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Chapter_5

Aaron L9/17/25About 8 minCausal inferencegpt-4.1response_md

Chapter 5: Interaction (from "Causal Inference: What If")

Main Content Summary / 主要内容总结

This chapter rigorously discusses interaction between two (or more) treatments both within the familiar counterfactual (potential outcomes) and the sufficient-component-cause frameworks. It defines interaction, distinguishes it from effect modification, gives formal criteria for detecting and identifying interaction under various conditions, and introduces the notion of mechanistic synergy and antagonism (sufficient cause interaction).

1. What is Interaction? / 何为交互作用？

Definition

There is interaction between two treatments AA and EE if the causal effect of AA on YY, after setting E=1E=1 via intervention, is different from the causal effect of AA on YY after setting E=0E=0 via intervention.

两个处理AA和EE间存在交互作用，如果在设定E=1E=1时，AA对YY的因果效应与设定E=0E=0时AA的因果效应不同。

Additive Scale (加法标度):

  • There is interaction on the additive scale if the following holds:

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)≠Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\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)

  • 如果以下不等式成立，则在加法标度上存在交互作用：

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)≠Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\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)

Equivalent Definition (等价定义):

  • On the additive scale, treatments AA and EE are symmetric in the definition:

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=1,e=0=1)≠Pr⁡(Ya=0,e=1=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=1,e=0}=1) \neq \Pr(Y^{a=0,e=1}=1) - \Pr(Y^{a=0,e=0}=1)

  • 在加法标度下，对AA和EE地位对称：

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=1,e=0=1)≠Pr⁡(Ya=0,e=1=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=1,e=0}=1) \neq \Pr(Y^{a=0,e=1}=1) - \Pr(Y^{a=0,e=0}=1)

2. Technical Point 5.1: Additive and Multiplicative Interaction / 技术要点 5.1：加法与乘法交互

Additive Interaction (加法交互)

  • No additive interaction iff:

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)=Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=0,e=1}=1) = \Pr(Y^{a=1,e=0}=1) - \Pr(Y^{a=0,e=0}=1)

  • Or equivalently:

    Pr⁡(Ya=1,e=1=1)=[Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)]+Pr⁡(Ya=0,e=1=1)\Pr(Y^{a=1,e=1}=1) = [\Pr(Y^{a=1,e=0}=1) - \Pr(Y^{a=0,e=0}=1)] + \Pr(Y^{a=0,e=1}=1)

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=0=1)=[Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)]+[Pr⁡(Ya=0,e=1=1)−Pr⁡(Ya=0,e=0=1)]\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=0,e=0}=1) = [\Pr(Y^{a=1,e=0}=1) - \Pr(Y^{a=0,e=0}=1)] + [\Pr(Y^{a=0,e=1}=1) - \Pr(Y^{a=0,e=0}=1)]

  • 无加法交互当且仅当：

    Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)=Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=0,e=1}=1) = \Pr(Y^{a=1,e=0}=1) - \Pr(Y^{a=0,e=0}=1)

  • 或者：

    Pr⁡(Ya=1,e=1=1)=[Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)]+Pr⁡(Ya=0,e=1=1)\Pr(Y^{a=1,e=1}=1) = [\Pr(Y^{a=1,e=0}=1) - \Pr(Y^{a=0,e=0}=1)] + \Pr(Y^{a=0,e=1}=1)

Superadditivity: (交互为超级加性时)

> in the above equations, ≠→>> \ \text{in the above equations},\ \neq \rightarrow >

Subadditivity: (交互为亚加性时)

> in the above equations, ≠→<> \ \text{in the above equations},\ \neq \rightarrow <

Multiplicative Scale (乘法交互)

  • There is interaction on the multiplicative scale if:

    Pr⁡(Ya=1,e=1=1)Pr⁡(Ya=0,e=0=1)≠Pr⁡(Ya=1,e=0=1)Pr⁡(Ya=0,e=0=1)×Pr⁡(Ya=0,e=1=1)Pr⁡(Ya=0,e=0=1)\frac{\Pr(Y^{a=1,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)} \neq \frac{\Pr(Y^{a=1,e=0}=1)}{\Pr(Y^{a=0,e=0}=1)} \times \frac{\Pr(Y^{a=0,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)}

  • 乘法标度上的交互定义：

    Pr⁡(Ya=1,e=1=1)Pr⁡(Ya=0,e=0=1)≠Pr⁡(Ya=1,e=0=1)Pr⁡(Ya=0,e=0=1)×Pr⁡(Ya=0,e=1=1)Pr⁡(Ya=0,e=0=1)\frac{\Pr(Y^{a=1,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)} \neq \frac{\Pr(Y^{a=1,e=0}=1)}{\Pr(Y^{a=0,e=0}=1)} \times \frac{\Pr(Y^{a=0,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)}

3. Distinguishing Interaction from Effect Modification / 区分交互和效应修饰

  • Interaction: Both AA and EE are manipulable and have equal status. The joint intervention is considered.
  • Effect modification: A variable VV modifies the effect of AA on YY, but VV itself is not (necessarily) manipulable or interpreted symmetrically with AA.

交互：AA和EE地位对等、均可人为干预，关注联合干预的效应。
效应修饰：某变量VV修饰AA对YY的效应，但VV本身通常不是干预变量，定义上关注AA。

4. Identifying Interaction / 交互作用的识别

  • All identification of interaction requires satisfied exchangeability, positivity, and consistency for both treatments (AA, EE).
  • If either AA or EE is randomized, these conditions are met for that variable, enabling estimation of joint and marginal risks via standardization or IP weighting.
  • If only AA is randomized, one can estimate effect modification by EE (i.e. estimate the causal effect of AA within strata of EE), but not the joint interaction (unless willing to impose extra untestable assumptions for EE).

对交互作用的识别需要“可交换性(Exchangeability)、正性(Positivity)、一致性(Consistency)”对联合处理(A,E)(A,E)均成立。

  • AA/EE中有一项完全随机分配，则对该变量成立。标准化/逆概率加权可用于交互作用估计。
  • 仅AA随机，则只能做效应修饰而非严格的交互作用识别。

5. Response Types and Interaction / 反应类型与交互

For two binary treatments (AA, EE) and binary outcome (YY):

  • Each subject has 4 counterfactuals: Ya=1,e=1,Ya=1,e=0,Ya=0,e=1,Ya=0,e=0Y^{a=1,e=1}, Y^{a=1,e=0}, Y^{a=0,e=1}, Y^{a=0,e=0}.
  • There are 16 possible response types (all combinations of outcomes to the 4 joint treatments).

These types can be grouped:

  • No additive interaction: If population only has certain types (see types 1, 4, 6, 11, 13, 16 in the table).
  • Additive interaction present: If some types are present where the effect of AA depends on the value of EE and vice versa (specific types, e.g., 7, 8, or 2, 3, 5, 9, 10, 12, 14, 15).

6. Technical Point 5.2: Monotonicity / 技术要点 5.2：单调性

  • If for all individuals, increasing AA or EE cannot prevent the outcome (i.e. Ya=1,e=1≥Ya=0,e=1Y^{a=1,e=1} \geq Y^{a=0,e=1}, Ya=1,e=0≥Ya=0,e=0Y^{a=1,e=0} \geq Y^{a=0,e=0}, etc.), the effects are monotonic.
  • Under monotonicity, certain types of sufficient causes (those with A=0A=0 or E=0E=0 present) cannot exist.

7. Sufficient Component Causes / 充要组分原因

  • Sufficient cause: minimal set of factors (“components”) that together inevitably bring about the outcome.
  • With two binary treatments, there are 9 possible sufficient causes:
    1. A=1A=1 only (regardless of EE)
    2. A=0A=0 only
    3. E=1E=1 only
    4. E=0E=0 only
    5. A=1A=1 AND E=1E=1
    6. A=1A=1 AND E=0E=0
    7. A=0A=0 AND E=1E=1
    8. A=0A=0 AND E=0E=0
    9. Non-treatment related cause (“doomed”)
  • 图示为“因果派/因果馅饼”（causal pies）。

8. Sufficient Cause Interaction (“Synergism/Antagonism”) / 充要原因交互作用（协同、拮抗）

  • Sufficient cause interaction: exists if some sufficient cause contains both A=1A=1 and E=1E=1 as components.
    • Synergism: A=1A=1 and E=1E=1 together cause the effect (e.g., a gene–gene or gene–environment “compositional epistasis”).
    • Antagonism: A=1A=1 and E=0E=0 (or vice versa) together cause the effect.
  • Empirical tests: Inequalities involving observed probabilities (Technical Point 5.1 and Fine Point 5.1, see below).

9. Fine Point 5.1: Sufficient/Empirical Test for Synergism / 协同的充分经验判据

A sufficient (not necessary) condition that there exist individuals for whom A=1A=1, E=1E=1 causes the outcome, but neither A=1A=1 alone nor E=1E=1 alone causes it (i.e., response types 7 and 8):

Pr⁡(Ya=1,e=1=1)−[Pr⁡(Ya=0,e=1=1)+Pr⁡(Ya=1,e=0=1)]>0\Pr(Y^{a=1, e=1}=1) - \left[\Pr(Y^{a=0, e=1}=1) + \Pr(Y^{a=1, e=0}=1)\right] > 0

Or equivalently:

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)>Pr⁡(Ya=1,e=0=1)\Pr(Y^{a=1, e=1}=1) - \Pr(Y^{a=0, e=1}=1) > \Pr(Y^{a=1, e=0}=1)

If monotonicity (no prevention possible) holds, weaken to:

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)>Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1, e=1}=1) - \Pr(Y^{a=0, e=1}=1) > \Pr(Y^{a=1, e=0}=1) - \Pr(Y^{a=0,e=0}=1)

判别类型7、8存在（协同）充要条件（但非必要）如下：

Pr⁡(Ya=1,e=1=1)−[Pr⁡(Ya=0,e=1=1)+Pr⁡(Ya=1,e=0=1)]>0\Pr(Y^{a=1, e=1}=1) - \left[\Pr(Y^{a=0, e=1}=1) + \Pr(Y^{a=1, e=0}=1)\right] > 0

若假定单调性，则弱化为：

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)>Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1, e=1}=1) - \Pr(Y^{a=0, e=1}=1) > \Pr(Y^{a=1, e=0}=1) - \Pr(Y^{a=0,e=0}=1)

10. Fine Point 5.4: Attributable Fraction for Multiple Causes / 多原因责任归属问题

  • The excess fractions for single treatments do not simply add for joint effects (may be >100%) since people may be "double-counted" if they are cases preventable by both AA and EE—only the joint intervention fraction is bounded by 100%.
  • 一个个体可能被AA和EE都归因，故两者责任归属和可超100%，但联合干预可归因不会超过100%。

11. Counterfactual vs. Sufficient-Component Frameworks / 反事实与充要组分因果框架

  • Counterfactual (Potential Outcomes) Framework: Focuses on the effects of a particular cause; answers "what would have happened if..."
  • Sufficient-Component-Cause Framework: Focuses on the causes of a particular effect; effective for thinking about mechanisms, interaction, synergism but is less generalizable (typically requires binary variables).
  • In practice, most statistical analysis uses the counterfactual framework due to generalizability and less data requirement.

Reference to Monotonicity and Sufficient Causes / 关于单调性与充要原因

  • If treatment effects are monotonic (no prevention possible), then certain sufficient causes (those involving A=0A=0 or E=0E=0) cannot exist.
  • 若效应单调，则涉及A=0A=0或E=0E=0的某些充要原因不成立。

(C) Technical and Fine Points / 技术与细致要点总结

(1) Additive/Multiplicative Definitions 切换 & 同等地位

  • Additive and multiplicative definitions are precisely delineated with explicit equations.
  • Treatments AA and EE are symmetrical.

(2) Monotonicity 单调性

  • Ya=1≥Ya=0Y^{a=1} \geq Y^{a=0} is monotonic increasing for AA, generalized to joint monotonicity for A,EA, E.

(3) Empirical sufficiency for synergism 协同判据

  • Inequalities for response types (Fine Point 5.1) allow empirical check for synergism.

(4) Sufficient causes mapping

  • One-to-one mapping between counterfactual types and combinations of sufficient causes.

(5) Non-additivity/Non-multiplicativity implies (or is implied by) interaction

  • Additivity implies no interaction; non-additivity (super-/sub-) implies interaction.

(6) Effect modification ≠ Interaction

  • Unless both variables subject to intervention & data support exchangeability/positivity/consistency for both.

(7) Excess fractions and “double counting”

  • Attributable fractions for AA, EE may sum over 100% due to overlap, only joint is meaningful.

公式列表/Formula Recap

Additive Interaction:

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)≠Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\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)

or

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=1,e=0=1)≠Pr⁡(Ya=0,e=1=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1,e=1}=1) - \Pr(Y^{a=1,e=0}=1) \neq \Pr(Y^{a=0,e=1}=1) - \Pr(Y^{a=0,e=0}=1)

Multiplicative Interaction:

Pr⁡(Ya=1,e=1=1)Pr⁡(Ya=0,e=0=1)≠Pr⁡(Ya=1,e=0=1)Pr⁡(Ya=0,e=0=1)×Pr⁡(Ya=0,e=1=1)Pr⁡(Ya=0,e=0=1)\frac{\Pr(Y^{a=1,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)} \neq \frac{\Pr(Y^{a=1,e=0}=1)}{\Pr(Y^{a=0,e=0}=1)} \times \frac{\Pr(Y^{a=0,e=1}=1)}{\Pr(Y^{a=0,e=0}=1)}

Synergism Sufficient Condition:

Pr⁡(Ya=1,e=1=1)−[Pr⁡(Ya=0,e=1=1)+Pr⁡(Ya=1,e=0=1)]>0\Pr(Y^{a=1, e=1}=1) - \left[\Pr(Y^{a=0, e=1}=1) + \Pr(Y^{a=1, e=0}=1)\right] > 0

or if monotonic:

Pr⁡(Ya=1,e=1=1)−Pr⁡(Ya=0,e=1=1)>Pr⁡(Ya=1,e=0=1)−Pr⁡(Ya=0,e=0=1)\Pr(Y^{a=1, e=1}=1) - \Pr(Y^{a=0, e=1}=1) > \Pr(Y^{a=1, e=0}=1) - \Pr(Y^{a=0, e=0}=1)

结论/Takeaway

  • Interaction is about how the effect of one treatment changes depending on the level of another—in principle, evaluated via joint interventions and counterfactual contrasts.
  • The counterfactual framework is statistically and empirically dominant, whereas the sufficient-component-cause framework aids in mechanistic understanding and philosophical clarity.
  • Proper identification of interaction requires strong assumptions for all involved treatments; effect modification is weaker and asymmetrical.
  • Carefully distinguish empirical definitions (risk difference/ratio) from mechanistic ones (sufficient causes); use the correct formula for your question.

Note:
If you need tables (response types etc.), or more formulaic worked examples, let me know!
如需反应类型表、公式实例或进一步深入，请告知。