Chapter 4 Effect Modification

4.1 Definition of effect modification
4.2 Stratification to identify effect modification
4.3 Why care about effect modification
4.4 Stratification as a form of adjustment
4.5 Matching as another form of adjustment
4.6 Effect modification and adjustment methods
Fine Point
Technical Point

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

4.1 Definition of effect modification

什么是 effect modification, 即causal effect在不同因素\(V\)下不同, 即

\[\mathbb{E} [Y^{a=1} - Y^{a=0}|V=1]
\not =
\mathbb{E} [Y^{a=1} - Y^{a=0}|V=0],
\]

或者

\[\frac{
\mathbb{E} [Y^{a=1}|V=1]
}{
\mathbb{E} [Y^{a=0}|V=1]
}
\not =
\frac{
\mathbb{E} [Y^{a=1}|V=0]
}{
\mathbb{E} [Y^{a=0}|V=0]
}.
\]

也就是说\(V\)这个因素会影响causal effect, 或许变好或许变差.

另外需要一提的是, additive effective modification 或许和 multiplicative effect modification 有偏差.

有可能前者显示\(V\)是一个effect modifier, 但是后者显示它不是.

所以一个因素是否是effect modifier还得依赖你所选的衡量指标.

4.2 Stratification to identify effect modification

\[\mathrm{Pr} [Y^{a=1}=1|V=1] - \mathrm{Pr} [Y^{a=0}=1|V=1], \\
\mathrm{Pr} [Y=1|A=1,V=1] - \mathrm{Pr} [Y=1|A=0,V=1], \\
\]

4.3 Why care about effect modification

可迁移性

4.4 Stratification as a form of adjustment

通过\(V\)将整个数据集分成子集, 并对每个子集计算相应的casual effect.

当然, 在此过程中我们往往也是需要条件可交换性的.

4.5 Matching as another form of adjustment

通过随机选择, 使得在不同子集中, 所关心元素的数量是一致的.

比如根据\(A\)划分treated 和 untreated, 通过随机选择使得\(L=l\)在两个子集中的数目是一样的.

此时,

\[\begin{array}{ll}
\mathrm{Pr}[Y^{a=1}]
& = \sum_l \mathrm{Pr} [Y^{a=1}|L=l] \mathrm{Pr}[L=l] \\
& = p \sum_l \mathrm{Pr} [Y|A=1,L=l] \\
& = \frac{1}{\mathrm{Pr}[A=1]} \sum_l \mathrm{Pr} [Y,A=1,L=l] \\
& = \mathrm{Pr} [Y|A=1]
\end{array}
\]

此时, 计算causal effect只需考虑\(\mathrm{Pr}[Y|A=a]\)即可.

4.6 Effect modification and adjustment methods

Standard, IP weighting, stratification, matching这几个方法的比较.

Fine Point

Effect in the treated

\[\mathrm{Pr} [Y=1|A=1]
\not =
\mathrm{Pr} [Y^{a=0}=1|A=1].
\]

Transportability

Collapsibility and the odds ratio

Technical Point

Computing the effect in the treated

计算\(\mathbb{E}[Y^a|A=a']\)只需要部分可交换性\(Y^a \amalg A|L\)即可.

Standard:

\[\sum_l \mathbb{E} [Y|A=a,L=l] \mathrm{Pr}[L=l|A=a'].
\]

IP weighting:

\[\frac{
\mathbb{E}[
\frac{I(A=a)Y}{f(A|L)}
\mathrm{Pr}[A=a'|L]
]
}
{
\mathbb{E}[
\frac{I(A=a)}{f(A|L)}
\mathrm{Pr}[A=a'|L]
]
}.
\]

注: 分母实际上是\(\mathrm{Pr}[A=a']\).

Pooling of stratum-specific effect measures

Relation between marginal and conditional risk ratios

\[\mathrm{Pr} [Y^{a=1}=1]
/
\mathrm{Pr} [Y^{a=0}=0] =
\sum_l
\frac{
\mathrm{Pr} [Y^{a=1}=1| L=l]
}
{
\mathrm{Pj} [Y^{a=0}=1|L=l]
}
w(l).
\]

其中,

\[w(l)
=
\frac{
\mathrm{Pr} [Y^{a=0}=1, L=l]
}
{
\mathrm{Pr} [Y^{a=0}=1]
}, \quad
\sum_l w(l)=1.
\]

巴特西