[{"source":{"position":32,"lines":["TE_{x,x'}(Y)=DE_{x,x'}(Y)-IE_{x,x'}(Y)"],"changePosition":null},"target":{"position":32,"lines":["TE_{x,x'}(Y)=DE_{x,x'}(Y)+IE_{x,x'}(Y)"],"changePosition":null},"type":"CHANGE"},{"source":{"position":34,"lines":["在实际的应用中$IE_{x,x'}(Y)$经常是负值，所以➖可以替换成➕。"],"changePosition":null},"target":{"position":34,"lines":[],"changePosition":null},"type":"DELETE"},{"source":{"position":41,"lines":["## 一个计算推导","当我们使用遇到链结构的因果推导中(A->B->C)这样的结构，如何计算因果效应呢？"],"changePosition":null},"target":{"position":40,"lines":["## 另一个视角看中介效应","
","上图中的M就是中介变量，X可以称为预前协变量。","### 因果中介效用","这里介绍如何识别T到Y的因果效应，首先$M_{i}(T_{i}=t)$表示当$T_{i}=t$的情况下M的值，可以将潜在的结果表示为一个中介变量和处理变量的函数，对应一个结果变量$Y_{i}(T_{i}=t,M_{i}=m)$，表示当单位i的处理变量取值为t的时候，中介变量为m的潜在结果。","对于二元处理变量$T \\in [0,1]$， 单位i的因果效应$\\sigma(t)$是一个处理变量的函数。"],"changePosition":null},"type":"CHANGE"},{"source":{"position":45,"lines":["ATE = (E[C|do(A=1),B=1] - E[C|do(A=0),B=1]) - (E[C|do(A=1),B=0] - E[C|do(A=0),B=0])"],"changePosition":null},"target":{"position":48,"lines":["\sigma_{i}(t)=Y_{i}(T=t, M_{i}=1)-Y_{i}(T=t, M_{i}=0) \tag{1.1}"],"changePosition":null},"type":"CHANGE"},{"source":{"position":47,"lines":["IE是直接因果效应，DE是间接因果效应。"],"changePosition":null},"target":{"position":50,"lines":["从上面的公式能够看出来，因果中介效应代表处理变量通过影响中介变量对结果变量的因果效应，这里会枚举T=1和T=0计算，最终将所有的$\\sigma_{i}(t)$求平均就是中介效用。因果中介效应$\\sigma_{i}(t)$也被称为间接效应。其实因果中介效应其实也包含一些假设，潜在的结果变量仅仅受到了处理变量和中介变量的影响。","通过上面的介绍，也比较容易推导出平均因果中介效应"],"changePosition":null},"type":"CHANGE"},{"source":{"position":49,"lines":["ATE = (E[C|do(A=1),B=1] - E[C|do(A=0),B=0]) - (E[C|do(A=0),B=1] - E[C|do(A=1),B=0]) \\","ATE = IE + DE"],"changePosition":null},"target":{"position":53,"lines":["\sigma(t)= E[\sigma_{i}(t)] \\","=E[Y_{i}(T=t, M_{i}=1)-Y_{i}(T=t, M_{i}=0)] \tag{1.1}"],"changePosition":null},"type":"CHANGE"},{"source":{"position":52,"lines":["这里的+号不用过于紧张， 括号内取反就可以。","(E[C|do(A=1),B=1] - E[C|do(A=0),B=0]) 表示 A 对 C 的直接效应，(E[C|do(A=0),B=1] - E[C|do(A=1),B=0]) 表示 A 对 C 的间接效应。"],"changePosition":null},"target":{"position":56,"lines":["### 总因果效应","还是假设二元处理变量$T \\in [0,1]$，单位i的总因果效应为$\\tau$ 定义如下","$$\begin{gathered}","\\tau_{i}=Y_{i}(T=1, M_{i}(1))-Y_{i}(T=1, M_{i}(0)) \\tag{2.1}","\end{gathered}$$","下面公式表示自然直接效应","$$\begin{gathered}","\\zeta(1-t)=Y_{i}(T=1-t,M_{i})-Y_{i}(T=0,M_{i}) \\tag{2.2}","\end{gathered}$$","而总因果效应就有了如下的表达","$$\begin{gathered}","\\tau_{i}=\\sigma_{i}(t)+\\zeta(1-t) \\tag{2.3}","\end{gathered}$$","看看我们如何理解上面这些关系，首先$\\zeta(t)$是一个处理变量t的函数。可以理解为处理变量不通过中介变量干预结果变量，这是直接对结果变量产生影响，所以这里固定了$M_{i}(t)$，改变处理变量对结果变量的影响。而公式2.3表示总因果效应是因果中介效应在处理变量t时的取值，与直接效应在处理变量为1-t的取值之和。"],"changePosition":null},"type":"CHANGE"},{"source":{"position":55,"lines":["",""],"changePosition":null},"target":{"position":71,"lines":[],"changePosition":null},"type":"DELETE"}]