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Lecture notes. Written from the material of Berkeley Stat 156, licensed CC BY-NC 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

4. Confounding, Backdoor Paths, M-Bias

What this covers

This is problem set 4 for Stat 156 (causal inference), due 2024-10-22. It supplies no exposition of its own — every question exercises material that lecture already covered: directed acyclic graphs (DAGs) as a language for causal structure, the backdoor criterion for identifying a causal effect, the ignorability assumption, and M-bias, the case where conditioning on a variable that looks like a safe control actually opens a path and creates bias. The set assumes the reader already has that vocabulary, together with potential-outcomes notation \(Y(1)\) and \(Y(0)\) for the outcome a unit would show under treatment and under control. Two of the problems also point to numbered exercises in the course text, A First Course in Causal Inference, which this chapter does not reproduce since it was not supplied.

Exercises

  1. Suppose you run a non-profit working to improve graduation rates in West Africa. You have read a study of a randomized controlled trial showing that direct cash transfers significantly improved graduation rates in Mexico. You are deciding whether your non-profit should roll out a direct-transfer program in West Africa instead. What further information would you want before making that decision? (There is no single right answer here — a thoughtful, well-reasoned answer earns full credit.)

  2. Problem 15.4 from A First Course in Causal Inference.

  3. Consider the causal graph below.

U X Z Y
The graph for problem 3: U is a common cause of X and Z, and both X and Z point into Y.

A. Describe the path or paths of association between \(Z\) and \(Y\). Which, if any, are causal paths? Which, if any, are backdoor paths? B. Which variable, or variables, must we condition on to achieve ignorability?

  1. Consider the causal graph below.
W U X Z Y
The graph for problem 4: W and U each point into X, W also points directly into Y, and U's other child Z points into Y.

A. Is there confounding between \(Z\) and \(Y\) in this graph — that is, do we need to adjust for any variable to identify the causal relationship between \(Z\) and \(Y\)? Explain why or why not. B. Can we identify the heterogeneous average causal effect \(E[Y(1) - Y(0) \mid X]\)? If so, give the identification; if not, explain why not.

  1. Give a real-life example of M-bias. Draw the M-bias graph structure and say what each node corresponds to in your example.

  2. Problem 16.2 from A First Course in Causal Inference.

  3. Problem 21.1 from A First Course in Causal Inference.

  4. Problem 21.8 from A First Course in Causal Inference.

  5. Problem 21.9 from A First Course in Causal Inference.

Sources

  • Problems 1, 3, 4, 5, and the textbook pointers for 2, 6, 7, 8, 9: HW4.md (berkeley-stat156, fall-2024, due 2024-10-22), itself a model reconstruction of HW4.pdf, a scanned PDF with no text layer, licensed CC BY-NC 4.0. Treat the wording of the problems as a paraphrase of the original and the one displayed equation, \(E[Y(1) - Y(0) \mid X]\), as unverified.
  • The two causal graphs (problems 3 and 4) are redrawn as SVG from the figures extracted from pages 1 and 2 of the same PDF (HW4/figures/p001-1.png, HW4/figures/p002-1.png); the node layout and edges follow those images.
  • No slides or transcript were supplied for this chapter — the source material is the problem set alone, so there is no lecture exposition to draw on beyond what the problems themselves state.
  • Problems 2, 6, 7, 8, and 9 refer to numbered problems (15.4, 16.2, 21.1, 21.8, 21.9) in the course text A First Course in Causal Inference. That text was not supplied, so its problem statements are not reproduced here.

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