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 \(B\):
  • \(W \rightarrow Z \rightarrow B\)
  • \(W \rightarrow Z \rightarrow A \rightarrow B\)

  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 \[ P(C\mid A, B, D) = P(C) \]

• \(B\) and \(C, D\) are independent, conditional on \(A\) \[ P(B\mid A, C, D) = P(B\mid A) \Longleftrightarrow B \perp C, D \mid A \]

• \(B\) and \(D\) are marginally dependent \[ P(B\mid D) \neq P(B) \]

DAG example 2

• \(A\) and \(B\) are independent, conditional on \(C\) and \(D\) \[ P(A\mid B, C, D) = P(A\mid C, D) \Longleftrightarrow A \perp B \mid C, D \]

• \(C\) and \(D\) are independent, conditional on \(A\) and \(B\) \[ P(D\mid A, B, C) = P(D\mid A, B) \Longleftrightarrow D \perp C \mid A, B \]

Decomposition of joint distributions

1. Start with roots (nodes with no parents)

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

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

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

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., \(P(A, B) \neq P(A) P(B)\)

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 (\(D\rightarrow E \rightarrow F\)) and the middle part is in \(C\), or

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

  • It contains an inverted fork (\(D\rightarrow E \leftarrow F\)), 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: \(A\rightarrow Y\) is a frontdoor path from \(A\) to \(Y\)

• Example: \(A\rightarrow Z \rightarrow Y\) 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, \(A \leftarrow X \rightarrow Y\) 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:

  • \(\{V\}\), or
  • \(\{W\}\), 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:
  • \(\{\}\), \(\{V\}\), \(\{W\}\), \(\{M, V\}\), \(\{M, W\}\), \(\{M, V, W\}\)
  • But not \(\{M\}\)

Backdoor path criterion: a multi backdoor paths example

• First path: \(A \leftarrow Z \leftarrow V \rightarrow Y\)

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

• Second path: \(A \leftarrow W \rightarrow Z \leftarrow V \rightarrow Y\)

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

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

• \(\{V\}\), \(\{Z, V\}\), \(\{Z, W\}\), or \(\{Z, V, W\}\)
  • But not \(\{Z\}\) or \(\{W\}\)

  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