Interaction Between Treatments

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What is Causal Interaction?

Causal interaction is a concept that emerges when we consider the combined effects of two or more interventions. Unlike effect modification, which examines whether a treatment effect varies across subgroups defined by a non-intervention variable, interaction specifically addresses how two interventions work together. Understanding interaction is critical for combination therapies, public health interventions, and understanding biological mechanisms.

Definition: Causal Interaction Causal interaction between two treatments A₁ and A₂ occurs when the combined effect of both treatments differs from the sum (or product, depending on scale) of their individual effects. In potential outcomes notation for the additive scale:

$E[Y^{a_1=1,a_2=1} - Y^{a_1=0,a_2=0}] \neq E[Y^{a_1=1,a_2=0} - Y^{a_1=0,a_2=0}] + E[Y^{a_1=0,a_2=1} - Y^{a_1=0,a_2=0}]$

This says the joint effect is not equal to the sum of the individual effects.

When two treatments are given together, three scenarios are possible:

Scenario	Description	Interpretation
Synergy	Joint effect > sum of individual effects	Treatments amplify each other
Antagonism	Joint effect < sum of individual effects	Treatments diminish each other
Additivity	Joint effect = sum of individual effects	No interaction

Example: Drug Combination Therapy Consider two antibiotics, Drug A and Drug B, for treating a bacterial infection. Individually, Drug A reduces infection risk by 30% and Drug B by 25%. If given together, the effect might be:

Synergistic: 70% reduction (greater than 30% + 25% = 55%)
Antagonistic: 40% reduction (less than expected 55%)
Additive: 55% reduction (exactly as expected from sum)

Reference: What If, Chapter 5 "Interaction between two treatments... exists when the causal effect of one treatment depends on the level of the other treatment... Interaction requires that both variables be interventions" (Hernán & Robins, p. 59).

The key conceptual point is that causal interaction requires imagining that we could intervene on both A₁ and A₂. This distinguishes it from effect modification, where the modifying variable V need not be something we can intervene on.

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