# Confounding

### Confounding

• Confounders: variables that affect both the treatment and the outcome

• If we assign treatment based on a coin flip, since the coin flip doesn’t affect the outcome, it’s not a confounder

• If older people are at higher risk of heart disease (the outcome) and are more likely to receive the treatment, then age is a confounder

• To control for confounders, we need to

1. Identify a set of variables $$X$$ that will make the ignorability assumption hold
• Causal graphs will help answer this question
1. Use statistical methods to control for these variables and estimate causal effects

# Causal Graphs

### Overview of graphical models

• Encode assumption about relationship among variables

• Tells use which variables are independent, dependent, conditionally independent, etc

## Terminologies of Directed Acyclic Graphs (DAGs)

### 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 $$D$$

## Relationship between DAGs and probability distributions

### 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, blocking, and colliders

### 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

### 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 and backdoor paths

### 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

• 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
1. 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
1. 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
1. 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
1. 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)