[(D_{11}=1)\cap (D_{10}=D_{01}=0)|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\times \hspace{0.17em}P(X_1=1,\hspace{0.17em}{X}_2=1|q),\hspace{0.17em}0\}P(q)\\ \ge \hspace{0.17em}{\sum }_q\text{max}\{0,\hspace{0.17em}\mathbb{\text{E}}[D_{11}-D_{10}-D_{01}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]P(X_1=1,\hspace{0.17em}{X}_2=1|q)\}P(q),\end{array}$$

from which it follows that (8) would then be a lower bound on the excess fraction for the sufficient cause A₅X₁X₂. This completes the proof.

Lack of Identification of Expected Counterfactual Outcomes Conditional on Exposures Under Time-Dependent Confounding

We use Theorem 3 of Shpitser and Pearl (2009) to show that $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ =1, Q = q] is not identified for arbitrary x₁ and x₂ in the causal directed acyclic graph constituted by Figure 1 which we will refer to as graph G. In Theorem 3 of Shpitser and Pearl (2009), we have for Figure 1 that X = {X₁, X₂}, Y = D, Z = {X₁, X₂}, W = ∅. Choose F = L and note that F is ancestral to Y ∪ Z = {D, X₁, X₂} in G_z where G_z is the graph G with the edges proceeding from Z = {X₁, X₂} removed. Now let W = X₁ and note that there are directed paths from W = X₁ to both X₂ ∈ Z and Y = D in the graph G_z\{W} = G_X2 i.e. in the graph G with the edges proceeding from X₂ removed. The directed path from W = X₁ to X₂ in G_X2 constituted by X₁ → L → X₂ has its first node in F = L. The directed path from W = X₁ to D in G_X2 constituted by X₁ → L → D has its first node in F = L. From Theorem 3 of Shpitser and Pearl (2009) it follows that $\mathbb{\text{E}}$[D_x1x2 |X₁ = 1, X₂ = 1, Q = q] is not identified for arbitrary x₁ and x₂.

Proof of Theorem 5

We will show that $\mathbb{\text{E}}$[D₁₁ – D₁₀ – D₀₁ + D₀₀|X₁ = 1, X₂ = 1, q] is identified. First note that

$$\begin{array}{l}\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}q]=\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]P(X_2=0|X_1=1,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]P(X_2=1|X_1=1,\hspace{0.17em}q)\end{array}$$

and thus

$$\begin{array}{l}\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]=\frac{1}{P(X_2=1|X_1=1,\hspace{0.17em}q)}\{\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}q]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\mathbb{\text{E}}[D|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]P(X_2=0|X_1=1,\hspace{0.17em}q)\}.\end{array}$$

Moreover, by (12), $\mathbb{\text{E}}$[D₁₀|X₁ = 1, q]

$$\begin{array}{l}={\sum }_{c,l}\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ ={\sum }_{c,l}\mathbb{\text{E}}[D_{10}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ ={\sum }_{c,l}\mathbb{\text{E}}[D|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q).\end{array}$$

Thus we see that $\mathbb{\text{E}}$[D₁₀|X₁ = 1, X₂ = 1, q] is identified and is given by

$$\begin{array}{l}\frac{1}{P(X_2=1|X_1=1,\hspace{0.17em}q)}\{{\sum }_{c,l}E[D|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}E[D|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]P(X_2=0|X_1=1,\hspace{0.17em}q)\}.\end{array}$$

Also, clearly $\mathbb{\text{E}}$[D₁₁|X₁ = 1, X₂ = 1, q] is identified since

$$\mathbb{\text{E}}[D_{11}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]=\mathbb{\text{E}}[D|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q].$$

It remains to show that $\mathbb{\text{E}}$[–D₀₁ + D₀₀|X₁ = 1, X₂ = 1, q] is identified. For $\mathbb{\text{E}}$[D₀₁ |X₁ = 1, X₂ = 1, q] and $\mathbb{\text{E}}$[D₀₀|X₁ = 1, X₂ = 1, q]

$$\begin{array}{ll}\mathbb{\text{E}}[D_{01}|X_1=1,q]& =\mathbb{\text{E}}[D_{01}|X_1=1,X_2=0,q]P(X_2=0|X_1=1,q)\\ & \hspace{0.17em}+\mathbb{\text{E}}[D_{01}|X_1=1,X_2=1,q]P(X_2=1|X_1=1,q)\end{array}$$ (A5)

and

$$\begin{array}{ll}\mathbb{\text{E}}[D_{00}|X_1=1,q]& =\mathbb{\text{E}}[D_{00}|X_1=1,X_2=0,q]P(X_2=0|X_1=1,q)\\ & \hspace{0.17em}+\mathbb{\text{E}}[D_{00}|X_1=1,X_2=1,q]P(X_2=1|X_1=1,q).\end{array}$$ (A6)

Solving (A5) for $\mathbb{\text{E}}$[D₀₁|X₁ = 1, X₂ = 1, q] and (A6) for $\mathbb{\text{E}}$[D₀₀|X₁ = 1, X₂ = 1, q] we obtain

$$\begin{array}{ll}& \mathbb{\text{E}}[D_{01}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]-\mathbb{\text{E}}[D_{00}|X_1=1,\hspace{0.17em}{X}_2=1,\hspace{0.17em}q]\\ =& \frac{\mathbb{\text{E}}[D_{01}|X_1=1,\hspace{0.17em}q]-\mathbb{\text{E}}[D_{00}|X_1=1,\hspace{0.17em}q]}{P(X_2=1|X_1=1,\hspace{0.17em}q)}\\ & -\frac{\mathbb{\text{E}}[D_{01}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]-\mathbb{\text{E}}[D_{00}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]}{P(X_2=1|X_1=1,\hspace{0.17em}q)/P(X_2=0|X_1=1,\hspace{0.17em}q)}.\end{array}$$

Now, under (12), $\mathbb{\text{E}}$[D₀₁|X₁ = 1, q] – $\mathbb{\text{E}}$[D₀₀|X₁ = 1, q] is identified and given by

$$\begin{array}{ll}& {\sum }_c\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,\hspace{0.17em}c]P(c|X_1=1,\hspace{0.17em}q)\\ =& {\sum }_c\mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,\hspace{0.17em}c]P(c|X_1=1,\hspace{0.17em}q)\\ =& {\sum }_{c,l}\mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ =& {\sum }_{c,l}\mathbb{\text{E}}[D_{01}|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\sum }_{c,l}\mathbb{\text{E}}[D_{00}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ =& {\sum }_{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\sum }_{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q).\end{array}$$

Furthermore, by (13), $\mathbb{\text{E}}$[D₀₁|X₁ = 1, X₂ = 0, q] − $\mathbb{\text{E}}$[D₀₀|X₁ = 1, X₂ = 0, q]

$$\begin{array}{l}=\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q]\\ =\hspace{0.17em}{\sum }_{c,l}\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ =\hspace{0.17em}{\sum }_{c,l}\mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ =\hspace{0.17em}{\sum }_{c,l}\mathbb{\text{E}}[D_{01}|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\sum }_{c,l}\mathbb{\text{E}}[D_{00}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ =\hspace{0.17em}{\sum }_{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(L=l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\sum }_{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q).\end{array}$$

Thus, $\mathbb{\text{E}}$[D₀₁ – D₀₀|X₁ = 1, X₂ = 1, q =

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{1}{P(X_2=1|X_1=1,\hspace{0.17em}q)}\{\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}q)\}\\ -\hspace{0.17em}\frac{P(X_2=0|X_1=1,\hspace{0.17em}q)}{P(X_2=1|X_1=1,\hspace{0.17em}q)}\{\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=1,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\sum _{c,l}\mathbb{\text{E}}[D|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]P(l|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c)P(c|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}q)\}\end{array}$$

Assumption 13 as a No-Interaction Assumption

We show that assumption (13) in the text, namely,

$$\mathbb{\text{E}}[D_{01}-D_{00}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]=\mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]$$

will hold for the causal directed acyclic graph given in Figure 1 if there is no additive-scale interaction between the effects of L and X₂ on D in the following sense. If Figure 1 constitutes a causal directed acyclic graph (Pearl, 1995) then let f(x₁, x₂, c, l, ∈_D) denote the non-parametric structural equation for D where ∈_D is the random term for D. We will argue that assumption (13) above will hold for the causal directed acyclic graph in Figure 1 if there is no interaction between the effects of L and X₂ on D in the sense that f (x₁, x₂, c, l, ∈_D) can be written as

$$f(x_1,x_2,c,l,{\varepsilon }_D)=f_1(x_1,c,l,{\varepsilon }_D)+f_2(x_1,x_2,c,{\varepsilon }_D).$$ (A7)

To see this note that under (A7) we have that $\mathbb{\text{E}}$[D₀₁ – D₀₀|X₁ = 1, X₂ = 0, c, l =

$$\begin{array}{ll}& \mathbb{\text{E}}[f_1(0,\hspace{0.17em}c,\hspace{0.17em}l,\hspace{0.17em}{\epsilon }_D)+f_2(0,\hspace{0.17em}1,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)-\{f_1(0,\hspace{0.17em}c,\hspace{0.17em}l,\hspace{0.17em}{\epsilon }_D)+f_2(0,\hspace{0.17em}0,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)\}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[f_2(0,\hspace{0.17em}1,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)-f_2(0,\hspace{0.17em}0,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l].\end{array}$$

Define F = f₂(X₁, X₂, C, ∈_D) then using the counterfactual graphs of Shpitser and Pearl (2007) it follows that $F_{x_1{x}_2}\coprod X_1|\{C,\hspace{0.17em}{X}_2,\hspace{0.17em}L\}$ and we thus have

$$\begin{array}{ll}& \mathbb{\text{E}}[f_2(0,\hspace{0.17em}1,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)-f_2(0,\hspace{0.17em}0,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[F_{x_1=0,\hspace{0.17em}{x}_2=1}-F_{x_1=0,\hspace{0.17em}{x}_2=0}|X_1=1,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[F_{x_1=0,\hspace{0.17em}{x}_2=1}-F_{x_1=0,\hspace{0.17em}{x}_2=0}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[f_2(0,\hspace{0.17em}1,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)-f_2(0,\hspace{0.17em}0,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[f_1(0,\hspace{0.17em}c,\hspace{0.17em}l,\hspace{0.17em}{\epsilon }_D)+f_2(0,\hspace{0.17em}1,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)-\{f_1(0,\hspace{0.17em}c,\hspace{0.17em}l,\hspace{0.17em}{\epsilon }_D)+f_2(0,\hspace{0.17em}0,\hspace{0.17em}c,\hspace{0.17em}{\epsilon }_D)\}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\\ =& \mathbb{\text{E}}[D_{01}-D_{00}|X_1=0,\hspace{0.17em}{X}_2=0,\hspace{0.17em}c,\hspace{0.17em}l]\end{array}$$

and this completes the proof.