Marginal independence means unconditional independence

  1. (15pts) For each of exercises below, draw a DAG that contains four variables A, B, C, and D. Each DAG should imply the (conditional) independencies listed in the corresponding exercise, and only these independencies Hint: Not all nodes needs to be connected to some other nodes. Hint: Marginal independence means unconditional independence. A C B D (a) DAG that implies A |=C | B and no other (conditional or marginal) independencies. A C B D (b) DAG that implies A |=C | B, D and no other (conditional or marginal) independencies. A C B D (c) DAG that implies A |=C | B, D and B |=D | A and no other (conditional or marginal) independencies. 2 2. (10pts) Find all pairwise marginal and conditional independencies in the DAG in Figure 1. A C B D Figure 1 3 3. (10pts) (Intrumental variables). Many observational studies suffer from confounding. In this exercise we investigate a method of “confounding adjustment” which, under certain assumptions, has the remarkable property of producing causal inference even in the presence of unmeasured confounding. Let A be the exposure of interest, let Y be the outcome of interest, and let U be all unmeasured variables (confounders) that affect both A and Y . Let Z be a measured variable which have the following properties: a) U does not affect Z, b) Z does not affect U, c) Z and U don’t have common causes, d) Z affects A, e) Z has no effect on Y , apart from an indirect effect mediated through A. A variable Z which have properties a)-e) is called an instrumental variable. (a) Draw a DAG that connects A, Y , U, and Z. (b) Show that an observed association between Z and Y implies that A has a causal effect on Y (that is, we can test whether A has a causal effect on Y by testing whether Z and Y are associated). Hint: You only need to explain using a graph; no formula is needed. 4 4. (20pts) Consider the following DAG. X R S T U V Y (a) List all pairs of variables that are independent conditional on the set {R, V }. Use R code to get the answer. (b) For each pair of nonadjacent variables, give a set of variables that, when conditioned on, renders that pair independent. Use R code to get the answer. Hint: You can include the empty set if the pair of nonadjacent variables are marginally independent. Hint: You only need to provide one possible set. They do not need to be the minimal set. (c) Suppose we generate data by the model described in the DAG, and we fit them with the linear equation Y = a + bX + cZ. Which of the variables in the model may be chosen for Z so as to guarantee that the slope b would be equal to zero? Hint: Recall, a non zero slope implies that Y and X are dependent given Z. Hint: Include all possible variables. Note also that Z is a single variable (not a combination of variables). (d) Suppose we fit the data with the equation: Y = a + bX + cR + dS + eT, which of the coefficients would be zero?