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Chapter_19

Aaron L9/17/25About 7 minCausal inferencegpt-4.1response_md

Chapter 19: Time-Varying Treatments (《因果推断：What If》第19章 时间变化处理)

This is the summary with all key points, technical details, fine points, and all formulas in Math + bilingual format.

19.1 The Causal Effect of Time-Varying Treatments

Key Ideas

  • Most of the book so far has considered static (time-fixed) treatments: AA set at t=0t=0 and outcome YY measured at a later time.
  • Time-varying treatments (AkA_k) can change at each time kk (k=0,1,...,Kk=0,1,...,K).
    • E.g., monthly ART (抗逆转录病毒治疗) therapy in HIV.
  • The treatment history is denoted A‾k=(A0,A1,...,Ak)\overline{A}_k = (A_0, A_1, ..., A_k).
  • For fixed treatments, average causal effect (ACE) is:

    E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}]

  • For time-varying treatments, the ACE is no longer uniquely defined as interventions are a vector/sequence.

中文

  • 之前讨论的均为时间固定处理（AA在t=0t=0设定，YY在未来测量）。
  • 时间变化处理（AkA_k）可以在每个时间点kk发生变化（k=0,1,...,Kk=0,1,...,K）。
    • 例如，HIV患者的每月抗病毒治疗。
  • 处理历史记为 A‾k=(A0,A1,...,Ak)\overline{A}_k = (A_0, A_1, ..., A_k)。
  • 固定处理的平均因果效应（ACE）：E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}]。
  • 时间变化处理的ACE不再唯一定义，因为干预是一个序列。

19.2 Treatment Strategies 策略

Definitions & Main Points

  • Treatment strategy (also: plan, policy, protocol, regime): rule assigning treatment at each time kk.
  • Examples:
    • "Always treat" ⇒ a‾=(1,1,...,1)=1‾\overline{a} = (1, 1, ..., 1) = \overline{1}
    • "Never treat" ⇒ a‾=(0,0,...,0)=0‾\overline{a} = (0, 0, ..., 0) = \overline{0}
  • ACE for strategies: contrast mean outcomes under two strategies:

    E[Ya‾=1‾]−E[Ya‾=0‾]E[Y^{\overline{a} = \overline{1}}] - E[Y^{\overline{a} = \overline{0}}]

  • Static strategy: treatment does not depend on time-varying covariates.
  • Dynamic strategy: treatment at time kk depends on the evolving covariate history (e.g. "Treat if CD4 drops below threshold").

Fine Point 19.1: Deterministic & Random Strategies

  • Deterministic dynamic strategy:

    g=(g0(a‾−1,l0),...,gK(a‾K−1,l‾K))g = \left( g_0(\overline{a}_{-1}, l_0), ..., g_K(\overline{a}_{K-1}, \overline{l}_K) \right)

  • Static deterministic strategy: gkg_k depends only on past treatments.
  • Random strategy: assigns a probability of treatment at each time (static or dynamic).
  • Optimal strategy: maximizes E[Yg]E[Y^g]; usually dynamic for drug interventions.

中文

  • 处理策略：给每个时间点kk分配处理的规则。
  • 举例：
    • 始终治疗：a‾=(1,1,...,1)\overline{a} = (1, 1, ..., 1)
    • 从不治疗：a‾=(0,0,...,0)\overline{a} = (0, 0, ..., 0)
  • 策略的平均因果效应：
    E[Ya‾=1‾]−E[Ya‾=0‾]E[Y^{\overline{a} = \overline{1}}] - E[Y^{\overline{a} = \overline{0}}]
  • 静态策略：处理与协变量无关。
  • 动态策略：处理取决于随时间变化的协变量，如CD4计数。

19.3 Sequentially Randomized Experiments (SRE) 顺序随机实验

Main Points

  • SRE: At each time kk, treatment AkA_k is randomly assigned, possibly depending on observed history only (A‾k−1,L‾k\overline{A}_{k-1}, \overline{L}_k).
  • Causal diagrams:
    1. No arrows from L‾k\overline{L}_k or U‾k\overline{U}_k to AkA_k—ideal randomization.
    2. Only from measured covariates—randomization conditional on observed history.
    3. Arrows from both measured and unmeasured—unmeasured confounding (typical in observational studies).
  • Sequential randomization eliminates time-varying confounding by unmeasured variables.

中文

  • 顺序随机实验（SRE）：每个时间点kk，AkA_k在观测历史下随机分配。
  • 三种因果图：
    1. 没有从L‾k\overline{L}_k或U‾k\overline{U}_k到AkA_k的箭头——理想随机化。
    2. 只有测量协变量——观测历史条件下的随机化。
    3. 有未测量协变量——有未测量混杂的观察性研究。
  • 顺序随机分配消除了未测变量带来的时间变化混杂。

19.4 Sequential Exchangeability 顺序可交换性

Main Points

  • Exchangeability (time-fixed): Ya⊥ ⁣ ⁣ ⁣⊥A∣LY^a \perp\!\!\!\perp A \mid L
  • Sequential conditional exchangeability (SCE): At each time, treatment is as if randomized given past (L‾k\overline{L}_k, A‾k−1\overline{A}_{k-1}).
    • Two time points:

      Yg⊥ ⁣ ⁣ ⁣⊥A0∣L0Yg⊥ ⁣ ⁣ ⁣⊥A1∣A0=g(L0),L0,L1\begin{align*} Y^g &\perp\!\!\!\perp A_0 \mid L_0 \\ Y^g &\perp\!\!\!\perp A_1 \mid A_0 = g(L_0), L_0, L_1 \end{align*}

    • General (for all kk and strategies gg):

      Yg⊥ ⁣ ⁣ ⁣⊥Ak∣A‾k−1=g(A‾k−2,L‾k−1),L‾kY^g \perp\!\!\!\perp A_k \mid \overline{A}_{k-1} = g(\overline{A}_{k-2}, \overline{L}_{k-1}), \overline{L}_k

  • Holds in SRE and some observational studies if treatment depends only on measured history.

Identification:

  • If SCE holds, means E[Yg]E[Y^g] is identified (using g-formula) for all (static/dynamic) gg.
  • Sequential positivity and consistency are also required (see below).

中文

  • 可交换性（固定处理）：Ya⊥ ⁣ ⁣ ⁣⊥A∣LY^a \perp\!\!\!\perp A | L
  • 顺序条件可交换性：每个时间点，处理在既往历史下如同随机。
    • 两时间点举例：
      Yg⊥ ⁣ ⁣ ⁣⊥A0∣L0Y^g \perp\!\!\!\perp A_0 | L_0
      Yg⊥ ⁣ ⁣ ⁣⊥A1∣A0=g(L0),L0,L1Y^g \perp\!\!\!\perp A_1 | A_0 = g(L_0), L_0, L_1
    • 一般化：
      Yg⊥ ⁣ ⁣ ⁣⊥Ak∣A‾k−1=g(A‾k−2,L‾k−1),L‾kY^g \perp\!\!\!\perp A_k | \overline{A}_{k-1} = g(\overline{A}_{k-2}, \overline{L}_{k-1}), \overline{L}_k
  • SRE和某些观察数据在满足条件下成立。
  • 只要顺序可交换性成立，就能识别所有（静态和动态）策略的E[Yg]E[Y^g]。

19.5 Positivity and Consistency for Time-varying Treatments

Technical Point 19.2:

Positivity

  • Fixed: fL(l)≠0  ⟹  fA∣L(a∣l)>0f_L(l) \neq 0 \implies f_{A|L}(a|l) > 0
  • Sequential: If fA‾k−1,L‾k(a‾k−1,l‾k)≠0f_{\overline{A}_{k-1}, \overline{L}_k}(\overline{a}_{k-1}, \overline{l}_k) \neq 0, then
    fAk∣A‾k−1,L‾k(ak∣a‾k−1,lk)>0f_{A_k|\overline{A}_{k-1}, \overline{L}_k}(a_k|\overline{a}_{k-1}, l_k) > 0 for all (a‾k,lk)(\overline{a}_k, l_k)

Consistency

  • Fixed: If A=aA=a for an individual, Ya=YY^a = Y for that person.
  • Sequential:

    Ya‾=Ya‾∗ if a‾∗=a‾Ya‾=Y if A=a‾Y^{\overline{a}} = Y^{\overline{a}^*} \text{ if } \overline{a}^* = \overline{a} \\ Y^{\overline{a}} = Y \text{ if } A = \overline{a}

  • For dynamic strategies, if at each kk, Ak=gk(Ak−1,Lk)A_k=g_k(A_{k-1}, L_k) for an individual, then Yg=YY^g = Y.

中文

  • 可积极性（positivity）：保证每个处理历史下各处理分配概率均大于零。
  • 一致性（consistency）：个体遵循所关注策略时，其反事实结果与观测结果一致。

19.6 Identifiability under Some but Not All Treatment Strategies

Key Technical Detail: SWIGs and Sequential Exchangeability

  • SWIG (Single World Intervention Graph): Used to directly check for sequential exchangeability and identification by d-separation.
  • Static Sequential Exchangeability:

    Ya‾⊥ ⁣ ⁣ ⁣⊥Ak∣A‾k−1=a‾k−1,L‾kY^{\overline{a}} \perp\!\!\!\perp A_k \mid \overline{A}_{k-1} = \overline{a}_{k-1}, \overline{L}_k

    Sufficient for static strategies.
  • Dynamic strategies identification requires stronger exchangeability. Not always possible if unmeasured common causes (e.g., "W") exist between A0A_0 and L1L_1 (see diagrams).
  • Fine Point: Even if L1L_1 is not a cause of YY, adjusting for L1L_1 may still be needed to block backdoor paths.

中文

  • SWIG图：可视化确认交换性与可识别性的重要工具。
  • 静态顺序可交换性：只对静态策略成立。
  • 动态策略 需更强的可交换性条件，若有未测共同原因则不能识别。

19.7 Time-varying Confounding and Time-varying Confounders

Definitions & Main Points

  • Time-varying confounders (L‾k\overline{L}_k): variables that:

    • Affect subsequent treatment and outcome (possibly via unmeasured UkU_k).
    • Change over time.

  • Need to adjust for L‾k\overline{L}_k at each kk to block backdoor path Ak←Lk←Uk→YA_k \leftarrow L_k \leftarrow U_k \rightarrow Y.

  • Definition (Fine Point 19.3):

    • There is confounding for E[Ya‾]E[Y^{\overline{a}}] if

      E[Ya‾]≠E[Y∣A=a‾]E[Y^{\overline{a}}] \neq E[Y|A=\overline{a}]

    • If
      E[Ya‾∣L0]=E[Y∣A=a‾,L0]E[Y^{\overline{a}}|L_0] = E[Y|A = \overline{a}, L_0]
      then confounding is time-fixed; otherwise mixing (time-varying) confounding is present.

  • Unmeasured confounding: Any violation of identifiability conditions or unmeasured common causes (e.g. UkU_k).

  • Time-varying confounders (also called time-dependent confounders): must be measured and adjusted via appropriate methods (see g-methods in next chapters).

中文

  • 时间变化混杂变量（L‾k\overline{L}_k）：影响未来处理及结局，并且随时间变化。
  • 每个时间点需调整L‾k\overline{L}_k以阻断反门路径。
  • 定义（细节19.3）：
    • 若 E[Ya‾]≠E[Y∣A=a‾]E[Y^{\overline{a}}] \neq E[Y|A = \overline{a}]，则存在混杂。
    • 若 E[Ya‾∣L0]=E[Y∣A=a‾,L0]E[Y^{\overline{a}}|L_0] = E[Y|A = \overline{a}, L_0]，则只有时间固定混杂。
  • 未测混杂：可识别性条件被破坏或存在未测共同原因。

Summary Table of Technical/Fine Points and Core Formulas

ConceptFormula (English)公式 (中文)
ACE (fixed treatment)E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}]E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}]
ACE (time-varying)E[Ya‾1]−E[Ya‾0]E[Y^{\overline{a}_1}] - E[Y^{\overline{a}_0}]时间序列两组策略的均值差
Sequential Exchangeability$Y^g \perp!!!\perp A_k \mid \overline{k-1}=g(\overline{A},\overline{L}_{k-1}),\overline{L}_k $顺序条件可交换性
Positivity (seq.)$f_{\overline{A}_{k-1},\overlinek}(\overline{a},\overline{l}k)\neq 0 \implies f{A_k\overline{A}_{k-1},\overline{L}_k}(a_k
Consistency (seq.)Ya‾=YY^{\overline{a}} = Y if A=a‾A=\overline{a}一致性
Time-varying confounding$E[Y^{\overline{a}}] \neq E[YA = \overline{a}]$
Static Seq. Exchangeability$Y^{\overline{a}} \perp!!!\perp A_k\overline{k-1}=\overline{a},\overline{L}_k $

In summary...（总结）

  • 核心思想：时间变化处理和动态策略背景下的因果推断需明确定义策略，关注顺序可交换性、可积极性和一致性。
  • 模型识别：是否能用观测数据识别因果效应，取决于所考虑策略、变量测量完整性与混杂结构。
  • 时间变化混杂变量的调整必须使用g-methods（第21章详述）而非普通回归调整。

The next chapter explains how to actually estimate these causal effects (g-methods).