CIS Primer Question 3.4.1

Here are my solutions to question 3.4.1 of Causal Inference in Statistics: a Primer (CISP). $$\DeclareMathOperator{\do}{do}$$

If we can only measure one additional variable to estimate the causal effect of $$X$$ on $$Y$$ in figure 3.8, then we should measure $$W$$. From question 3.3.1 we see that no single variable satisfies the backdoor criteria. Moreover, visual inspection of the graph verifies that $$W$$ satisfies the frontdoor criteria:

1. it intercepts all (the only) directed paths from $$X$$ to $$Y$$;
2. there is no unblocked path from $$X$$ to $$W$$; and
3. all backdoor paths from $$W$$ to $$Y$$ are blocked by $$X$$.

To illustrate this, lets simulate the causal effect in 3 separate ways:

1. by intervention,
2. via the backdoor, and
3. via the frontdoor.

Here are the data. Note that we have created functions for $$W$$ and $$Y$$ for use later.

N <- 100000

W <- function(x) {
N <- length(x)
rbinom(N, 1, inv_logit(-x))
}

Y <- function(d, w, z) {
N <- length(d)
rbinom(N, 1, inv_logit(-d - w + 3*z))
}

df <- tibble(id = 1:N) %>%
mutate(
b = rnorm(N, 0, 1),
a = b + rnorm(N, 0, 0.1),
c = rnorm(N, 0, 1),
d = rbinom(N, 1, inv_logit(-1 + c)),
z = rbinom(N, 1, inv_logit(-2 + 2*b + c)),
x = rbinom(N, 1, inv_logit(a + z)),
w = W(x),
y = Y(d, w, z)
)
Simulated data for figure 3.8
id b a c d z x w y
1 0.3641297 0.3917626 1.0369530 1 1 0 0 1
2 0.0287563 0.0397299 0.5736271 0 0 1 1 0
3 -0.7727052 -0.5993870 -0.5179657 0 1 0 1 1
4 0.4107888 0.5737898 1.2586840 0 1 1 0 1
5 2.3512417 2.1631719 0.6746523 1 1 1 0 1

Intervention

In order to simulate an intervention, we assign values to $$X$$ randomly, then assign new values for all its descendents. After intervention, the causal effect of $$X$$ on $$Y$$ is simply $$\mathbb P(Y \mid X)$$.

intervention <- df %>%
# intervene on x
mutate(
x = rbinom(n(), 1, 0.5),
w = W(x),
y = Y(d, w, z)
) %>%
# model P(y | do(x))
glm(
formula = y ~ x,
family = binomial(),
data = .
) %>%
# predict
augment(
newdata = tibble(x = 0:1),
type.predict = 'response'
) 
P(Y | do(X))
x .fitted .se.fit
0 0.4637566 0.0022239
1 0.5072714 0.0022422

We can compare this causal effect to the simple statistical effect to see the difference.

noncausal <- df %>%
# model P(y | x)
glm(
formula = y ~ x,
family = binomial(),
data = .
) %>%
# predict
augment(
newdata = tibble(x = 0:1),
type.predict = 'response'
) 
P(Y | X) ≠ P(Y | do(X))
x .fitted .se.fit
0 0.3797282 0.0022595
1 0.5759927 0.0021293

Backdoor

Since $$\{X, Z\}$$ satisfies the backdoor criteria, we can use it to apply the backdoor adjustment. First we’ll need $$\mathbb P(D, Z)$$.

# P(d, z)
p_d_z <- df %>%
group_by(d, z) %>%
count() %>%
ungroup() %>%
mutate(p_d_z = n / sum(n)) 

Now we model $$\mathbb P(Y \mid X, D, Z)$$, multiply it by $$\mathbb P(D, Z)$$, then take the sum for each value of $$X$$.

backdoor <- formula(y ~ 1 + x + z + d) %>%
# model P(y | x, d, z)
glm(
family = binomial(),
data = df
) %>%
# predict
augment(
type.predict = 'response',
newdata =
crossing(
d = c(0, 1),
x = c(0, 1),
z = c(0, 1)
)
) %>%
# get P(d, z)
mutate(p_y_given_d_x_z = .fitted) %>%
inner_join(p_d_z, by = c('d', 'z')) %>%
# backdoor adjustment over d, z
group_by(x) %>%
summarise(p_y_given_do_x = sum(p_y_given_d_x_z * p_d_z))
Backdoor estimates for P(Y | do(X))
x p_y_given_do_x
0 0.4681530
1 0.5033398

Note that the backdoor adjusted estimates are similar to the estimates from intervention.

Frontdoor

To apply the frontdoor adjustment with $$W$$, we’ll need $$\mathbb P(W \mid X)$$, $$\mathbb P(X^\prime)$$, and $$\mathbb P(Y \mid X, W)$$.

p_w_given_x <- df %>%
group_by(x, w) %>%
count() %>%
group_by(x) %>%
mutate(p_w_given_x = n / sum(n)) %>%
ungroup()

p_xprime <- df %>%
group_by(xprime = x) %>%
count() %>%
ungroup() %>%
mutate(p_xprime = n / sum(n))

p_y_given_xprime_w <- formula(y ~ 1 + x + w) %>%
glm(
family = binomial(),
data = df
) %>%
augment(
newdata = crossing(x = 0:1, w = 0:1),
type.predict = 'response'
) %>%
transmute(
xprime = x,
w,
p_y_given_xprime_w = .fitted
)

Now we apply the frontdoor adjustment:

$\mathbb P (Y \mid \do(X)) = \sum_{x^\prime, w} \mathbb P(x^\prime) \cdot \mathbb P(w \mid x) \cdot \mathbb P (y \mid x^\prime, w) .$

frontdoor <- p_w_given_x %>%
inner_join(p_y_given_xprime_w, by = 'w') %>%
inner_join(p_xprime, by = 'xprime') %>%
group_by(x) %>%
summarise(sum(p_w_given_x * p_y_given_xprime_w * p_xprime))
Frontdoor estimates of P(Y | do(X))
x sum(p_w_given_x * p_y_given_xprime_w * p_xprime)
0 0.4623710
1 0.5041105

Our frontdoor estimates of $$\mathbb P(Y \mid \do(X))$$ are very similar to the intervention and backdoor estimates.