Appendix A — Appendix: Stochastic direct and indirect effects

A.1 Definition of the effects

Consider the following directed acyclic graph.

A.2 Motivation for stochastic interventions

• So far we have discussed controlled, natural, and interventional (in)direct effects
• These effects require that $0<\mathsf{P}(A=1\mid W)<1$
• 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

A.3 Definition of stochastic effects

There 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

A.3.1 Example: incremental propensity score interventions (IPSI)

See (1)

Definition of the intervention

• Assume $A$ is binary, and $\mathsf{P}(A=1\mid W=w)=g(1\mid w)$ is the propensity score
• Consider an intervention in which each individual receives the intervention with probability $g_{\delta }(1\mid w)$, equal to $$g_{\delta }(1\mid w)=\frac{\delta g(1\mid w)}{\delta g(1\mid w)+1-g(1\mid w)}$$
• 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 $$\delta =\frac{\text{odds}(A_{\delta }=1\mid W=w)}{\text{odds}(A=1\mid W=w)}$$
• 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

A.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 $g_{\delta }(1\mid w)$

A.3.2 Example: modified treatment policies (MTP)

See (2)

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 $P{M}_{2.5}$ and you are interested in an intervention that reduces $P{M}_{2.5}$ concentration by some amount $\delta$

A.3.3 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:

$$\psi =\mathbb{E}[Y_{A_{\delta }}-Y]$$

• Recall the NPSEM

• From this we have

• 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:

A.4 Identification assumptions

• Confounder assumptions:
  • $A\perp \perp Y_{a,m}\mid W$
  • $M\perp \perp Y_{a,m}\mid W,A$
• No confounder of $M\to Y$ affected by $A$
• Positivity assumptions:
  • If $g_{\delta }(a\mid w)>0$ then $g(a\mid w)>0$
  • If $\mathsf{P}(M=m\mid W=w)>0$ then $\mathsf{P}(M=m\mid A=a,W=w)>0$

Under these assumptions, stochastic effects are identified as follows

• The indirect effect can be identified as follows

• The direct effect can be identified as follows

• Let’s dissect the formula for the indirect effect in R:

  n <- 1e6
  w <- 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 $(M,A,W)$:

  fit_y1 <- lm(y ~ m + a + w)
  fit_y2 <- lm(y ~ a + w)

• Get predictions fixing $A=a$ for all possible values $a$

  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 [] for each value $a$

  pseudo_a1 <- pred_y2_a1 - pred_y1_a1
  pseudo_a0 <- pred_y2_a0 - pred_y1_a0

• Estimate the propensity score $g(1\mid w)$ and evaluate the post-intervention propensity score $g_{\delta }(1\mid w)$

  pscore_fit <- glm(a ~ w, family = binomial())
  pscore <- predict(pscore_fit, type = 'response')
  ## How do the intervention vs observed propensity score compare
  pscore_delta <- 2 * pscore / (2 * pscore + 1 - pscore)

• What do the post-intervention propensity scores look like?

  plot(pscore, pscore_delta, xlab = 'Observed prop. score',
       ylab = 'Prop. score under intervention')
  abline(0, 1)

  

A.5 What are the odds of exposure under intervention vs real world?

odds <- (pscore_delta / (1 - pscore_delta)) / (pscore / (1 - pscore))
summary(odds)

#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>       2       2       2       2       2       2

• Compute the sum

$$\sum _a\{\mathbb{E}(Y\mid A=a,W)-\mathbb{E}(Y\mid A=a,M,W)\}{g}_{\delta }(a\mid W)$$

indirect <- pseudo_a1 * pscore_delta + pseudo_a0 * (1 - pscore_delta)

• The average of this value is the indirect effect

  ## E[Y(Adelta) - Y(Adelta, M)]
  mean(indirect)

  #> [1] 0.1092928

• The direct effect is

• Which can be computed as

  direct <- (pred_y1_a1 - y) * pscore_delta +
         (pred_y1_a0 - y) * (1 - pscore_delta)
  mean(direct)

  #> [1] 0.1092246

A.6 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 (3)
  • R implementation to be released soon…stay tuned!

1.

Kennedy EH. Nonparametric causal effects based on incremental propensity score interventions. Journal of the American Statistical Association Taylor & Francis; 2018;

2.

Dı́az I, Hejazi NS. Causal mediation analysis for stochastic interventions. Journal of the Royal Statistical Society: Series B (Statistical Methodology) Wiley Online Library; 2020;82:661–83.

3.

Hejazi NS, Rudolph KE, Laan MJ van der, Dı́az I. Nonparametric causal mediation analysis for stochastic interventional (in) direct effects. Biostatistics [Internet] Oxford University Press; 2022;(in press). Available from: https://arxiv.org/abs/2009.06203