Skip to main content

Chapter_2

Aaron L9/17/25About 3 minCausal inferenceDeepSeek-R1-0528-2ndresponse_md
### Chapter 2: RANDOMIZED EXPERIMENTS **Introduction** - Causal questions involve an action (treatment), outcome, and specific population. - Example: Investigating whether looking up causes others to look up via a coin-flip experiment (55% looked up when treated vs. 1% untreated). - **Randomized experiment**: Investigator assigns treatment via random mechanism (e.g., coin flip). - **Nonrandomized experiment**: Treatment assignment follows deterministic rules (e.g., based on gender), which may introduce comparability issues.

2.1 Randomization

Subsection Content

  • In real-world studies, only one potential outcome ((Y^a)) is observed per individual, depending on received treatment (A).
  • Counterfactual outcomes are missing, creating challenges for effect estimation.
  • Randomization ensures missing counterfactual data occurs by chance, enabling consistent effect estimation.

Technical Points

  • Exchangeability ((Y^a \perp!!!\perp A) for all (a)):
    • Treated and untreated groups have identical counterfactual risks under the same treatment:
      [
      \Pr[Y^a = 1 \mid A = 1] = \Pr[Y^a = 1 \mid A = 0] = \Pr[Y^a = 1]
      ]
    • Implies treatment (A) does not predict counterfactual outcome (Y^a).
    • Full exchangeability: Joint independence (Y^A \perp!!!\perp A) (stronger than exchangeability).
    • Mean exchangeability: (E[Y^a \mid A = a'] = E[Y^a]) (sufficient for identifying (E[Y^a] = E[Y \mid A = a])).
  • Ideal randomized experiments:
    • No loss to follow-up, full adherence, single treatment version, double-blind assignment.
    • Association equals causation: Causal risk ratio = Associational risk ratio.

Fine Points

  • Crossover experiments:
    • Sequential treatment assignment (e.g., measuring blood pressure under treatment/no treatment).
    • Requires strong assumptions:
      1. No carryover effects ((Y^{a_0,a_1}{t=1} = Y^{a_1})).
      2. Time-invariant individual causal effects ((Y^{a=1}{it} - Y^{a=0} = \alpha_i)).
      3. Time-invariant counterfactual outcomes under no treatment ((Y^{a=0}_{it} = \beta_i)).
    • Inapplicable to irreversible outcomes (e.g., death).

Key Example

  • Heart transplant study (Table 2.1):
    • Without full counterfactual data, exchangeability cannot be verified.
    • Small samples or non-exchangeable assignments (e.g., conditional randomization) may still be valid experiments.

2.2 Conditional Randomization

Subsection Content

  • Marginally randomized experiments: Single randomization probability for all individuals (e.g., 65% treated).
  • Conditionally randomized experiments: Randomization probabilities depend on covariates (L) (e.g., 75% treated if (L=1), 50% if (L=0)).

Technical Points

  • Conditional exchangeability ((Y^a \perp!!!\perp A \mid L) for all (a)):
    • Holds within strata of (L) (e.g., critical/noncritical condition).
    • Does not imply marginal exchangeability.
  • Missing data mechanisms:
    • MCAR: Missingness independent of all variables (holds in marginal randomization).
    • MAR: Missingness depends only on observed data (holds in conditional randomization).

Key Example

  • Heart transplant data (Table 2.2):
    • Imbalance in (L) (69% treated vs. 43% untreated in critical condition) indicates lack of marginal exchangeability.
    • Valid as conditionally randomized experiment.

2.3 Standardization

Subsection Content

  • Method to compute causal effects in conditionally randomized experiments by standardizing stratum-specific risks.

Technical Points

  • Standardized risk:
    [
    \Pr[Y^a = 1] = \sum_l \Pr[Y = 1 \mid L = l, A = a] \Pr[L = l]
    ]
  • Identification: Counterfactual risk is identified if expressible via observed data probabilities.
  • Causal risk ratio:
    [
    \frac{\Pr[Y^{a=1} = 1]}{\Pr[Y^{a=0} = 1]} = \frac{\sum_l \Pr[Y=1 \mid L=l, A=1] \Pr[L=l]}{\sum_l \Pr[Y=1 \mid L=l, A=0] \Pr[L=l]}
    ]

Fine Points

  • Risk periods:
    • Risks are period-specific (e.g., 5-day mortality risk).
    • Causal effects may vary by risk period (e.g., antibiotics delay but do not prevent 100-year mortality).

Key Example

  • Heart transplant study:
    • Standardized risk under treatment = (0.5), under no treatment = (0.5) → Causal risk ratio = (1.0).

2.4 Inverse Probability Weighting

Subsection Content

  • Alternative to standardization; weights individuals by inverse probability of received treatment.

Technical Points

  • IP weights: (W^A = 1 / f[A \mid L])
    • (f[A \mid L]): Conditional probability/density of treatment given covariates.
  • Pseudo-population:
    • Simulated population where treatment is marginally randomized.
    • Associational risk ratio in pseudo-population equals causal risk ratio.
  • Equivalence to standardization: Mathematically identical under positivity.

Key Example

  • Heart transplant study:
    • IP weights: (2.0) for untreated with (L=0), (1.33) for treated with (L=1).
    • Causal risk ratio = (1.0) (matches standardization).

Chapter Conclusion

  • Randomized experiments (marginal or conditional) enable causal inference via standardization or IP weighting.
  • Limitations: Often unethical, impractical, or untimely (e.g., heart transplants).
  • Observational studies are frequently necessary alternatives.

Note: All summaries are based solely on the provided text. Figures (2.1–2.3) were referenced but not included; their descriptions were inferred from context.