d-seperation

D-separation

Definition

Two variables are said to be d-separated if all paths between them are blocked (otherwise they are d-connected). Two sets of variables are said to be d-separated if each variable in the first set is d-separated from every variable in the second set.

Rules: When is a path open or blocked?

1. If there are no variables being conditioned on, a path is blocked if and only if two arrowheads on the path collide at some variable on the path.

library(ggdag)


Attaching package: 'ggdag'

The following object is masked from 'package:stats':

    filter

dag1_confound_ggdag <- dagify(
 C ~ A + B
)
# Plot the DAG
ggdag(dag1_confound_ggdag) + theme_dag_blank()

Beispiel:

D=Drivingskills
A=Alkohol
U=Unfall

Der Pfad zwischen den Drivingskills und dem Alkoholkonsum ist durch den Collider Unfall blockiert.

dag2_confound_ggdag <- dagify(
  U ~ D + A
)
# Plot the DAG
ggdag(dag2_confound_ggdag, layout = "circle") + theme_dag_blank()

Regression:

# set number of observations
n <- 10000

D <- rnorm(n, 0, 1)
A <- rbinom(n, 1, 0.05)
U <- 0.01 + 0.3*A - 0.1 * D + rnorm(n, 0, 1)

lm(U ~ A + D)


Call:
lm(formula = U ~ A + D)

Coefficients:
(Intercept)            A            D  
    0.02153      0.29753     -0.09725

2. Any path that contains a noncollider that has been conditioned on is blocked.
The path between A and C is blocked, conditioning on B.

dag3_confound_ggdag <- dagify(
C ~ B,
B ~ A,
coords = list(x = c(A = 1, B = 2, C = 3),
                y = c(A = 1, B = 1, C = 1))
)
# Plot the DAG
ggdag(dag3_confound_ggdag) + theme_dag_blank()

Beispiel:

F=Feuer
R=Rauch
A=Alarm

Feuer und Alarm sind unabhängig, bedingt auf Rauch.

dag4_confound_ggdag <- dagify(
  A ~ R,
  R ~ F,
   coords = list(x = c(F = 1, R = 2, A = 3),
                y = c(F = 1, R = 1, A = 1))
)
# Plot the DAG
ggdag(dag4_confound_ggdag) + theme_dag_blank()

3. A collider that has been conditioned on does not block a path. If we condition on C, the path between A and B is not blocked. They are d-connected.

dag5_confound_ggdag <- dagify(
 C ~ A + B
)
# Plot the DAG
ggdag(dag5_confound_ggdag, layout = "circle") + theme_dag_blank()

Durch die Konditionierung auf “Unfall” ist der Pfad zwischen Drivingskills und Alkohol nicht mehr blockiert.

dag6_confound_ggdag <- dagify(
  U ~ D + A
)
# Plot the DAG
ggdag(dag6_confound_ggdag, layout = "circle") + theme_dag_blank()

4. A collider that has a descendant that has been conditioned on does not block a path.

library(ggdag)
dag7_confound_ggdag <- dagify(
 C ~ A + B,
 D ~C,
 coords = list(x = c(B = 1, C = 2, D = 2, A = 3),
                y = c(B = 3, C = 2, D = 1, A = 3))
)
# Plot the DAG
ggdag(dag7_confound_ggdag) + theme_dag_blank()

Beispiel

Neue Variable S=Schäden am Auto
Die Variable “Schäden am Auto” ist ein Nachfolger des Colliders “Unfall”. D und A waren d-seperated durch Collider U, aber sind d-connected, sobald wir auf S bedingen.

library(ggdag)
dag8_confound_ggdag <- dagify(
 U ~ D + A,
 S ~ U,
 coords = list(x = c(D = 1, U = 2, S = 2, A = 3),
                y = c(D = 3, U = 2, S = 1, A = 3))
)
# Plot the DAG
ggdag(dag8_confound_ggdag) + theme_dag_blank()

Excercise

What sets of variables d-seperate the nonadjacent nodes in this DAG?

dag9_confound_ggdag <- dagify(
 X ~ Z1 + Z3,
 Z3 ~ Z1 + Z2,
 W ~ X,
 Y ~ W + Z2 + Z3,
 coords = list(x = c(X = 1, Z1 = 1, Z3 = 2, W = 2, Z2 = 3, Y = 3),
                y = c(X = 1, Z1 = 3, Z3 = 2, W = 1, Z2 = 3, Y = 1))
)
# Plot the DAG
ggdag(dag9_confound_ggdag) + theme_dag_blank()