Chapter_5

Aaron L9/17/25About 3 min

### Chapter 5: INTERACTION

Introduction

Introduces the concept of interaction between two treatments (e.g., "looking up at the sky" and "being dressed/naked"). Defines interaction as occurring when the causal effect of one treatment differs across levels of another treatment. Highlights the importance of interaction for implementing effective interventions and outlines the chapter's focus on counterfactual and sufficient-component-cause frameworks.

Technical Points:

Interaction is formally defined as a difference in the causal effect of treatment (A) when (E=1) vs. (E=0): $\left( \Pr[Y^{a=1,e=1}=1] - \Pr[Y^{a=0,e=1}=1] \right) \neq \left( \Pr[Y^{a=1,e=0}=1] - \Pr[Y^{a=0,e=0}=1] \right)$
Treatments (A) and (E) have equal status in interaction definitions (symmetry property).

Details Points:

Example: Randomized experiment with heart transplant ((A)) and vitamins ((E)) illustrates interaction when causal risk differences differ (e.g., 0.1 vs. 0.2).
Distinction from effect modification: Effect modification involves varying effects of (A) across strata of (V) (e.g., sex), while interaction involves joint interventions on both (A) and (E).

5.1 Interaction Requires a Joint Intervention

Defines counterfactual outcomes under joint interventions ((Y^{a,e})) and emphasizes that interaction assessment requires manipulating both treatments. Uses a heart transplant/vitamins example to formalize counterfactuals.

Technical Points:

Joint counterfactual (Y^{a,e}): Outcome when (A=a) and (E=e) are set simultaneously.
Consistency: (Y = Y^{A,E}) links observed outcomes to counterfactuals.

Details Points:

Four treatment groups: ((A=1,E=1)), ((A=1,E=0)), ((A=0,E=1)), ((A=0,E=0)).
Counterfactual outcomes are deterministic in this section.

5.2 Identifying Interaction

Describes identifiability conditions (exchangeability, positivity, consistency) for interaction. Explains equivalence between interaction and effect modification when (E) is randomized.

Technical Points:

Identifiability requires exchangeability, positivity, and consistency for both treatments.
When (E) is randomized, interaction on additive scale equals effect modification: $\left( \Pr[Y^{a=1}=1|E=1] - \Pr[Y^{a=0}=1|E=1] \right) \neq \left( \Pr[Y^{a=1}=1|E=0] - \Pr[Y^{a=0}=1|E=0] \right)$
Non-random (E): Standardization/IP weighting adjusts for covariates.

Details Points:

(AE) can be viewed as a combined treatment with four levels.
Effect modification without interaction is possible (e.g., (V) acts as a proxy for an unmeasured interactor).

5.3 Counterfactual Response Types and Interaction

Classifies individuals into 16 deterministic response types for two binary treatments. Links response types to additive interaction.

Technical Points:

16 response types based on counterfactual outcomes ((Y^{1,1}, Y^{0,1}, Y^{1,0}, Y^{0,0})).
No additive interaction if all individuals belong to types {1, 4, 6, 11, 13, 16} (effects of (A) and (E) independent).
Additive interaction implies presence of specific types (e.g., types 7, 8, 10, 12, 14, 15).

Details Points:

Table 5.2: Lists all 16 response types (e.g., Type 1: always dies; Type 16: always survives).
Monotonicity: Assumes treatments cannot prevent outcomes (e.g., no "helped" individuals).

5.4 Sufficient Causes

Introduces sufficient-component-cause framework to represent causal mechanisms. Uses background factors ((U)) to define minimal sufficient causes.

Technical Points:

Sufficient cause: A set of factors (e.g., (A=1 \land U_1=1)) that inevitably produces outcome.
For two treatments, 9 possible sufficient causes (e.g., by (A) only, by (A \land E), by neither).

Details Points:

Figure 5.1: Graphical "causal pies" for single treatment (3 sufficient causes).
Figure 5.2: Graphical "causal pies" for two treatments (9 sufficient causes).
Effect modification: Distribution of (U) (e.g., allergy prevalence) affects causal effect magnitude.

5.5 Sufficient Cause Interaction

Defines "sufficient cause interaction" (synergism/antagonism) and links it to counterfactual response types.

Technical Points:

Synergism: (A=1 \land E=1) in a sufficient cause (e.g., individuals with (U_5=1)).
Antagonism: (A=1 \land E=0) (or similar) in a sufficient cause.
Empirical conditions for synergism (e.g., under monotonicity): $\Pr[Y^{1,1}=1] - \Pr[Y^{0,1}=1] > \Pr[Y^{1,0}=1] - \Pr[Y^{0,0}=1]$

Details Points:

Fine Point 5.3: "Biologic interaction" may not imply physical interaction (e.g., gene alleles).
Rothman's synergism/antagonism framework is based on sufficient causes.

5.6 Counterfactuals or Sufficient-Component Causes?

Compares counterfactual and sufficient-cause frameworks, emphasizing their complementary roles.

Technical Points:

Counterfactual framework: Answers "what happens?" under interventions.
Sufficient-cause framework: Answers "how does it happen?" via mechanisms.
Limitations: Sufficient-cause framework is deterministic and limited to binary variables.

Details Points:

Fine Point 5.4: Attributable fractions can exceed 100% due to overlapping sufficient causes (e.g., Zeus case).
Technical Point 5.3: Monotonicity restricts possible sufficient causes (e.g., no (A=0) components if (A) never prevents outcome).
Figure 5.3: Sufficient causes under monotonic effects (crossed-out components).

Chapter Summary

Technical Focus:

Interaction requires joint interventions and is defined via counterfactual contrasts.
Identifiability relies on exchangeability/positivity/consistency for both treatments.
Sufficient-cause interaction reveals mechanistic synergism/antagonism.

Key Distinctions:

Interaction ≠ effect modification (latter ignores interventions on (V)).
Counterfactuals (effects of causes) vs. sufficient causes (causes of effects).

Limitations:

Sufficient-cause framework is less applicable to continuous variables.
Counterfactual framework remains primary for causal effect estimation.

Note: Image content (Figures 5.1–5.3) is summarized based on textual descriptions; direct analysis is not possible. Mathematical formulas are preserved in LaTeX.