Instrumental Variables Baiocchi Chen Small 2014 Instrumental Variables
Instrumental Variables Baiocchi, Chen, Small (2014)
Instrumental Variables • No unmeasured confounding between IV and outcome. (i. e. , ignorable treatment assignment) • IV must affect the treatment of interest. (i. e. , E[D(1)-D(0)] not equal to 0) • No direct effect of IV on outcome. (i. e. , exclusion restriction assumption)
Potential Instrumental Variables • • Randomized encouragement trials Distance to specialty care provider Preference-based IVs Calendar time Genes Timing of admission (e. g. , day of the week) Insurance plan
IV Assumptions • A 1. SUTVA • A 2. IV is positively correlated with treatment received • A 3. IV is independent of unmeasured confounders (conditional on covariates) • A 4. Exclusion restriction • One of the following: – A 5. Monotonicity – No current treatment value interaction E[Y(1)-Y(0)|D=1, Z=1, X] = E[Y(1)-Y(0)|D=1, Z=0, X] Under A 5: “Average treatment effect same among always takers and compliers conditional on X. ”
Estimators • {Eh(Y|Z=1)–Eh(Y|Z=0)}/{Eh(D|Z=1)–Eh(D|Z=0)} – Wald estimator – Also known as two stage least squares estimator • Regress D on Z to get Eh(D|Z=z) and then regress Y on Eh(D|Z=z) – Can also be written as Covh(Y, Z)/Covh(D, Z) • Same two stage procedure can be used when covariates are included • Variance estimates need to account for the first stage estimation. This can be done by stacking estimating equations and using a sandwich variance. • It’s even possible to compute Eh(D|Z=z) from a different sample. – Two sample IV
Instrumental Variables (linear models; econometrics literature) • y i = b 0 + b 1 xi + e i • x i = g 0 + g 1 zi + n i • Most common IV estimator of b: (Z’X)-1 Z’y = (Z’X)-1 Z’(Xb+e) = (Z’X)-1 Z’Xb+(Z’X)-1 Z’e b. This is the same as Covhat(Y, Z) / Covhat(X, Z). (not being careful with notation. ) • Or 2 -stage least squares: – Estimate g as (Z’Z)-1 Z’X – Regress predicted X on Y instead of X: Xhat=Z(Z’Z)-1 Z’X (Xhat’ Xhat)-1 Xhat’Y={[Z(Z’Z)-1 Z’X]’[Z(Z’Z)-1 Z’X]}-1[Z(Z’Z)-1 Z’X]’Y =(Xhat’ X)-1 Xhat’Y
Econometrics Lingo • Endogenous: – Correlated with ei • Exogenous: – Uncorrelated with ei • If there is confounding, then xi is correlated with ei, so it is therefore endogenous. • The idea is to replace actual values of xi (endogenous) with predicted values of xi (exogenous but still related to actual xi).
Compliance Example • • ITT estimate As-treated estimate Per protocol estimate IV estimate • Discuss each of these and what it estimates, strengths, and weaknesses.
What we’re estimating • IV approach estimates LATE (also called CACE) • If we’re interested in ATE (also called ACE), we can only come up with bounds using IV approach (assumptions A 1 -A 5). • Bounds are based on subject matter knowledge regarding how different ATE are for never-takers and always-takers than they are for compliers. • We can estimate – E[Y(1)|D(1)=D(0)=1] with estimate of E[Y|Z=0, D=1], – E[Y(1)|D(1)=D(0)=0] with estimate of E[Y|Z=1, D=0] • But only half of what we need.
Who are the compliers? • Never can know because not assigned both treatment. • However, we can estimate characteristics of subjects who are likely to be compliers. • Section 5. 2 describes an approach to compute expectation of X among compliers, which can be compared with the overall population.
Section 6. 1 • A 2 is easily tested with the data. • Discusses various heuristic ways to test assumptions A 3 -A 4. • Apparently assessing the direct effect of Z on Y conditioning on X and D is not a good test of A 4.
Sensitivity Analyses • A 3: No unmeasured confounders: – Two parameter sensitivity analysis. – Put in a confounder with an assumed relationship with Y and an assumed relationship with Z. – Perform analyses for various levels. • A 4: Exclusion restriction: – Single sensitivity parameter. – Interaction term between Z and D in the outcome model, with the sensitivity parameter assumed. • A 5: Monotonicity assumption: – Two sensitivity parameters. – One parameter sets proportion of defiers – Second parameter fixes ACE among defiers.
Weak and Strong Instruments • Proportion compliers: – ∑{E[Di|Zi=1, Xi]-E[Di|Zi=0, Xi]}/N • Small proportion indicates weak instrument. • Another way to measure IV strength is Corr(Z, D). • Problems with weak instrument: – Large variance – Two-stage least squares estimation may be biased and asymptotic variance under-estimated – Very sensitive to unmeasured confounding
Variance of IV • Under some simplifying assumptions, “the sample size needed for an IV study is approximately equal to that needed for a conventional observational study to detect the same magnitude of effect divided by the proportion of compliers squared. ” • That can be a much larger sample size.
Reporting an IV Analysis • Describe theoretical basis for choice of IV. • Report strength of IV and results from first-stage model. • Report distribution of measured confounders across levels of IV and treatment. • Explore concomitant treatments. • Discuss the interpretation of treatment effects estimated by the IV. • Report a sensitivity analysis.
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