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The goal is to estimate the potential outcome, Y^{*}\bigl(a\bigr), that would be observed if the subject were assigned treatment a. Then compare the mean outcome if all patients in the population were assigned either treatment: \mu_{a} = \mathbb{E}Y^{*}(a). We want to estimate \mu_a using observed data \{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}.

The goal is to estimate the potential outcome, Y^{*}\bigl(a\bigr), that would be observed if the subject were assigned treatment a. Then compare the mean outcome if all patients in the population were assigned either treatment: \mu_{a} = \mathbb{E}Y^{*}(a). We want to estimate \mu_a using observed data \{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}.

假设观测数据是{ bigl (xi，a _ i，y _ i bigr)} ^ { n }{ i = 1}从未知分布 p 中抽取的 i.d (独立同分布) ，其中

假设观测数据是<math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math>，这些数据是从未知的分布中抽取出来的独立同分布数据，其中

* x 在{0,1}中的数学{ r } ^ { p }协变量

* <math>X \in \mathbb{R}^{p}</math> 为协变量；

* a 中是两个可能的处理。我们不假设治疗是随机分配的。目标是估计潜在的结果，y ^ {  

* <math>A \in \{0, 1\}</math> 是两个可能的处理；

* } bigl (a bigr) ，如果给受试者分配治疗 a，可以观察到这个结果。然后比较平均结果，如果所有患者在人口分配任一治疗: mu _ { a } = mathbb { e } y ^ {

* <math>Y \in \mathbb{R}</math> 为响应量；

* }(a)。我们想用观测数据{ bigl (xi，a _ i，y _ i bigr)} ^ { n }{ i = 1}来估计 mu _ a。

* 我们不假设治疗是随机分配的。

=== Estimator Formula ===

目标是估计潜在结果<math>Y^{*}\bigl(a\bigr)</math>，这个结果可以在给受试者分配治疗 <math>a</math>的情况下观测到。然后比较所有患者在总体中被分配为任一治疗方法的平均结果: <math>\mu_{a} = \mathbb{E}Y^{*}(a)</math>。我们想用观测数据<math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math>来估计 <math>\mu_a</math> 。

 

=== 估计器公式 ===

<blockquote><math>\hat{\mu}^{IPWE}_{a,n} = \frac{1}{n}\sum^{n}_{i=1}Y_{i} \frac{\mathbf 1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}</math></blockquote>

<blockquote><math>\hat{\mu}^{IPWE}_{a,n} = \frac{1}{n}\sum^{n}_{i=1}Y_{i} \frac{\mathbf 1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}</math></blockquote>

\hat{\mu}^{IPWE}_{a,n} = \frac{1}{n}\sum^{n}_{i=1}Y_{i} \frac{\mathbf 1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}

\hat{\mu}^{IPWE}_{a,n} = \frac{1}{n}\sum^{n}_{i=1}Y_{i} \frac{\mathbf 1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}

=== Estimator Formula ===

==== 构建 IPWE ====

 

==== Constructing the IPWE ====

# <math>\mu_{a} = \mathbb{E}\frac{\mathbf{1}_{A=a} Y}{p(A|X)}</math>  where <math>p(a|x) = \frac{P(A=a,X=x)}{P(X=x)}</math>

# <math>\mu_{a} = \mathbb{E}\frac{\mathbf{1}_{A=a} Y}{p(A|X)}</math>  where <math>p(a|x) = \frac{P(A=a,X=x)}{P(X=x)}</math>

# construct <math>\hat{p}_{n}(a|x)</math> or <math>p(a|x)</math> using any propensity model (often a logistic regression model)

# construct <math>\hat{p}_{n}(a|x)</math> or <math>p(a|x)</math> using any propensity model (often a logistic regression model)