CS6480 Assignment 3 Solution

Instructions
Questions
1. (10 points) Define Collider-bias, M-bias and Selection-bias and illustrate using causal graphs and adjustment sets (6 points). What kind of bias will be observed if we condition on W while estimating the causal effect of X on Y in the following graph (1 point)? How do we estimate the causal effect of X on Y for a given value of W, i.e. how do you find p(Y = y|do(X = x),W = w)? Your expression should contain only statistical quantities but not any do notations. (3 points)

2. (5 Points) The table below lists the results of a hypothetical experiment on 200 people. W,Y are treatment and response variables. X is a covariate. Each row identifies a category of people with the same values of X,W,Ydo(W=0),Ydo(W=1), and Y obs. For example, there are 30 people in category 1 and each of these people has X = 0,W = 0,Ydo(W=0) = 4,Ydo(W=1) = 6 and because they are in the control group we have Y obs = 4.
Category Number of People X W Ydo(W=0) Ydo(W=1) Yobs
1 30 0 0 4 6 4
2 30 0 1 4 6 6
3 10 1 0 4 6 4
4 30 1 1 4 6 6
5 20 0 0 10 12 10
6 20 0 1 10 12 12
7 15 1 0 10 12 10
8 45 1 1 10 12 12
(a) Do you believe that treatment assignment is unconfounded given X for these data? Justify your answer. (2 points)
(b) Assuming the table above reflects the population of interest, what is the propensity score for X = 0 and X = 1? (2 points)
3. (6 points) Define and give examples for the following assumptions of causal inference used in the study of instrumental variables.
• Exclusion Restriction (2 points)
• Relevance (2 points)
• Exogeneity (2 points)
4. (5 points) The following table represents the joint distribution over three random variables X,Y,Z which are related as shown in the causal graph below. Using inverse probability weights guided by propensity scores, calculate E(Y |do(X = 1))−E(Y |do(X = 0)). You can either solve this problem using Python or can be done using pen and paper.
X Y Z p(X,Y,Z)
1 1 1 0.116
1 1 0 0.274
1 0 1 0.009
1 0 0 0.101
0 1 1 0.334
0 1 0 0.079
0 0 1 0.051
0 0 0 0.036

5. (5 points) Consider that we are estimating ψ = E[Y (1)−Y (0)]. The direct way of estimating ψ is as follows:
µˆ0 = EX[Y |X = x,W = 0] µˆ1 = EX[Y |X = x,W = 1]
ψ = µˆ1 − µˆ0
where W is a treatment variable and X is the set of covariates. Starting with ψ = EX[µˆ1(X)− µˆ0(X)], derive an expression that is equivalent to finding ψ through inverse probability weighting (IPW) guided by propensity score P(W = 1|X = x).
6. (4 points) Consider the below structural equations of a causal model:
X = U1
Z = aX + U2 Y = bZ
(b) If we represent the counterfactual value of Y had X been x as YX=x, Fill the below table of counterfactual quantities.(3 points)
u1 u2 XU1=u1,U2=u2 ZU1=u1,U2=u2 YU1=u1,U2=u2 ZX=0,U1=u1,U2=u2 ZX=1,U1=u1,U2=u2 YX=0,U1=u1,U2=u2 YX=1,U1=u1,U2=u2
0 0
0 1
1 0
1 1
7. (5 points) Let X,Y are treatment and response variables. If we represent the counterfactual value of Y had X been x as YX=x, prove that the counterfactual query of the form E(YX=x|e) is identifiable in linear models given the evidence e.(Assume that e is obtained through the Abduction step of counterfactual inference.)