On two-stage estimation of structural instrumental variable models.

Confounding is a leading cause of bias in observational studies. Regression adjustment or propensity score methods may be used to overcome confounding, but they require that all confounders be observed. Using econometric terminology, the exposure variable is said to be endogenous when the exposure is correlated with an error term due to sharing unmeasured confounders. Endogeneity may also occur in randomized trials when there is noncompliance, which may be related to unobserved variables that are risk factors for the outcome. In such cases, the usual regression estimators may not be consistent. An instrumental variable methodology yields an unbiased estimator of the effect of an endogenous exposure. Although the requirements of an instrumental variable depend on the particular analytical method, the following three conditions are commonly used Brookhart et al., 2010: (i) an instrumental variable has a causal effect on an exposure ; (ii) affects an outcome only through the exposure ; (iii) is unrelated to measured and unmeasured confounders and . In randomized trials, a randomization assignment indicator is often used as an instrumental variable (Robins & Tsiatis, 1991; Loeys & Goetghebeur, 2003; Loeys et al., 2005; Nie et al., 2011) to estimate causal treatment effects. Structural linear equation modelling is popular for estimating causal effects in the instrumental variable setting and provides the foundation for our instrumental variable method. The outcome model of interest relates to and via a linear model, and the exposure model relating to and is also specified via a linear model. The regression parameters in the outcome model are identified by the instrumental variable. In the absence of measured confounders, an instrumental variable estimator may be obtained as the ratio of two covariance estimators. In the case where there are confounders, the generalized method of moments (Hansen, 1982) or two-stage least squares estimation (Anderson & Rubin, 1950) may be used with complete data. Our approach builds upon these earlier methods to accommodate missing data set-ups. Considerable work has been done in settings where both the outcome and the exposure are fully observed and assumed to satisfy either semiparametric linear or nonlinear or nonparametric structural equation models with unspecified error distributions (Anderson & Rubin, 1950; Amemiya, 1974; Newey, 1990; Newey & Powell, 2003). The popular two-stage least squares estimator has an explicit form, with a well-characterized sampling distribution and plug-in variance estimation, making inference straightforward (Bollen, 1996). However, if either outcome or exposure is incompletely observed, such methods are not applicable. There has been limited work addressing two-stage instrumental variable estimation in such settings. While the two-stage methods we propose are generally applicable to cases with incomplete data, in this article our focus is on right-censored data. There have been attempts to extend classical two-stage instrumental variable estimators. Robins & Tsiatis (1991) developed instrumental variable methods for correcting noncompliance in randomized trials by using a rank-preserving structural failure time model. This model is an alternative to the usual two-stage model, which is tailored to failure time data. A limitation is that the semiparametric estimation methods require that censoring times always be known, as with fixed follow-up, and hence censoring due to drop-out is not permitted. Brannas (2000) considered ad hoc two-stage instrumental variable estimators for linear structural equation models, which are adaptations of a symmetric trimmed least squares estimator (Powell, 1986) and a Buckley & James (1979) estimator for right-censored data. However, the theoretical properties of these procedures were not investigated and a rigorous investigation of two-stage instrumental variable estimation in linear models with right censoring does not seem to be available in the literature. Loeys & Goetghebeur (2003) proposed instrumental variable methods for the effect of treatment received in a randomized trial with all-or-nothing compliance based on a proportional hazards model. These methods were extended to permit other covariates in the proportional hazards model (Loeys et al., 2005). Li et al. (2015) and Tchetgen Tchetgen et al. (2015) developed two-stage least squares estimators for an additive hazard model. Nie et al. (2011) proposed an efficient instrumental variable estimator for comparing survival probabilities in randomized trials with noncompliance and administrative censoring, which is an extension of the estimator of Baker (1998). In § 2, we propose a general framework for two-stage instrumental variable estimation of semiparametric linear structural equation models for outcome and exposure which accommodates incomplete data. The main requirement is that there exist semiparametric methods for fitting linear models to the outcome and exposure. Such methods are well developed for truncated and censored time-to-event outcomes. At stage 1, we fit the exposure model and a reduced form of the outcome model conditional on the instrumental variable. At stage 2, we estimate the regression parameters in the true outcome model using a weighted minimum distance method based on the stage 1 results. This yields a closed-form estimator, for which a particular choice of weight leads to the standard two-stage least squares estimator with fully observed data. For the case of right censoring, the procedure does not require that the censoring time be always observed. We prove that our estimators are consistent and asymptotically normal, and provide a theoretically justified resampling technique for making inferences. The optimal weight is characterized, resulting in a minimum variance estimator which may be superior to the usual two-stage estimator. In § 3, we discuss details related to the implementation of our semiparametric estimator when either outcome or exposure may be censored, employing existing estimators for accelerated failure time models with right-censored event times. These methods perform well in simulations reported in § 4, where naive estimation that ignores the unmeasured confounders may produce severely biased estimates of exposure effects. The practical utility of the proposed methods is illustrated by a study of the comparative effectiveness of colon cancer treatments.