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)
```


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# 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.


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# 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.


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# 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.


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# 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.


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# Exercise 5

Discuss this Tweet (https://twitter.com/Lizstuartdc/status/1302604052753244161) 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?



