Chapter 4 introduced effect modification: the effect of a single treatment \(A\) varying across levels of another variable \(V\). Many causal questions, however, concern two or more treatments applied together. If the causal effect of one treatment depends on the value we set for the other, the two treatments interact.
This chapter defines interaction between two treatments in two frameworks:
Extend the heart transplant example with a second treatment:
Each individual now has four counterfactual outcomes, \(Y^{a=1,e=1}\), \(Y^{a=1,e=0}\), \(Y^{a=0,e=1}\), and \(Y^{a=0,e=0}\).
Definition 1 (Joint Intervention and Joint Counterfactual) A joint intervention sets two or more treatments at once. The counterfactual outcome \(Y^{a,e}\) is the outcome that would have been observed had we intervened to set \(A\) to \(a\) and \(E\) to \(e\).
Definition 2 (Interaction on the Additive Scale) There is interaction between \(A\) and \(E\) on the additive scale in the population if the causal risk difference for \(A\) when everybody receives \(E\) differs from the causal risk difference for \(A\) when nobody receives \(E\):
\[ \Pr[Y^{a=1,e=1}=1] - \Pr[Y^{a=0,e=1}=1] \neq \Pr[Y^{a=1,e=0}=1] - \Pr[Y^{a=0,e=0}=1]. \]
Example 1 (Transplant and Vitamins (Hernán and Robins 2020, 61–62)) Suppose the causal risk difference for transplant is
Because \(0.1 \neq 0.2\), transplant and vitamins interact on the additive scale.
Write \(p_{ae} \stackrel{\text{def}}{=}\Pr[Y^{a,e}=1]\). Moving terms across the inequality in Definition 2 shows that the same condition can be stated with the roles of \(A\) and \(E\) swapped:
\[ \begin{aligned} p_{11} - p_{01} &\neq p_{10} - p_{00} \\ p_{11} - p_{01} - p_{10} + p_{00} &\neq 0 && \text{(subtract } p_{10} - p_{00} \text{ from both sides)} \\ p_{11} - p_{10} &\neq p_{01} - p_{00} && \text{(add } p_{01} - p_{00} \text{ to both sides)} \end{aligned} \]
The last line says that the causal risk difference for vitamins \(E\) differs between “everybody transplanted” and “nobody transplanted”.
Technical Point 5.1: Additive and Multiplicative Scales
No interaction on the additive scale means the joint effect is the sum of the two separate effects:
\[ p_{11} - p_{00} = (p_{10} - p_{00}) + (p_{01} - p_{00}). \]
When the two sides differ, the interaction is
On the multiplicative scale (causal risk ratios), \(A\) and \(E\) interact if
\[ \frac{p_{11}}{p_{00}} \neq \frac{p_{10}}{p_{00}} \times \frac{p_{01}}{p_{00}}, \]
with supermultiplicative (\(>\)) and submultiplicative (\(<\)) interaction defined analogously.
| Effect modification by \(V\) | Interaction between \(A\) and \(E\) | |
|---|---|---|
| Counterfactuals involved | \(Y^a\) | \(Y^{a,e}\) |
| Interventions | on \(A\) only | joint, on \(A\) and \(E\) |
| Status of the two variables | unequal: \(V\) is not intervened on | equal |
Interaction concerns a joint effect, so identifying it requires exchangeability, positivity, and consistency for both treatments.
If vitamins \(E\) are randomly and unconditionally assigned, then for each \(a\) and \(e\)
\[ \begin{aligned} \Pr[Y^{a,e}=1] &= \Pr[Y^{a,e}=1 \mid E=e] && \text{(exchangeability for } E\text{: } Y^{a,e} \perp\!\!\!\perp E\text{)} \\ &= \Pr[Y^{a,E}=1 \mid E=e] && \text{(}Y^{a,e} = Y^{a,E}\text{ when } E=e\text{)} \\ &= \Pr[Y^{a}=1 \mid E=e] && \text{(recursive substitution, } Y^a = Y^{a,E}\text{)}. \end{aligned} \]
Substituting into Definition 2, interaction on the additive scale becomes
\[ \Pr[Y^{a=1}=1 \mid E=1] - \Pr[Y^{a=0}=1 \mid E=1] \neq \Pr[Y^{a=1}=1 \mid E=0] - \Pr[Y^{a=0}=1 \mid E=0], \]
which is the definition of additive effect modification of \(A\) by \(E\).
We still need the four marginal risks \(\Pr[Y^{a,e}=1]\). Under the usual identifying assumptions for both treatments, they can be computed by standardization or IP weighting over the measured covariates.
Equivalently, treat \(AE\) as one treatment with four levels (11, 01, 10, 00). Identifying interaction is then the familiar problem of identifying the effect of a single treatment, with more treatment values and more counterfactual outcomes.
If we are willing to assume exchangeability for \(A\) but not for \(E\) (e.g., \(A\) is randomized and we look at subgroups defined by \(E\)), we can assess effect modification by \(E\) but generally not interaction between \(A\) and \(E\): computing the effect of \(A\) within strata of \(E\) requires no assumptions about the effect of \(E\).
| Type | \(Y^{a=0}\) | \(Y^{a=1}\) | Zeus’s family (Table 1.1) |
|---|---|---|---|
| Doomed | 1 | 1 | Artemis, Athena, Persephone, Ares |
| Helped | 1 | 0 | Hebe, Kronos, Poseidon, Apollo, Hermes, Dionysus |
| Hurt | 0 | 1 | Rheia, Leto, Aphrodite, Zeus, Hephaestus, Polyphemus |
| Immune | 0 | 0 | Demeter, Hestia, Hera, Hades |
With two dichotomous treatments each individual has four counterfactual outcomes, so there are \(2^4 = 16\) response types (Miettinen 1982).
| Type | \(Y^{1,1}\) | \(Y^{0,1}\) | \(Y^{1,0}\) | \(Y^{0,0}\) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 0 |
| 3 | 1 | 1 | 0 | 1 |
| 4 | 1 | 1 | 0 | 0 |
| 5 | 1 | 0 | 1 | 1 |
| 6 | 1 | 0 | 1 | 0 |
| 7 | 1 | 0 | 0 | 1 |
| 8 | 1 | 0 | 0 | 0 |
| 9 | 0 | 1 | 1 | 1 |
| 10 | 0 | 1 | 1 | 0 |
| 11 | 0 | 1 | 0 | 1 |
| 12 | 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 1 | 1 |
| 14 | 0 | 0 | 1 | 0 |
| 15 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
In six types the effect of each treatment does not depend on the other:
If everyone has one of types 1, 4, 6, 11, 13, 16, there is no interaction between \(A\) and \(E\) on the additive scale.
Additive interaction implies that some individuals belong to at least one of three classes (Greenland and Poole 1988), each invariant to recoding \(A\) and \(E\):
No additive interaction implies either that nobody is in these classes or that equal deviations of opposite sign cancel exactly (e.g., equal proportions of types 7 and 10, or of types 8 and 12).
Example 2 (Cancellation of Interaction Types) Let a fraction \(q\) of the population be type 8, a fraction \(q\) be type 12, and the rest type 16. Reading the risks off Table 2: \(p_{11} = q\) (type 8), \(p_{01} = q\) (type 12), \(p_{10} = 0\), \(p_{00} = 0\). Then
\[ \begin{aligned} p_{11} - p_{01} - p_{10} + p_{00} &= q - q - 0 + 0 \\ &= 0, \end{aligned} \]
so there is no interaction on the additive scale in the population, even though every type-8 and type-12 individual has an effect of \(A\) that depends on \(E\).
Technical Point 5.2: Monotonicity
For one treatment, \(Y^{a=0} > Y^{a=1}\) only for the “helped”. If nobody is helped, every individual has \(Y^{a=1} \geq Y^{a=0}\), and the causal effect of \(A\) on \(Y\) is monotonic.
For two treatments, the effects of \(A\) and \(E\) are monotonic if every \(Y^{a,e}\) is nondecreasing in both \(a\) and \(e\), i.e., nobody has
Fine Point 5.1: Detecting Synergistic Types
Do individuals exist who develop the outcome under both treatments but under neither alone, i.e., with \(Y^{a=1,e=1}=1\) and \(Y^{a=0,e=1}=Y^{a=1,e=0}=0\) (types 7 and 8)? A sufficient condition (VanderWeele and Robins 2007a, 2008) is
\[ p_{11} - p_{01} - p_{10} > 0, \quad\text{equivalently}\quad p_{11} - p_{01} > p_{10}. \]
If the effects are monotonic, a weaker sufficient condition is superadditive interaction:
\[ p_{11} - p_{01} > p_{10} - p_{00}, \]
which then implies type 8 exists (monotonicity rules out type 7).
The variety of response types shows that \(A\) is not the only determinant of \(Y\). The sufficient-component-cause framework represents the other determinants as background factors.
Definition 3 (Minimal Sufficient Cause) A sufficient cause of \(Y\) is a set of conditions that together inevitably produce \(Y\). It is minimal if it contains the smallest set of background factors that, with the treatment component, suffices. Its elements are its component causes; “sufficient-component causes” refers to both.
In the oversimplified heart transplant example (Hernán and Robins 2020, 66–67):
| Sufficient cause | Treatment component | Background factor (example) |
|---|---|---|
| \(A=1\) with \(U_1=1\) | transplant | allergy to anesthesia |
| \(A=0\) with \(U_2=1\) | no transplant | ejection fraction below 20% |
| \(U_0=1\) | none | pancreatic cancer at study start |
Figure 5.1 of the book draws each sufficient cause as a circle (“causal pie”) divided into its components.
Compare two populations, identical except that \(U_1=1\) (allergy) has prevalence
Randomize half of each population to \(A=1\). The average causal effect of transplant on death is greater in the second population, because the sufficient cause “\(A=1\) plus \(U_1=1\)” is ten times more common there.
With treatments \(A\) and \(E\) there are 9 possible sufficient causes (Greenland and Poole 1988), with treatment components
Each also contains background factors from \(U_1, \ldots, U_8\) and \(U_0\) (Figure 5.2 of the book).
The counterfactual definition of interaction (Definition 2) is a contrast of counterfactual risks. It can be identified in an ideal randomized experiment on \(A\) and \(E\) without any knowledge of the mechanisms by which the treatments act.
A second concept of interaction refers to mechanisms directly.
Definition 4 (Sufficient Cause Interaction) “A sufficient cause interaction between \(A\) and \(E\) exists in the population if \(A\) and \(E\) occur together in a sufficient cause” (Hernán and Robins 2020, 69).
Example 3 (Background Factor \(U_5\)) Suppose individuals with \(U_5=1\) develop the outcome when receiving both vitamins and transplant but not when receiving only one of them. Then a sufficient cause interaction exists if anyone has \(U_5=1\). Hence, if some individual has \(Y^{a=1,e=1}=1\) and \(Y^{a=0,e=1}=Y^{a=1,e=0}=0\), a sufficient cause interaction between \(A\) and \(E\) is present (Hernán and Robins 2020, 69–70).
Antagonism between \(A\) and \(E\) can be viewed as synergism between \(A\) and “no \(E\)” (or between “no \(A\)” and \(E\)).
Sufficient cause interaction is defined through mechanisms, yet sometimes it can be detected with no knowledge of them: if the inequalities of Fine Point 5.1 hold, synergism between \(A\) and \(E\) exists.
Fine Point 5.2: From Response Types to Component Causes
| Type | \(Y^{a=0}\) | \(Y^{a=1}\) | Component causes |
|---|---|---|---|
| Doomed | 1 | 1 | \(U_0=1\) or \(\{U_1=1\) and \(U_2=1\}\) |
| Helped | 1 | 0 | \(U_0=0\), \(U_1=0\), \(U_2=1\) |
| Hurt | 0 | 1 | \(U_0=0\), \(U_1=1\), \(U_2=0\) |
| Immune | 0 | 0 | \(U_0=0\), \(U_1=0\), \(U_2=0\) |
Each combination of component causes gives exactly one response type, but a response type can arise from several combinations (e.g., “doomed” arises from any combination with \(U_0=1\), or with \(U_1=1\) and \(U_2=1\)).
Fine Point 5.3: “Biologic” Interaction
Sufficient cause interaction is often called biologic interaction (Rothman et al. 1980), but it need not involve treatments acting on each other.
Example 4 (Two Alleles (Hernán and Robins 2020, Fine Point 5.3, p. 71)) Let \(A\) and \(E\) indicate a deleterious mutation in each of the two alleles of a gene that produces an essential protein (VanderWeele and Robins 2007a). Infants with both mutations (\(A=1\), \(E=1\)) lack the protein and die within a week; those with one or no mutation survive. There is synergism, since a sufficient cause of death contains \(A=1\) and \(E=1\), yet the alleles arguably do not physically act on each other.
The two frameworks answer different questions:
| Sufficient-component causes | Counterfactuals | |
|---|---|---|
| Starts from | a particular effect | a particular cause or intervention |
| Asks | “how does it happen?” | “what happens?” |
| Describes mechanisms | yes | no |
For estimating average causal effects of hypothetical interventions, the subject of the book, the counterfactual framework is the natural one.
Useful for teaching, because they illustrate
Limited for data analysis, because in its classical form the framework
Fine Point 5.4: Attributable Fractions Do Not Add
Suppose the excess fraction (Fine Point 3.5) is 75% for \(A\) and 50% for \(E\). A joint intervention cannot prevent \(75\% + 50\% = 125\%\) of cases: no intervention, single or joint, can prevent more than 100%.
The resolution: an individual like Zeus with \(U_5=1\), treated with \(A=1\) and \(E=1\), would not have been a case had either treatment been withheld, so Zeus is counted in both the 75% and the 50%.
Technical Point 5.3: Monotonicity Rules Out Some Sufficient Causes
If smoking (\(A=1\)) never prevents heart disease and physical inactivity (\(E=1\)) never prevents heart disease, then no sufficient cause can contain \(A=0\) or \(E=0\).