In dhessing/ipcalg: Causal discovery for IDAGs

library(tidyverse)
library(ipcalg)
library(pcalg)

Definition

We will denote a causal effect of $A$ on $Y$ by $\Delta Y_A$. We refer to a graph including $\Delta Y_A$ as an IDAG. If there is an interaction between some variable and A, there is a directed arrow (or path) from this variable to $\Delta Y_A$. In contrast, effect measure modification only corresponds to an association between some variable and $\Delta Y_A$. [Nilsson]

Types of effect modification.

According to [VanderWeele], there are four types of effect modification. In these examples we have intervention $A$ and outcome variable $Y$. $Q$ and $A$ have a direct effect on $Y$, and there is an interaction effect between $Q$ and $A$.

No effect modification

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  Q = rnorm(n),
  Y = A + Q + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot()

Direct effect modification

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  Q = rnorm(n),
  Y = A + Q + A * Q + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot()

We call $Q$ a direct effect modifier for the causal effect of $A$ on $Y$, because $A$ is a direct cause of $Y$. This is represented in the IDAG as an arrow from $Q$ to $\Delta Y_A$.

Indirect effect modification

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  C = rnorm(n),
  Q = C + rnorm(n),
  Y = A + Q + A * Q + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot(2)

We say that $C$ is an indirect effect modifier for the causal effect of $A$ on $Y$, since $C$ affects $Y$ indirectly through $Q$. This is represented in the IDAG as an arrow from $C$ to $Q$.

Effect modification by proxy

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot(1)

$X$ is an effect modifier for the causal effect of $A$ on $Y$ because $X$ contains information about $Q$ (which serves a direct effect modifier for the causal effect of $A$ on $Y$). However, because $R$ is not a cause of $A$, we say that $X$ is an effect modifier by proxy. This is represented in the IDAG as an arrow from $X$ to $R$.

Effect modification by common cause

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  C = rnorm(n),
  X = C + rnorm(n),
  Q = C + rnorm(n),
  Y = A + Q + A * Q + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot(2)

Because $X$ contains information about $C$ which is a cause of the direct effect modifier $Q$, we call $X$ an effect modifier by common cause of the effect of $A$ on $Y$.

Total effect modification

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  Q = rnorm(n),
  C = rnorm(n),
  X = A + A * Q + Q + rnorm(n),
  Y = X + C + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot()

set.seed(42)
n <- 1000
alpha <- 0.01
data <- tibble(
  A = rnorm(n),
  B = rnorm(n),
  C = rnorm(n),
  Y = A + B + C + A*B + rnorm(n),
)

pc(suffStat = list(C = cor(data), n = nrow(data)),
  indepTest = gaussCItest,
  alpha = alpha,
  labels = colnames(data)) %>%
  ipc(data, x = 'A', y = 'Y', alpha = alpha) %>%
  plot()