Chapter_3

Aaron L9/17/25About 9 min

Chapter 3 Observational Studies — Key Points (中英文对照)

核心概念概览

This chapter discusses under what circumstances observational studies can deliver valid causal inferences, focusing on three key identifiability conditions:

Consistency (一致性)
Exchangeability (可交换性)
Positivity (正的概率性/可行性)

These conditions are collectively required to identify causal effects in observational data, allowing us to treat the data as if they arose from a (conditionally) randomized experiment.

本章重点讨论在何种条件下，观察性研究可以得出有效的因果推断，核心在于三个“可识别性假设”，即一致性、可交换性与正则性。

1. Identifiability Conditions (可识别性条件)

英文总结

Identifiability: A causal effect is identifiable if the observed data, together with the assumptions, imply exactly one value for the causal effect. Otherwise, it's not identifiable.
In randomized experiments, identifiability is ensured by design (due to randomization).
In observational studies, identifiability requires the investigator to assume these conditions, since treatment assignment is not random.

The Three Conditions

Consistency:
- The treatment values must represent actual, specific, and well-defined interventions, consistent with what's present in the data.
Exchangeability (Conditional):
- The distribution of counterfactual outcomes is the same between treated and untreated, conditional on measured covariates $L$ , i.e.,
$Y^a \perp\!\!\!\perp A \mid L$
Positivity:
- Every subject must have a positive probability of receiving each treatment, given their covariates:
$\Pr[A = a \mid L = l] > 0 \quad \text{for all } a, \text{ and } l \text{ with } \Pr[L = l] \neq 0$

中文总结

可识别性：若在一组假设下，观测到的数据分布仅支持某一特定的因果效应数值，则称该因果效应（非参数）可识别；反之则不可识别。
在随机试验中，可识别性由设计保证（原因是随机分配）。
在观察性研究中，需依赖研究者的假设前提，因为处理的分配非随机。

三个条件

一致性：
- 比较的处理值必须代表实际、具体和明确定义的干预，与数据中实际的处理版本一致。
（条件）可交换性：
- 条件于观测协变量 $L$ ，处理组与对照组的反事实结果分布相同，即
$Y^a \perp\!\!\!\perp A \mid L$
正则性（正的概率性）：
- 每个受试者在其协变量 $L$ 下，接受各种处理的概率均大于0：
$\Pr[A = a \mid L = l] > 0 \quad \text{对于所有满足 } \Pr[L = l] \neq 0 \text{ 的 } l$

2. Exchangeability (可交换性)

英文要点

Meaning: If treated individuals had (counterfactually) not been treated, they would have had the same expected outcome as the untreated (conditional on $L$ ).
Formally: $Y^a \perp\!\!\!\perp A | L$
In practice, requires that all confounders (variables affecting both treatment and outcome) are controlled for by $L$ . If unmeasured confounders $U$ exist, exchangeability fails.
Impossible to empirically test whether exchangeability holds, because we can't observe counterfactuals ( $Y^a$ for those with $A \neq a$ ).
Fine Point 3.1: Identifiability depends on exchangeability; without it, observed associations do not necessarily reflect causal effects.
Even measuring more variables cannot guarantee exchangeability; it depends on substantive knowledge, hence always involves some unverifiable assumptions.

中文要点

含义：若对照组和处理组在给定 $L$ 后反事实预期结果一致，则两组可交换。
形式化表达： $Y^a \perp\!\!\!\perp A | L$
需要 $L$ 包含所有影响处理和结局的混杂因素。若存在未测量变量 $U$ （未收集到的混杂因素），则可交换性不成立。
可交换性无法通过观测数据检验，因为不能同时观测到某个个体的 $A=a$ 和 $A \neq a$ 时的 $Y^a$ 。
精细点3.1：因果效应的可识别性依赖于可交换性，否则观测关联无法反映因果效应。
即使收集更多变量，也不能绝对保证可交换性，需依赖领域知识和假设。

3. Positivity (正则性/可行性)

英文要点

Definition: For every subgroup defined by $L=l$ , there must be at least some individuals who received every treatment ( $A$ ) of interest, i.e. $\Pr[A = a \mid L = l] > 0$ for all $a$ and for all $l$ such that $\Pr[L = l] \neq 0$ .
Positivity can sometimes be checked in data (for measured variables $L$ ).
If positivity fails, neither standardization nor IP weighting can validly estimate average causal effects.
Technical Point 3.1: When positivity fails, the IP-weighted mean and standardized mean are undefined or lack causal interpretation because the groups being contrasted are different (don't overlap in $L$ ).

中文要点

定义：对于协变量 $L=l$ 定义的每个亚组，必须至少有一部分个体接受所有关心的处理（ $A$ ），即 $\Pr[A = a \mid L = l] > 0$ 对于所有 $a$ 以及所有 $\Pr[L = l] \neq 0$ 的 $l$ 。
正则性在观测数据（对于已测量的 $L$ ）时有时可以检查。
若正则性不成立，标准化和IP加权都无法得出有效的因果估计。
技术要点3.1：正则性不满足时，IP加权和标准化均不具备原本的因果效应含义，因为参与比较的组发生了变化（不同组的 $L$ 值没有交集）。

4. Consistency (一致性)

英文要点

Consistency: For people with $A = a$ , the observed outcome $Y$ equals the counterfactual outcome under $a$ , i.e., $Y^a = Y$ for those with $A = a$ .
Two aspects:
1. Well-defined counterfactuals: The intervention $a$ must be precise, with all relevant details specified.
2. Linkage: The versions of treatment in the data must match those for which the causal effect is defined.
Consistency can fail:
- If the definition of treatment is too vague (e.g., "weight gain" without specifying how).
- If the treatment of interest isn't even present in observed data (no one received $a$ ).
Treatment Variation Irrelevance: Sometimes it may be justified to group near-identical versions of $a$ into one treatment, but this is an assumption.

中文要点

一致性：对于 $A=a$ 的个体，观测到的 $Y$ 等于在 $a$ 下的反事实结果 $Y^a$ ，即 $Y^a = Y$ 对于 $A = a$ 成立。
两大方面：
1. 反事实 $Y^a$ 的定义必须明确，干预 $a$ 需具体描述所有重要细节。
2. 需确保数据中的处理与定义的干预吻合，二者可链接。
一致性可能失效的两种情形：
- 干预定义不明确，如只说“体重增加”而未说明具体方式。
- 数据中未包含所关心的干预版本（如所有人接收到的 $a$ 都不是我们想研究的 $a$ ）。
治疗版本无关性假设：有时可合理地将几种近似处理合并，但这是一种额外假设。

5. The Target Trial (目标试验)

英文要点

When using observational data to answer causal questions, we should explicitly specify the target trial—a hypothetical randomized experiment we wish we could conduct.
Emulating a target trial makes:
- The causal question precise and interventions well-defined.
- It easier to check whether necessary conditions (exchangeability, etc.) can plausibly hold.
- The analysis more transparent and interpretable for decisions.
Not all causal questions can be turned into target trials; if you can't clearly define the intervention, stick to prediction rather than causal inference.

中文要点

在用观察性数据推断因果时，应该明确指定“目标试验”——你希望可以做的假想随机试验。
目标试验的拟合（emulation）可以：
- 明确干预和因果问题；
- 更容易检查是否满足可交换性等必要条件；
- 提升分析的透明度和可解释性，便于决策。
并非所有因果问题能转化为目标试验。如果无法清晰、具体地定义干预，则更适合做预测而非因果推断。

6. Technical Point & Fine Points 总结

Technical Point 3.1 (标准化/加权的正则性)

The standardized mean: $\sum_l E[Y|A = a, L = l] \Pr[L = l]$ is only defined when $\Pr[A = a | L = l] > 0$ for all observed $l$ .
If positivity fails, IP weights and standardization lose their causal meaning.
Under positivity, with exchangeability, the difference in IP weighted means is the causal effect.

技术要点3.1（标准化/加权的正则性）

标准化均值： $\sum_l E[Y|A = a, L = l] \Pr[L = l]$ 仅当所有观测到的 $l$ 有 $\Pr[A = a | L = l] > 0$ 时才可计算。
若正则性不成立，IP加权和标准化无法给出因果效应。
满足正则性和可交换性时，IP加权均值的差异等于平均因果效应。

Fine Point 3.2 (Crossover designs)

Key point: For crossover experiments estimating individual causal effects, "no carryover effect" and other stringent conditions are required; in practice, these often don't hold.

要点：交叉实验要识别个体因果效应需极强假设（比如没有残留效应），实际中往往无法满足。

Fine Point 3.3 (States vs. interventions)

Variables like "obesity" or "socioeconomic status" can be ill-defined as interventions.
Must specify how the change occurs to render $Y^a$ well-defined.
Otherwise, the "causal effect of being obese" is vague; only the effect of a specific, well-specified intervention is meaningful.
“肥胖”等状态变量通常缺乏良好定义的干预，需要具体说明变化方式，否则反事实 $Y^a$ 本身不明确。

Fine Point 3.4 (Protocol interpretation)

Even randomized experiments may yield different results, if the protocol omits important elements (e.g., surgeon experience).
Thus, the same "treatment" label $a$ can have different meanings and effects across studies, threatening transportability.
即便随机试验，如果方案未细化（比如手术医生经验），不同实验得到的“因果效应”会不同。同一 $a$ 标签在不同研究中含义可能不同，影响结果的可移植性。

Fine Point 3.5 (Possible worlds)

Philosophers' "closest possible world" view always gives consistency ( $Y^a = Y$ for $A=a$ ), but counterfactuals are ill-defined when $A\neq a$ .
Modern causal inference insists counterfactuals must be constructed via specific, well-defined interventions.
哲学上的“最近可能世界”总让 $Y^a = Y$ （对 $A=a$ ），但 $A\neq a$ 时，反事实总是含糊的。现代因果推断强调反事实需由具体、明确定义的干预来决定。

7. Attributable Fraction (可归因分数)

Measures the proportion of cases in the observed data that "would not have occurred" had everyone received a reference treatment $a=0$ :
$\text{Attributable Fraction} = \frac{\Pr[Y = 1] - \Pr[Y^{a=0} = 1]}{\Pr[Y = 1]}$
This is different from the "etiologic fraction" (proportion of cases mechanically caused by the exposure).
描述在观测数据中，若每个人都接受参考处理( $a=0$ )，可避免的发病比例：
$\text{可归因分数} = \frac{\Pr[Y = 1] - \Pr[Y^{a=0} = 1]}{\Pr[Y = 1]}$

8. When Target Trial Cannot Be Emulated (目标试验无法实现时)

When interventions can't be well-defined, and target trial can't be emulated, even approximate inference is impossible; only predictive (associational) statements remain.
若干预无法明确定义或目标试验无法拟合，只能做预测（相关），而非因果推断（反事实预测）。

结论

Randomized experiments provide causal effect estimates by design.
Observational studies require that the three identifiability conditions (consistency, exchangeability, positivity) hold as assumptions to draw causal conclusions.
Explicitly specifying a target trial clarifies causal questions and the plausibility of identifiability assumptions.
If interventions cannot be well defined, restrict to prediction rather than causal inference.

全章公式小结（Summary of Key Formulas）

1. Conditional Exchangeability (条件可交换性):

Y^a \perp\!\!\!\perp A \mid L

2. Positivity (正则性):

\Pr[A = a \mid L = l] > 0 \quad \forall \ a, \ l : \Pr[L = l] \neq 0

3. Standardized Mean (标准化均值，可用于估计反事实均值):

\sum_l E[Y | A = a, L = l] \Pr[L = l]

4. Attributable Fraction (可归因分数):

\text{Attributable Fraction} = \frac{\Pr[Y = 1] - \Pr[Y^{a=0} = 1]}{\Pr[Y = 1]}

[For further technical details, see the respective Fine and Technical Points as outlined above.]