Brief introduction to instrumental variables. IV Workshop, Bristol, Miguel A. Hernán Department of Epidemiology Harvard School of Public Health

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1 Brief introduction to instrumental variables IV Workshop, Bristol, 2008 Miguel A. Hernán Department of Epidemiology Harvard School of Public Health

2 Goal: To consistently estimate the average causal effect of exposure A on outcome Y Effect measures Causal risk ratio: Pr[Y a=1 =1] / Pr[Y a=0 =1] Causal risk difference: Pr[Y a=1 =1] Pr[Y a=0 =1] Etc... Assume no selection bias, no measurement error 19-Jun-09 Instrumental variables 2

3 All available methods Including Stratification Regression Matching Standardization Inverse probability weighting G-estimation (more later) require one unverifiable condition 19-Jun-09 Instrumental variables 3

4 Exchangeability conditional on the measured covariates L Y a A L l for all a Equivalent expressions exposure is randomized no unmeasured confounding the risk in the exposed if unexposed is the same as the risk in the unexposed, and vice versa within levels of the covariates in L 19-Jun-09 Instrumental variables 4

5 The assumption of exchangeability is Necessary for all the methods above Empirically unverifiable in observational studies Even if there is conditional exchangeability, there is no way we can know it with certainty Data necessary to test this condition is, by definition, unavailable 19-Jun-09 Instrumental variables 5

6 Instrumental variables (IV) methods Unlike all other methods, IV methods can be used to consistently estimate causal effects in the absence of exchangeability i.e., in the presence of unmeasured confounding 19-Jun-09 Instrumental variables 6

7 Overview of this lecture 1. IV in randomized experiments Standard IV estimator 2. IV in observational studies 3. Problems of IV methods 4. Conclusion 19-Jun-09 Instrumental variables 7

8 Definition of an instrument Z (provisional) i. Z has a causal effect on the exposure A ii. Z affects the outcome Y only through A no direct effect of Z on Y iii. Z does not share common causes with the outcome Y no confounding for the effect of Z on Y Note: what follows is a conceptual discussion more rigorous discussion in Hernán and Robins Epidemiology Jun-09 Instrumental variables 8

9 Instrument Z U Z A Y Example: Randomized experiment Z: assigned treatment (dichotomous) A: actual treatment (dichotomous) Y: outcome U: unmeasured factors 19-Jun-09 Instrumental variables 9

10 Effects that can be computed (risk difference scale) U Z A Y average effect of Z on Y PrY 1 Z 1 PrY 1 Z 0 average effect of Z on A PrA 1 Z 1 PrA 1 Z 0 19-Jun-09 Instrumental variables 10

11 IV estimator for the average effect of A on Y U Z A Y effect of Z on Y effect of Z on A PrY 1 Z 1 PrY 1 Z 0 PrA 1 Z 1 PrA 1 Z 0 19-Jun-09 Instrumental variables 11

12 Standard IV estimator EY a1 EY a0 EY Z 1 EY Z 0 EA Z 1 EA Z 0 Intention-to-treat effect in the numerator is inflated by a denominator that decreases with the degree of noncompliance Equal to intent-to-treat effect in the absence of noncompliance Can also be used in observational settings 19-Jun-09 Instrumental variables 12

13 Definition of an instrument Z (final) i. Z and exposure A are associated because Z has a causal effect on A Z and A share common causes ii. Z affects the outcome Y only through A no direct effect of Z on Y iii. Z does not share common causes with the outcome Y no confounding for the effect of Z on Y 19-Jun-09 Instrumental variables 13

14 Instrument Z U U* A Y Z 19-Jun-09 Instrumental variables 14

15 Overview of this lecture 1. IV in randomized experiments Standard IV estimator 2. IV in observational studies 3. Problems of IV methods 4. Conclusion 19-Jun-09 Instrumental variables 15

16 Examples of instruments in observational studies: Access Patients with myocardial infarction A: Invasive procedures Y: Mortality Z: Distance to hospital with capability for invasive procedures McClellan et al Jun-09 Instrumental variables 16

17 Examples of instruments in observational studies: Access Patients with myocardial infarction A: Cigarette smoking Y: Health outcome (physical functional status) Z: Cigarette price Leigh and Schembri Jun-09 Instrumental variables 17

18 Examples of instruments in observational studies: Genes Mendelian randomization A: Alcohol intake Y: Coronary heart disease Z: genetic variants associated with alcohol metabolism, e.g., ADH3 polymorphic forms: slow/fast oxidizers Katan 1986, Davey Smith and Ebrahim 2004, 19-Jun-09 Instrumental variables 18

19 Examples of instruments in observational studies: Genes A: Dietary calcium Y: Osteoporosis U*: Lactose intolerance gene (unmeasured) Z: Self-reported lactose intolerance 19-Jun-09 Instrumental variables 19

20 Examples of instruments in observational studies: Preference Pharmacoepidemiology / Outcomes research A: Type of drug (e.g., chemotherapy) Y: Outcome (e.g., mortality in lung cancer patients) Z: Physician s preference e.g., Korn and Baumrind 1998, Craig et al 2001, Brooks et al 2003, Brookhart et al 2006, 19-Jun-09 Instrumental variables 20

21 Examples of instruments in observational studies: Preference A: Type of drug (e.g., chemotherapy) Y: Outcome (e.g., mortality in lung cancer patients) U*: Physician s preference (unmeasured) Z: Proxy for preference, e.g., proportion of patients receiving A=1 19-Jun-09 Instrumental variables 21

22 Isn t it amazing? After all, we can consistently estimate causal effects in observational studies in the presence of unmeasured confounding If we find an instrument conditional exchangeability of the exposed and unexposed is not necessary for causal inference from observational data! Why were we wasting our time with methods that require exchangeability for the exposure? 19-Jun-09 Instrumental variables 22

23 Overview of this lecture 1. IV in randomized experiments Standard IV estimator 2. IV in observational studies 3. Problems of IV methods 4. Conclusion 19-Jun-09 Instrumental variables 23

24 The problems of IV methods 1. It is impossible to verify that Z is an instrument A non instrument introduces bias 2. A weak instrument Z blows up the bias 3. Instruments are insufficient to estimate causal effects Additional unverifiable assumptions are required 4. Standard IV methods deal poorly with time-varying exposures 19-Jun-09 Instrumental variables 24

25 Problem 1: Conditions (ii) and (iii) are not empirically verifiable ii. Z affects the outcome Y only through A no direct effect of Z on Y iii. Z does not share common causes with the outcome Y no confounding for the effect of Z on Y IV estimate may be more biased than the unadjusted estimate 19-Jun-09 Instrumental variables 25

26 Problem 1 effect of Z on Y effect of Z on A PrY 1 Z 1 PrY 1 Z 0 PrA 1 Z 1 PrA 1 Z 0 19-Jun-09 Instrumental variables 26

27 Problem 2: A weak instrument amplifies the bias i. Z and A are associated because Z has a causal effect on A Z and A share common causes If Z weakly associated with A Small denominator of the IV estimator Biases that affect the numerator (unmeasured confounding for Z, direct effect of Z) or small sample bias in the denominator will be exaggerated IV estimate may be more biased than the unadjusted estimate 19-Jun-09 Instrumental variables 27

28 Problems 1 and 2 Effects of exposure estimated by IV methods may be much larger than effects estimated by conventional adjustment methods or randomized experiments Because violation of the IV assumptions or small sample size may result in large bias that exaggerates the effect estimate This may be counterintuitive to epidemiologists used to conventional methods 19-Jun-09 Instrumental variables 28

29 Problem 3: An instrument is not enough An instrument only allows to compute bounds for, not a point estimate of, the causal effect Bounds may be too wide: not very informative In addition, 95% confidence intervals around the bounds Robins 1989 But what about the standard IV estimator? 19-Jun-09 Instrumental variables 29

30 Problem 3: An instrument is not enough EY a1 EY a0 EY Z 1 EY Z 0 EA Z 1 EA Z 0 The standard IV estimate provides a point estimate, not bounds The above equality only true under additional identifying assumptions All unverifiable 19-Jun-09 Instrumental variables 30

31 Problem 3: An instrument is not enough Some examples of assumptions a) No between-subject heterogeneity b) No interaction between instrument and exposure Additive or multiplicative 19-Jun-09 Instrumental variables 31

32 Unverifiable assumption a): No between-subject heterogeneity The effect of A on Y is the same for every individual implicit in econometric methods (e.g., 2-stage least squares) along with modeling assumptions Special case: sharp null hypothesis In general, biologically implausible for continuous exposures and logically impossible for dichotomous exposures Too strong an assumption the extreme no interaction assumption 19-Jun-09 Instrumental variables 32

33 Unverifiable assumption b): No interaction No effect modification by Z (or U*) of the effect of A on Y If no effect modification on the additive scale: estimator of E[Y a=1 ] E[Y a=0 ] is the standard IV estimator multiplicative scale: estimator of E[Y a=1 ] E[Y a=0 ] is another IV estimator See Theorem 4 in Hernán and Robins Jun-09 Instrumental variables 33

34 Unverifiable assumption b.1): No interaction on the additive scale The effect of A on Y on the risk difference scale is the same in treated subjects with Z=1 as in treated subjects with Z=0 and similarly among untreated subjects Such effect modification may occur if unmeasured factors U interact with A on an additive scale to cause outcome Y 19-Jun-09 Instrumental variables 34

35 Such effect modification is expected in many studies e.g., in a study of lung cancer patients A: Type of chemotherapy Y: Mortality U: Past response to treatment Z: Physician s preference Effect modification in the treated if the risk difference for the effect of chemotherapy A on mortality Y was modified by past response to treatment U 19-Jun-09 Instrumental variables 35

36 An alternative to the previous unverifiable assumptions Monotonicity: For all subjects, the level of treatment A that the subject would take if given a level of the instrument Z is a monotone increasing function of the level of Z See Hernán and Robins 2006 for definition of monotonicity for continuous instruments Also an unverifiable assumption Let us see some examples 19-Jun-09 Instrumental variables 36

37 Monotonicity Randomized trial Instrument Z is randomized assignment 4 types of people: Always takers: A z=0 =1, A z=1 =1 Never takers: A z=0 =0, A z=1 =0 Compliers (cooperative): A z=0 =0, A z=1 =1 Defiers: A z=0 =1, A z=1 =0 Monotonicity if no defiers plausible 19-Jun-09 Instrumental variables 37

38 Monotonicity Observational study Instrument Z is not randomized assignment Still 4 types of people: A z=0 =1, A z=1 =1 A z=0 =0, A z=1 =0 A z=0 =0, A z=1 =1 A z=0 =1, A z=1 =0 Monotonicity holds if there are no subjects of the type (A z=0 =1, A z=1 =0) We do not refer to subjects (A z=0 =1, A z=1 =0) as defiers because there is no assignment to be defied 19-Jun-09 Instrumental variables 38

39 Monotonicity: Observational study of chemotherapy and lung cancer Instrument Z is physician s preference If only 2 doctors in the study: doctor Z=1 prefers A=1, and doctor Z=0 prefers A=0 Monotonicity if no patient would receive A=0 if treated by doctor Z=1 and A=1 if treated by doctor Z=0 If many doctors in the study: Monotonicity if no patient would receive A=0 if treated by any of the doctors Z=1 (those who prefer A=1) and A=1 if treated by any of the doctors Z=0 (those who prefer A=0) Plausible? 19-Jun-09 Instrumental variables 39

40 Under the monotonicity assumption The standard IV estimator does not estimate the average effect of A on Y in the population but in a subset of the population: the compliers A z=0 =0, A z=1 =1 Angrist, Imbens, Rubin 1986 Problem: the compliers cannot be identified the subset of the population the effect estimate refers to is unknown 19-Jun-09 Instrumental variables 40

41 Who are the compliers? In the randomized trial example, the subset of the population who would comply with whichever treatment is assigned to them In the preference example, the subset of the population who would be treated with A=1 by all doctors who prefer A=1 with A=0 by all doctors who prefer A=0 Less clear if exposure is continuous 19-Jun-09 Instrumental variables 41

42 Problem 4: Time-varying exposures Most epidemiologic exposures are timevarying The standard IV estimator estimates the effect of a non time-varying exposure using a non time-varying instrument What if time-varying exposures and/or instruments? G-estimation of structural nested models Standard IV estimator is a particular case of a structural nested model 19-Jun-09 Instrumental variables 42

43 Conclusions (I) IV methods require FOUR assumptions The first 3 (instrumental) assumptions are nonnegotiable association between instrument and exposure no direct effect of the instrument on outcome no unmeasured confounding (conditional exchangeability) for the instrument Wide variety of fourth assumptions no interaction, monotonicity, etc Different assumptions result in the estimation of different causal effects 19-Jun-09 Instrumental variables 43

44 Conclusions (II) IV methods replace one unverifiable assumption no unmeasured confounding (conditional exchangeability) for the exposure By other unverifiable assumptions no unmeasured confounding (conditional exchangeability) for the instrument no direct effect of the instrument no interaction, monotonicity, etc The fundamental problem of causal inference is not solved but simply shifted Need to decide which assumptions are more likely to hold in each particular example 19-Jun-09 Instrumental variables 44

45 Conclusions (III) IV methods are not a magic bullet for unmeasured confounding reliance on different unverifiable assumptions compared with IPW, etc. IV methods are underutilized deserve greater attention in epidemiology But users of IV methods need to be aware of the limitations of these methods otherwise, conflicting estimates may result counterintuitive 19-Jun-09 Instrumental variables 45

46 References Hernán MA, Robins JM. Instruments for causal inference: an epidemiologist s dream? Epidemiology 2006; 17: Plus all the other papers the organizers considered good enough to them to you 19-Jun-09 Instrumental variables 46