潜在结果框架

Quote

Two roads diverged in a yellow wood And sorry I could not travel both

Robert Frost, The Road Not Taken

个体 $i$ 的处理分配（treatment assignment） $D_{i}$ 为二元时，结果变量 $Y_{i}$ 的潜在结果定义为

Y_{i} = {\begin{cases} Y_{i} (0), if D_{i} = 0 \\ Y_{i} (1), if D_{i} = 1 \end{cases}

一般地，个体 $i$ 的处理分配 $D_{i}$ 为多元乃至连续时，结果变量 $Y_{i}$ 的潜在结果定义为

Y_{i} = Y_{i} (d_{i}), where d_{i} \in D_{i}

更一般地，个体 $i$ 的结果变量 $Y_{i}$ 还可能受到其他个体的处理分配 $D_{- i}$ 的影响。但是，我们通常假设这种影响不存在，这被称为 SUTVA 假设：

Y_{i} (d_{i}, d_{- i}) = Y_{i} (d_{i})

以二元处理分配的情形为例，潜在结果的定义可以总结为

\begin{aligned} Y_{i} & = D_{i} Y_{i} (1) + (1 - D_{i}) Y_{i} (0) \\ = Y_{i} (0) + [Y_{i} (1) - Y_{i} (0)] D_{i} \end{aligned}

其中， $Y_{i} (1) - Y_{i} (0)$ 就是处理分配 $D_{i}$ 对个体 $i$ 产生的处理效应。

然而，在个体层面，我们至多只能观察到一个实现了的潜在结果，其他潜在结果永远不得而知。也就是说，个体处理效应是根本不可识别的（fundamentally unidentifiable）。因此，我们退而求其次，将个体潜在结果（常数）视为来自总体潜在结果（随机变量）的简单随机抽样，转而寻求识别平均处理效应（ATE）:

A T E = E [Y (1)] - E [Y (0)]

我们有时也可能按实际处理分配分组，寻求识别组平均处理效应（ATT & ATU）

\begin{aligned} A T T & = E [Y (1) - Y (0) ∣ D = 1] \\ A T U & = E [Y (1) - Y (0) ∣ D = 0] \end{aligned}

独立性假设

要想识别平均处理效应，一个自然的想法是直接比较分组结果变量均值之差

Δ = E [Y ∣ D = 1] - E [Y ∣ D = 0]

显然，只有满足

\begin{aligned} E [Y (1) ∣ D = 1] & = E [Y (1)] \\ E [Y (0) ∣ D = 0] & = E [Y (0)] \end{aligned}

也就是说，只有处理组 $Y (1)$ 的均值、控制组 $Y (0)$ 的均值和总体一致时， $A T E = Δ$ 才成立；换句话说，如果有任何因素使得分组过程并不随机，处理组和控制组并不可比，均值之差就不等于平均处理效应，这被称为选择偏差（selection bias）。可惜的是，由于选择偏差包含未知的 $Y (1)$ 和 $Y (0)$ ，同样是不可识别的。我们只好暂且幻想能够随机分组，也就是满足独立性（independence）假设：

{Y (0), Y (1)} ⊥ ⊥ D

这样直接比较分组均值就能得到处理效应估计值了：

\hat{A T E} = \frac{\sum_{i = 1}^{n} Y_{i} \cdot D_{i}}{\sum_{i = 1}^{n} D_{i}} - \frac{\sum_{i = 1}^{n} Y_{i} \cdot (1 - D_{i})}{\sum_{i = 1}^{n} (1 - D_{i})}

（编写这样复杂的解析式似乎没有代码 mean(Y[D==1])=mean(Y[D==0]) 直观）

条件独立性 - 共同支撑假设

引入条件独立性（conditional independence）假设，又称 selection on observables

{Y (0), Y (1)} ⊥ ⊥ D ∣ X

其中， $X$ 是同时影响处理分配和潜在结果的可观测协变量集合，简称 confounders

为了识别条件处理效应，还要求满足共同支撑（common support）假设

0 < P (D = 1 ∣ X) < 1

其中， $P (D = 1 ∣ X)$ 常简记为 $p (X)$ ，称为倾向得分（propensity score）

Definition Treatment effect of unit $i$

T E_{i} \equiv Y_{i} (0) - Y_{i} (1)

Definition Average treatment effect on the treated

A T T \equiv E [Y_{i} (1) - Y_{i} (0) ∣ D_{i} = 1]

Definition Average treatment effect on the untreated

A T U \equiv E [Y_{i} (1) - Y_{i} (0) ∣ D_{i} = 0]

Definition Average Treatment Effect

A T E \equiv E [Y_{i} (1) - Y_{i} (0)] = p A T T + (1 - p) A T U

Potential Outcome	Observed Outcome	Counterfactual Outcome
Treated	$Y_{i} (1)$	$Y_{i} (0)$
Untreated	$Y_{i} (0)$	$Y_{i} (1)$

Average Potential Outcome	Observed Outcome	Counterfactual Outcome
Treated	$T_{1} = E [Y_{i} (1) ∣ D_{i} = 1]$	$T_{0} = E [Y_{i} (0) ∣ D_{i} = 1]$
Untreated	$C_{0} = E [Y_{i} (0) ∣ D_{i} = 0]$	$C_{1} = E [Y_{i} (1) ∣ D_{i} = 0]$

Compare the Naive estimator with the true effects:

\begin{aligned} T_{1} - C_{0} & = T_{1} - T_{0} + T_{0} - C_{0} = A T T + T_{0} - C_{o} \\ T_{1} - C_{0} & = C_{1} - C_{0} + T_{1} - C_{1} = A T U + T_{1} - C_{1} \\ T_{1} - C_{0} & = p (T_{1} - T_{0}) + (1 - p) (C 1 - C 0) + (T_{0} - C_{0}) \\ + (1 - p) [(T_{1} - T_{0}) - (C_{1} - C_{0})] \\ = A T E + (T_{0} - C_{0}) + (1 - p) (A T T - A T U) \end{aligned}

$T_{0} - C_{0}$ is called selection bias,代表两组处理前特征上的不同；

$(1 - p) (A T T - A T U)$ is called heterogeneous treatment effect bias,代表两组处理后影响效果的不同。

实验研究

For unit $i$ , we have pretreatment covariates $X_{i}$ , a binary treatment indicator $D_{i}$ , and an observed outcome $Y_{i}$ with two potential outcomes $Y_{i} (1)$ and $Y_{i} (0)$ under the treatment and control, respectively. For simplicity, we assume

{D_{i}, X_{i}, Y_{i} (0), Y_{i} (1)}_{i = 1}^{n} \overset{i i d}{\sim} {D, X, Y (0), Y (1)}

So we can drop the subscript $i$ for quantities of this population.

Definition Potential outcome of $Y$

Y = {\begin{cases} Y (0), if D = 0 \\ Y (1), if D = 1 \end{cases}

Definition Treatment effect of unit $i$

T E \equiv Y (0) - Y (1)

Definition Average treatment effect on the treated

A T T \equiv E [T E ∣ D = 1] = E [Y (1) - Y (0) ∣ D = 1]

Definition Average treatment effect on the untreated

A T U \equiv E [T E ∣ D = 0] = E [Y (1) - Y (0) ∣ D = 0]

Definition Average Treatment Effect

A T E \equiv E [T E] = E [Y (1) - Y (0)] = A T T \times P (D = 1) + A T U \times P (D = 0)

Definition 差分被估量 (naive estimand)

Δ = E [Y ∣ D = 1] - E [Y ∣ D = 0] = E [Y (1) ∣ D = 1] - E [Y (0) ∣ D = 0]

比较 $Δ$ 和 $A T T$ 、 $A T U$ 的差异

\begin{aligned} Δ - A T T & = E [Y (0) ∣ D = 1] - E [Y (0) ∣ D = 0] \\ Δ - A T U & = E [Y (1) ∣ D = 1] - E [Y (1) ∣ D = 0] \end{aligned}

这说明如果处理组和对照组的潜在结果不同，估计组平均处理效应时就会存在选择偏差。

比较 $Δ$ 和 $A T E$ 的差异

\begin{aligned} Δ - A T E & = Δ - A T T \times P (D = 1) - A T U \times P (D = 0) \\ = (Δ - A T T) + (A T T - A T U) \times P (D = 0) \\ = (Δ - A T U) + (A T U - A T T) \times P (D = 1) \end{aligned}

两个等式对应两种分解方法，分解为相应组的选择偏差和异质处理效应。

引入独立性假设（Independence Assumption, IA）

{Y (0), Y (1)} ⊥ ⊥ D

即是否接受处理是随机的，则有

Δ = A T T = A T U = A T E

此时，选择偏差和异质处理效应全部消失。

观察研究

Definition Conditional average treatment effect on the treated

A T T (X) \equiv E [Y (1) - Y (0) ∣ D = 1, X]

Definition Conditional average treatment effect on the untreated

A T U (X) \equiv E [Y (1) - Y (0) ∣ D = 0, X]

Definition Conditional average Treatment Effect

A T E (X) \equiv E [Y (1) - Y (0) ∣ X] = A T T (X) \times P (D = 1 ∣ X) + A T U (X) \times P (D = 0 ∣ X)

Definition 条件差分被估量 (naive estimand)

Δ (X) = E [Y ∣ D = 1, X] - E [Y ∣ D = 0, X] = E [Y (1) ∣ D = 1, X] - E [Y (0) ∣ D = 0, X]

比较 $Δ (X)$ 和 $A T T (X)$ 、 $A T U (X)$ 的差异

\begin{aligned} Δ (X) - A T T (X) & = E [Y (0) ∣ D = 1, X] - E [Y (0) ∣ D = 0, X] \\ Δ (X) - A T U (X) & = E [Y (1) ∣ D = 1, X] - E [Y (1) ∣ D = 0, X] \end{aligned}

比较 $Δ (X)$ 和 $A T E (X)$ 的差异

\begin{aligned} Δ (X) - A T E (X) & = Δ (X) - A T T (X) \times P (D = 1 ∣ X) - A T U (X) \times P (D = 0 ∣ X) \\ = [Δ (X) - A T T (X)] + [A T T (X) - A T U (X)] \times P (D = 0 ∣ X) \\ = [Δ (X) - A T U (X)] + [A T U (X) - A T T (X)] \times P (D = 1 ∣ X) \end{aligned}

引入条件独立性假设（Conditional Independence Assumption, CIA）

{Y (0), Y (1)} ⊥ ⊥ D ∣ X

即给定 $X$ 后是否接受处理是随机的，则有

Δ (X) = A T T (X) = A T U (X) = A T E (X)

此时，选择偏差和异质处理效应全部消失。

根据定义易知

\begin{aligned} A T T & = \sum A T T (X = x) P (X = x ∣ D = 1) \\ A T U & = \sum A T U (X = x) P (X = x ∣ D = 0) \\ A T E & = \sum A T E (X = x) P (X = x) \end{aligned}

因此，在 CIA 假设下使用 $Δ (X)$ 即可估计任意因果效应。

Attention

条件差分被估量 $Δ (X)$ 的存在要求 $X$ 划分任意的层同时包含处理组和对照组，这被称为共同支撑条件 (common support condition)，记作 $0 < P (D = 1 ∣ X) < 1$ 。事实上，共同支撑条件在 $X$ 维数很高时难以满足，这为倾向得分的出现埋下了伏笔。