Chapter_6

Aaron L9/17/25About 10 min

Chapter 6: Graphical Representation of Causal Effects

6.1 Causal Diagrams

ENGLISH

Causal inference often relies on expert knowledge and untestable assumptions about the causal network connecting variables such as treatment (A), outcome (Y), and covariates (L).
As problems get more complex, it becomes crucial to explicitly represent our assumptions about causal relationships among variables.
Causal diagrams (also known as Directed Acyclic Graphs, DAGs) are graphical tools to represent qualitative knowledge and assumptions about causal structure.
- Directed: edges (arrows) have a specific direction, e.g., ( L \to A ).
- Acyclic: No cycles allowed (i.e., a node cannot be its own ancestor).
KEY CONVENTION:
- An arrow from (V) to (W) indicates a direct causal effect of (V) on (W).
- Absence of an arrow means no direct causal effect for any individual in the population.
- The diagram does not specify whether effects are harmful or protective, or encode interactions between causes.

中文

因果推断通常依赖于专家知识和关于变量（如处理A，结果Y，协变量L）之间因果网络的不可检验假设。
随着问题复杂度提升，明确表达我们关于变量间因果关系的假设变得非常重要。
因果图（有向无环图，DAG）是一种用于表达因果结构中的定性知识和假设的图形工具。
- 有向：箭头有方向，例如 ( L \to A )。
- 无环： 不允许有回路（即节点不能成为自身祖先）。
核心约定：
- 从 (V) 指向 (W) 的箭头表示 (V) 对 (W) 有直接因果作用。
- 没有箭头表示在整个人群中不存在这种直接因果作用。
- 图中不会区分这种作用是有害还是保护，也不编码多因子的交互作用。

Technical Point 6.1: Causal DAGs 数学定义

ENGLISH

A DAG ( G ) has nodes (random variables) ( V = (V_1,\ldots,V_M) ).
Parents: ( PA_m ) is the set of nodes with arrows into ( V_m ).
Markov Factorization:
The joint distribution ( f(v) ) factors according to the DAG if: $f(v) = \prod_{j=1}^M f(v_j | pa_j)$
Causal DAG Requirements:
1. Absence of an arrow from ( V_j ) to ( V_m ) implies no direct causal effect.
2. All common causes (even unmeasured) of any two variables must be on the graph.
3. Every variable is a cause of its descendants.
Causal Markov Assumption: Conditional on its direct causes (parents), a variable is independent of any variable it does not cause.

中文

一个DAG（有向无环图）( G ) 的节点为随机变量 ( V = (V_1,\ldots,V_M) )。
父节点（Parents）： ( PA_m ) 表示所有指向 ( V_m ) 的节点集合。
马尔可夫分解： $f(v) = \prod_{j=1}^M f(v_j | pa_j)$
因果DAG要求：
1. 没有箭头 ( V_j \to V_m ) 表示 ( V_j ) 对 ( V_m ) 没有直接因果作用。
2. 任何两个变量的共同原因（即使未被观测）都必须包含在图中。
3. 任一变量都是其后代的原因。
因果马尔可夫假设： 在已知其直接原因的条件下，一个变量与任何不是其结果的变量独立。

6.2 Causal Diagrams and Marginal Independence

ENGLISH

Causal diagrams also encode expectations about associations (correlations) as well as causation.
Key Results:
1. If (A) has a causal effect on (Y), they are generally associated:
  - ( \Pr[Y^{a=1}=1] \neq \Pr[Y^{a=0}=1] \implies \Pr[Y=1|A=1] \neq \Pr[Y=1|A=0] )
2. If (A) has no direct effect but shares a common cause (L) with (Y), (A) and (Y) are associated due to (L).
3. If (A) and (Y) only share a common effect (collider) (L), they are marginally independent; the collider blocks association.

中文

因果图除了表达因果关系，还隐含了变量间相关性（关联）的期望。
主要结论：
1. 如果 (A) 对 (Y) 有因果作用，则两者一般相关： $\Pr[Y^{a=1}=1] \neq \Pr[Y^{a=0}=1] \implies \Pr[Y=1|A=1] \neq \Pr[Y=1|A=0]$
2. 若 (A) 与 (Y) 没有直接作用，但有共同原因 (L)，则 (A) 与 (Y) 由于 (L) 而相关。
3. 如果 (A) 和 (Y) 只在一个共同结果（碰撞点，collider）(L) 处关联，则它们在边际上独立，碰撞点阻断了相关性。

Technical Point 6.2: DAGs and Counterfactual Models

ENGLISH

Each causal DAG ( G ) represents an underlying counterfactual model.
Nonparametric Structural Equation Model (NPSEM):
- Each node ( V_m ) is determined by a function ( f_m(pa_m, \epsilon_m) ).
- NPSEM implies the existence of counterfactuals for every variable.
Factual and counterfactual values are constructed recursively.
Different Models:
- NPSEM-IE: Assumes all errors ( \epsilon_m ) are mutually independent.
- FFRCISTG: Fully randomized model with fewer independence assumptions, implies causal Markov under weaker assumptions.

中文

每个因果DAG ( G ) 对应一个反事实模型。
非参数结构方程模型（NPSEM）：
- 每个变量 ( V_m ) 由 ( f_m(pa_m, \epsilon_m) ) 决定。
- 该模型假定每个变量的反事实值都存在。
实际值与反事实值递归构建。
不同类型模型：
- NPSEM-IE： 假设所有误差项 ( \epsilon_m ) 相互独立。
- FFRCISTG： 完全随机化结构图，独立假设更弱，但足以推导因果马尔可夫性。

Technical Point 6.3: Independencies in Counterfactual Models

ENGLISH

Key Idea: NPSEM-IE implies stronger independence than FFRCISTG.
Exchangeability:
- Under NPSEM-IE: ( (Y^{a=0}, Y^{a=1}) \perp!!!\perp A ) (full exchangeability)
- Under FFRCISTG: ( Y^a \perp!!!\perp A ) for ( a\in{0,1} ) (marginal exchangeability)
Unless otherwise stated, a DAG represents FFRCISTG "as detailed as the data".

中文

核心思想： NPSEM-IE 推导的独立性比 FFRCISTG 更强。
可交换性：
- NPSEM-IE下：( (Y^{a=0}, Y^{a=1}) \perp!!!\perp A )（完全可交换）
- FFRCISTG下：( Y^a \perp!!!\perp A )（边际可交换性）
默认情况下，DAG代表“与数据细节一致”的FFRCISTG。

6.3 Conditional Independence in Causal Diagrams

ENGLISH

Marginal (unconditional) association changes when conditioning on variables.
General Heuristic Rules:
- Conditioning on a mediator blocks the association between treatment and outcome.
  - Eg: If (A) causes (B), which causes (Y), then (A) and (Y) are independent given (B).
- Conditioning on a common cause blocks spurious association:
  - If (L) causes both (A) and (Y), conditioning on (L) makes (A \perp!!!\perp Y | L ).
- Conditioning on a collider or its descendant opens an otherwise blocked path, producing spurious association:
  - If (A \to L \leftarrow Y), then (A \perp!!!\perp Y); but ( A \not\perp!!!\perp Y|L ).

中文

边际（无条件）关联在给定其他变量后可能发生变化（条件独立性）。
一般图解规则：
- 条件在中介变量上，阻断了处理与结果间的关联。
  - 如：若 (A\to B\to Y)，则有 (A \perp!!!\perp Y | B )。
- 条件在共同原因上，阻断伪相关：
  - 若 (L) 同时作用于 (A, Y)，则 (A \perp!!!\perp Y | L )。
- 条件在碰撞点或其后代上，打开原本被阻断的路径，引入伪相关：
  - 如 (A \to L \leftarrow Y)，则 (A \perp!!!\perp Y)；但 (A \not\perp!!!\perp Y|L)。

Fine Point 6.1: d-Separation

A systematic summary of graphical rules that determine conditional independence:

ENGLISH
A path between two nodes is blocked if and only if:

It contains a collider that is not conditioned on (nor any of its descendants).
It contains a non-collider node that has been conditioned on.

d-separation Definition:

Two sets of variables (A) and (B) are d-separated by (C) if all paths between a variable in (A) and a variable in (B) are blocked by (C).
Pearl's theorem: The causal Markov assumption implies:
If (A) is d-separated from (B) given (C), then (A) is statistically independent of (B) given (C).

中文
d-分离法则：
对于路径，只如果以下成立，则路径“被阻断”：

路径上存在碰撞点，且未对其（或其后代）进行条件限制。
路径上存在非碰撞点，且对其进行了条件限制。

d-分离定义：

如果变量集A与B之间所有路径都被C阻断，则称A与B在C条件下d-分离。
Pearl定理： 如果A与B在C下d-分离，则A与B在C条件下统计独立。

Fine Point 6.2: Faithfulness

ENGLISH

Faithfulness assumption: If (A \perp B \mid C) holds statistically, then (A) is d-separated from (B) by (C) in the DAG.
Unfaithful distributions can arise e.g. from effect modification with perfect cancellation (so average effect is zero but individuals have effects).
Violated by design in certain matched studies (associational paths cancel).
In practice, faithfulness is generally assumed unless deterministic relationships or design cause violations.

中文

忠实性假设： 若(A)在(C)下与(B)统计独立，则在DAG中A与B在C下d-分离。
不忠实常因交互作用恰好抵消或设计（如匹配研究）产生。
实务中一般假定忠实性，除非存在确定性关系或实验设计导致违背。

6.4 Positivity and Consistency in Causal Diagrams

ENGLISH

Positivity:
- For every possible set of covariates, there is a positive probability of receiving every treatment.
- DAGs cannot encode most violations of positivity (except in deterministic cases, depicted by bold arrows).
Consistency:
- Arrows from treatment to outcome must represent well-defined, unambiguous interventions.
- Interventions must be meaningfully defined; otherwise, "the effect of (A) on (Y)" becomes ambiguous (e.g., 'weight loss' by different methods leads to different implications).

中文

可积极性(positivity)：
- 对于每组协变量，接受所有处理选项的概率都必须大于零。
- DAG通常无法表达可积极性被破坏的情形（除非因果关系是确定性的，可以用粗体箭头表示）。
一致性(consistency)：
- 处理到结果的箭头必须代表清晰、明确的干预。
- 如果干预不明确，“A对Y的作用”就变得模糊（如：不同方式减重影响不同）。

Fine Point 6.3: Discovery of Causal Structure

ENGLISH

Possible to infer causal structure from data under faithfulness and with knowledge of time ordering, but often impossible in practice due to equivalence classes of DAGs.
In some scenarios, conditional independencies and association patterns can allow unique identification (e.g., in chains (Z\to A\to Y) with certain associations).

中文

如满足忠实性假设且有时序信息，在某些情形下（如三节点链）可通过数据唯一确定部分因果结构。
但通常，由于等价类DAG的存在，仅凭相关/独立结构难以唯一确定因果结构。

6.5 Structural Classification of Bias

ENGLISH

Systematic bias: Data do not permit identification of the causal effect even with infinite data.
Types of bias (source by DAG structure):
1. Confounding (common causes):
  - Treatment and outcome share a common cause, association ≠ effect.
2. Selection bias (conditioning on common effects):
  - Conditioning on a collider or its descendant generates spurious association.
Formal definitions:
- Unconditional bias: $\Pr[Y^{a=1}=1] - \Pr[Y^{a=0}=1] \neq \Pr[Y=1|A=1] - \Pr[Y=1|A=0]$ Occurs when ( Y^{a} \not\perp!!!\perp A ).
- Conditional bias: $\Pr[Y^{a=1}=1|L=l] - \Pr[Y^{a=0}=1|L=l] \neq \Pr[Y=1|A=1,L=l] - \Pr[Y=1|A=0,L=l]$ Occurs if ( Y^{a} \not\perp!!!\perp A|L=l ).

中文

系统性偏倚（bias）： 即使拥有无限大样本，数据依然无法识别因果效应。
偏倚分类（按因果结构）：
1. 混杂（共同原因）：
  - 处理与结果有共同原因，导致观测关联≠实际因果效应。
2. 选择偏倚（共同结果/碰撞点）：
  - 条件在碰撞点或其后代上，引入伪相关。
形式化定义：
- 无条件偏倚： $\Pr[Y^{a=1}=1] - \Pr[Y^{a=0}=1] \neq \Pr[Y=1|A=1] - \Pr[Y=1|A=0]$ 若( Y^{a} \not\perp!!!\perp A )。
- 有条件偏倚： $\Pr[Y^{a=1}=1 | L=l] - \Pr[Y^{a=0}=1 | L=l] \neq \Pr[Y=1 | A=1, L=l] - \Pr[Y=1 | A=0, L=l]$ 若( Y^{a} \not\perp!!!\perp A|L=l )。

6.6 Structure of Effect Modification

ENGLISH

Effect modification: The magnitude or direction of treatment effect varies by levels of a third variable (effect modifier).
Causal diagrams are less helpful for illustrating modification, since arrows simply indicate presence of direct effects, not their magnitude or sign.
Two types of effect modifiers:
- Causal effect modifier: Has a direct effect on the outcome.
- Surrogate effect modifier: Associated with the causal effect modifier (by cause, shared causes, or conditioning).
Causal DAGs distinguish associations, but not the type of interaction or the direction of effect modification.

中文

效应修饰(effect modification)： 处理效应的大小或方向随第三变量（效应修饰因子）变化。
因果图对效应修饰的表达有限，只表示直接作用的存在，无法表达其方向或大小。
效应修饰因子的分类：
- 因果修饰因子（causal effect modifier）： 对结果有直接作用。
- 代理修饰因子（surrogate effect modifier）： 与因果修饰因子有关，可以因果、共同原因或条件化产生关联。
因果DAG可以区分相关，不区分效应修饰的类型或方向。

Summary Table

Key Concept (English)	中文对应术语	Key Equation / Principle
Direct Causal Effect in DAG	直接因果作用	Arrow from ( V \to W ) shows direct effect
Markov Factorization	马尔可夫分解	( f(v) = \prod_{j=1}^M f(v_j
Causal Markov Assumption	因果马尔可夫假设	Conditional independence on direct causes
d-separation	d-分离	Path blocked: collider not conditioned, or non-collider conditioned
Faithfulness	忠实性	( A\perp B
Systematic bias	系统性偏倚	( \Pr[Y^{a=1}=1] - \Pr[Y^{a=0}=1] \ne \Pr[Y=1
Effect Modification	效应修饰	Size/direction of effect varies by modifier variable

Key Takeaways

Causal DAGs are essential for representing and reasoning about complex causal assumptions.
The causal Markov assumption and faithfulness link graphical structure and statistical independence.
d-separation provides systematic rules for reading conditional independence from a DAG.
Systematic bias arises mainly from confounding (common causes) and selection bias (conditioning on common effects).
Effect modification is not encoded in the structure of a DAG, only the existence of direct effects, not their form or variation.

(If you want specific illustrations for key DAGs or more practical examples in each category, please specify!)