Chapter_15

Aaron L9/17/25About 7 min

Below is a comprehensive, step-by-step summary of Chapter 15: Outcome Regression and Propensity Scores from “Causal Inference: What If”, with all major points, technical details, fine points, and all formulas included.

The summary is presented in both English and Chinese, with formulas in standard LaTeX markdown ( $...$ blocks). Each section is organized for clarity and completeness.

Chapter 15: Outcome Regression and Propensity Scores

第十五章：结果回归与倾向性评分

Overview 概述

Outcome regression and propensity score analyses are the most commonly used parametric methods in causal inference.
The chapter discusses applicability, strengths, and limitations, especially compared to more robust g-methods (IP-weighting, standardization, and g-estimation).
该章介绍了结果回归与倾向性评分法的适用性、优劣势，尤其与更稳健的g方法（IP加权、标准化、g估计）进行对比。

15.1 Outcome Regression 结果回归

1. Structural Model for Outcome Regression 结构性模型

The average causal effect of treatment $A$ (e.g., smoking cessation) on outcome $Y$ (e.g., weight gain), conditional on covariates $L$ :

E[Y^{a,c=0} \mid L] = \beta_0 + \beta_1 a + \beta_2 aL + \beta_3 L

$\beta_1$ , $\beta_2$ capture the causal effects; $\beta_3$ captures the association between $L$ and $Y$ .
其中 $\beta_1$ 和 $\beta_2$ 是因果参数， $\beta_3$ 仅描述协变量 $L$ 对 $Y$ 的关联，不一定有因果解释。

2. Nuisance Parameters and Model Specification 滞余参数（干扰参数）与模型设定

If $\beta_0 + \beta_3L$ mis-specifies $E[Y^{a=0, c=0} | L]$ , estimates for $\beta_1$ , $\beta_2$ may be biased.
滞余参数（如 $\beta_0$ 、 $\beta_3$ ）若模型错设，主效应估计会有偏。

Fine Point 15.1:

In outcome regression, nuisance parameters are $\beta_0,\beta_3$ .
In g-estimation (structural nested model), nuisance parameters come from the propensity model $Pr(A=1|L)$ (e.g., parameters $\alpha$ in $\logit Pr(A=1|L) = \alpha_0 + \alpha_1 L$ ).
Which method to use depends on which nuisance model you trust more.
Doubly robust methods can be preferred.
结果回归法的滞余参数是 $\beta_0,\beta_3$ ，g估计法的滞余参数在倾向模型中。建议使用更稳健的双稳健法。

3. Estimation and Assumptions 估计与假定

Key assumptions: Exchangeability, positivity, consistency, correct model specification of $E[Y | A, C=0, L]$ .
主要假设：可交换性、阳性、干扰一致性、模型设定正确。
Estimation is performed by fitting the regression model:

E[Y | A, C = 0, L] = \alpha_0 + \alpha_1 A + \alpha_2 A L + \alpha_3 L

可以用普通最小二乘法拟合该模型。
If all confounders are adjusted and model is correct, $\alpha_j = \beta_j$ .
For binary outcomes, use $\logit Pr(Y=1|A, C=0, L)$ .

4. Product Terms 交互作用项

Adding $A \cdot L$ (product/interactions) allows for effect modification.
不带交互项时， $\hat\beta_1$ 可看作条件效应和边际效应的估计。

15.2 Propensity Scores 倾向性评分

1. Definition and Estimation 定义与估计

The propensity score is $\pi(L) = Pr(A=1|L)$ .
倾向性评分定义为 $A=1$ 的条件概率，给定协变量 $L$ 。
Estimated via (e.g.) logistic regression.

2. Properties 性质

In RCT, $\pi(L)=0.5$ for everyone.
In observational studies, $\pi(L)$ must be estimated.
Balancing property: For the same value of $\pi(L)$ , the distribution of $L$ is the same for $A=1$ and $A=0$ : $A \perp\!\!\!\perp L \mid \pi(L)$ .
倾向性评分具备"平衡"性质，即给定 $\pi(L)$ ， $L$ 在处理组与对照组中的分布一致。

3. Use and Key Result 关键结论

Exchangeability given $L$ $\implies$ exchangeability given $\pi(L)$ :
$Y^a \perp\!\!\!\perp A \mid L \implies Y^a \perp\!\!\!\perp A \mid \pi(L)$
Positivity holds within levels of $\pi(L)$ iff it holds within $L$ .

4. Balancing Scores & Prognostic Scores 平衡评分 & 预后评分

Technical Point 15.1:

A balancing score $b(L)$ is any function such that $A \perp\!\!\!\perp L | b(L)$ .
Adjustment for $b(L)$ is sufficient if $L$ is sufficient.
Prognostic score $s(L)$ : $Y^{a=0} \perp\!\!\!\perp L | s(L)$ ; methods using $s(L)$ require stronger assumptions and do not easily generalize to time-varying treatments.

15.3 Propensity Stratification and Standardization 倾向评分分层与标准化

1. Conditional Effect 条件效应

For a given propensity score $s$ :
$ACE(s) = E[Y^{a=1,c=0} | \pi(L)=s ] - E[Y^{a=0,c=0} | \pi(L)=s ]$ $= E[Y | A=1, C=0, \pi(L)=s ] - E[Y | A=0, C=0, \pi(L)=s ]$
Typically, create strata (e.g. deciles) of $\pi(L)$ and estimate effects within each stratum.
常见做法：将样本按 $\pi(L)$ 分为若干层，分别估计效应。

2. Issues 问题

Since $\pi(L)$ is continuous, exact matching is rare—approximate via strata.
Within each stratum, exchangeability may fail if $\pi(L)$ distributions differ between $A=1,0$ .
使用分层法时，层内 $\pi(L)$ 分布不一致会破坏组间可交换性。
Regression on $\pi(L)$ as a continuous variable (possibly nonlinear, e.g. splines) is an alternative.

3. Standardization 标准化

To estimate population marginal effects, standardize conditional means $E[Y|A,C=0,\pi(L)]$ by the distribution of $\pi(L)$ in the sample.
方法同第13章，只是使用 $\pi(L)$ 代替 $L$ 。

15.4 Propensity Matching 倾向评分匹配

1. Motivation and Process 动机与过程

Match treated and untreated units with similar values of $\pi(L)$ .
Propensity matched samples estimate the effect in the matched population.
倾向评分匹配后，配对样本的效应估计可近似作为匹配样本的因果效应。
Matching criteria (“closeness”) influence the bias-variance tradeoff:
- Too loose: loss of exchangeability.
- Too tight: lose observations, wider confidence intervals.

2. Positivity (Overlap) and Target Population 阳性（重叠性）与目标人群

Matching guarantees “positivity” by restricting to overlapping support, but the resulting population may not be well-characterized in terms of $L$ .
用倾向分数组检测重叠范围对保证阳性有效，但实际可推广性受限。
限制研究对象范围时，更好的办法是使用实际变量（如年龄、吸烟年数），而不是倾向分数。

3. Effect of Matching on Interpretation 匹配影响解释

Fine Point 15.2:

Matched estimates may be closer to effects in the treated due to exclusion of unmatched controls.
If there is effect modification by $\pi(L)$ , interpretation of results becomes less policy relevant, since statements are about $\pi(L)$ , not about measured attributes.

15.5 Propensity Models, Structural Models, and Predictive Models

倾向模型、结构模型与预测模型

1. Distinction 区别

Propensity model: $Pr(A=1|L)$
- Used to achieve exchangeability (adjust for confounding).
- Parameters are nuisance; no direct causal interpretation.
- Example: logistic regression of treatment on $L$ .
Structural model: Relationship between $A$ and potential outcomes (e.g., $E[Y^{a,c=0}|L]$ or marginally $E[Y^{a}]$ ).
- Parameters directly encode causal effects.
Outcome regression: Can be used for both causal inference (with correct conditions and proper selection of $L$ ) and prediction (where causality isn’t the focus).
- Predictive use does not require or impart causal interpretation.

2. Variable Selection for Causal vs Predictive Modelling

因果建模与预测建模的变量选择

Common Misunderstandings 常见误区

Including variables that strongly predict $A$ but aren’t needed for exchangeability increases variance or can introduce bias.
E.g., including a “Hospital” variable that nearly perfectly predicts $A$ but isn’t related to outcome increases variance, possibly with no gain in bias reduction.
在倾向评分建模中，只需纳入保证可交换性所需的协变量，而非所有可预测 $A$ 的变量。
Including colliders or instruments can induce bias.

Best Practice 最佳实践

For causal inference:
- Focus only on covariates that ensure exchangeability.
- Model misspecification can be reduced by using flexible approaches (e.g., splines).
- 必须满足：可交换性、阳性、干预一致性、模型正确设定。

Summary Table (Key Formulas and Principles) 章节总结

Concept	English Formula & Meaning	中文公式与含义
Outcome regression	$E[Y^	L] = \beta_0 + \beta_1 a + \beta_2 aL + \beta_3 L$
Propensity score	$\pi(L) = Pr(A=1	L)$
Balancing property	$A \perp!!!\perp L	\pi(L)$
Exchangeability	$Y^a \perp!!!\perp A	L \implies Y^a \perp!!!\perp A
Matching population	Effect applies to matched set, overlapping $\pi(L)$ support	匹配后效应只针对有重叠的样本人群
Causal v. predictive	Causal models focus on exchangeability, prediction models focus on predictive accuracy	因果建模重协变量控制，预测建模重预测准确性，不等价
Positivity	$0 < Pr(A=a	\pi(L)=s) < 1$ (for all $s$ with $Pr[\pi(L)=s]>0$ )

Practical Recommendations 实务建议

Use outcome regression with caution: model must be correctly specified, and products/interactions included only as required.
倾向分数法适合用于简化多维协变量控制，并可通过分层、标准化、匹配等方法估计因果效应。
Variable selection for causal inference should not be based on predictive model procedures.
采用双稳健方法（既建模结果，又建模处理）更为安全可靠。

References:

Rosenbaum PR, Rubin DB. The central role of the propensity score in observational studies for causal effects. Biometrika. 1983.
Abadie A, Imbens GW. Large sample properties of matching estimators for average treatment effects. Econometrica. 2006.

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