Burgess, Davies, and Thompson1 have presented a method for using instrumental variables (IVs) to investigate whether an exposure-outcome relation is nonlinear and for estimating its shape. As noted by Burgess et al, most methods for estimating causal effects using IVs have assumed that the exposure-outcome relation is linear. I congratulate Burgess and colleagues for focusing attention on the important problem of estimating nonlinear exposure-outcome relations using IV methods and contributing an interesting and stimulating paper. In this commentary, I make 2 points. First, the localized average causal effect curve estimated by Burgess et al reflects a combination of how the exposure’s effect varies across the range of the exposure for a specific subject and how the effect of the exposure varies among groups of subjects with different IV-free exposure levels. This curve can provide insight into the linearity vs. nonlinearity of the subject-specific exposure-outcome relation, but it is different and needs to be interpreted carefully. Second, the localized average causal effect curve estimation method proposed by Burgess et al is sensitive to the assumption that the effect of the IV on exposure is not correlated with potential outcomes; sensitivity analysis methods that allow for violations of this assumption are available.
Following the notation of Burgess et al, let Yi be the observed outcome, Xi the observed exposure, and Gi the observed IV for subject i. Let Yi (x) be the potential outcome that subject i would have if she were to have exposure level x and let Xi (g) be the potential exposure level that subject i would have if she were to have IV level g. Consider the simple setting that G is a binary IV and that there is no heterogeneity in the effect of the IV on the exposure,
Burgess et al stratify subjects by their estimated IV-free exposure level,
, where
is the coefficient on G in the linear regression of X on G, and they estimate the localized average causal effect (LACE) by applying the ratio method (2-stage least squares) to each stratum. They propose a method to estimate the localized average causal effects across the range of the IV-free exposure by using a sliding window. Under the no heterogeneity in the effect of the IV on the exposure equation 1 and smoothness conditions, the estimated localized average causal effect at IV-free exposure level a will converge to
LACE (a) is the per-unit effect of an increase in exposure from a to a + γ for subjects with IV-free exposure level a. Following the terminology of Burgess et al, we call the curve of LACE (a) for the range of possible exposures a the “localized average causal effect (LACE) curve”; this curve is the estimand of Burgess et al’s method.
For assessing whether the exposure’s effect is nonlinear, the curve we are interested in is the “subject-specific effect curve (SSEC)” which shows how the effect of a fixed increase in the exposure (say of size γ) changes the potential outcome for a given subject. We will analyze the relation between the subject-specific effect curve and the localized average causal effect curve under some simplifying assumptions. Suppose that the subject-specific effect curve is the same for all subjects with the same IV-free exposure level a,
but the subject-specific effect curve may differ for subjects with different IV-free exposure levels. Furthermore, suppose that although the level of the subject specific effect curve may differ for subjects with different IV-free exposure levels, the curves are parallel,
where f (0) = 0 and the term f (a) is the difference between the level of the subject-specific effect curve for subjects with IV-free exposure a and IV-free exposure 0. If l (x) is constant in x, then there is a linear exposure-outcome relation for each given subject; if l (x) is not constant then the exposure-outcome relation is nonlinear. Under equation 4, the localized average causal effect curve is
If f ′(a) > 0, which means that the level of the subject-specific effect curve is increasing as the IV-free exposure increases, then the localized average causal effect curve will have a steeper ascent than the subject-specific effect curves. This is illustrated in the Figure in which f (a) = a and in part A of the Figure, l (x) = 0.2 + x and in part B of the Figure, l (x) = 0.2. In part A of the Figure, the subject-specific effect is increasing in the exposure, but it is not increasing as fast as the localized average causal effect curve; in part B of the Figure, the subject-specific effect is not changing as the exposure changes, but the localized average causal effect curve is increasing because the level of the subject-specific effect curves is higher among subjects with a higher IV-free exposure. In summary, a change in the localized average causal effect curve as the exposure increases does not necessarily mean that there is a nonlinear exposure-outcome relation; it could just mean that the subject-specific effect curve is higher among subjects with a higher IV-free exposure.
Although the localized average causal effect curve does not measure how the exposure effect changes as the baseline exposure changes, it can provide useful information about this relation when supplemented with additional knowledge. For example, if the localized average causal effect curve has a steep slope and the slope for the subject-specific effect curve is not thought to be as steep (eg, in part A of the Figure, f′ (a) is thought to be less than the slope of the localized average causal effect curve), this would provide evidence for a nonlinear exposure-outcome relation. Also, the localized average causal effect curve is of interest for uses in addition to assessing whether there is a nonlinear exposure-outcome relation, such as for making personalized treatment decisions. Suppose a person with genes Gi and estimated IV-free exposure
is trying to decide how much she will benefit from reducing her BMI; the localized average causal effect at the person’s IV-free exposure estimates the benefit. Shinohara and Frangakis have investigated using Mendelian randomization to aid decisions about treating gout in this manner (Shinohara T, personal communication, 2014).
My second point concerns the robustness of the method proposed by Burgess et al to the assumption that the effect of IV on the exposure is not correlated with potential outcomes and alternative methods which relax this assumption. Their method provides a consistent estimate of the localized average causal effect curve when there is no heterogeneity in the effect of the IV on the exposure. In their online supplementary materials, Burgess et al have explored the robustness of their method to violations of the no heterogeneity assumption, finding that their method is robust to heterogeneity when the effects of the IV on the exposure and of the exposure on the outcome are not correlated. One situation not considered in the supplementary materials is that the effect of the IV might be correlated with unmeasured confounders that affect the outcome, e.g., the people whose BMI is more affected by their BMI-associated genetic variants might be people who exercise less. Consider the same setting as in equation 2 of their supplementary materials except that we now let
have a bivariate normal distribution with means (0.25, 0), variances (0.12, 1), and correlation 0.5, where
. Then, for the linear model f1 (xi) = 0.4xi, the mean estimates of the localized average causal effect in the 4 strata of IV-free exposure (below 1, 1–2, 2–3, and above 3) in 1000 simulations using their method were 0.08, 0.55, 0.59, and 0.70. Even though the subject-specific effect curves are flat for all subjects (ie, the exposure-outcome relation is linear), the method proposed by Burgess et al suggests that there is a nonlinear effect because the effect of the IV is correlated with unmeasured confounders. The IV-free exposure level is an example of a coarsened principal strata.2 The assumption that there is no heterogeneity in the effect of the IV on the exposure corresponds to there being a perfect correlation between the components of the principal strata; here the components of the principal strata are potential exposure levels under different levels of the IV. Methods have been developed that estimate treatment effects within principal strata allowing for there to be less than perfect correlation between the components of the principal strata.3–6 The correlation between the components of the principal strata cannot be fully identified from the data and these methods have sensitivity parameters for how correlated the components are (here, how correlated the potential exposure levels are for different levels of the IV).
In summary, I congratulate Burgess and colleagues for a valuable and important paper on using IV analysis to investigate a nonlinear exposure-outcome relation. The localized average causal effect curve estimated by Burgess et al reflects a combination of how nonlinear the exposure-outcome relation is and also how much the level of the exposure-outcome relation varies between groups of subjects with different IV-free exposure levels. Because the localized average causal effect curve combines these 2 different relations, I suggest that careful thought needs to be given to understanding the implications of the localized average causal effect curve for nonlinearity of the exposure-outcome relation in a given empirical setting. I also suggest that if there is the possibility of the effect of the IV being correlated with potential outcomes, sensitivity analysis methods such as3–6 should be used to examine the potential impact of this correlation on estimates of the localized average causal effect curve.
REFERENCES
1. Burgess S, Davies N, Thompson S. Instrumental variable analysis with a nonlinear exposure-outcome relationship. Epidemiology.
2. Frangakis C, Rubin D. Principal stratification in causal inference. Biometrics. 2002;58:21–29
3. Jin H, Rubin D. Principal stratification for causal inference with extended partial compliance. J Am Stat Assoc. 2008;103:101–111
4. Bartolucci F, Grilli L. Modeling partial compliance through copulas in the princi-pal stratification framework. J Am Stat Assoc. 2011;106:469–479
5. Schwartz S, Li F, Mealli F. A bayesian semiparametric approach to intermediate variables in causal inference. J Am Stat Assoc. 2011;106:1331–1344
6. Ma Y, Roy J, Marcus B. Causal models for randomized trials with two active treatments and continuous compliance. Stat Med. 2011;30:2349–2362
