Kennedy, Edward H. 2018. “Nonparametric Causal Effects Based on Incremental Propensity Score Interventions.”Journal of the American Statistical Association, no. just-accepted.
Definition of the intervention
Assume \(A\) is binary, and \(\P(A=1\mid W=w) = g(1\mid w)\) is the propensity score
Consider an intervention in which each individual receives the intervention with probability \(g_\delta(1\mid w)\), equal to \[\begin{equation*}
g_\delta(1\mid w)=\frac{\delta g(1\mid w)}{\delta g(1\mid w) +
1 - g(1\mid w)}
\end{equation*}\]
e.g., draw the post-intervention exposure from a Bernoulli variable with probability \(g_\delta(1\mid w)\)
The value \(\delta\) is user given
Let \(A_\delta\) denote the post-intervention exposure distribution
Some algebra shows that \(\delta\) is an odds ratio comparing the pre- and post-intervention exposure distributions \[\begin{equation*}
\delta = \frac{\text{odds}(A_\delta = 1\mid W=w)}
{\text{odds}(A = 1\mid W=w)}
\end{equation*}\]
Interpretation: what would happen in a world where the odds of receiving treatment is increased by \(\delta\)
Let \(Y_{A_\delta}\) denote the outcome in this hypothetical world
8.3.1.1 Illustrative application for IPSIs
Consider the effect of participation in sports on children’s BMI
Mediation through snacking, exercising, etc.
Intervention: for each individual, increase the odds of participating in sports by \(\delta=2\)
The post-intervention exposure is a draw \(A_\delta\) from a Bernoulli distribution with probability \(g_\delta(1\mid w)\)
Dı́az, Iván, and Nima S Hejazi. 2020. “Causal Mediation Analysis for Stochastic Interventions.”Journal of the Royal Statistical Society: Series B (Statistical Methodology) 82 (3): 661–83.
Definition of the intervention
Consider a continuous exposure \(A\) taking values in the real numbers
Consider an intervention that assigns exposure as \(A_\delta = A - \delta\)
Example: \(A\) is pollution measured as \(PM_{2.5}\) and you are interested in an intervention that reduces \(PM_{2.5}\) concentration by some amount \(\delta\)
8.3.2 Mediation analysis for stochastic interventions
The total effect of an IPSI can be computed as a contrast of the outcome under intervention vs no intervention: \[\begin{equation*}
\psi = \E[Y_{A_\delta} - Y]
\end{equation*}\]
Recall the NPSEM \[\begin{align*}
W & = f_W(U_W)\\
A & = f_A(W, U_A)\\
M & = f_M(W, A, U_M)\\
Y & = f_Y(W, A, M, U_Y)
\end{align*}\]
From this we have \[\begin{align*}
M_{A_\delta} & = f_M(W, A_\delta, U_M)\\
Y_{A_\delta} & = f_Y(W, A_\delta, M_{A_\delta}, U_Y)
\end{align*}\]
Thus, we have \(Y_{A_\delta} = Y_{A_\delta, M_{A_\delta}}\) and \(Y =
Y_{A,M_{A}}\)
Let us introduce the counterfactual \(Y_{A_\delta, M}\), interpreted as the outcome observed in a world where the intervention on \(A\) is performed but the mediator is fixed at the value it would have taken under no intervention: [Y_{A_, M} = f_Y(W, A_, M, U_Y)]
Then we can decompose the total effect into: \[\begin{align*}
\E[Y&_{A_\delta,M_{A_\delta}} - Y_{A,M_A}] = \\
&\underbrace{\E[Y_{\color{red}{A_\delta},\color{blue}{M_{A_\delta}}} -
Y_{\color{red}{A_\delta},\color{blue}{M}}]}_{\text{stochastic natural
indirect effect}} +
\underbrace{\E[Y_{\color{blue}{A_\delta},\color{red}{M}} -
Y_{\color{blue}{A},\color{red}{M}}]}_{\text{stochastic natural direct
effect}}
\end{align*}\]
8.4 Identification assumptions
Confounder assumptions:
\(A \indep Y_{a,m} \mid W\)
\(M \indep Y_{a,m} \mid W, A\)
No confounder of \(M\rightarrow Y\) affected by \(A\)
Positivity assumptions:
If \(g_\delta(a \mid w)>0\) then \(g(a \mid w)>0\)
If \(\P(M=m\mid W=w)>0\) then \(\P(M=m\mid A=a,W=w)>0\)
Under these assumptions, stochastic effects are identified as follows
The indirect effect can be identified as follows \[\begin{align*}
\E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\
&\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, W)
-\E(Y\mid A=a, M, W)\}}g_\delta(a\mid W)}\right]
\end{align*}\]
The direct effect can be identified as follows \[\begin{align*}
\E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\
&\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, M, W)
- Y\}}g_\delta(a\mid W)}\right]
\end{align*}\]
Let’s dissect the formula for the indirect effect in R:
Code
n <-1e6w <-rnorm(n)a <-rbinom(n, 1, plogis(1+ w))m <-rnorm(n, w + a)y <-rnorm(n, w + a + m)
First, fit regressions of the outcome on \((A,W)\) and \((M,A,W)\):
Code
fit_y1 <-lm(y ~ m + a + w)fit_y2 <-lm(y ~ a + w)
Get predictions fixing \(A=a\) for all possible values \(a\)
Estimate the propensity score \(g(1\mid w)\) and evaluate the post-intervention propensity score \(g_\delta(1\mid w)\)
Code
pscore_fit <-glm(a ~ w, family =binomial())pscore <-predict(pscore_fit, type ="response")## How do the intervention vs observed propensity score comparepscore_delta <-2* pscore / (2* pscore +1- pscore)
What do the post-intervention propensity scores look like?
Code
plot(pscore, pscore_delta,xlab ="Observed prop. score",ylab ="Prop. score under intervention")abline(0, 1)
8.5 What are the odds of exposure under intervention vs real world?
Compute the sum \[\begin{equation*}
\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, W) -
\E(Y\mid A=a, M, W)\}}g_\delta(a\mid W)}
\end{equation*}\]
The direct effect is \[\begin{align*}
\E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\
&\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, M,
W) - Y\}}g_\delta(a\mid W)}\right]
\end{align*}\]
Still require the absence of intermediate confounders
But, compared to the NDE and NIE, we can design a randomized study where identifiability assumptions hold, at least in principle
There is a version of these effects that can accommodate intermediate confounders (Hejazi et al. 2022)
R implementation to be released soon…stay tuned!
Hejazi, Nima S, Kara E Rudolph, Mark J van der Laan, and Iván Dı́az. 2022. “Nonparametric Causal Mediation Analysis for Stochastic Interventional (in) Direct Effects.”Biostatistics (in press). https://doi.org/10.1093/biostatistics/kxac002.
Source Code
# Appendix: Stochastic direct and indirect effects {#stochastic}```{r}#| label: load-renv#| echo: false#| message: falserenv::autoload()library(here)```## Definition of the effectsConsider the following directed acyclic graph.```{tikz}#| fig-cap: Directed acyclic graph under no intermediate confounders of the mediator-outcome relation affected by treatment\dimendef\prevdepth=0\pgfdeclarelayer{background}\pgfsetlayers{background,main}\usetikzlibrary{arrows,positioning}\tikzset{>=stealth',punkt/.style={rectangle,rounded corners,draw=black, very thick,text width=6.5em,minimum height=2em,text centered},pil/.style={->,thick,shorten <=2pt,shorten >=2pt,}}\newcommand{\Vertex}[2]{\node[minimum width=0.6cm,inner sep=0.05cm] (#2) at (#1) {$#2$};}\newcommand{\VertexR}[2]{\node[rectangle, draw, minimum width=0.6cm,inner sep=0.05cm] (#2) at (#1) {$#2$};}\newcommand{\ArrowR}[3]{ \begin{pgfonlayer}{background}\draw[->,#3] (#1) to[bend right=30] (#2);\end{pgfonlayer}}\newcommand{\ArrowL}[3]{ \begin{pgfonlayer}{background}\draw[->,#3] (#1) to[bend left=45] (#2);\end{pgfonlayer}}\newcommand{\EdgeL}[3]{ \begin{pgfonlayer}{background}\draw[dashed,#3] (#1) to[bend right=-45] (#2);\end{pgfonlayer}}\newcommand{\Arrow}[3]{ \begin{pgfonlayer}{background}\draw[->,#3] (#1) -- +(#2);\end{pgfonlayer}}\begin{tikzpicture} \Vertex{-4, 0}{W} \Vertex{0, 0}{M} \Vertex{-2, 0}{A} \Vertex{2, 0}{Y} \Arrow{W}{A}{black} \Arrow{A}{M}{black} \Arrow{M}{Y}{black} \ArrowL{W}{Y}{black} \ArrowL{A}{Y}{black} \ArrowL{W}{M}{black}\end{tikzpicture}```## Motivation for stochastic interventions- So far we have discussed controlled, natural, and interventional (in)direct effects- These effects require that $0 < \P(A=1\mid W) < 1$- They are defined only for binary exposures- _What can we do when the positivity assumption does not hold or the exposure is continuous?_- Solution: We can use stochastic effects## Definition of stochastic effectsThere are two possible ways of defining stochastic effects:- Consider the effect of an intervention where the exposure is drawn from a distribution - For example incremental propensity score interventions- Consider the effect of an intervention where the post-intervention exposure is a function of the actually received exposure - For example modified treatment policies- In both cases $A \mid W$ is a non-deterministic intervention, thus the name _stochastic intervention_### Example: incremental propensity score interventions (IPSI) [@kennedy2018nonparametric] {#ipsi}#### Definition of the intervention {.unnumbered}- Assume $A$ is binary, and $\P(A=1\mid W=w) = g(1\mid w)$ is the propensity score- Consider an intervention in which each individual receives the intervention with probability $g_\delta(1\mid w)$, equal to \begin{equation*} g_\delta(1\mid w)=\frac{\delta g(1\mid w)}{\delta g(1\mid w) + 1 - g(1\mid w)} \end{equation*}- e.g., draw the post-intervention exposure from a Bernoulli variable with probability $g_\delta(1\mid w)$- The value $\delta$ is user given- Let $A_\delta$ denote the post-intervention exposure distribution- Some algebra shows that $\delta$ is an odds ratio comparing the pre- and post-intervention exposure distributions \begin{equation*} \delta = \frac{\text{odds}(A_\delta = 1\mid W=w)} {\text{odds}(A = 1\mid W=w)} \end{equation*}- Interpretation: _what would happen in a world where the odds of receiving treatment is increased by $\delta$_- Let $Y_{A_\delta}$ denote the outcome in this hypothetical world#### Illustrative application for IPSIs- Consider the effect of participation in sports on children's BMI- Mediation through snacking, exercising, etc.- Intervention: for each individual, increase the odds of participating in sports by $\delta=2$- The post-intervention exposure is a draw $A_\delta$ from a Bernoulli distribution with probability $g_\delta(1\mid w)$### Example: modified treatment policies (MTP) [@diaz2020causal] {.unnumbered}#### Definition of the intervention {.unnumbered}- Consider a continuous exposure $A$ taking values in the real numbers- Consider an intervention that assigns exposure as $A_\delta = A - \delta$- Example: $A$ is pollution measured as $PM_{2.5}$ and you are interested in an intervention that reduces $PM_{2.5}$ concentration by some amount $\delta$### Mediation analysis for stochastic interventions- The total effect of an IPSI can be computed as a contrast of the outcome under intervention vs no intervention: \begin{equation*} \psi = \E[Y_{A_\delta} - Y] \end{equation*}- Recall the NPSEM \begin{align*} W & = f_W(U_W)\\ A & = f_A(W, U_A)\\ M & = f_M(W, A, U_M)\\ Y & = f_Y(W, A, M, U_Y) \end{align*}- From this we have \begin{align*} M_{A_\delta} & = f_M(W, A_\delta, U_M)\\ Y_{A_\delta} & = f_Y(W, A_\delta, M_{A_\delta}, U_Y) \end{align*}- Thus, we have $Y_{A_\delta} = Y_{A_\delta, M_{A_\delta}}$ and $Y = Y_{A,M_{A}}$- Let us introduce the counterfactual $Y_{A_\delta, M}$, interpreted as the outcome observed in a world where the intervention on $A$ is performed but the mediator is fixed at the value it would have taken under no intervention:\[Y_{A_\delta, M} = f_Y(W, A_\delta, M, U_Y)\]- Then we can decompose the total effect into: \begin{align*} \E[Y&_{A_\delta,M_{A_\delta}} - Y_{A,M_A}] = \\ &\underbrace{\E[Y_{\color{red}{A_\delta},\color{blue}{M_{A_\delta}}} - Y_{\color{red}{A_\delta},\color{blue}{M}}]}_{\text{stochastic natural indirect effect}} + \underbrace{\E[Y_{\color{blue}{A_\delta},\color{red}{M}} - Y_{\color{blue}{A},\color{red}{M}}]}_{\text{stochastic natural direct effect}} \end{align*}## Identification assumptions- Confounder assumptions: + $A \indep Y_{a,m} \mid W$ + $M \indep Y_{a,m} \mid W, A$- No confounder of $M\rightarrow Y$ affected by $A$- Positivity assumptions: + If $g_\delta(a \mid w)>0$ then $g(a \mid w)>0$ + If $\P(M=m\mid W=w)>0$ then $\P(M=m\mid A=a,W=w)>0$Under these assumptions, stochastic effects are identified as follows- The indirect effect can be identified as follows \begin{align*} \E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\ &\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, W) -\E(Y\mid A=a, M, W)\}}g_\delta(a\mid W)}\right] \end{align*}- The direct effect can be identified as follows \begin{align*} \E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\ &\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, M, W) - Y\}}g_\delta(a\mid W)}\right] \end{align*}- Let's dissect the formula for the indirect effect in R:```{r}n <-1e6w <-rnorm(n)a <-rbinom(n, 1, plogis(1+ w))m <-rnorm(n, w + a)y <-rnorm(n, w + a + m)```- First, fit regressions of the outcome on $(A,W)$ and $(M,A,W)$:```{r}fit_y1 <-lm(y ~ m + a + w)fit_y2 <-lm(y ~ a + w)```- Get predictions fixing $A=a$ for all possible values $a$```{r}pred_y1_a1 <-predict(fit_y1, newdata =data.frame(a =1, m, w))pred_y1_a0 <-predict(fit_y1, newdata =data.frame(a =0, m, w))pred_y2_a1 <-predict(fit_y2, newdata =data.frame(a =1, w))pred_y2_a0 <-predict(fit_y2, newdata =data.frame(a =0, w))```- Compute \[\color{ForestGreen}{\{\E(Y\mid A=a, W)-\E(Y\mid A=a, M, W)\}}\] for each value $a$```{r}pseudo_a1 <- pred_y2_a1 - pred_y1_a1pseudo_a0 <- pred_y2_a0 - pred_y1_a0```- Estimate the propensity score $g(1\mid w)$ and evaluate the post-intervention propensity score $g_\delta(1\mid w)$```{r}pscore_fit <-glm(a ~ w, family =binomial())pscore <-predict(pscore_fit, type ="response")## How do the intervention vs observed propensity score comparepscore_delta <-2* pscore / (2* pscore +1- pscore)```- What do the post-intervention propensity scores look like?```{r}plot(pscore, pscore_delta,xlab ="Observed prop. score",ylab ="Prop. score under intervention")abline(0, 1)```## What are the odds of exposure under intervention vs real world?```{r}odds <- (pscore_delta / (1- pscore_delta)) / (pscore / (1- pscore))summary(odds)```- Compute the sum \begin{equation*} \color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, W) - \E(Y\mid A=a, M, W)\}}g_\delta(a\mid W)} \end{equation*}```{r}indirect <- pseudo_a1 * pscore_delta + pseudo_a0 * (1- pscore_delta)```- The average of this value is the indirect effect```{r}## E[Y(Adelta) - Y(Adelta, M)]mean(indirect)```- The direct effect is \begin{align*} \E&(Y_{A_\delta} - Y_{A_\delta, M}) =\\ &\E\left[\color{Goldenrod}{\sum_{a}\color{ForestGreen}{\{\E(Y\mid A=a, M, W) - Y\}}g_\delta(a\mid W)}\right] \end{align*}- Which can be computed as```{r}direct <- (pred_y1_a1 - y) * pscore_delta + (pred_y1_a0 - y) * (1- pscore_delta)mean(direct)```## Summary- Stochastic (in)direct effects - Relax the positivity assumption - Can be defined for non-binary exposures - Do not require a cross-world assumption- Still require the absence of intermediate confounders - But, compared to the NDE and NIE, we can design a randomized study where identifiability assumptions hold, at least in principle - There is a version of these effects that can accommodate intermediate confounders [@hejazi2020nonparametric] - `R` implementation to be released soon...stay tuned!
