Chapter_16
9/17/25About 4 min
### Chapter 16: INSTRUMENTAL VARIABLE ESTIMATION **Introduction** - Causal inference methods previously discussed require measuring all confounders, but residual bias persists if unmeasured confounders exist. - Instrumental variable (IV) estimation offers an alternative approach under different assumptions, avoiding the need to measure all confounders.
16.1 The three instrumental conditions
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- Causal diagram (Figure 16.1): Depicts a double-blind randomized trial where:
- (Z): Randomization assignment (1: treatment, 0: placebo).
- (A): Actual treatment received (non-adherence possible).
- (Y): Outcome.
- (U): Unmeasured factors affecting both (A) and (Y).
- Problem: Standard methods (IP weighting, standardization, etc.) require adjusting for (U) to block backdoor path (A \leftarrow U \rightarrow Y).
- IV solution: Uses an instrumental variable (Z) meeting three conditions:
(i) (Z) is associated with (A) (relevance).
(ii) (Z) affects (Y) only through (A) (exclusion restriction).
(iii) (Z) and (Y) share no common causes (exchangeability). - Observational example: Cigarette price as a candidate instrument for smoking cessation ((A)) and weight change ((Y)). Only condition (i) is empirically verifiable.
Technical Points (Technical Point 16.1)
- Formal definitions:
- (i) Relevance: (Z \not!\perp!!!\perp A) (non-null association).
- (ii) Exclusion restriction: (Y^{z,a} = Y^{a}) (no direct effect of (Z) on (Y)).
- (iii) Exchangeability: (Y^{a,z} \perp!!!\perp Z) (marginal or joint).
- Instrument types:
- Causal instrument: (Z) directly affects (A) (Figure 16.1).
- Surrogate instrument: Proxy for unmeasured causal instrument (U_Z) (Figures 16.2, 16.3).
Fine Points (Fine Point 16.1)
- Candidate instruments in observational studies:
- Genetic factors: e.g., ALDH2 polymorphism for alcohol effects (Mendelian randomization).
- Preference: Physician’s prescribing preference (e.g., last prescription issued).
- Access: Physical distance to facility or calendar period.
- Weak instrument: Defined by small (Z)-(A) association (e.g., risk difference = 6%).
16.2 The usual IV estimand
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- IV estimand for dichotomous (Z):
[
\frac{E[Y|Z=1] - E[Y|Z=0]}{E[A|Z=1] - E[A|Z=0]}
] - Intuition: Numerator = intent-to-treat effect of (Z) on (Y); denominator = effect of (Z) on (A) (adherence).
- Estimation:
- Standard IV estimator: Ratio of sample averages.
- Two-stage least squares:
- Fit (E[A|Z] = \alpha_0 + \alpha_1 Z), predict (\hat{A}).
- Fit (E[Y|Z] = \beta_0 + \beta_1 \hat{A}); (\hat{\beta}_1) = IV estimate.
- Example: Smoking cessation ((A)) and weight gain ((Y)) with price instrument ((Z)):
- Estimate = 2.4 kg (95% CI: -36.5 to 41.3), indicating weak instrument issues.
Technical Points (Technical Point 16.2, 16.3, 16.4)
- Partial identification: Bounds for causal effects can be narrowed using IVs but often remain wide/uninformative.
- Structural mean models:
- Additive: (E[Y - Y^{a=0}|A,Z] = A(\beta_0 + \beta_1 Z)).
- Usual IV estimand = (\beta_0) under no effect modification by (Z) ((\beta_1=0)).
- Multiplicative: Identifies causal risk ratios under multiplicative homogeneity.
- Additive: (E[Y - Y^{a=0}|A,Z] = A(\beta_0 + \beta_1 Z)).
16.3 A fourth identifying condition: homogeneity
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- Homogeneity conditions (iv):
- Constant treatment effect across individuals (implausible).
- Equal average effect across (Z) in treated/untreated.
- No additive effect modification by unmeasured (U).
- Constant (Z)-(A) association across (U).
- General condition:
[
\text{Cov}\left[E[Y^{a=1} - Y^{a=0}|U], E[A|Z=1,U] - E[A|Z=0,U]\right] = 0.
] - Challenges: Homogeneity is often implausible (e.g., smoking cessation effect varies with unmeasured smoking intensity).
Technical Point (Technical Point 16.5)
- Proof: IV estimand equals (E[Y^{a=1} - Y^{a=0}]) under general homogeneity.
16.4 An alternative fourth condition: monotonicity
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- Compliance types: Always-takers, never-takers, compliers, defiers.
- Monotonicity (iv): No defiers ((A^{z=1} \geq A^{z=0})).
- IV estimand under monotonicity:
[
E[Y^{a=1} - Y^{a=0} | \text{compliers}].
] - Criticisms:
- Compliers are unobservable and instrument-dependent.
- Monotonicity may not hold in observational studies (e.g., physician preference instruments).
- Interpretation unclear for surrogate instruments.
Technical Points (Technical Point 16.6, 16.7)
- Structural mean models with covariates: Adjust for pre-instrument covariates (V).
- Proof: IV estimand = complier average causal effect (CACE) under monotonicity.
16.5 The three instrumental conditions revisited
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- Weak instruments (Fine Point 16.2):
- Substantive: Small (Z)-(A) association.
- Statistical: First-stage F-statistic < 10.
- Problems:
- Wide confidence intervals.
- Amplifies bias from violations of (ii)/(iii).
- Finite-sample bias even with valid instruments.
- Violations of conditions:
- (ii) Direct effect of (Z) on (Y) (e.g., from coarsening treatment).
- (iii) Confounding for (Z)-(Y) effect (Figure 16.10).
- Mitigation: Adjust for measured covariates (V), but balance checks may be misleading.
16.6 Instrumental variable estimation versus other methods
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- Key differences from IP weighting/standardization:
- Replaces conditional exchangeability with IV assumptions.
- Minor violations can cause large biases.
- Best suited for point interventions with strong instruments.
- Regression discontinuity (Fine Point 16.3): Estimates causal effects near a cutoff (e.g., age 65 for treatment eligibility) under continuity assumptions.
- Triangulation: Combine IV with other methods for robustness.
Figures Summary
- Figure 16.1: Causal instrument in a randomized trial.
- Figures 16.2–16.3: Surrogate instruments.
- Figures 16.4–16.7: Compliance types (always-takers, compliers, etc.).
- Figures 16.8–16.10: Violations of IV conditions (direct effects, confounding).
Key Limitations
- Conditions (ii)/(iii) untestable; homogeneity/monotonicity often implausible.
- Weak instruments amplify bias.
- CACE may lack practical relevance.