Chapter_16

Aaron L9/17/25About 4 min

### Chapter 16: INSTRUMENTAL VARIABLE ESTIMATION

技术重点 (Technical Points):

核心理论:
- IV estimation addresses unmeasured confounding by leveraging an instrumental variable (Z) that meets three conditions:
  (i) (Z) is associated with treatment (A) (relevance condition).
  (ii) (Z) affects outcome (Y) only through (A) (exclusion restriction).
  (iii) (Z) and (Y) share no common causes (exchangeability).
- In randomized trials, (Z) (randomization assignment) is a valid instrument:
  - Condition (i) holds as assignment influences treatment receipt.
  - Condition (ii) holds under double-blind design.
  - Condition (iii) holds due to random assignment.
重要概念:
- Causal instrument: Directly affects treatment (e.g., Figure 16.1).
- Surrogate instrument: Proxy for an unmeasured causal instrument (U_Z) (e.g., Figures 16.2–16.3).
应用场景:
- Observational studies (e.g., using cigarette price as (Z) for smoking cessation effect on weight).

细节要点 (Fine Points):

辅助信息:
- Only condition (i) is empirically verifiable (e.g., Pr([A=1|Z=1] - \text{Pr}[A=1|Z=0] > 0)).
- Conditions (ii) and (iii) are untestable; violations may arise from unblinding (condition ii) or unmeasured confounders (condition iii).
案例分析:
- Cigarette price instrument: Weak association (risk difference = 6%), termed a weak instrument.
相关背景:
- Proposed instruments in observational studies include genetic factors (Mendelian randomization), physician preference, and access measures (e.g., distance to facilities).

技术重点 (Technical Points):

关键公式:
- For dichotomous (Z), the IV estimand for (E[Y^{a=1}] - E[Y^{a=0}]) is: $\frac{E[Y|Z=1] - E[Y|Z=0]}{E[A|Z=1] - E[A|Z=0]}$
- Continuous (Z): Estimand = (\frac{\text{Cov}(Y,Z)}{\text{Cov}(A,Z)}).
核心理论:
- Numerator: Intention-to-treat effect of (Z) on (Y).
- Denominator: Measures adherence (strength of (Z)-(A) association).
- Low adherence inflates the estimand.
应用场景:
- Randomized trials with non-adherence; observational studies with surrogate instruments.

细节要点 (Fine Points):

辅助信息:
- Standard IV estimator: Ratio of sample estimates (e.g., 2.4 kg weight gain in smoking cessation example).
- Two-stage-least-squares:
  1. Fit (E[A|Z] = \alpha_0 + \alpha_1 Z), predict (\hat{E}[A|Z]).
  2. Fit (E[Y|Z] = \beta_0 + \beta_1 \hat{E}[A|Z]); (\hat{\beta}_1) equals IV estimate.
案例分析:
- Weak instruments yield wide confidence intervals (e.g., 95% CI: -36.5 to 41.3 kg) and amplify bias.

技术重点 (Technical Points):

核心理论:
- Three homogeneity conditions for IV estimand to equal (E[Y^{a=1}] - E[Y^{a=0}]):
  1. Constant treatment effect (implausible; implies rank preservation).
  2. Equal average effect across (Z) in treated/untreated:
    (E[Y^{a=1} - Y^{a=0}|Z=1, A=a] = E[Y^{a=1} - Y^{a=0}|Z=0, A=a]).
  3. No additive effect modification by unmeasured (U):
    (E[Y^{a=1}|U] - E[Y^{a=0}|U] = E[Y^{a=1}] - E[Y^{a=0}]).
- General condition: (\text{Cov}[e(U), t(U)] = 0), where:
  - (e(U) = E[Y^{a=1} - Y^{a=0}|U]) (effect modification by (U)).
  - (t(U) = E[A|Z=1,U] - E[A|Z=0,U]) (modification of (Z)-(A) association).
重要概念:
- Additive structural mean model:
  (E[Y - Y^{a=0}|A,Z] = A(\beta_0 + \beta_1 Z)); (\beta_0) is IV estimand if (\beta_1=0).

细节要点 (Fine Points):

辅助信息:
- Homogeneity conditions are untestable and often implausible (e.g., smoking cessation effect varies with prior smoking intensity).
相关背景:
- Covariates can be incorporated via structural mean models to allow effect variation.

技术重点 (Technical Points):

核心理论:
- Monotonicity: No defiers (i.e., (A^{z=1} \geq A^{z=0}) for all individuals).
- Under monotonicity, IV estimand equals the complier average causal effect (CACE):
  (E[Y^{a=1} - Y^{a=0} | A^{z=1}=1, A^{z=0}=0]).
- Proof relies on:
  - Zero effect of (Z) on (Y) in always-takers/never-takers.
  - No defiers.
重要概念:
- Compliance types: Always-takers, never-takers, compliers, defiers (principal strata).

细节要点 (Fine Points):

案例分析:
- Smoking cessation: Compliers quit under high cigarette price ((Z=1)) but not under low ((Z=0)).
辅助信息:
- CACE is a local average treatment effect (LATE), not the population effect.
- Monotonicity is plausible in trials but questionable in observational studies (e.g., physician preference instruments).
相关背景:
- Surrogate instruments (e.g., Figures 16.2–16.3) complicate CACE interpretation if (U_Z) is continuous.

技术重点 (Technical Points):

核心理论:
- Weak instruments (small (Z)-(A) association):
  - Amplify bias from violations of (ii)/(iii).
  - Introduce finite-sample bias.
- Condition (ii) violations: Direct (Z \rightarrow Y) effect (e.g., from dichotomizing continuous treatments).
- Condition (iii) violations: Unmeasured confounding (e.g., Figure 16.10).
应用场景:
- Adjusting for covariates (V) may improve plausibility of exchangeability.

细节要点 (Fine Points):

辅助信息:
- Weak instrument definitions:
  1. Substantive: Small true (Z)-(A) association.
  2. Statistical: First-stage F-statistic < 10.
- Selection bias: Excluding treatment levels (e.g., (A=0)) invalidates IV estimates.

技术重点 (Technical Points):

核心理论:
- IV vs. IP weighting/standardization:
  - Replaces conditional exchangeability with IV conditions.
  - More sensitive to minor assumption violations.
- Ideal use cases:
  - Strong causal instrument.
  - Dichotomous, time-fixed treatment.
  - Homogeneity or monotonicity holds.
重要概念:
- Regression discontinuity design: Estimates effect near a threshold (e.g., age 65 for treatment eligibility).

细节要点 (Fine Points):

辅助信息:
- IV estimates often have wide confidence intervals, limiting practical utility.
- Transparency in reporting assumptions (e.g., via sensitivity analyses) is critical.
相关背景:
- Triangulation with other methods (e.g., difference-in-differences) recommended.

Figures and Images:

Figures 16.1–16.10 illustrate causal diagrams (e.g., randomized trials, surrogate instruments, violations). Descriptions are contextually summarized above; specific content requires original diagrams.