Efficient Nonparametric Causal Inference with Missing Exposure Information

1 Introduction

It is very common for there to be missing data in observational studies where causal effects are of interest. In this paper we consider studies where there is substantial missingness in an exposure variable. This is a very common feature of observational studies, and examples abound in the literature. For example, Zhang et al. [1] described the Consortium on Safe Labor observational study, where the goal was to estimate effects of mothers’ body mass index on infant birthweight. There, covariate and outcome information was essentially always available, but body mass index data was only available for about half of the mothers. Shortreed and Forbes [2] used the Framingham Heart Study data to assess effects of physical activity on cardiovascular disease and mortality, but up to 30 % of subjects were missing physical activity information. Similarly, Ahn et al. [3] used the Molecular Epidemiology of Colorectal Cancer study to estimate effects of physical activity on colorectal cancer stage, but 20 % of subjects were missing physical activity data. Shardell and Hicks [4] described an analysis of the Baltimore Hip Studies involving older adults with hip fractures, where the goal was to assess effects of perceived mobility recovery on survival. However, this self-reported mobility measure was unavailable for 27 % of subjects. Molinari [5] and Mebane Jr and Poast [6] give numerous other examples of studies with missing exposure information, particularly in survey settings, e. g. from the National Longitudinal Survey of Youth, and the Health and Retirement Study. This is certainly a prevalent problem.

In fact the problem is even more widespread, since in instrumental variable studies one can view the instrument as a type of exposure (e. g. for the purpose of estimating intention-to-treat-style effects, as well as other instrumental variable estimands that require estimating instrument effects on both treatment and outcome). And it is similarly common for instrument values to be missing. For example, in a Mendelian randomization context Burgess et al. [7] used genetic variants as instrumental variables to study the effect of C-reactive protein on fibrinogen and coronary heart disease. However, data on these variants was missing for up to 10 % of subjects, due to difficulty in interpreting output from genotyping platforms. Mogstad and Wiswall [8] and Chaudhuri and Guilkey [9] give further examples of missing instruments from economics.

Although missing exposures and instruments are a prevalent problem, the proposed methods for dealing with this issue have relied on potentially restrictive modeling assumptions, and have been somewhat ad hoc in not considering optimal efficiency. For example Williamson et al. [10] and Zhang et al. [1] propose interesting semiparametric estimators, but they rely on parametric models for nuisance functions, and do not consider the question of efficiency in either nonparametric or semiparametric models. Zhang et al. [1] also only considers binary outcomes. Chaudhuri and Guilkey [9] discusses (semiparametric) efficiency theory, but only for a finite-dimensional parameter in a population moment condition depending on a known function. This means their results apply to classical linear models, but not to the fully nonparametric setting pursued here. Kennedy and Small [11] consider nonparametric efficiency theory in missing instrumental variable problems, but only in simpler settings with one-sided noncompliance and no covariates.

Thus we fill these gaps by giving a new identifying expression for average treatment effects of multivalued discrete exposures in the presence of complex confounding and missing exposure values, deriving the efficient influence function and corresponding nonparametric efficiency bounds, and constructing nonparametric estimators that can be n-consistent and asymptotically normal, even if nuisance functions are estimated at slower rates via nonparametric machine learning tools. Finally we apply these general results to also address the problem of causal inference with a partially missing instrumental variable. Throughout we make use of a missing at random assumption used by previous authors, allowing exposure missingness to depend on post-exposure outcome information.

2 Missing exposures

In this section we consider the general problem of identification and efficient estimation of average treatment effects, when exposure values are missing at random, allowing the missingness mechanism to depend on both covariates and post-treatment outcome information.

2.1 Setup

Suppose we observe an iid sample (O1,...,On)∼P where

O=(X,R,RZ,Y)

for X∈Rd denoting covariate information, Z∈{z1,...,zk} a discrete treatment or exposure, R ∈ {0, 1} an indicator for whether Z is observed or not, and Y=(Y1,...,Yp)T∈Rp a vector of p outcomes of interest. In general we use script characters to denote the support of a random variable, e. g. X∈X⊆Rd. For notational simplicity we further define the nuisance functions

μ(y∣x)=P(Y≤y∣X=x)π(x,y)=P(R=1∣X=x,Y=y)λz(x,y)=P(Z=z∣X=x,Y=y,R=1).

Note that µ is the cumulative distribution function of the outcome given covariates, π can be viewed as the missingness propensity score, and λ the regression on covariates and outcomes of treatment among those for whom it is measured. We further define

βz(x)=E{Yλz(X,Y)∣X=x}=∫Yyλz(x,y)dμ(y∣x)γz(x)=E{λz(X,Y)∣X=x}=∫Yλz(x,y)dμ(y∣x).

The quantity βz(x)={βz1(x),...,βzp(x)}T is a vector of the same dimension as Y. We will see shortly that, under missing at random assumptions, γ_z equals the propensity score, while βz equals the product of the propensity score and outcome regression.

Our goal is to estimate the mean ψz=E(Yz)={E(Y1z),...,E(Ypz)}T∈Rp of the outcomes that would have been observed under treatment level z∈Z. It is well-known that this equals

(1)ψz=∫XE(Y∣X=x,Z=z)dP(x)

under the following standard causal assumptions:

(2)(Consistency.)Y=YZ

(3)(Z−Positivity.)P{ϵ<P(Z=z∣X)<1−ϵ}=1forallz∈Z

(4)(Z−Exchangeability.)Z⊥⊥Yz∣Xforallz∈Z

These assumptions have been discussed extensively in the literature [12, 13], so we refer the reader elsewhere for more details.

Crucially, when treatment Z is missing for some subjects, expression (1) is still not identified even under (2)–(4), since Z is not observed unless R = 1. We consider identification under missing at random conditions used for example by Chaudhuri and Guilkey [10], Williamson et al. [9] and Zhang et al. [1], which are:

(5)(R−Exchangeability.)R⊥⊥Z∣X,Y

(6)(R−Positivity.)P{ϵ<P(R=1∣X,Y)<1−ϵ}=1

Note that the missing at random condition (5) allows missingness in treatment Z to depend on the post-treatment outcome Y; this will be important if the outcome captures some information about the missingness mechanism beyond the covariates. An alternative missing-at-random assumption would be R⊥⊥(Z,Y)∣X; note however that this implies our R-exchangeability assumption, as well as the further testable implication that R⊥⊥Y∣X. Therefore our assumption is strictly weaker.

Figure 1 uses directed acyclic graphs to illustrate two different data generating processes that satisfy exchangeability conditions (4) and (5). The first represents a process where missingness occurs prior to the outcome (e. g. subjects miss a visit when they would have contributed treatment information); the second represents a process where missingness occurs after the outcome (e. g. survey non-response or data corruption after measurement).

Figure 1:

Two directed acyclic graphs for which the required exchangeability assumptions (4) and (5) hold, where (U1,U2) are unmeasured and Z is only observed when R = 1. In graph (a) missingness can be viewed as occurring prior to the outcome, while in (b) it can be viewed as occurring after. The variable U₂ can be represented as the potential outcome Y0.

3 Application to missing instruments

Here we apply the theory from the previous section to identify and efficient estimate the local average treatment effect in instrumental variable studies with missing instrument values.

It is quite common for some instrument values to be missing in instrumental variable studies [8, 9, 11]. This setup fits in the proposed framework from the previous section as follows. We have O=(X,R,RZ,Y) where Z  ∈ {0, 1} is now an instrument, and Y=(A,Y) for A ∈ {0, 1} a binary treatment and Y∈R an outcome of interest. Here the outcome Y∈R is a scalar, but a multivariate outcome presents no additional complications. Note also our slight abuse of notation in using bold Y=(A,Y) to denote a vector that contains the scalar outcome Y. Then we can write

where βzt(x)=E{Tλz(X,A,Y)∣X=x}.

In addition to the causal assumptions (2)–(4) and missing at random assumptions (5)–(6) from before, we further make the instrumental variable assumptions:

Our first result identifies the local average treatment effect under the assumptions above.

Although we focus on the local average treatment effect, the same observed data functional can represent other treatment effects under varying assumptions (e. g. the effect on the would-be-treated under a no-effect-modification assumption as discussed for example by Hernán and Robins [27]). Thus our results equally apply to these other settings.

Now we go on to use the theory from the previous section to construct an efficient estimator of the local average treatment effect θ. As with βz(x), we can decompose the efficient influence function ϕz from the previous section as

for the two outcomes (A,Y)∈Y. As before we write ϕz=ϕz(O;P) and ϕˆz=ϕz(O;Pˆ) to ease notation, and suppose Pˆ is constructed from an independent sample. The proposed estimator is given by

This simply takes the ratio of the corresponding estimators for the effects of Z on A and Y, respectively.

The next result describes the asymptotic properties of the estimator θˆ, and gives conditions under which it is n-consistent and asymptotically normal, akin to the earlier Theorem 2 for a general ψˆz.

As before, the estimator θˆ has a fast convergence rate that is second-order involving products of nuisance errors, so that under for example n^– 1/4-type rates the estimator will be n-consistent, asymptotically normal, and efficient. It is also doubly robust, as pointed out in the next corollary.

To summarize, the above results extend the work of Chaudhuri and Guilkey [8], Mogstad and Wiswall [9], and Kennedy and Small [11], by providing an efficient nonparametric estimator of the instrumental variable estimand when some instrument values are missing, allowing adjustment for complex confounding via flexible data-adaptive estimators of the nuisance functions.

4 Discussion

In this paper we filled a gap in the literature by considering nonparametric identification, efficiency theory, and estimation of average treatment effects in the presence of complex confounding and missing exposure values, where the exposure missingness can depend not only on the covariates but also the outcome information. We derived the efficient influence function for the average treatment effect and corresponding nonparametric efficiency bounds, and constructed nonparametric estimators can attain these efficiency bounds under weak rate conditions on the nuisance estimators. This allows one to incorporate modern flexible regression and machine learning tools. We also apply our general results to the problem of causal inference with a partially missing instrumental variable, yielding a new estimator and efficiency bound in this problem as well.

There are several important avenues for future work. First, it will be useful to study finite-sample properties of the estimators proposed here, in comparison to the more parametric estimators proposed in earlier work. Relatedly, it would be useful to construct an efficient plug-in estimator using targeted maximum likelihood [28, 29], which would respect bounds on the parameter space, e. g. when Y is bounded. Second, we restricted study to possibly multi-valued but discrete point treatments; it would be of interest to extend to treatments that are continuous [30, 31] or time-varying [32, 33]. This would also be useful for continuous instrumental variable problems [34] with instrument missingness. Further, identification, efficiency theory, and estimation are all more complicated in settings where there is simultaneous missingness in covariates, treatment, and outcome [35]; however, this also occurs often in practice and deserves deeper investigation. Lastly, we assumed exchangeability in the sense of the missing indicator R being conditionally independent of the underlying exposure Z given both covariates X and outcome Y; it would be of interest to consider the case where we only assume R⊥⊥Z∣X. However, there average treatment effects are no longer point identified, and so one would need to consider bounds and/or sensitivity analysis.