2 Causal Inference Potpourri A research team wants to estimate the effectiveness of a new veterinary drug for sick seals. They ask aquariums across the country to volunteer their sick seals for the experiment. Since the team offers monetary compensation for volunteering, zoos with less income decide to volunteer their sick seals, whereas zoos with more income are less compelled to volunteer their seals. It turns out that zoos with less income feed their seals less nutritious diets (regardless of whether they are sick or healthy), due to budgetary constraints. Less nutritious diets prevent seals from recovering as effectively. (a) (2 points) Draw a causal graph between variables X, Y, I and N which denote receiving the drug, recovering, the income level of the zoo, and how nutritious a seal's diet is, respectively. Justify each edge in your graph. (b) (3 points) We saw in lecture that if we can identify and condition on (adjust for) all confounding variables, then we can use the unconfoundedness assumption to compute the average treatment effect (ATE). The backdoor criterion provides a way to determine which variables are confounders. In particular, we simply need to \"block\" all the confounding pathways in the graphical model between X and Y. In a causal graph, we dene a path between two nodes X and Y as a sequence of nodes beginning with X and ending with Y, where each node is connected to the next by an edge (pointed in either direction). Given an ordered pair of variables (X, Y), a set of variables 3 satises the backdoor criterion relative to (X, Y) if no node in S is a descendant of X (to prevent us from conditioning on colliders), and S blocks every path between X and Y that contains an arrow into X. Using the causal graph in the previous part, determine all possible sets of vari- ables that satisfy the backdoor criterion relative to (X, Y)