Unit 4.8 · about 6 h
79 / 324Causal claims, causal graphs, adjustment and identification
Prediction describes observations; causal identification asks what would happen under an intervention. A precise association cannot replace the assumptions that connect those two questions.
- explain
- derive
You can now
- Distinguish predictive, associational and causal claims by stating the intervention and potential-outcome estimand.
- Draw a DAG, recognize confounders, colliders and mediators, and use the backdoor criterion to select or reject an adjustment set.
- Derive the adjustment formula using consistency, conditional exchangeability and positivity, and check a confounded example by hand and in NumPy.
- For the optional extension, distinguish subgroup average treatment effects from individual effects and prognostic differences.
Covers
- Predictive, associational and causal claims
- Causal graphs, confounders, colliders and mediators
- Interventions, adjustment and identification assumptions
- Heterogeneous treatment effects
Study, in this order
4 steps
- 14.310x Lecture 14: potential outcomes and assignmentMIT 14.310x - Data Analysis for Social Scientists
Slides 1 to 14: causal statements, predictive questions, potential outcomes, SUTVA, assignment and the selection problem
- What If: define and identify the effectCausal Inference: What If
August 19, 2026 edition: Sections 1.1 and 1.2 (pp. 3 to 6), Section 1.5 (pp. 9 to 11), Section 2.3 (pp. 19 to 20), and Section 3.1 (pp. 27 to 29)
Depth · Keep individual versus average effects, consistency, exchangeability, positivity and standardization. Skip alternative effect scales and the chapter's longer historical discussion.
- What If: paths, confounding and valid adjustmentCausal Inference: What If
Sections 6.1 to 6.3 (pp. 75 to 81), Section 7.3 (pp. 95 to 98), and Section 8.1 (pp. 109 to 110)
Depth · Work the chain, fork and collider rules and the backdoor criterion. Skip Technical Points 6.2 and 6.3 on structural models and cross-world independence, and Fine Point 6.2 on faithfulness. Do not proceed to SWIGs, longitudinal treatment or semiparametric estimation.
- Optional: heterogeneous average effectsCausal Inference: What If
Sections 4.1 and 4.2, pp. 47 to 51, heterogeneity and stratification to identify effect modification
Depth · Optional topic only. Compare identified effects within pre-treatment strata; stop before transportability and matching. Do not build a treatment-effect learner here.
Focus on
- For a well-defined treatment \(A\), potential outcomes \(Y(1)\) and \(Y(0)\) define \(\text{ATE} = \mathbb{E}[Y(1) - Y(0)]\). Consistency relates observed \(Y\) to \(Y(A)\). No interference is needed for this simple unit-level treatment notation; otherwise others' assignments must enter the potential outcomes.
- A predictive target uses observed information; an associational contrast conditions on \(A\); an intervention sets \(A\) and removes its incoming causal arrows. A DAG records substantive causal assumptions, including omitted common causes, not just observed correlations.
- A backdoor adjustment set contains no descendants of \(A\) and blocks every path from \(A\) to \(Y\) that starts with an arrow into \(A\). Conditioning blocks noncollider paths but can open a collider or a collider's descendant. For the total effect, conditioning on a mediator generally removes part of the effect rather than repairing confounding.
- Under consistency, \(Y(a)\) independent of \(A\) given \(L\), and \(P(A = a \mid L) > 0\) on the target population's support, \(\mathbb{E}[Y(a)] = \sum_l \mathbb{E}[Y \mid A = a, L = l] P(L = l)\). Graphical blocking supports exchangeability only if the assumed causal graph is appropriate. Optional subgroup effects also require adequate adjustment and overlap within the subgroup.
Reproduce closed-book
float64 without random draws: P = numpy.array([[0.4, 0.1], [0.1, 0.4]], dtype=numpy.float64), rows L = 0, 1 and columns A = 0, 1; Y = numpy.array([[0, 2], [1, 3]], dtype=numpy.float64). Compute observed = (P*Y).sum(axis=0)/P.sum(axis=0) and intervention = P.sum(axis=1) @ Y. Compare with [0.2, 2.8] and [0.5, 2.5]; require maximum absolute error below 1e-12. These are population identities, not fitted in-sample scores. Explain why adjusting for \(L\) closes the backdoor path, while making \(A\) deterministic given \(L\) would destroy the observational overlap needed for this formula.Traps
- Adjusting for every measured predictor. A collider or post-treatment variable can introduce bias or change the effect being estimated.
- Thinking a randomized assignment fixes interference or treatment-dependent missing outcomes. Randomization addresses baseline assignment confounding, not every validity failure.
- Calling a subgroup with high untreated outcomes a subgroup with large treatment benefit. Prognosis and the contrast \(Y(1) - Y(0)\) are different quantities.
Practice
The section mini-project covers this: define a potential-outcome estimand, draw a confounding/selection DAG, compare observed and intervention means under a known generator, and demonstrate one invalid adjustment and one overlap failure. Optional: report pre-treatment subgroup average effects without claiming individual-effect identification.
Self-test
-
Prove the adjustment formula and name the role of each assumption.
For integrable potential outcomes, total expectation gives \(\mathbb{E}[Y(a)] = \sum_l \mathbb{E}[Y(a) \mid L = l] P(L = l)\). Conditional exchangeability replaces the inner mean by \(\mathbb{E}[Y(a) \mid A = a, L = l]\). Positivity ensures that observed treatment stratum exists for each \(l\) with positive target mass. Consistency replaces \(Y(a)\) with observed \(Y\) within \(A = a\). This gives the adjustment formula. The simple \(Y(a)\) notation presumes a well-defined intervention and no interference. -
In \(L \to A\), \(L \to Y\), \(A \to M \to Y\), should you adjust for \(L\) or \(M\) to estimate the total effect of \(A\)?
Adjusting for \(L\) blocks the backdoor fork \(A \leftarrow L \to Y\). \(M\) is a descendant of \(A\) on the causal pathway, so it is not an ordinary backdoor adjustment variable for the total effect. Conditioning on \(M\) changes the contrast and does not by itself identify a direct effect. -
Why can conditioning on \(C\) in \(A \to C \leftarrow U \to Y\) create bias even if \(A\) was randomized?
The \(A\)-to-\(Y\) path through \(C\) is blocked marginally because \(C\) is a collider. Conditioning on \(C\) opens it and associates \(A\) with \(U\) within \(C\) strata, so randomization's marginal exchangeability need not survive. Adjustment for predictive variables is not automatically safe. -
What fails if \(A = L\) in the four-cell model, even when you know the DAG?
\(P(A = 1 \mid L = 0)\) and \(P(A = 0 \mid L = 1)\) are zero. The missing treatment-stratum means cannot be recovered by standardization from observed data alone. The structural equation could supply extrapolation, but then identification depends on that additional modeling knowledge, not the observed backdoor adjustment formula. -
Optional: does identifying \(\mathbb{E}[Y(1) - Y(0) \mid L]\) identify each individual's treatment effect?
No. It identifies a subgroup average under the relevant assumptions, not the joint pair of potential outcomes for an individual. Usually only one potential outcome is observed per individual. Different individual-effect distributions can share the same identifiable subgroup means.
Completion criterion
Stop when
You reconstruct the adjustment proof, pass the four-cell NumPy check, and select or reject adjustment sets in a new fork, mediator and collider graph while naming untestable assumptions. The heterogeneous-effects extension is optional.
Mark passed, then open 5.1 Only mark this when you can do it without notes.Leads to
- 4.4 Uncertainty, robustness and honest claims Keep causal identification separate from predictive evaluation claims.
- 5.1 Holdout, cross-validation and resampling by the right unit Keep identification separate from sampling uncertainty and population selection.
Resources for this unit
- MIT 14.310x - Data Analysis for Social Scientists Follow lectures 11-15, 17-20 and 22 with matching slides and problem sets for inference, regression, causality and experiment design.
- Causal Inference: What If Read Chapters 1-3 and 6-8 for causal effects, DAGs, confounding, selection bias and identification reasoning.
Also in this section
- MIT 18.05 - Inference practice Use classes 17-20, 22-24 and 26 with their problem sets for tests, intervals, bootstrap and regression.
- Patterns of trustworthy experimentation Read after the course to connect health checks, guardrails and statistical interpretation to production launch decisions.
- Always Valid Inference Use the introduction and worked motivation to understand optional-stopping failure. Reproducing the theory is outside this gate.
- Probabilistic Machine Learning: An Introduction Use the April 18, 2025 online edition: Sections 4.2.3 to 4.2.5, 4.5.1 and 4.6.4.1 for Bernoulli and Gaussian MLE/MAP; the definition and central intervals in 4.6.6; 2.1.2 and 11.7.4 for predictive uncertainty. Exercises 4.1 and 4.2 supply derivation practice.
- Permutation nulls and false discovery control Read
permutation_testNotes, Unpaired statistics, for exchangeability and exact partitions;false_discovery_controlNotes and the named thrombolysis example for Bonferroni and Benjamini-Hochberg. Use as an independent implementation oracle, not as a substitute for stating assumptions. - Probability calibration: the statistical property Read the opening calibration definition, the proper-scoring-rule note, and Section 1.16.1, Calibration curves. Stop before recalibration methods; the later Classical ML unit implements them.