Homework 11: Use and Abuse of Conditioning
36-402, Advanced Data Analysis
Due at the start of class, 26 April 2011
1. (30 points) Refer to figure 1 in Homework 10.
(a) (5 points) Using the back door criterion, describe a way to estimate
the causal effect of smoking on cancer.
(b) (5 points) Using the front door criterion, describe a different way to
estimate the causal effect of smoking on cancer.
(c) (5 points) Is there a way to use instrumental variables to estimate
the causal effect of smoking on cancer in this model? Explain.
(d) (5 points) Using your back-door identification strategy and the data
file from last time, estimate Pr (cancer = 1|do(smoking = 1.5)).
(e) (5 points) Repeat this using your front-door identification strategy.
(f) (5 points) Do your two estimates of the casual effect match? Explain.
2. (25 points)Take the model in Figure 1. Suppose that X∼ N (0,1), Y=
αX +and Z=β1X+β2Y+η, where and ηare mean-zero Gaussian
noise with common variance σ2. Set this up in R and regress Ytwice,
once on Xalone and once on Xand Z. Can you find any values of the
parameters where the coefficient of Xin the second regression is even
approximately equal to α? (It’s possible to solve this problem exactly
through linear algebra instead.)
3. (25 points) Take the model in Figure 2 and parameterize it as follows:
U∼ N (0,1), X=α1U+,Z=βX +η,Y=γZ +α2U+ξ, where , η, ξ
are independent Gaussian noises with mean zero and common variance
σ2. If you regress Yon Z, what coefficient do you get, on average? If
you regress Yon Zand X? If you do a back-door adjustment for X?
(Approach this either analytically or through simulation, as you like.)
4. (20 points) Continuing in the set-up of the previous problem, what coef-
ficient do you get for Xwhen you regress Yon Zand X? Now compare
this to the front-door adjustment for the effect of Xon Y.
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