Terminology of graphs

• Directed graph: shows that $A$ affects $Y$

• Undirected graph: $A$ and $Y$ are associated with each other

• Nodes or vertices: $A$ and $Y$

  • We can think of them as variables

• Edge: the link between $A$ and $Y$

• Directed graph: all edges are directed

• Adjacent variables: if connected by an edge

Paths

• A path is a way to get from one vertex to another, traveling along edges

• There are 2 paths from $W$ to $B$:
  • $W\to Z\to B$
  • $W\to Z\to A\to B$

  Directed Acyclic Graphs (DAGs)

  • No undirected paths

• No cycles

• This is a DAG

More terminology

• $A$ is $Z$’s parent
• $D$ has two parents, $B$ and $Z$
• $B$ is a child of $Z$
• $D$ is a descendant of $A$
• $Z$ is a ancestor of $D$

DAG example 1

• C is independent of all variables $$P(C\mid A,B,D)=P\left(C\right)$$

• $B$ and $C,D$ are independent, conditional on $A$ $$P(B\mid A,C,D)=P(B\mid A)⟺B\perp C,D\mid A$$

• $B$ and $D$ are marginally dependent $$P(B\mid D)\ne P\left(B\right)$$

DAG example 2

• $A$ and $B$ are independent, conditional on $C$ and $D$ $$P(A\mid B,C,D)=P(A\mid C,D)⟺A\perp B\mid C,D$$

• $C$ and $D$ are independent, conditional on $A$ and $B$ $$P(D\mid A,B,C)=P(D\mid A,B)⟺D\perp C\mid A,B$$

Decomposition of joint distributions

1. Start with roots (nodes with no parents)

2. Proceed down the descendant line, always conditioning on parents

• $P(A,B,C,D)=P\left(C\right)P\left(D\right)P(A\mid D)P(B\mid A)$

• $P(A,B,C,D)=P\left(D\right)P(A\mid D)P(B\mid D)P(C\mid A,B)$

Compatibility between DAGs and distributions

• In the above examples, the DAGs admit the probability factorizations. Hence, the probability function and the DAG are compatible

• DAGs that are compatible with a particular probability function are not necessarily unique

• Example 1:

• Example 2:

• In both of the above examples, $A$ and $B$ are dependent, i.e., $P(A,B)\ne P\left(A\right)P\left(B\right)$

Types of paths

• Forks

• Chains

• Inverted forks

When do paths induce associations?

• If nodes $A$ and $B$ are on the ends of a path, then they are associated (via this path), if

  • Some information flows to both of them (aka Fork), or
  • Information from one makes it to the other (aka Chain)

• Example: information flows from $E$ to $A$ and $B$

• Example: information from $A$ makes it to $B$

Paths that do not induce association

• Information from $A$ and $B$ collide at $G$

• $G$ is a collider

• $A$ and $B$ both affect $G$:

  • Information does not flow from $G$ to either $A$ or $B$
  • So $A$ and $B$ are independent (if this is the only path between them)

• If there is a collider anywhere on the path from $A$ to $B$, then no association between $A$ and $B$ comes from this path

Blocking on a chain

• Paths can be blocked by conditioning on nodes in the path

• In the graph below, $G$ is a node in the middle of a chain. If we condition on $G$, then we block the path from $A$ to $B$

• For example, $A$ is the temperature, $G$ is whether sidewalks are icy, and $B$ is whether someone falls
  • $A$ and $B$ are associated marginally
  • But if we conditional on the sidewalk condition $G$, then $A$ and $B$ are independent

Blocking on a fork

• Associations on a fork can also be blocked

• In the following fork, if we condition on $G$, then the path from $A$ to $B$ is block

No need to to block a collider

• The opposite situation occurs if a conllider is blocked

• In the following inverted fork

  • Originally $A$ and $B$ are not associated, since information collides at $G$
  • But if we condition on $G$, then $A$ and $B$ become associated

• Example: $A$ and $B$ are the states of two on/off switches, and $G$ is whether the lightbulb is lit up.

  • The two switches $A$ and $B$ are determined by two independent coin flips

  • $G$ is lit up only if both $A$ and $B$ are in the on state

  • Conditional on $G$, the two switches are not independent: if $G$ is off, then $A$ must be off if $B$ is on

d-separation

• A path is d-separated by a set of nodes $C$ if

  • It contains a chain ($D\to E\to F$) and the middle part is in $C$, or

  • It contains a fork ($D\leftarrow E\to F$) and the middle part is in $C$, or

  • It contains an inverted fork ($D\to E\leftarrow F$), and the middle part is not in $C$, nor are any descendants of it

• Two nodes, $A$ and $B$, are d-separated by a set of nodes $C$ if it blocks every path from $A$ to $B$. Thus $$A\perp B\mid C$$

• Recall the ignorability assumption $$Y^0,Y^1\perp A\mid X$$

Confounders on paths

• A simple DAG: $X$ is a confounder between the relationship between treatment $A$ and outcome $Y$

• A slightly more complicated graph

  • $V$ affects $A$ directly
  • $V$ affects $Y$ indirectly, through $W$
  • Thus, $V$ is a confounder

Frontdoor paths

• A frontdoor path from $A$ to $Y$ is one that begins with an arrow emanating out of $A$

• We do not worry about frontdoor paths, because they capture effects of treatment

• Example: $A\to Y$ is a frontdoor path from $A$ to $Y$

• Example: $A\to Z\to Y$ is a frontdoor path from $A$ to $Y$

Do not block nodes on the frontdoor path

• If we are interested in the causal effect of $A$ on $Y$, we should not control for (aka block) $Z$

  • This is because controlling for $Z$ would be controlling for an affect of treatment

• Causal mediation analysis involves understanding frontdoor paths from $A$ and $Y$

Backdoor paths

• Backdoor paths from treatment $A$ to outcome $Y$ are paths from $A$ to $Y$ that travels through arrows going into $A$

• Here, $A\leftarrow X\to Y$ is a backdoor path from $A$ to $Y$

• Backdoor paths confound the relationship between $A$ and $Y$, so they need to be blocked!

• To sufficiently control for confounding, we must identify a set of variables that block all backdoor paths from treatment to outcome

  • Recall the ignorability: if $X$ is this set of variables, then $Y^0,Y^1\perp A\mid X$

Criteria

• Next we will discuss two criteria to identify sets of variables that are sufficient to control for confounding

  • Backdoor path criterion: if the graph is known
  • Disjunctive cause criterion: if the graph is not known

Backdoor path criterion

• Backdoor path criterion: a set of variables $X$ is sufficient to control for confounding if

  • It blocks all backdoor paths from treatment to the outcome, and
  • It does not include any descendants of treatment

• Note: the solution $X$ is not necessarily unique

Backdoor path criterion: a simple example

• There is one backdoor path from $A$ to $Y$

  • It is not blocked by a collider

• Sets of variables that are sufficient to control for confounding:

  • $\left\{V\right\}$, or
  • $\left\{W\right\}$, or
  • $\{V,W\}$

Backdoor path criterion: a collider example

• There is one backdoor path from $A$ to $Y$

  • It is blocked by a collider $M$, so there is no confounding

• If we condition on $M$, then it open a path between $V$ and $W$

• Sets of variables that are sufficient to control for confounding:
  • $\left\{\right\}$, $\left\{V\right\}$, $\left\{W\right\}$, $\{M,V\}$, $\{M,W\}$, $\{M,V,W\}$
  • But not $\left\{M\right\}$

Backdoor path criterion: a multi backdoor paths example

• First path: $A\leftarrow Z\leftarrow V\to Y$

  • No collider on this path
  • So controlling for either $Z$, $V$, or both is sufficient

• Second path: $A\leftarrow W\to Z\leftarrow V\to Y$

  • $Z$ is a collider
  • So controlling $Z$ opens a path between $W$ and $V$
  • We can block $\left\{V\right\}$, $\left\{W\right\}$, $\{Z,V\}$, $\{Z,W\}$, or $\{Z,V,W\}$

• To block both paths, it’s sufficient to control for

• $\left\{V\right\}$, $\{Z,V\}$, $\{Z,W\}$, or $\{Z,V,W\}$
  • But not $\left\{Z\right\}$ or $\left\{W\right\}$

  Disjunctive cause criterion

  Disjunctive cause criterion

  • For many problems, it is difficult to write down accurate DAGs

• In this case, we can use the disjunctive cause criterion: control for all observed causes of the treatment, the outcome, or both

• If there exists a set of observed variables that satisfy the backdoor path criterion, then the variables selected based on the disjunctive cause criterion are sufficient to control for confounding

• Disjunctive cause criterion does not always select the smallest set of variable to control for, but it is conceptually simple

Example

• Observed pre-treatment variables: $\{M,W,V\}$

  • Unobserved pre-treatment variables: $\{U_1,U_2\}$

  • Suppose we know: $W,V$ are causes of $A$, $Y$ or both

• Suppose $M$ is not a cause of either $A$ or $Y$

  • Comparing two methods for selecting variables

1. Use all pre-treatment covariates: $\{M,W,V\}$
2. Use variables based on disjunctive cause criterion: $\{W,V\}$

Example continued: hypothetical DAG 1

1. Use all pre-treatment covariates: $\{M,W,V\}$

• Satisfy backdoor path criterion? Yes

2. Use variables based on disjunctive cause criterion: $\{W,V\}$

• Satisfy backdoor path criterion? Yes

Example continued: hypothetical DAG 2

1. Use all pre-treatment covariates: $\{M,W,V\}$

• Satisfy backdoor path criterion? Yes

2. Use variables based on disjunctive cause criterion: $\{W,V\}$

• Satisfy backdoor path criterion? Yes

Example continued: hypothetical DAG 3

1. Use all pre-treatment covariates: $\{M,W,V\}$

• Satisfy backdoor path criterion? No

2. Use variables based on disjunctive cause criterion: $\{W,V\}$

• Satisfy backdoor path criterion? Yes

Example continued: hypothetical DAG 4

1. Use all pre-treatment covariates: $\{M,W,V\}$

• Satisfy backdoor path criterion? No

2. Use variables based on disjunctive cause criterion: $\{W,V\}$

• Satisfy backdoor path criterion? No

References

• Coursera class: “A Crash Course on Causality: Inferring Causal Effects from Observational Data”, by Jason A. Roy (University of Pennsylvania)

  • https://www.coursera.org/learn/crash-course-in-causality