Graphing Causal Models

Illustrating Causality using DAGs

• A DAG is a theoretical representation of the knowledge about a studied phenomena
• In reality, causality can run in multiple directions

• However, causality runs in one direction using DAG notation

  • Specifically, it only runs forward in time
  • Meaning, there are no cycles in a DAG

• DAGs are useful for explaining causality in terms of counterfactuals

  • A causal effect is defined as a comparison between two states:

    • One state that actually happened when some intervention took on some value
    • And another state that didn’t happen (i.e. the counterfactual) under some other intervention
• We can think of a DAG as a graphical representation of a chain of causal effects

Defining Notation for DAGs

• DAGs consist of a combination of nodes and arrows

• Nodes represent random variables

  • These random variables are created by a data-generating process

• Arrows represent a causal effect between two random variables

  • The direction of the arrow captures the direction of causality

• Causal effects can happen in two related ways:

  1. Direct relationship: $T \to Y$
  2. Indirect relationship: $T \to X \to Y$

Defining Bi-Directional and Circular Paths

• A circular path can't be represented using DAGs
• However, circular paths appear all the time in reality

• For example, there is a causal effect of learning resources on IQ

  • Also, there is a causal effect of IQ on income
  • And, there is a causal effect of income on learning resources
• Similarly, bi-directional paths can't be represented using DAGS
• Bi-directional paths are very similar to circular relationships

• For example, there is a causal effect of chances of getting an interview on amount of experience

  • Also, there is a causal effect of amount of experience on chances of getting an interview

Illustrating a Simple DAG

• The below DAG has three random variables: $D$, $X$, and $Y$

• Here, there is a direct path $T \to Y$

  • This represents a causal effect

• Also, there is a backdoor path $T \gets X \to Y$

  • This isn't a causal effect
  • This path creates spurious correlations between $D$ and $Y$
  • Open backdoor paths are a common source of bias

• In this example, $X$ is known as a confounder

  • This is because $X$ jointly determines $T$ and $Y$
  • So, $X$ confounds our ability to determine the effect of $T$ and $Y$ in naive comparisons

Defining a Causal Path

• The below DAG illustrates a causal path
• Here, we see a treatment $T$ linked to an outcome $Y$
• Notice, there is a mediating variable $X$ between them
• Mediating variables are those which form part of a causal pathway

• For example, sexual promiscuity is a risk factor for HPV

  • Also, HPV is a risk factor for cervical cancer
  • So in this case, HPV is a mediating variable

Defining a Confounding Path

• The confounder is a parent of both treatment $T$ and outcome $Y$
• Here, $X$ is known as the confounder

• For example, there is a causal effect of weather on ice cream sales

  • Also, there is a causal effect of weather on sunburns
  • As a result, there are spurious correlations created between ice cream sales and sunburns
• Every open backdoor path has a confounder, but not all confounders indicate a backdoor path is open

Defining a Colliding Path

• A collider is a child of both treatment $T$ and outcome $Y$
• Here, $X$ is known as the collider

• For example, there is a causal effect of pneumonia on hospital admittance

  • Also, there is a causal effect of a stroke on hospital admittance
• Colliders close a backdoor path when they're excluded from a model

Defining a Backdoor Path

• A backdoor path is a non-causal path from a node $X$ to node $Y$ that would remain if any arrows pointing out of $X$ were removed

  • These removed arrows are potentially causal paths

• The most common example of a backdoor path is a confounding path

  • But, not all confounding paths are backdoor paths

• The following DAG is an example of an open backdoor path:

  • There is a causal effect of smoking on obesity
  • There is a causal effect of smoking on mortality
  • These is a causal effect of obesity on mortality

Defining an Open Backdoor Path

• A backdoor path is open if the following are true:

  • There is a causal effect of $X$ on $Y$
  • There is a common ancestor of $X$ and $Y$

• An open backdoor path is the most common source of bias

  • Thus, our goal is to close backdoor paths
  • Every open backdoor path has a confounder, but not all confounders indicate a backdoor path is open

Creating Bias with Open Backdoor Paths

• There are three reasons a backdoor path can be open:

  1. We could be conditioning on a collider
  2. We could be conditioning on a mediator to a collider
  3. We may not be capturing or controlling for an unobserved confounder

• In summary, there are three basic types of open backdoor paths

  1. Confounding bias
  2. Selection bias due to conditioning on a collider

  3. Selection bias due to conditioning on a mediator

     • A mediator is a variable between the treatment $T$ and outcome $Y$

Closing Open Backdoor Paths

• There are two ways to close an open backdoor path:

  1. Conditioning on a confounder

     • Obviously, we can only do this if a confounder exists on an open backdoor path
     • Conditioning on a variable is equivalent to fixing (or including) a variable in our regression model

  2. Not conditioning on a collider or its mediators

     • Not conditioning on a collider always closes a backdoor path
     • Not conditioning on a mediator to a collider always closes a backdoor path
     • Not conditioning on a variable is equivalent to excluding a variable from our regression model
• Both methods must be enforced in order to close all open backdoor paths

References

• Lecture about Causality and DAGS
• Lecture about Colliders
• Textbook about DAGs with Examples
• Causal Inference Textbook
• Python Causality Handbook
• Comprehensive Causal Inference Textbook
• Controlling for Variables in Regression