Chapter 8 Selection Bias

8.1 The structure of selection bias
8.2 Examples of selection bias
8.3 Selection bias and confounding
8.4 Selection bias and censoring
8.5 How to adjust for selection bias
8.6 Selection without bias
Fine Point
- Selection bias in case-control studies
- The strength and direction of selection bias
Technical Point
- THe built-in selection bias of hazard bias
- Multiplicative survival model

Hern\(\'{a}\)n M. and Robins J. Causal Inference: What If.

上一章讲了confounding, 这种bias来源于treatment和outcome受同一个未观测的cause影响, 但是这种bias在随机实验中可以避免.

这一章要讲的试selection bias, 即便是在随机试验中, 也无法避免.

8.1 The structure of selection bias

这里, 作者给出了一个非常好的例子, 如上图所示:

\(A \in \{0, 1\}\) 表示是否注射叶酸, 而\(Y \in \{0, 1\}\) 表示胎儿是否心脏畸形, 而\(C \in \{0, 1\}\)则表示是否死亡.

这里, 虽然是否注射叶酸, 我们是随机选择的, 但是在实际调查中, 只有顺利出生(\(C=0\))的才会被记录是否心脏畸形.

所以, 我们必须在\(C=0\)的条件下估计causal effect.

但是注意到, \(A\)存在指向\(C\)的箭头(即\(A=1\)会降低死亡的风险).

此时, \(Y^a\)和\(A\)在给定\(C=1\)的条件下并不独立.

这就是本章讲的selection bias.

8.2 Examples of selection bias

8.3 Selection bias and confounding

8.4 Selection bias and censoring

虽然我们只有\(C=0\)的情况, 我们可以把\(C\)也看出一个treatment, 则我们只需要关注

\[Y^{a, c=0},
\]

即可.

8.5 How to adjust for selection bias

如何计算\(\mathbb{E}[Y^{a, c=0}]\), 这一节给出的是一种特殊的IP weighting的方法, 说实话没怎么看懂, 这里以上图为例给出我自己的理解.

\[\mathbb{E}[\frac{I(A=a, C=0)Y}{f(C|A,L)}] \\
=
\sum_l \sum_y \frac{I(A=a, C=0)Y}{f(C=0|A=a,L=l)} \mathrm{Pr}[Y|a,c,l] \mathrm{Pr}[C=0|a,l] \mathrm{Pr}[A=a, L=l] \\
=
\sum_l \sum_y I(A=a, C=0)Y^{a,0} \mathrm{Pr}[Y^{a,0}|l] \mathrm{Pr}[A=a, L=l] \\
=
\mathbb{E} [Y^{a, c=0}] \mathrm{Pr}[A=a].
\]

其实, 个人感觉如果是

\[\mathbb{E}[\frac{I(A=a, C=0)Y}{f(C,A|L)}] ,
\]

就直接可以得出结果了.

8.6 Selection without bias

这一节讲了给定\(Y\)的情况下, \(A, E\)产生关联的不同情况.

Fine Point

Selection bias in case-control studies

The strength and direction of selection bias

Technical Point

THe built-in selection bias of hazard bias

Multiplicative survival model

\[\mathrm{Pr}[Y=0|E=e, A=a] = g(e)h(a), \\
\mathrm{Pr}[Y=1|E=e, A=a] = 1 - g(e)h(a). \\
\]

巴特西