Topic 5 Key Structures in Causal Graphs

Pre-class work

Videos/slides

Checkpoint questions: Link to Moodle checkpoint

  1. Suppose that the variables A, B, and C in a causal graph are connected as such: A – B – C without knowledge of the arrow directions. If we only know that A and C are marginally dependent, which of the following structures are possible?
    1. A –> B –> C
    2. A <– B <– C
    3. A <– B –> C
    4. A –> B <– C
  2. If we know that A and C are conditionally dependent given B, which of the following structures are possible?
    1. A –> B –> C
    2. A <– B <– C
    3. A <– B –> C
    4. A –> B <– C
  3. If we know that A and C are marginally dependent and that A and C are conditionally independent given B, which of the following structures are possible?
    1. A –> B –> C
    2. A <– B <– C
    3. A <– B –> C
    4. A –> B <– C
  4. How do chains, forks, and colliders relate to the concepts of exchangeability and causal effects? Explain in a few sentences.





Learning Goals

  • DSEP3: Simulate data from causal graphs under linear and logistic regression structural equation models to check d-separation properties through regression modeling and visualization.





Exercises

Solutions to these exercises are available on Moodle.
Navigate to PollEverywhere for some warm-up exercises.



You can download a template RMarkdown file to start from here.

In these exercises, you’ll be practicing simulating data from structural equation models and verifying marginal and conditional (in)dependence properties in DAG structures.

  • Always use a regression model as a check.
  • If the situation readily corresponds to a plot, also make a plot as a check.

Coding note: When you simulate binary variables and store them in a dataset, it is useful to store them explicitly as categorical as below. (This is most helpful for plotting.)

# X is binary. Y and Z are quantitative.
sim_data <- data.frame(X = factor(X), Y, Z)


Exercise 1

Simulate a chain X -> Y -> Z where all three variables are quantitative. (Use a sample size of 10,000 and a significance level of 0.05 throughout these exercises.)

Use appropriate check(s) to verify the conditional relation in this structure.



Exercise 2

Simulate a fork X <- Y -> Z where X and Z are quantitative, and Y is binary.

Use appropriate check(s) to verify the conditional relation in this structure.



Exercise 3

Simulate a collider X -> Y <- Z where Y also has a child A (Y -> A). Let all 4 variables be binary.

Use appropriate check(s) to verify the marginal and conditional relations between X and Z in this structure.



Exercise 4

Can we extend building block thinking to longer, more complex structures? Let’s investigate here (conceptually, no simulation).

  1. Consider the longer structure A <- B <- C -> D. What do you expect about marginal/conditional (in)dependence of A and D? Explain.

  2. Consider the longer structure A -> B <- C <- D -> E. What do you expect about marginal/conditional (in)dependence of A and E? Explain.



Exercise 5

Discuss the Tweet below with your group.

  • What does Liz Stuart mean by a “stark effect”? What might the trends look like if the effect weren’t so stark?
  • What is the treatment? What evidence does this analysis provide? Is it strong evidence? Do you have any concerns?



Take a few minutes to reflect on today’s ideas by filling out an exit ticket.