Illustrating Backdoor Paths

Creating Bias with Open Backdoor Paths

• 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

• 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

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

     • 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

Breaking Down Types of Biases

• There are two sources of bias:

  1. Confounding bias
  2. Selection bias

• Confounding bias happens when we aren't conditioning on a confounder

  • It arises when we control for fewer variables than we should

• Selection bias happens when we are conditioning on a collider

  • Selection bias is a type of collider bias
  • It arises when we control for more variables than we should
  • We observe this if the treatment and potential outcome are independent, but dependent once we condition on a collider

Illustrating Confounding Bias

• Suppose we're interested in measuring the causal effect of education on wage

  • Here, education is our treatment
  • And, wage is our outcome
  • Most likely, intelligence is a confounder

• Since education and wage share the same cause, estimating the causal effect of education on wage becomes difficult

  • For example, someone could argue any causal effects are mostly contributed by intelligence rather than education

• We must close the backdoor path by controlling for intelligence

  • In other words, we can keep intelligence fixed when comparing different levels of education
  • Said another way, we can compare people with the same level of intelligence, but different levels of education

Illustrating Collider Bias

• Suppose we're interested in measuring the causal effect of beauty on success

  • Here, beauty is our treatment
  • And, talent is some covariate
  • Then, success is our outcome and collider
• By conditioning on celebrity success, you are opening a second path between the treatment and the outcome
• Which, will make it harder to measure the direct effect

• One way to think about this is that by fixing success, you are looking at small groups of the population where success is the same

  • Then, finding the effect of beauty on those groups
  • But, by doing so, you are also indirectly and inadvertently not allowing success to change much
  • As a result, you won’t be able to see how beauty changes success effectively
  • Because, you are not allowing success to change as it should

Examples of Opening and Closing Backdoors

• For the following examples, $\perp$ represents independence

  • And, $\not \perp$ represents dependence
  • Whereas, $|$ represents conditioning on a variable

• $\bold{X \perp T}$

  • Since $Y$ is a collider that hasn't been conditioned on

• $\bold{X \not \perp T | Y}$

  • Since $Y$ is a collider that has been conditioned on

• $\bold{X \not \perp T | W}$

  • Since $Y$ is a collider with a descendent that has been conditioned on

• $\bold{Y \perp G}$

  • Since $U$ is a collider that hasn't been conditioned on

• $\bold{Y \not \perp G | U}$

  • Since $U$ is a collider that has been conditioned on

• $\bold{Y \perp G | U,T}$

  • Since $T$ is a confounder that has been conditioned on
  • Conditioning on $U$ opens the path, but conditioning on $T$ closes the path

Describing Berkson's Paradox as Collider Bias

• Suppose that beauty and talent are uncorrelated in the population

• But, suppose that beauty and talent are correlated in a sample only containing celebrities

  • Sampling on celebrities could lead someone to wrongly infer that talent is correlated with beauty for the entire population

• And, beauty and talent can both cause celebrity success

  • So, success clearly is a collider variable
• As a result, conditioning on the collider will only focus on the relationship between beauty and talent amongst celebrities

• Under this incorrect model of success:

  • Knowing that a talentless person is a successful actor would imply that the person must be beautiful
  • This is the source of collider bias

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
• Wiki of Relation between Berkson's Paradox and Collider Bias
• Illustrating Collider Bias with Hollywood Actors