CS6480: Causal Inference and Learning
Instructions
Questions
(a) Name all the parents of Z
(b) Name all the ancestors of Z
(c) Name all the children of W
(d) Name all the descendants of W
(e) Identify all the paths(not necessarily direct) between X and T
(f) Identify all directed paths from X to T
(g) Are W and T marginally independent? Explain.
(h) Use the product decomposition of probability distributions to identify the joint probability distribution of the five variables represented as nodes (T,W,X,Y,Z).
(i) Are W and T conditionally independent given Z? If yes, prove that p(W,T|Z)=p(W|Z)p(T|Z). If not, explain why.
(j) Are W and T conditionally independent given Z,Y ? If yes, prove that:
p(W,T|Z,Y ) = p(W|Z,Y )p(T|Z,Y )
If not, explain why.
(k) Find a set of variables that d-separates X and T. Explain your reasoning. What does this tell us about the conditional independence of X and T in the data (i.e., when are they conditionally independent)?
2. (6 points) The table below gives data for 8 fast food restaurants that were part of a study of the effect of raising the minimum wage in New Jersey (NJ). The “treatment” group is the 2 restaurants in NJ and the “control” group is a set of 6 restaurants in Pennsylvania (PA) (where the minimum wage was not raised). The outcome Y obs is the number of people employed (including part-time employees) at the end of the year. There are two covariates – Xi1, the identity of the fast food chain (Burger King or Kentucky Fried Chicken) and Xi2, duration of employment at the end of the year prior to the increase in the minimum wage.
Observation Treatment Restaurant Chain Initial Employment Final Employment
i Wi Xi1 Xi2 Yiobs
1 NJ BK 22.5 30.0
2 NJ KFC 14.0 12.5
3 PA KFC 13.8 17
4 PA BK 26.5 18.5
5 PA BK 20.0 19.5
6 PA BK 13.5 21
7 PA BK 32.5 26.5
8 PA KFC 21.0 23.0
Now, answer the following questions:
(a) We want to use matching to estimate the effect of raising the minimum wage assuming that unconfoundedness holds. We will match a single control unit with each treatment unit (without replacement). Our distance measure is D(i,j) = 100×I(Xi1 6= Xj1)+|Xi2−Xj2| where the indicator I is 1 if the two units are different chains and 0 if they are the same chain. Identify the matches for the 2 treatment units. (2 points)
(b) Define Average Treatment Effect on the Treated (ATT) (Hint: This is ATE restricted to the treated!). Under which causal inference assumption is ATT equal to Average Treatment Effect (ATE) ? Explain (2 points)
(c) Find ATT from the above table. (2 points)
3. (12 points) In an observational study, when we observe all the relevant features/attributes, we can use adjustment formula to identify the causal effect (e.g., using regression analysis as we saw in class). In this question, we assume that we have all the information required for identifying the causal effect. Now, consider the following equations that generate the data. All noise random variables are normally distributed and are independent of each other. T is the treatment variable and Y is the outcome/target variable.
• Draw the causal graph looking at the equations. Show noise variables in the graph explicitly in the causal graph. Represent noise variables as nodes and draw an edge connecting noise variables and the features(W,T,Y,Z) (1 point).
• Generate 10,000 data points using these set of equations. Make the continuous treatment variable T binary by re-assigning its value to 1 if its original value is greater than 3.5 else to 0 (2 points).
• Write code to implement linear regression which returns regression coefficients (2 points).
• Using the regressor implemented above, write code to evaluate E(Z|do(T = t))−E(Z|do(T = t0)) (1 point)
• Now, use your code to evaluate the following quantities. Report the numbers and a discussion on your findings (6 points).
– E(Y |T = 1) − E(Y |T = 0)
– E(Y |do(T = 1)) − E(Y |do(T = 0)) with {W} as adjustment set(i.e., regress Y on T,W with do(T = t))
– E(Y |do(T = 1)) − E(Y |do(T = 0)) with {W,Z} as adjustment set(i.e., regress Y on T,W,Z with do(T = t))
4. (5 points) Answer the questions below:
(a) What is the mean independence assumption?(1 point)
(b) How it is useful in claiming that the regression coefficient of x in y = β0 + β1x + u, β1 is the same as the causal parameter? u in the equation above represents an error term that is usually assumed to include unobserved variables(2 points).
(c) Explain your reasoning by relating with Conditional Average Treatment Effect (CATE) and Exchangeability (2 points).
5. (2 points) If we assume that the distribution of covariates X is same across treatment groups T, prove that p(Y = y|do(T = t)) = p(Y = y|T = t).
