Chapter 19 Time-varying Treatments

文章目录

19.1 The causal effect of time-varying treatments
19.2 Treatment strategies
19.3 Sequentially randomized experiments
19.4 Sequential exchangeability
19.5 Identifiability under some but not all treatment strategies
19.6 Time-varying confounding and time-varying confounders
Fine Point
- Deterministic and random treatment strategies
- Per-protocol effects to compare treatment strategies
- Dynamic strategies that depend on baseline covariates
- A definition of time-varying confounding
Technical Point
- On the definition of dynamic strategies
- Positivity and consistency for time-varying treatments
- The many forms of sequential exchangeability

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这一章开始介绍treatments如果是随时间变化的情况.

19.1 The causal effect of time-varying treatments

Aˉk=(A0,A1,⋯,Ak),\bar{A}_k = (A_0, A_1, \cdots, A_k), Aˉk=(A0,A1,⋯,Ak),
即在不同的时间段可能采取不同的方案AiA_iAi.

我们可以考察
E[Yaˉ],\mathbb{E} [Y^{\bar{a}}], E[Yaˉ],
这里aˉ\bar{a}aˉ是如上的一种配置.

19.2 Treatment strategies

除了上面的有限的组合外(2K2^K2K?), 还有额外的特别的strategy.
比如, A1=1A_1=1A1=1的时候, A2A_2A2有50%的概率为1, A1=0A_1=0A1=0的时候, A2A_2A2有70%的概率为1.
显然这种情况无法用任意固定的aˉ\bar{a}aˉ表示, 所以我们也称aˉ\bar{a}aˉ为静态的治疗策略, 而上述的为动态的治疗策略, 用g=gk(aˉk−1,lˉk)g=g_k(\bar{a}_{k-1}, \bar{l}_{k})g=gk(aˉk−1,lˉk).
其中lkl_klk是一些其他的变量, 比如一些confounders.

19.3 Sequentially randomized experiments

如上图所示, 图片19.1, 19.2可以表示序列随机实验(自然也可以表示为观测数据), 因为每一个阶段AkA_kAk的选择只依赖前面的aˉk−1\bar{a}_{k-1}aˉk−1(19.1)或者加上当前状态lkl_klk(19.2).
图19.3因为每一个阶段的treatment还与未观测变量有关, 所以不可能是序列随机实验.

19.4 Sequential exchangeability

Yg⨿Ak∣Aˉk−1=g(Aˉk−2,Lˉk−1),Lˉk,∀g,k=0,1,⋯,K.Y^g \amalg A_k | \bar{A}_{k-1} = g(\bar{A}_{k-2}, \bar{L}_{k-1}), \bar{L}_k, \quad \forall g, k=0,1,\cdots, K. Yg⨿Ak∣Aˉk−1=g(Aˉk−2,Lˉk−1),Lˉk,∀g,k=0,1,⋯,K.

19.5 Identifiability under some but not all treatment strategies

利用SWIG图来判断序列可交换性是非常有效的.

图19.5中的情况,
Ya0,a1⨿A0,Ya0,a1⨿A1∣A0=a0,L1,Y^{a_0, a_1} \amalg A_0, Y^{a_0, a_1} \amalg A_1|A_0=a_0, L_1, Ya0,a1⨿A0,Ya0,a1⨿A1∣A0=a0,L1,

图19.6虽然多了W0W_0W0, 但是序列可交换性是一样的(注意这里研究的是静态的情况).
这些可由下面的SWIG图推得.

当采用动态的治疗策略的时候, 情况就要复杂一些:

图19.9的可交换性没有变, 但是19.10的就变了, Ya0,a1Y^{a_0, a_1}Ya0,a1与A0A_0A0不再独立, 因为此时二者存在一个backdoor.

究其原因就是, 因为是动态策略, 所以g1g_1g1不再是常数, 是一个和L1L_1L1有关联的变量, 所以打通了backdoor.
这是序列可交换性的特别之处.

19.6 Time-varying confounding and time-varying confounders

和普通的概念是类似的.

Fine Point

Deterministic and random treatment strategies

Per-protocol effects to compare treatment strategies

Dynamic strategies that depend on baseline covariates

A definition of time-varying confounding

Technical Point

On the definition of dynamic strategies

Positivity and consistency for time-varying treatments

iffAˉk−1,Lˉk(aˉk−1,lˉk)≠0,thenfAk∣Aˉk−1,Lˉk(ak∣aˉk−1,lˉk)>0,∀(aˉk,lˉk).\mathrm{if}\: f_{\bar{A}_{k-1}, \bar{L}_k}(\bar{a}_{k-1}, \bar{l}_k) \not = 0, \mathrm{then}\: f_{A_k|\bar{A}_{k-1}, \bar{L}_k}(a_k|\bar{a}_{k-1}, \bar{l}_k) > 0, \forall (\bar{a}_k, \bar{l}_k). iffAˉk−1,Lˉk(aˉk−1,lˉk)=0,thenfAk∣Aˉk−1,Lˉk(ak∣aˉk−1,lˉk)>0,∀(aˉk,lˉk).

The many forms of sequential exchangeability