Chapter_3

### Chapter 3: Observational Studies

Introduction

Observational studies involve observing and recording data without investigator intervention, contrasting with randomized experiments. The chapter introduces a thought experiment about whether looking up at the sky causes others to do the same, highlighting how unmeasured confounders (e.g., thunderous noises) can invalidate causal claims. While randomized experiments are ideal for causal inference, observational studies (e.g., in evolution, climate science) remain essential when experiments are impractical. The chapter outlines conditions under which observational studies can yield valid causal inferences.

3.1 Identifiability Conditions

Technical Points:

• Observational studies can emulate conditionally randomized experiments if three conditions hold:
  1. Consistency: Treatment values correspond to well-defined interventions matching the data.
  2. Exchangeability: Treatment assignment depends only on measured covariates $L$ (i.e., $Y^a \perp\!\!\!\perp A \mid L$).
  3. Positivity: Probability of receiving every treatment level given $L$ is positive: $\Pr[A=a \mid L=l] > 0$ for all $a$ where $\Pr[L=l] \neq 0$.
• If these hold, methods like IP weighting or standardization (from Chapter 2) identify causal effects.

Fine Points:

• Identifiability Definition: An effect is identifiable if assumptions imply a unique value for the effect measure. Without exchangeability, data may be consistent with multiple causal risk ratios (e.g., $<1$, $>1$, or $=1$ depending on unmeasured risk factors).
• Disciplinary Preferences: Epidemiology often assumes conditional exchangeability, while economics favors instrumental variables when exchangeability is implausible.

3.2 Exchangeability

Technical Points:

• Marginal Exchangeability: In randomized experiments, treated and untreated groups are exchangeable due to balanced outcome predictors.
• Conditional Exchangeability: In observational studies, exchangeability given $L$ ($Y^a \perp\!\!\!\perp A \mid L$) is assumed but not guaranteed. Unmeasured confounders (e.g., smoking status $U$) can violate this assumption.
• Confounding: Variables like $L$ (critical condition) that affect treatment and outcome are confounders. Adjusting for $L$ enables causal estimation, but unmeasured $U$ (e.g., smoking) can bias results.

Fine Points:

• Verification Impossibility: Conditional exchangeability cannot be empirically verified because counterfactual outcomes under unobserved treatments are missing.
• Crossover Experiments: Require strong assumptions (e.g., no carryover effects) to estimate causal effects without randomization.

3.3 Positivity

Technical Points:

• Definition: Positivity requires $\Pr[A=a \mid L=l] > 0$ for all treatment levels $a$ and covariate values $l$ present in the population.
• Consequences: Violations (e.g., no heart transplants for critical patients) make standardized/IP-weighted risks undefined. IP-weighted means remain defined but lose causal interpretation.

Fine Points:

• Empirical Verification: Positivity can sometimes be checked (e.g., confirming all $L$ strata include treated/untreated individuals).
• Scope: Only applies to covariates $L$ needed for exchangeability (e.g., "blue eyes" irrelevant if not a confounder).

3.4 Consistency: Defining Counterfactual Outcomes

Technical Points:

• Well-Defined Interventions: Counterfactuals $Y^a$ require precise specification of interventions (e.g., heart transplant protocol details). Vague treatments (e.g., "weight gain") lead to ill-defined causal effects.
• Treatment Variation Irrelevance: Different versions of treatment (e.g., surgeon experience) may be assumed irrelevant, but this risks bias if versions affect outcomes.

Fine Points:

• States vs. Interventions: Causal questions about states (e.g., obesity) are ill-defined without specifying interventions (e.g., diet/exercise).
• Protocol Ambiguity: Identical protocols can yield different effects if implementation varies (e.g., surgeon experience).

3.5 Consistency: Linking Counterfactuals to Data

Technical Points:

• Linking Requirement: Observed outcome $Y$ must equal $Y^a$ for individuals with $A=a$. This fails if the intervention in the data mismatches the target intervention (e.g., surgeons with <10 transplants vs. required ≥10).
• Data Requirements: Detailed treatment data are needed to assess consistency. Unmeasured versions (e.g., missing surgeon experience) compromise validity.

Fine Points:

• Possible Worlds Philosophy: Counterfactuals are defined in "closest possible worlds," but interventions clarify ambiguity in observational settings.

3.6 The Target Trial

Technical Points:

• Emulation Framework: Observational analyses should emulate a hypothetical target trial with specified:
  • Eligibility criteria
  • Interventions
  • Treatment assignment
  • Outcomes
  • Follow-up period
• Role of Identifiability Conditions: Exchangeability, positivity, and consistency ensure observational data mirror the target trial.

Fine Points:

• Attributable Fraction: Measures excess cases due to treatment: $\frac{\Pr[Y=1] - \Pr[Y^{a=0}=1]}{\Pr[Y=1]}$. Differs from etiologic fraction (mechanically caused cases).
• Vague Interventions: Unspecified treatments (e.g., "weight loss") risk impractical assumptions (e.g., instantaneous liposuction). Explicit target trials enhance relevance and avoid hidden biases.

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Key Takeaways:

• Observational studies require exchangeability, positivity, and consistency to estimate causal effects.
• Exchangeability is unverifiable and relies on measuring confounders.
• Positivity ensures sufficient data across treatment-covariate combinations.
• Consistency demands well-defined interventions and alignment with observed treatments.
• The target trial framework structures causal questions, improving transparency and reducing vagueness.