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.)

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


<br>


# 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.)

```{r}

```

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

b. 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.

```{r}

```

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

b. 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.

```{r}

```

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

b. 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).

## Part a

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

## Part b

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





