A Note on the Control Function Approach with an Instrumental Variable and a Binary Outcome

1 Review of control function for continuous treatment

Suppose that one has observed a rare binary outcome Y, a continuous exposure A,and a binary IV Z. Throughout, U will refer to an unmeasured continuous confounder of the A–Y causal association. A standard formulation of a data generating model for the control function approach assumes the outcome is generated from the log-linear model

with βA the log risk ratio causal association between A and Y conditional on U (see for example, Palmer et al., 2011). The model rules out the possibility of latent effect heterogeneity of A wrt U on the multiplicative scale. For continuous A, the model further posits the following regression for the exposure:

where γ2≠0. In addition, the model assumes that U and Z are independent, as would be the case for a valid IV. Note that the above model encodes explicitly the assumption that Z is a valid IV, which satisfies the following conditions:

1. Z only affect Y through its association with A, which is encoded by the fact that although Z appears in the conditioning event on the left-hand side of eq. [1], it does not appear on the right-hand side of the equation.

2. The unmeasured confounder of the exposure effect on the outcome is independent of the IV, thus Z is independent of U.

3. The IV is relevant for the exposure, i.e. Z predicts A and thus γ1≠0.

where

The above equation gives a simple parametrization for the log-linear regression of Y on (A,Z), which allows one to recover under the stated assumptions, the causal log risk ratio association βA between A and Y. Estimation typically proceeds in two stages. In the first stage, one fits a standard linear regression to estimate the exposure model

by ordinary least squares (OLS), which in turn is used to estimate the residual Δ with Δ⌢=A−(α⌢0+α⌢1Z), where (a⌢0,a⌢1) are OLS estimates. Then in a second stage, one regresses Y on (A,Δ⌢) using standard logistic regression, as a suitable approximation for the log-linear model [4]. The regression coefficient for the exposure in the second stage logistic regression will then be approximately unbiased for βA. We will refer to the above two stage procedure as the linear-logistic control function approach. The large sample variance of the resulting estimator of βA must acknowledge the first stage estimation of EA|Z which is easily obtained from standard M-estimation theory (see the appendix). Alternatively, when convenient, one could also use the nonparametric bootstrap to obtain confidence intervals.

One can assess the extent of unobserved confounding, by evaluating the strength of association between Y and Δ, which can be performed with a test of the null hypothesis that α1=0.

2 Control function for binary treatment

Now, suppose that A is dichotomous, then as noted by Didelez et al. (2010), assumption [2] cannot be satisfied for binary exposure. Thus, we will consider an alternative formulation, whereby assumption [2] is replaced by the following location shift model for U:

The assumption would hold if U were normally distributed given A,Z with homoscedastic variance; however, this is not strictly required and the model allows for an arbitrary distribution for δ.

Result 1: Under assumptions [1], [3], and the assumption that U and Z are independent, we have that

where(β0∗,ω1,ω2)are defined in the Appendix.

Result 1 provides formal justification for a generalization of the control function approach in the context of a binary treatment, with the standard control function approach recovered by setting ω2=0. Interestingly, unlike the standard formulation for continuous treatment, the regression model [4] formally allows for heterogeneity in the degree of selection bias due to confounding if ω2≠0.

Implementation of the approach in practice is fairly straightforward. The main adjustment to account for binary treatment is in the first stage estimation of the treatment model, whereby OLS estimation of a linear model for continuous treatment can be replaced with maximum likelihood estimation (MLE) of a logistic regression for binary treatment:

to produce an estimated propensity score π⌢(Z)=Pr^(A=1|Z) using the MLE (a⌢0,a⌢1). The second stage of the approach proceeds by estimating the logistic regression of Y on (A,Δ⌢,ZΔ⌢), upon redefining the estimated residual as Δ⌢=A−π⌢(Z). The resulting estimator of the regression coefficient for the exposure in the second stage logistic regression will be approximately unbiased for βA provided that the assumptions of Result 1 hold. For inference, one may use M-estimation theory to derive the large sample variance of the estimator provided in the Appendix; alternatively, one can proceed with the nonparametric bootstrap.

3 Control function without rare disease assumption

If Y is not rare in the target population, one may adopt one of several existing methods to estimate the risk ratio regression [4] that do not require the rare disease assumption, including the log-binomial model of Wacholder (1986), the Poisson regression approach of Zou (2004), and the semiparametric locally efficient approach of Tchetgen Tchetgen (2013a).

4 Control function under case–control sampling

Case–control studies are a common design in epidemiologic practice, particularly is settings where Y is rare in the population, or measuring Z or A is costly. Accounting for case–control ascertainment is fairly straightforward in the case of a rare outcome, since logistic regression, which appropriately accounts for the sampling design, can continue to be used in the second stage; however, the first stage regression model must be modified to account for possible selection bias. A simple strategy entails restricting estimation of the first stage regression of A on Z, to the subset of controls with Y=0, which should yield a reasonable approximation of the population regression model. This approach may however be inefficient, since it does not make use of the exposure and IV measured among cases. Under certain assumptions, it may be possible to improve the efficiency of the first stage regression which in turn may lead to a more efficient second stage estimator of the treatment effect. This can be achieved by using all available information on both cases and controls, and by adjusting for case–control status in estimating the first stage regression model. For instance, for continuous A,one may modify the first stage regression and instead estimate:

which involves adjusting for ascertainment by directly conditioning on case–control status in the regression model. Under the rare disease assumption, the above model would in principle recover an unbiased estimator (α˜0,α˜1) of α0,α1 provided the degree of ascertainment bias (here encoded by a non-null value of α2) does not vary with Z (Tchetgen Tchetgen et al., 2013; Tchetgen Tchetgen, 2013b). It is important to note that some care is needed in forming the residual used in the second stage logistic regression, which must reflect the residual value for A in the underlying population and is therefore obtained by evaluating the predicted mean of A under the above estimated mean model, after setting Y=0 for both cases and controls (Tchetgen Tchetgen et al., 2013), i.e.

5 Adjusting for covariates

Here we consider a straightforward generalization to allow for the presence of covariates C such that Z is a valid IV conditional on C but not necessarily so upon marginalizing over any component of C. Assuming standard prospective sampling, in order to incorporate such covariates, it suffices to modify regression models used in the first and second stages, such that in the case of continuous exposure, the first stage regression further adjusts for C, e.g.

and likewise, for binary A, one could specify

The second stage regression in the rare outcome situation could also be modified accordingly, e.g.

with Δ⌢ the estimated residual A−E⌢(A|Z,C), and analogous adjustments can be made to the risk ratio regression approach recommended for non-rare outcomes, as well as under case–control sampling.

6 Conclusion

In this note, an alternative framework is proposed to motivate the control function IV approach in the context of binary outcome, with binary treatment. Although emphasis is given to binary treatment, the approach can be modified to handle other types of discrete treatments, without much difficulty. The approach can also be used with a continuous IV without further difficulty. In addition, unlike previous formulations of the control function approach, the proposed framework allows for heterogeneity in the magnitude of selection bias (on the risk ratio scale) with respect to the IV. Such heterogeneity may reflect latent heterogeneity in the degree of association of the IV with the treatment, the presence of which cannot be ruled out with certainty in practice. Ignoring such heterogeneity when present may invalidate the commonly used control function approach, and therefore the proposed methods provide a valuable framework for relaxing this assumption.