Topic 5 Key Structures in Causal Graphs

Learning Goals

  • Explain the intuition behind marginal and conditional (in)dependence relations in chains, forks, and colliders in causal graphs
  • Simulate data from causal graphs under linear and logistic regression structural equation models to check these properties through regression modeling and visualization


Slides from today are available here.




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.)

  1. What conditional relationship is implied by this chain structure? What is the intuition behind/rationale for this relationship?

  2. Use appropriate check(s) to verify this conditional relationship.



Exercise 2

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

  1. What conditional relationship is implied by this fork structure? What is the intuition behind/rationale for this relationship?

  2. Use appropriate check(s) to verify this conditional relationship.



Exercise 3

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

  1. What marginal and conditional relationships between X and Z are implied by this collider structure? What is the intuition behind/rationale for this relationship?

  2. Use appropriate check(s) to verify these relationships.



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.



Extra

If you would like to delve more into the probability theory behind these ideas, try the following exercise.

Use the Causal Markov assumption/product decomposition rule to prove:

  1. The conditional independence relations in forks and chains
  2. The marginal independence and conditional dependence relations in colliders