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Only one of each potential outcome (PO) can be realized, the other cannot, for the same assignment to condition, so when we try to estimate treatment effects, we need something to replace the fully contrary-to-fact ones with observables (or estimate them). When ignorability/exogeneity holds, like when people are randomized to be treated or not, we can ‘replace’ *''Y''₀¹ with its observable counterpart Y₁¹, and *Y₁⁰ with its observable counterpart ''Y''₀⁰, not at the individual level Y_i’s, but when it comes to averages like E[''Y''_''i''¹ – ''Y''_''i''⁰], which is exactly the causal treatment effect (TE) one tries to recover.

Only one of each potential outcome (PO) can be realized, the other cannot, for the same assignment to condition, so when we try to estimate treatment effects, we need something to replace the fully contrary-to-fact ones with observables (or estimate them). When ignorability/exogeneity holds, like when people are randomized to be treated or not, we can ‘replace’ *''Y''₀¹ with its observable counterpart Y₁¹, and *Y₁⁰ with its observable counterpart ''Y''₀⁰, not at the individual level Y_i’s, but when it comes to averages like E[''Y''_''i''¹ – ''Y''_''i''⁰], which is exactly the causal treatment effect (TE) one tries to recover.

对于相同的处理分配条件，每个潜在结果(PO)中只有一个是实际发生可观测的，而另一个不会发生也无法观测，所以当我们尝试估计处理效应时，需要用可观测值(或估计值)来替代无法观测的反事实结果。当可忽略性/外生性成立时，例如个体是否接受处理是随机的，此时可利用已观测的 Y₁¹'替换'*''Y''₀¹，利用已观测的 Y₀⁰'替换'*''Y''₁⁰，不是个人层面的Y_i，而是从平均角度出发，如 E[''Y''_''i''¹ – ''Y''_''i''⁰]，这正是人们尝试获取的因果处理效应（TE）。

对于相同的处理分配条件，每个潜在结果(PO)中只有一个是实际发生可观测的，而另一个不会发生也无法观测，所以当我们尝试估计处理效应时，需要用可观测值(或估计值)来替代无法观测的反事实结果。当可忽略性/外生性成立时，例如个体是否接受处理是随机的，此时可利用已观测的 Y₁¹'替换'*''Y''₀¹，利用已观测的 Y₀⁰'替换'*''Y''₁⁰，不是个人层面的Y_i，而是从平均角度出发，如 E[''Y''_''i''¹ – ''Y''_''i''⁰]，这正是大家尝试获取的因果处理效应（TE）。

现在，我们通过简单的加减相同的完全反事实量 *Y₁⁰ 得到:

现在，我们通过简单的加减相同的完全反事实量 *Y₁⁰ 得到:

E[Y_i1¹ – Y_i0⁰] = E[Y_i1¹ –*Y₁⁰  +*Y₁⁰ - Y_i0⁰] = E[Y_i1¹ –*Y₁⁰] + E[*Y₁⁰ - Y_i0⁰] = ATT + {选择性偏差},

E[Y_i1¹ – Y_i0⁰] = E[Y_i1¹ –*Y₁⁰  +*Y₁⁰ - Y_i0⁰] = E[Y_i1¹ –*Y₁⁰] + E[*Y₁⁰ - Y_i0⁰] = ATT + {选择性偏差},

其中，第一项 ATT = 处理组的平均处理效应<ref>{{cite journal|last1=Imai|first1=Kosuke|title=Misunderstandings between experimentalists and observationalists about causal inference|journal=Journal of the Royal Statistical Society, Series A (Statistics in Society)|date=2006|volume=171|issue=2|pages=481–502|doi=10.1111/j.1467-985X.2007.00527.x|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695}}</ref>，第二项是当个体可选择属于“处理”组或“控制”组而非完全随机分配时引入的偏差。

其中，第一项 ATT = 处理组的平均处理效应<ref>{{cite journal|last1=Imai|first1=Kosuke|title=Misunderstandings between experimentalists and observationalists about causal inference|journal=Journal of the Royal Statistical Society, Series A (Statistics in Society)|date=2006|volume=171|issue=2|pages=481–502|doi=10.1111/j.1467-985X.2007.00527.x|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695}}</ref>，第二项是当个体可选择属于“处理”组或“控制”组而非完全随机分配时引入的偏差。