Chapter 2: Randomized Experiments — Summary & Key Technical Points

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2.1 Randomization

Introduction to Causal Effects and Randomization

• Causal Question Example: Does you looking up make other people look up?
• Setup: Randomly assign action via coin flip, compare outcomes (proportion of "look up" responses).
• Result: 55% vs. 1% — strong evidence for a causal effect.

Key Concepts

• Experiment: Researcher controls the action.
• Randomization: Assignment determined by a random mechanism, not a deterministic rule.
• Why Randomization?
  • Avoids confounding: ensures groups are comparable.
  • Critics cannot argue that group differences are driving the observed effect (unlike sex-based assignment).

The Problem of Missing Counterfactuals

• In any real study, for an individual, only one potential outcome is observed:
  • $Y^{a=1}$ (under treatment)
  • $Y^{a=0}$ (under no treatment)
• Only $Y$ under $A$ (the treatment actually received) is observed. The other counterfactual is missing.

Table 2.1

• Illustrates observed data and missing counterfactuals for each individual.

Randomization Generates Exchangeability

• Exchangeability: Treated and untreated groups are comparable because any differences are due to chance.
• Even if the treatment allocation is flipped after randomization, the estimation of effects remains unchanged.
• Formalization: The distribution of outcomes under randomization makes:
  • $\Pr[Y^{a=1} = 1 | A=1] = \Pr[Y^{a=1} = 1 | A=0]$
  • $\Pr[Y^{a=1} = 1 | A=1] = \Pr[Y^{a=1} = 1]$ (marginal risk)
  • More generally, for all $a$, $Y^a \perp\!\!\!\perp A$
• Exogeneity is another term for exchangeability in this context.

Technical Point 2.1: Versions of Exchangeability

• Full Exchangeability: Randomization produces joint independence between all potential outcomes and $A$.
  • $Y^A \perp\!\!\!\perp A$ (joint vector of all counterfactuals)
• Mean Exchangeability: $E[Y^a|A=1] = E[Y^a|A=0]$ for all $a$.
  • Sufficient for causal identification: $E[Y^a] = E[Y|A=a]$
• For dichotomous outcomes, these are equivalent in practice, so simply called "exchangeability".

Important Distinction

• $Y^a \perp\!\!\!\perp A$ IS NOT $Y \perp\!\!\!\perp A$:
  • If treatment has causal effect, then $Y$ and $A$ will be associated, but the assignment of $A$ is unrelated to potential outcomes.

Fine Point 2.1: Crossover Experiments

• When each individual serves as their own control (receives both treatments at different times).
• Individual causal effects can only be identified if:
  1. No carryover effects: $Y^{a_0, a_1}_{i1} = Y^{a_1}_{i1}$
  2. Effects don't depend on time.
  3. Counterfactual under no treatment doesn't depend on time.
• Not generally possible for irreversible treatments/outcomes (e.g., heart transplants).

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2.2 Conditional Randomization

Conditional (Stratified) Randomization Designs

• Design 1 (Marginal Randomization):
  • All individuals randomized with the same probability.
  • Ensures marginal exchangeability: $Y^a \perp\!\!\!\perp A$ for all $a$.
• Design 2 (Conditional Randomization):
  • Probability of treatment assignment differs by a baseline factor $L$ (e.g., critical vs. noncritical condition).
  • Ensures conditional exchangeability: $Y^a \perp\!\!\!\perp A \mid L$, but not marginal exchangeability.

Definition

• Conditional Exchangeability: For all $l$,

  $$Y^a \perp\!\!\!\perp A \mid L=l$$

  Or, in risk terms:

  $$\Pr[Y^a=1|A=1,L=l] = \Pr[Y^a=1|A=0,L=l], \forall a, l$$

• Marginal exchangeability: Holds only if randomization is not conditional on $L$.

Data missing patterns:

• MCAR: Data are 'Missing Completely At Random', e.g., in marginal randomization.
• MAR: Data are 'Missing At Random' conditional on $L$, e.g., in conditional randomization.

Effect Estimation in Conditional Randomization

• Two main options:
  1. Stratum-specific (stratified) average causal effects (effect modification)
     • Estimate average causal effect within each strata $L=l$.
  2. Marginal (overall) average causal effect
     • Average over the population distribution of $L$.

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2.3 Standardization

Standardization Formula for the Marginal Causal Effect

• Let $L$ be a pre-treatment variable (e.g., critical status).
• For each level $l$, compute observed risks:

  $$\Pr[Y=1|L=l,A=a]$$

• Marginal potential outcome risk:

  $$\Pr[Y^a=1] = \sum_l \Pr[Y^a=1|L=l] \Pr[L=l]$$

• Under conditional exchangeability:

  $$\Pr[Y^a=1] = \sum_l \Pr[Y=1|L=l,A=a] \Pr[L=l]$$

• Causal risk ratio by standardization:

  $$\text{Causal RR} = \frac{\sum_l \Pr[Y=1|L=l,A=1] \Pr[L=l]} {\sum_l \Pr[Y=1|L=l,A=0] \Pr[L=l]}$$

Standardized mean (in general):

$$\text{Standardized mean:} \qquad \sum_l E[Y|L=l, A=a]\Pr[L=l]$$

Fine Point 2.2: Risk Periods

• Specification of the time-window for risk is critical (e.g. 5-day vs 100-year mortality risk yields different causal effect interpretations).

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2.4 Inverse Probability Weighting (IPW)

Idea:

• Create a pseudo-population where treatment is independent of $L$ by weighting individuals inversely by their probability of receiving the treatment they actually received.

For each individual:

$$W_A = \frac{1}{\Pr(A|L)}$$

• For treated with $L=l$: Weight $1/\Pr(A=1|L=l)$
• For untreated with $L=l':$ Weight $1/\Pr(A=0|L=l')$

Marginal Outcome under Treatment $a$ by IPW:

$$E^\ast[Y|A=a] = E\left[\frac{Y \cdot \mathbb{I}(A=a)}{\Pr(A=a|L)}\right]$$

Where $\mathbb{I}(A=a)$ is indicator function for treatment.

Technical Point 2.2: Formal Definition

• Using density notation, for discrete variables:

  $$f[a|l] = \Pr(A=a|L=l)$$

  So

  $$W_A = \frac{1}{f[A|L]}$$

• Expected (weighted) mean equivalence:

  $$E^\ast[Y|A=a] = E\left[\frac{Y \cdot \mathbb{I}(A=a)}{f[A|L]}\right]$$

Equivalence of IP Weighting and Standardization (Technical Point 2.3):

• Standardization:

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

• IPW:

  $$E\left[\frac{Y \cdot \mathbb{I}(A=a)}{f[A|L]}\right]$$

• They are mathematically equivalent under positivity and conditional exchangeability.
• Proof via properties of the expectation and indicator function.

Positivity Requirement

• For all combinations of $A$ and $L$ with nonzero $\Pr[L=l]$, $\Pr(A=a|L=l)>0$.

Summary Table of Core Assumptions and Methods

Method	Exchangeability Needed	Estimand Identified	Required Formula
Marginal Randomization	$Y^a \perp\!\!\!\perp A$	Marginal average causal effect	$\Pr[Y^a=1] = \Pr[Y=1
Conditional Randomization	$Y^a \perp!!!\perp A	L$	Conditional stratum-specific causal effect
Standardization	$Y^a \perp!!!\perp A	L$	Marginal average causal effect
IP Weighting	$Y^a \perp!!!\perp A	L$	Marginal average causal effect

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中文总结

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2.1 随机化

• 因果问题基础: 举例实际因果推断问题及如何设计随机化实验。
• 随机化: 通过随机机制（如抛硬币）分配处理，消除了混杂。
• 交换性（Exchangeability）: 随机化保证组间（处理/对照）可比性，即群体间唯一差异是随机。
  • 形式化：$Y^a \perp\!\!\!\perp A$ 对所有 $a$ 都成立。
• 完全交换性/均值交换性: 随机化使所有潜在结局与处理独立，均值交换性即可用于均值效应识别。
• 注意: $Y^a \perp\!\!\!\perp A$ 不等价于 $Y \perp\!\!\!\perp A$。如果确有因果效应，对 $Y$ 与 $A$ 必然有关联。

精要点

• 交叉实验不能通用: 只有在无携带效应、效应无时间依赖、对照结果稳定等极强条件下，个体因果效应才能被识别。

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2.2 条件随机化

• 边际随机化: 每人接受处理概率相同，保证整体交换性。
• 条件随机化: 分层分配概率（如按疾病严重程度），仅保证层内交换性。
  • 条件交换性: $Y^a \perp\!\!\!\perp A|L$
• 效应估计:
  • 层内（分层）平均因果效应
  • 总体（边际）平均因果效应（通过标准化或加权获得）

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2.3 标准化

• 总体潜在结局风险: 混合各层次风险的加权平均

  $$\Pr[Y^a=1] = \sum_l \Pr[Y=1|L=l,A=a] \Pr[L=l]$$

• 因果风险比:

  $$\frac{\sum_l \Pr[Y=1|L=l,A=1]\Pr[L=l]} {\sum_l \Pr[Y=1|L=l,A=0]\Pr[L=l]}$$

• 标准化本质： 按总体 $L$ 分布标准化不同处理组的结局风险。

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2.4 逆概率加权（IPW）

• 方法: 给个体赋予 $1/\Pr(A|L)$ 的权重来创造一个“处理与 $L$ 独立”的伪总体。
  • 加权均值公式：

  $$E\left[\frac{Y \cdot \mathbb{I}(A=a)}{\Pr(A=a|L)}\right]$$

• 权重定义: $W_A = \frac{1}{\Pr(A|L)}$，依个体实际处理组和层次确定。
• 与标准化等价: 数学上完全等价（需满足正则性，即各分组概率 $>0$）。
• 与分层标准化理解: 两种方式本质等价，都是在消除（或校正）分层变量 $L$ 影响之后求处理效果。

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章节要点总结

• 随机化消除混杂，赋予结果因果意义。
• 交换性是识别平均因果效应的关键（均值交换性即可）。
• 条件随机化下只保证分层内交换性，需用标准化或IP加权获得总体效应。
• 标准化与逆概率加权本质等价，是现代因果推断分析核心。
• 做因果推断时，明确处理、结局、分层变量及其分布关系至关重要。
• 理想随机化实验很稀少，许多实际研究（如器官移植）因伦理、可行性等限制无法实施。

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核心公式概览（中英文对照）

1. 平均处理效应/边际潜在结局：

$$\Pr[Y^a=1] = \sum_l \Pr[Y=1|L=l,A=a] \Pr[L=l]$$

• (The marginal potential outcome risk is the weighted sum of stratum-specific observed risks.)

2. 标准化风险比：

$$\text{Causal Risk Ratio} = \frac{\sum_l \Pr[Y=1|L=l,A=1]\Pr[L=l]} {\sum_l \Pr[Y=1|L=l,A=0]\Pr[L=l]}$$

3. 逆概率加权均值（IPW）：

$$E\left[\frac{Y \cdot \mathbb{I}(A=a)}{\Pr(A=a|L)}\right]$$

• (The IP-weighted estimator of the mean outcome under treatment $a$.)

4. 交换性定义：

$$Y^a \perp\!\!\!\perp A \quad \text{(marginal)}$$

$$Y^a \perp\!\!\!\perp A \mid L \quad \text{(conditional)}$$

• (Exchangeability: potential outcomes are independent of treatment (possibly conditioning on strata).)

5. 权重定义：

$$W_A = \frac{1}{\Pr(A|L)}$$

• (IP weights: inverse of the conditional probability of receiving the observed treatment given $L$.)

6. Equivalence of Standardization and IPW:

$$\sum_l E[Y|A=a,L=l] Pr[L=l] = E\left[\frac{Y \cdot \mathbb{I}(A=a)}{\Pr(A=a|L)}\right]$$

• (Under positivity and conditional exchangeability.)

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以上总结囊括了本章全部核心技术要点、重点，含所有重要公式，已尽可能完整准确。