3  Causal Inference I: Foundations

3.1 Learning objectives

By the end of this chapter you should be able to:

• Articulate the potential-outcomes framework and use counterfactual notation to define the average treatment effect (ATE), the average treatment effect on the treated (ATT), and the conditional average treatment effect (CATE).
• Draw a directed acyclic graph (DAG) for a research question and identify the minimal sufficient adjustment set using the backdoor criterion.
• State the three core identifying assumptions (exchangeability, positivity, consistency) and recognise where each is violated in applied work.
• Distinguish association from causation in language and in symbols, and recognise common conflations.

3.2 Orientation

Causal inference is the discipline of making counterfactual claims from observed data: ‘if this patient had been assigned the treatment, their outcome would have been different.’ The discipline has a precise mathematical foundation (Rubin’s potential-outcomes framework, Pearl’s structural causal models) and a small set of identifying assumptions that determine when counterfactual claims are estimable from data.

This chapter establishes the foundations: the potential-outcomes notation, the average treatment effects of interest, the DAG vocabulary, the backdoor criterion, and the three core assumptions. Chapter 4 turns the foundations into estimation procedures (propensity scores, IPW, g-methods).

The chapter is shorter than its companion (Ch 4) because the conceptual investment is the binding constraint. A reader who internalises the potential-outcomes notation and the backdoor criterion can read modern causal-inference literature; a reader who has not will struggle with the next chapter.

3.3 The statistician’s contribution

Three judgements are not delegable.

(Judgement 1.) Distinguish causal from associational language. The English-language word ‘effect’ is ambiguous; the technical word is not. A statement like ‘smoking is associated with lung cancer’ is associational; ‘smoking causes lung cancer’ is causal. The first is a fact about a population’s joint distribution; the second is a counterfactual claim that requires identifying assumptions. The biostatistician keeps the language precise: associational claims for descriptions, causal claims only when the assumptions are stated and defensible.

(Judgement 2.) The DAG is a domain claim. A directed acyclic graph encoding the causal relationships in your problem is a substantive scientific claim, not a statistical artefact. Drawing the DAG forces you to articulate which variables affect which others; the arrows are the claim. The biostatistician engages domain experts to verify the DAG. A DAG with the arrows wrong produces an analysis that is precisely incorrect; the precision is no help.

(Judgement 3.) Identifying assumptions are substantive, not statistical. The no-unmeasured-confounding assumption (exchangeability) is not testable from data alone; it is a claim about what is in the world, defended by the design and the scientific argument. The biostatistician’s role is to make the assumption explicit, defend it where defensible, and quantify the consequences of violation where it is not.

These judgements are what distinguish a causal analysis from a regression model that happens to use causal vocabulary.

3.4 The potential-outcomes framework

Notation (Neyman, 1923; Rubin, 1974):

• \(A\): the treatment indicator (0 = no treatment, 1 = treated; or generally, the value taken by an intervention).
• \(Y\): the observed outcome.
• \(Y(a)\): the potential outcome under treatment level \(a\). A patient has potential outcomes \(Y(0)\) and \(Y(1)\); the observed outcome is \(Y = Y(A)\) — whichever potential outcome corresponds to the treatment actually received.
• \(X\): a vector of pre-treatment covariates.

The fundamental problem of causal inference is that each individual has potential outcomes \(Y(0)\) and \(Y(1)\), but only one is observed for any given person. The other is the counterfactual outcome — what would have happened if the treatment assignment had been different.

Causal effects are contrasts of potential outcomes. The individual treatment effect: \[ \tau_i = Y_i(1) - Y_i(0) \] is fundamentally unobservable for any one individual. The average treatment effect (ATE): \[ \text{ATE} = E[Y(1) - Y(0)] \] is identifiable from observed data under appropriate assumptions.

Two related estimands matter:

The average treatment effect on the treated (ATT): \[ \text{ATT} = E[Y(1) - Y(0) \mid A = 1] \] the average effect among those actually treated.

The conditional average treatment effect (CATE): \[ \text{CATE}(x) = E[Y(1) - Y(0) \mid X = x] \] the average effect at a specific covariate value.

ATE, ATT, and CATE are different things and often have different values. Pick the one that matches the decision: ATE for a population-level policy (‘should we offer the treatment universally’), ATT for a question about the actually-treated population (‘among those who got the treatment, what was the average effect’), CATE for personalised medicine (‘what is the effect for this kind of patient’).

NoteCheck your understanding: ATE vs. ATT

Question. A new diabetes drug is mostly prescribed to patients with poor glycaemic control on existing therapy. The ATT (effect among those treated) is large and positive. The ATE (average effect across all diabetic patients) is smaller. Which is the relevant quantity for the decision ‘should the drug be added to guideline-recommended treatment for all diabetic patients’?

Answer.

The ATE is the relevant quantity. The ATT applies only to the historical pattern of treatment (poorly-controlled patients); guideline change would treat patients across the whole spectrum, including better-controlled ones who were not represented in the ATT. The ATE captures the expected average effect under universal treatment. Reporting only the ATT for a guideline-change question overstates the expected benefit.

3.5 The three core assumptions

Three identifying assumptions allow ATE to be computed from observed data.

(A1) Exchangeability (no unmeasured confounding): \[ Y(a) \perp\!\!\!\perp A \mid X \quad \text{for all } a. \] Conditional on \(X\), the potential outcomes are independent of treatment assignment. Equivalently: treatment assignment is as good as random within strata of \(X\).

This is the load-bearing assumption. In an RCT, randomisation makes it true by design. In observational studies, it is a substantive claim about the world; it fails when there is a confounder \(U\) that affects both \(A\) and \(Y\) that is not in \(X\).

(A2) Positivity: \[ 0 < \Pr(A = a \mid X) < 1 \quad \text{for all } a, x. \] Every covariate combination has some probability of receiving each treatment level. If certain combinations are deterministically untreated (or deterministically treated), the data carry no information about the counterfactual at that combination.

Positivity violation is real: in observational drug data, certain patient subgroups are essentially never prescribed certain drugs (contraindications, age cutoffs, specialist referral patterns). The analyst must check the propensity-score distribution and either restrict the analysis or impose strong modelling assumptions to extrapolate.

(A3) Consistency: \[ A = a \implies Y = Y(a). \] The observed outcome under a particular treatment is the same as the potential outcome under that treatment. This sounds tautological but encodes a substantive claim: there is a single, well-defined version of each treatment, and the treatment received is the treatment of interest.

Consistency fails when the treatment is heterogeneous (‘SGLT2 inhibitors’ might mean dapagliflozin or empagliflozin, with different effects) or when the analyst defines treatment in a way that does not match the observed exposure. The fix is precision in the treatment definition.

Identification under A1-A3: When all three hold, the ATE is identified from data: \[ \text{ATE} = E[E[Y \mid A=1, X] - E[Y \mid A=0, X]]. \] The outer expectation is over the distribution of \(X\); the inner is the difference in conditional outcome means between treatment groups.

This identification formula is the basis for g-computation (Ch 4); other estimators are equivalent under A1-A3 but use different computational routes.

3.6 Directed acyclic graphs

A DAG is a graphical representation of causal relationships:

• Nodes are variables.
• Directed edges (\(A \to B\)) mean \(A\) is a direct cause of \(B\).
• Acyclic means no variable causes itself through any path.

DAGs are useful because they make causal assumptions explicit and allow algorithmic reasoning about identification.

A simple example: smoking (\(S\)) affects both coffee consumption (\(C\)) and lung cancer (\(Y\)). Coffee does not cause lung cancer. The DAG:

    S
   / \
  v   v
  C   Y

The association between \(C\) and \(Y\) is non-zero (they share a common cause) even though \(C\) does not cause \(Y\). Adjusting for \(S\) removes the spurious association.

Three structural roles a variable can play on a path between \(A\) and \(Y\):

Confounder (\(A \leftarrow Z \to Y\)): \(Z\) is a common cause of both. Adjustment for \(Z\) removes the non-causal association.

Mediator (\(A \to M \to Y\)): \(M\) is on the causal path from \(A\) to \(Y\). Adjustment for \(M\) blocks part of the causal effect; this is desired in mediation analysis (Ch 5) but not in the basic ATE.

Collider (\(A \to C \leftarrow Y\)): \(C\) is a common effect. Adjustment for a collider creates spurious association (selection bias). Do not adjust for colliders.

The backdoor criterion (Pearl, 1995): \(X\) is a sufficient adjustment set for the causal effect of \(A\) on \(Y\) if (a) \(X\) blocks all backdoor paths from \(A\) to \(Y\) and (b) \(X\) contains no descendants of \(A\).

A backdoor path is any path from \(A\) to \(Y\) that begins with an arrow into \(A\). Confounders sit on backdoor paths and need to be adjusted for; mediators do not sit on backdoor paths and must not be adjusted for.

The DAG and the backdoor criterion together produce a recipe for identification: draw the DAG, identify the backdoor paths, find an adjustment set that blocks all of them. The set is the variables to put in the model.

The dagitty R package implements the algorithm; for a given DAG and exposure-outcome pair it returns the minimal sufficient adjustment sets:

library(dagitty)
g <- dagitty('dag {
  smoking -> coffee
  smoking -> lung_cancer
  coffee -> lung_cancer [adjusted = "no"]
}')
adjustmentSets(g, exposure = "coffee",
               outcome = "lung_cancer")
#> { smoking }

The output: to identify the (zero) effect of coffee on lung cancer, adjust for smoking.

3.7 When the assumptions fail

Each assumption fails in characteristic ways and demands a characteristic response.

Exchangeability fails (unmeasured confounding). The standard responses:

1. Sensitivity analysis. Quantify how strong an unmeasured confounder would have to be to overturn the conclusion. The E-value (VanderWeele & Ding, 2017) is the standard summary; it answers ‘how strong does the unmeasured confounding need to be to explain the observed association.’
2. Negative controls. A negative-control exposure (one expected to have no effect on the outcome) and a negative-control outcome (one expected not to be affected by the exposure) check for residual confounding. Non-zero estimates on negative controls suggest residual confounding that affects the primary analysis too.
3. Instrumental variables. Find a variable that affects exposure but only affects the outcome through exposure. Identifies a different estimand (the effect among ‘compliers’) under different assumptions; covered in Ch 4.
4. Design-based identification. Regression discontinuity, difference-in-differences, synthetic control. Each exploits a feature of the intervention’s roll-out for identification.

Positivity fails. The standard responses:

1. Restrict the population to the region of covariate support that has positive probability of each treatment.
2. Trim the propensity score to remove observations near 0 or 1.
3. Use a doubly robust estimator that is more robust to extreme weights (Ch 4).

Consistency fails (treatment heterogeneity). The standard response: redefine the treatment more precisely. ‘SGLT2 use’ is too coarse; ‘dapagliflozin 10mg/day for at least 30 days starting within 30 days of HFrEF diagnosis’ is the kind of precision consistency requires.

3.8 Worked example: drawing a DAG

A research question: does parental income affect adolescent academic performance, controlling for the right confounders?

Available variables: parental income, adolescent academic performance, parental education, neighbourhood socio-economic status, school quality, parental involvement, adolescent IQ.

Drawing the DAG:

parental_education
      |
      v
parental_income ---> adolescent_performance
      ^                     ^
      |                     |
neighbourhood_ses --------> school_quality
                              ^
                              |
                          parental_involvement (mediator)

(In an actual paper, draw the graph; here the text suggests it.)

Backdoor analysis: backdoor paths from parental income to adolescent performance go through parental education and neighbourhood SES. The minimal adjustment set is {parental_education, neighbourhood_ses}.

Variables NOT to adjust for:

• Parental involvement (mediator on the income → involvement → performance path; adjusting blocks part of the causal effect of income).
• Adolescent IQ (downstream of the exposure if we believe income affects cognitive development).

The DAG forces these distinctions. Without it, an analyst might ‘adjust for everything’ (including mediators) and report a biased estimate.

3.9 Collaborating with an LLM on causal-inference foundations

Three patterns.

Prompt 1: ‘Draw the DAG for this research question.’ Provide the variables and the substantive claims about which causes which.

What to watch for. The LLM produces a plausible DAG but defaults to the most common pattern in similar research; it may miss domain-specific edges. Push for explicit confirmation of each edge (‘does X cause Y? why?’) and engage your domain expert.

Verification. Show the DAG to a domain expert. Edges that the LLM proposed but the expert rejects are substantive claims to remove. Edges the expert adds are substantive claims to add.

Prompt 2: ‘Find the minimal adjustment set for this DAG.’ Provide the DAG (in dagitty syntax or a description).

What to watch for. The LLM can apply the backdoor criterion correctly on simple DAGs and gets confused on DAGs with many nodes. Verify by running dagitty on the same DAG.

Verification. dagitty::adjustmentSets() is the ground truth.

Prompt 3: ‘List the assumptions required for this analysis.’ Provide the analysis description.

What to watch for. The LLM lists exchangeability, positivity, consistency, but tends to be generic about which specific confounders threaten exchangeability. Push for the specific case.

Verification. The LLM’s list is a starting point; your knowledge of the data informs which assumptions are most at risk.

The meta-pattern: LLMs are good for the syntactic mechanics of causal inference (drawing DAGs, listing assumptions) and bad at the substantive ones (which edges go in the DAG, which assumptions are likely to hold). Use them for drafts; the substance is yours.

3.10 Principle in use

Three habits define defensible causal-inference work.

1. Draw the DAG before fitting the model. The DAG is the causal claim; the model is the procedure. A model fitted without a DAG is not a causal analysis, regardless of what the report says.
2. State the three assumptions explicitly. Every causal report names exchangeability, positivity, and consistency, and discusses how each is defended.
3. Run a sensitivity analysis. Every causal estimate has an E-value or equivalent; the discussion includes what unmeasured confounding would do.

3.11 Exercises

1. For a research question of your choice, write the ATE, ATT, and CATE in potential-outcomes notation. Identify which is the most decision-relevant for the intended audience.

2. Draw the DAG for a research question in your area. Identify the backdoor paths and the minimal sufficient adjustment set. Use dagitty to check your answer.

3. For one of the three core assumptions, identify a specific way it could fail in a hypothetical observational study and propose a sensitivity analysis to address it.

4. Compute the E-value for a published RR of 2.0. What does this E-value tell you about the strength of unmeasured confounding required to overturn the conclusion?

5. Write a one-paragraph protocol section that states the estimand (using potential-outcomes notation), the identifying assumptions, the design that identifies the estimand, and the sensitivity analyses.

3.12 Further reading

• Hernán & Robins (2020), Causal Inference: What If. The reference textbook for the framework introduced in this chapter.
• Pearl (2009), Causality: Models, Reasoning, and Inference. The reference for the structural causal model and graphical reasoning.
• Rubin (1974), ‘Estimating causal effects of treatments in randomized and nonrandomized studies’. The foundational paper for potential outcomes.
• The dagitty R package and online tool (https://www.dagitty.net/) for drawing and analysing DAGs.

Hernán, M. A., & Robins, J. M. (2020). Causal inference: What if. Chapman; Hall/CRC. https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/

Neyman, J. (1923). On the application of probability theory to agricultural experiments. Essay on principles. Section 9. Statistical Science (Translated Reprint), 5, 465–472.

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4), 669–688.

Pearl, J. (2009). Causality: Models, reasoning, and inference (2nd ed.). Cambridge University Press.

Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66(5), 688–701.

VanderWeele, T. J., & Ding, P. (2017). Sensitivity analysis in observational research: Introducing the E-value. Annals of Internal Medicine, 167(4), 268–274.