Summary
Robins (1998) introduced marginal structural models, a general class of counterfactual models for the joint effects of time-varying treatments in complex longitudinal studies subject to time-varying confounding. Robins (1998) established the identification of marginal structural model parameters under a sequential randomization assumption, which rules out unmeasured confounding of treatment assignment over time. The marginal structural Cox model is one of the most popular marginal structural models for evaluating the causal effect of time-varying treatments on a censored failure time outcome. In this paper, we establish sufficient conditions for identification of marginal structural Cox model parameters with the aid of a time-varying instrumental variable, in the case where sequential randomization fails to hold due to unmeasured confounding. Our instrumental variable identification condition rules out any interaction between an unmeasured confounder and the instrumental variable in its additive effects on the treatment process, the longitudinal generalization of the identifying condition of Wang & Tchetgen Tchetgen (2018). We describe a large class of weighted estimating equations that give rise to consistent and asymptotically normal estimators of the marginal structural Cox model, thereby extending the standard inverse probability of treatment weighted estimation of marginal structural models to the instrumental variable setting. Our approach is illustrated via extensive simulation studies and an application to estimating the effect of community antiretroviral therapy coverage on HIV incidence.
Keywords: Causal inference, Instrumental variable, Marginal structural model, Observational study, Survival analysis, Unmeasured confounding
1. Introduction
Robins (1998, 2000) introduced a new class of counterfactual models, known as marginal structural models, that encode the joint causal effects of time-varying treatments subject to time-varying confounding. Marginal structural models are particularly powerful, as they estimate the causal effects of time-dependent treatments in the presence of time-dependent confounders that are affected by prior treatments. For identification of marginal structural model parameters, Robins (1998, 2000) relied on a sequential randomization assumption, also known as sequential exchangeability, which rules out unmeasured confounding of treatment assignment over time. Applications of marginal structural models abound in the health and social sciences (see, e.g., Robins et al., 2000; Hernán et al., 2001; Cole & Hernán, 2008; Cerdá et al., 2010).
Right-censored data are of common occurrence in epidemiological studies where, due to censoring, the clinical outcome may not always be observed. In such settings, marginal structural models extend to time-dependent Cox models. Unlike standard time-dependent Cox models, marginal structural Cox models (Hernán et al., 2000) allow for adjustment of time-varying confounders through the use of inverse probability of treatment weighting, and thus can be employed to estimate the causal effects of time-varying treatments in the presence of time-varying confounders; see de Keyser et al. (2014), Karim et al. (2014) and Ali et al. (2016) for recent applications of marginal structural Cox models in clinical studies.
However, sequential randomization can seldom be guaranteed in observational studies, even if one adjusts for a large number of covariates in an effort to make the assumption credible. The instrumental variable method (Goldberger, 1972; Imbens & Angrist, 1994; Angrist et al., 1996; Wooldridge, 2010) is a well-known approach to estimating causal effects subject to unmeasured confounding in observational studies. An instrumental variable is defined as a pre-treatment variable that is independent of all unmeasured confounders, and does not have a direct causal effect on the outcome other than through the treatment. In a double-blind placebo-controlled randomized trial, random assignment is a common example of an ideal instrumental variable for the causal effect of treatment when some patients fail to comply with the assigned treatment, provided that double-blinding is maintained. To our knowledge, Michael et al. (2020) were the first to consider identification and estimation of marginal structural mean models in the context of a time-varying treatment and a time-varying instrumental variable. However, additional challenges arise with censored survival data, which were not addressed by Michael et al. (2020).
The first formal instrumental variable approach for right-censored survival outcomes was proposed by Robins & Tsiatis (1991), who parameterized the treatment effect under a structural accelerated failure time model. The approach, which applies to both point and time-varying treatments, can be challenging to implement in practice, because of the need for artificial censoring in order to estimate the structural model parameters (Robins & Tsiatis 1991; Joffe et al., 2012). More recently, Li et al. (2015) and Tchetgen Tchetgen et al. (2015) considered estimating the conditional hazard difference under an additive hazard model (Aalen, 1989). Tchetgen Tchetgen et al. (2015) also considered instrumental variable estimation for a Cox structural model under rare disease. MacKenzie et al. (2014) considered instrumental variable estimation of a marginal structural Cox model for point exposure, but their estimator generally fails to be consistent as their proposed estimating equation fails to be unbiased. Martinussen et al. (2017b) developed instrumental variable estimators under a semiparametric structural cumulative survival model, which is closely related to the additive hazard model of Li et al. (2015) and Tchetgen Tchetgen et al. (2015). In contrast, Martinussen et al. (2017a) and Sørensen et al. (2019) considered estimating the causal hazard ratio among the treated. The literature on instrumental variable methods for complier causal effect is also well developed for survival data (e.g., Loeys et al., 2005; Cuzick et al., 2007; Nie et al., 2011; Yu et al., 2015; Kianian et al., 2019). However, to date none of these approaches has been extended to time-varying settings, although some progress was made by Yende-Zuma et al. (2019) under somewhat restricted conditions, including that the unmeasured confounder cannot be an effect of prior treatment.
In fact, many instrumental variables that have been used in point exposure studies are often valid time-varying instrumental variables in settings where longitudinal data are available. For instance, in many well-established longitudinal observational studies such as the Multicenter AIDS Cohort Study and the Women’s Interagency HIV Study, both HIV treatment assignment information and treatment adherence information between follow-up visits are routinely collected, and therefore treatment assignment is a potential time-varying instrumental variable for the causal effects of time-varying antiretroviral therapy actually taken on HIV-related outcomes. Validity of treatment assignment as an instrumental variable relies on the assumption that confounding by indication can be fully accounted for, an assumption which is well justified in the literature (Robins et al., 2000; Hernán et al., 2001). Nowadays, even if not measured directly, adherence information is routinely inferred from pharmacy data in large electronic medical records when studying the causal effects of time-varying treatments on a variety of disease outcomes beyond HIV. Other examples of a common instrumental variable that is also inherently time-varying include physician treatment preference as physician preferences evolve with clinical practice and needs (Brookhart & Schneeweiss), distance from nearest college as individuals move over time, distance to nearest needle exchange program (Frangakis et al., 2004), calendar period (Cain et al., 2009), and differential distance between the nearest low-level and nearest high-level neonatal intensive care units in analyses of the causal effects of delivery at a high- versus low-level neonatal intensive care unit on birth outcomes (Zubizarreta et al., 2013; Yang et al., 2014). We acknowledge that sometimes instrumental variables may be related to initiating a treatment, but not adhering to it, so one should exercise caution in selecting a valid time-varying instrumental variable.
In this paper, we adopt an instrumental variable approach and establish sufficient conditions for identification of marginal structural Cox model parameters encoding the joint effects of binary time-varying treatments by leveraging a binary time-varying instrumental variable when the sequential randomization assumption fails to hold. Our identifying conditions extend those of Wang et al. (2022) to the longitudinal treatment setting, and require that in each risk set no unobserved confounder interacts with the instrumental variable in its additive effects on the treatment process. Our proposed semiparametric estimators extend standard inverse probability of treatment weighted estimation, which is the most popular method of estimating marginal structural Cox models under the sequential randomization assumption, by incorporating time-varying instrumental variables through a modified set of weights. We formally establish identification of our modified weighted estimating equations, and we provide asymptotic theory that allows, in the absence of model misspecification, valid inference about the marginal structural Cox model parameters.
2. Marginal structural Cox models
2.1. Notation
We first introduce some notation. Continuous time is denoted by 
and is measured in weeks or months since the beginning of a subject’s follow-up. The index 
is often used to indicate an integer number of weeks or months, and 
denotes the administrative end of follow-up. Let 
and 
denote, respectively, a binary treatment taken by a subject and a vector of relevant prognostic factors for the survival outcome in the time interval (
, where 
. We assume that 
temporally precedes 
. Although time 
is treated as continuous, we assume that recorded data on the treatment and prognostic factors do not change except at integer times. For any time-dependent variable 
, we let 
denote the history of that variable up to time 
. For example, the covariate process up to 
is 
, where 
denotes the smallest integer greater than or equal to 
. Furthermore, let 
be the censoring time, 
the observed time and 
the censoring indicator, where 
. Throughout the paper, except where necessary, we will suppress the subscript indexing the individual because we assume that the observed data of each subject are drawn independently from a distribution common to all subjects.
Figure 1 gives a causal graph representation of longitudinal confounding. The Markov property encoded in this graph implies that a node is independent of nondescendants conditional on its parents. Bi-directed edges into the covariate process 
and outcome 
indicate the possibility that there are unmeasured common causes confounding their association. As pointed out by Robins (1997), a standard Cox model of the joint effects of treatment on time to event, which would typically condition on 
, is subject to collider bias, and therefore bound to incorrectly report a causal effect linking treatment to the event time even under the sharp null hypothesis of no causal effect. This potential flaw of standard regression models motivated Robins (1998) to develop a marginal structural model as a principled way of circumventing this difficulty. Importantly, he assumed that there are no unmeasured common causes of the treatment process and the outcome 
, hence encoding sequential randomization, which we formally define in § 2. Thus, under a data-generating mechanism consistent with Fig. 1, the joint causal effect of treatment on 
would in fact be point-identified under a nonparametric model for the observed data using the g-formula of Robins (1997).
Fig. 1.
A causal graph of longitudinal confounding with bi-directed arrows.
We conclude this subsection by introducing counterfactuals which are key to defining marginal structural models. Neyman (1923) proposed using counterfactual outcomes to define the causal effects of time-independent treatments in randomized experiments. Rubin (1974) further used potential outcomes in the analysis of causal effects of time-independent treatments in observational studies. Robins (1986, 1987) proposed a formal counterfactual theory of causal inference that extends Neyman’s time-independent treatment theory to longitudinal studies with both direct and indirect effects, and time-varying treatments and confounders. For a specific fixed treatment history 
, 
is defined to be the random vector representing a subject’s covariate process had the subject been treated with the particular treatment regime 
rather than with their observed treatment history; 
is defined to be the subject’s time to death had the subject been treated with the particular treatment regime 
. Throughout the paper, we assume that the future cannot cause the past; for example, 
and 
.
2.2. Identification of marginal structural Cox models under the sequential randomization assumption
Suppose that we are interested in estimating the parameter 
indexing a marginal structural Cox model which encodes the causal effect of all potential treatment histories,
 |
(1) |
where 
is a known function satisfying 
is an unspecified baseline hazard function and 
are baseline covariates. Suppose that we observe data 
. We denote by 
and 
the treatment history and covariate history up to failure, respectively. In this section, we assume an independent censoring mechanism, i.e., 
. Three important assumptions are sufficient for the identification of 
from the observed data in marginal structural Cox models.
First, we make the standard consistency assumption that 
almost surely, which means that the observed failure time corresponds to the potential failure time under a potential intervention that sets the treatment process to the observed treatment history.
The next assumption is the well-known sequential randomization assumption proposed in Robins (1998),
 |
(2) |
where the term 
is defined as an empty set throughout the paper. This assumption generalizes the assumption of ignorable treatment assignment (Rosenbaum & Rubin, 1983) to longitudinal studies with time-varying treatments and confounders. It says that conditional on the treatment history up to time 
and recorded covariates up to time 
, the treatment at time 
is independent of the counterfactual outcome 
. The sequential randomization assumption holds if all common causes of 
and 
are included in 
, thereby ruling out unmeasured confounding of the treatment process as encoded in graph of Fig. 1.
Finally, we make the following treatment positivity assumption:
 |
for 
. This assumption states that, conditional on the observed history, there is a positive probability of receiving either treatment value at any given time. This assumption makes it possible to draw inferences about longitudinal treatment comparisons encoded in the marginal structural models.
We define the time-varying weights
 |
(3) |
where 
is a user-specified function of the treatment process 
. Robins (1998) showed that 
in (1) is the solution to the weighted estimating equation
 |
where 
is the counting process for the outcome variable, and 
is a vector-valued function which has the same dimension as the causal parameter of interest 
. In particular, given the marginal structural Cox model
 |
(4) |
and taking 
, 
is the solution to the weighted estimating equation
 |
The weights 
are generally unknown and need to be estimated from the observed data. Hernán et al. (2001) proposed using pooled logistic regression to estimate a model for the treatment process in both the numerator and the denominator of 
. The estimator obtained by substituting 
for 
and evaluating expectations under the empirical distribution, is 
-consistent and asymptotically linear under standard regularity conditions, including 
-consistency of an estimator for the treatment process 
.
3. Instrumental variable identification and inference of marginal structural Cox models
3.1. Identification
It is not always possible to ensure that a sufficiently rich set of variables 
was collected for sequential randomization to hold. With this in mind, suppose that 
denotes an unobserved common cause of 
and 
, such that (2) fails. Therefore, the treatment process is endogenous, i.e., subject to unmeasured confounding. On the other hand, suppose that one has observed a time-varying instrumental variable 
which satisfies the instrumental variable conditions described below. We develop a method of leveraging such an instrumental variable process to identify and estimate the marginal structural Cox model parameter 
.
A variety of instrumental variable models have been discussed in the literature on point treatment cases; see Swanson et al. (2018) for a comprehensive review. In the following, we adopt the latent counterfactual instrumental variable model described in Swanson et al. (2018) for the proposed method. Specifically, we consider a setting like the one depicted in Fig. 2, where the sequential randomization assumption fails to hold, as we allow unmeasured time-varying covariates that confound the treatment process. Suppose that 
is a common cause of 
and 
, where 
. We make the following assumption of latent sequential randomization.
Fig. 2.
A causal graph of longitudinal confounding with unobserved confounders.
Assumption 1
(Latent sequential randomization). We have that
.
Denote the observed data by 
. A binary time-varying instrumental variable 
is observed just prior to 
, but after 
and satisfies the following time-varying instrumental variable conditions for 
.
Assumption 2
(Instrumental variable relevance). We have that
with the history process defined as
.
Assumption 3
(Exclusion restriction). We have that
Assumption 4
(Instrumental variable independence). We have that
Assumption 5
(Instrumental variable positivity). We have that
Assumptions 2–4 are core instrumental variable conditions, while Assumption 5 is needed for nonparametric identification (Greenland, 2000; Hernán & Robins, 2006). Assumption 2 requires that the instrumental variable be associated with the treatment conditional on the history process. Assumption 2 does not rule out confounding of the 
-
association by an unmeasured factor; however, if present, such a factor must be independent of 
. We will refer to 
as causal instrumental variables when no such confounding is present. Assumption 3 states that there can be no direct causal effect of 
on 
, 
and 
not mediated by 
. Assumption 4 essentially says that the null direct causal effect of 
on 
, 
and 
would be identified conditional on the history process if one could intervene and set 
.
Finally, we also require the following condition that there is no additive interaction between 
and 
in a model for 
given 
, 
and 
.
Assumption 6
(Independent compliance type). We have that
(5)
This assumption is a longitudinal generalization of the assumption made by Wang & Tchetgen Tchetgen (2018) and Wang et al. (2022) in the case of point exposure.
To interpret Assumption 6, suppose that the causal effect of 
on 
is unconfounded given 
and 
, i.e.,
 |
for 
. Then
 |
The causal interpretation is that, while the unmeasured confounders 
may confound the causal effects of 
does not predict compliance type expressed in terms of an individual’s potential treatment status under hypothetical instrumental variable interventions 
at any 
. Figure 3 gives a causal graph representation of longitudinal confounding with unobserved confounders 
and causal instrumental variables 
. In principle, we only require (5) so that 
need not be a causal instrumental variable, i.e., the association between 
and 
may be subject to uncontrolled confounding, provided the confounder is independent of 
.
Fig. 3.
A causal graph of longitudinal confounding with unobserved confounders and instrumental variables.
Finally, we make a positivity assumption regarding censoring, namely 
, and the following standard independent censoring assumption.
Assumption 7.
We have that
.
It may not always be reasonable to make such an independence assumption about loss-to-follow-up censoring; in fact, in § 5 we implement inverse probability of censoring weights that appropriately account for dependent censoring by time-varying covariates under the weaker condition 
.
3.2. Identification of marginal structural Cox model parameters
Before presenting the main identification result, the following lemma states that under Assumptions 4 and 6, 
is empirically identified from the observed data.
Lemma 1.
Under Assumptions 4 and 6, for
we have that
Next, we define the novel time-varying weights
 |
(6) |
where
 |
with 
being user-specified, possibly empirically determined. Given that these weights can be identified from the observed data, we have the following theorem, which establishes that 
in marginal structural Cox model (1) is identified.
Theorem 1.
Suppose that consistency, positivity and Assumptions 1–7 hold. Then
solves the population moment equation
with
where
and
is a
-dimensional vector-valued function such that
is invertible.
3.3. Instrumental variable-based weighted estimator and large-sample properties
Theorem 1 motivates a weighted estimating equation of the parameter 
for the marginal structural Cox model defined in (1) in the presence of unmeasured confounding, i.e., 
is the solution to the population estimating equation
 |
(7) |
However, the weights 
are unknown and need to be estimated from the observed data. We propose using various parametric models to estimate these densities. For example, one can estimate the densities 
, 
and 
with the logistic regressions
 |
estimated by standard maximum likelihood. Let 
and 
denote the maximum likelihood estimates of 
and 
, respectively, and let 
. The compliance type 
can then be estimated by
 |
We denote the estimated weights by
 |
Our final estimator 
solves an empirical version of (7), where 
is replaced by 
. By standard M-estimation theory, 
is asymptotically linear with first-order expansion
 |
where
 |
, 
is the influence function of 
and 
is the baseline cumulative hazard function. Confidence intervals can then be constructed, either using an estimate of the asymptotic variance of 
based on the influence function representation given above, or via the nonparametric bootstrap.
4. Simulation studies
4.1. Preliminaries
In this section we conduct simulation studies in a setting with 
to compare the proposed estimator with existing estimators. As a benchmark, we consider an oracle weighted estimator of marginal structural Cox models, which uses the correctly specified weight 
instead of 
in (3). This oracle estimator is clearly not feasible in practice because 
would not be observed. In addition, we implement both a marginal structural Cox model estimated via inverse probability of treatment weighting, incorrectly assuming the sequential randomization assumption given the 
process, and a time-varying Cox model which directly adjusts for 
in the regression model.
Generating failure time outcomes under a specific marginal structural Cox model is not straightforward. We adopt the approach developed in the 2006 PhD thesis of E. Tchetgen Tchetgen from Harvard University, which we outline in the next subsection.
4.2. Generating potential outcomes under a marginal structural Cox model
Let 
, with 
, denote a person’s set of potential outcomes. Under our identifying assumptions and the further condition that
 |
the full data 
have a joint likelihood that factorizes as
 |
Suppose that we wish to generate 
under the marginal structural Cox model
 |
(8) |
where 
if 
and 
if 
. In the Supplementary Material it is shown that, by generating 
from an exponential density function with constant hazard 
and then defining 
under the accelerated failure time model
 |
we obtain the marginal structural Cox model in (8). We further specify
 |
Upon generating 
, we simulate the processes 
and 
. The censoring time is generated independently, and the observed time and censoring indicator are defined as 
and 
, respectively.
4.3. Simulation settings
We consider two settings, with 
for the first scenario and 
for the second scenario. In both scenarios, the random errors 
are generated from 
. The baseline hazard is 
. For convenience, and with a slight abuse of notation, we omit the input argument of 
and write simply 
hereafter. The data-generating mechanism for the treatment and survival time is as follows: 
, where 
; 
, where 
; 
, where 
is the cumulative distribution function of the standard normal distribution; 
, where 
; 
, where 
; 
, where 
; 
; and 
, where 
.
The censoring time is generated by 
where 
follows 
, yielding a censoring rate of approximately 40
for the first scenario and 30
for the second scenario. We also perform a sensitivity analysis for different censoring rates in each scenario; the results are presented in the Supplementary Material. We consider sample sizes 
. Each simulation is repeated 500 times. We further perform sensitivity analyses for violation of various instrumental variable assumptions in each scenario; these results can be found in the Supplementary Material. In addition, we examine the empirical coverage of 95
confidence intervals of 
for both scenarios at sample size 
. The confidence intervals are obtained by nonparametric bootstrap with 
replications.
Both the standard inverse probability of treatment weighted estimator and our proposed weighted estimator of the marginal structural Cox model (4) require estimation of weights. The density functions 
, and 
in (3) and (6) are estimated via maximum likelihood of time-specific logistic regression models,
 |
The conditional density of 
is estimated via maximum likelihood estimation using the model
 |
where the 
link function is the inverse standard normal cumulative distribution function. The data-generating mechanism specifies a model 
. Estimation of the compliance type is therefore performed via maximum likelihood estimation of the Bernoulli probability mass function for observations still at risk at time 
,
 |
for 
. Thus we obtain the estimated compliance type 
, where 
is the maximizer of 
.
4.4. Simulation results
We plot the squared bias and mean squared error of the estimated causal parameter in Fig. 4 for the first scenario and Fig. 5 for the second scenario.
Fig. 4.
Scenario 1 simulation results: (a) Monte Carlo squared bias and (b) mean squared error of 
. The following methods are compared: a standard time-varying Cox model that adjusts for 
directly in the regression (red); the proposed instrumental variable estimator (green); standard inverse probability of treatment weighted estimation of the marginal structural Cox model under the sequential randomization assumption (blue); and oracle inverse probability of treatment weighted estimation of the marginal structural Cox model which includes 
and 
in the treatment model (purple).
Fig. 5.
Scenario 2 simulation results: (a) Monte Carlo squared bias and (b) mean squared error of 
. The methods are the same as in Fig. 4.
From Fig. 4(a) and Fig. 5(a) we see that the Cox proportional hazard model and standard weighted estimation of the marginal structural Cox model have severe bias, the former because there exist time-dependent confounders that are affected by previous treatments and unmeasured confounding, and the latter because of unmeasured confounding. Our proposed estimator outperforms both the Cox proportional hazard model and standard weighted estimation of the marginal structural Cox model, in terms of bias and mean squared error. The oracle weighted estimator of the marginal structural Cox model performs as well as the proposed method, and both the bias and the mean squared error converge to zero as the sample size increases. This confirms that our proposed instrumental variable approach performs nearly as well in terms of bias, as the infeasible inverse probability of treatment weighted estimator had 
been observed, and can outperform the latter in terms of efficiency.
Table 1 reports empirical coverage percentages of 95
nonparametric bootstrap confidence intervals. The proposed instrumental variable method and the oracle inverse probability of treatment weighted estimator achieve the nominal coverage. However, the confidence intervals of both the standard weighted estimation of the marginal structural Cox model and the standard time-varying Cox proportional hazard model fail to attain nominal coverage.
Table 1.
Coverage 
of 
confidence intervals
| iv | sra | sra.o | Cox | |
|---|---|---|---|---|
Scenario 1 ( ) |
96.4 | 87.4 | 93.6 | 67.2 |
Scenario 2 ( ) |
97.2 | 87.2 | 94.2 | 66.4 |
iv, the proposed instrumental variable estimator; sra, standard inverse probability of treatment weighted estimation of the marginal structural Cox model; sra.o, oracle inverse probability of treatment weighted estimation of the marginal structural Cox model; Cox, a standard time-varying Cox model.
5. Estimating the effect of community antiretroviral therapy coverage on HIV acquisition
We apply the proposed method to an HIV study analysed in Tanser et al. (2013), which found evidence that significant reduction of HIV incidence can be achieved by nurse-led, devolved, public-sector antiretroviral therapy, or ART, programmes in rural sub-Saharan African settings, where complete coverage of therapy under existing treatment guidelines has not yet been attained. The analysis in Tanser et al. (2013) was based on a standard time-varying Cox proportional model. Our goal is to examine whether unmeasured confounding biased the reported association between high coverage of ART and the decline in risk of HIV acquisition in rural KwaZulu-Natal, South Africa.
We reanalysed the dataset considered in Tanser et al. (2013), which comes from one of Africa’s largest population-based prospective cohort studies to follow up individuals who were HIV-uninfected at baseline. Our analysis is restricted to 6093 individuals who were enrolled in the study, were known to be HIV-negative on 5 June 2008, and had complete covariate and instrumental variable data. The objective of our analysis is to determine the joint effects of living in a high-coverage community at two time-points, 
5 June 2008 and 
1 January 2011, on HIV incidence. The overall cumulative proportion of events for the outcome was 6.3
. We consider the following six time-varying covariates: number of partners in the past 12 months; current marital status; wealth index in terms of quintile; age and gender; location of residence; and community HIV prevalence. ART coverage is defined as the proportion of all HIV-infected individuals receiving ART at every location (Tanser et al., 2013). HIV prevalence and ART coverage of an individual’s surrounding community were determined for each year of observation. ART coverage and HIV prevalence around each individual were measured by means of a moving two-dimensional Gaussian kernel with a search radius of 3 km for each year of observation (Tanser et al., 2013). Here ART coverage is dichotomized at 30
, i.e., 
if ART coverage is 30
or more and 
otherwise, such that 40
of 
equals 1 over person-time.
Our instrumental variable is defined in terms of travel distance to the nearest ART facility: 
if the distance is less than 3.8 km and 
otherwise, such that 65
of 
equals 1 over person-time. The travel distance to the closest ART facility is found to be strongly associated with ART coverage. The adjusted log odds ratios and 95
confidence intervals for the association between travel distance to the closest ART facility and community ART coverage are 
, and 
at 
and 
, respectively, which justify instrumental variable relevance, Assumption 2. Furthermore, it is reasonable to assume that the mechanism by which the local density of ART clinics, and therefore travel distance to the nearest ART clinic, affects HIV incidence is primarily through ART coverage, and so the exclusion restriction, Assumption 3, holds.
We specify the marginal structural model (4) and estimate the various models needed to construct standard inverse probability of treatment weighted weights, as well as our proposed instrumental variable weights under the following model specification:
 |
In the Supplementary Material we also consider incorporating weights that account for dependent censoring, by adapting the approach of Robins & Rotnitzky (1992) to our method under the assumption 
. This entails multiplying 
by an estimate of
 |
The results with and without censoring weights are similar, indicating no evidence of dependent censoring. We obtained point estimates and 95
confidence intervals of the hazard ratio, using the nonparametric bootstrap with 1000 replications for standard weighted estimation and instrumental variable estimation of the marginal structural Cox model. To attenuate the impact of extreme values of weights, we truncated the 2.5th and 97.5th percentiles of the weights for the proposed estimator (Cole & Hernán, 2008). The histogram of weights and the results for the untruncated weights are reported in the Supplementary Material. The untruncated instrumental variable weighted estimator yields a similar point estimate to that of the truncated instrumental variable weighted estimator, but wider confidence intervals, due to outliers in the distribution of estimated weights. As can be seen from Table 2, the instrumental variable point estimate is much smaller, which we interpret, under our instrumental variable assumptions, as appropriately accounting for unmeasured confounding; this suggests that Tanser et al. (2013) might have underestimated the true effect of ART coverage.
Table 2.
Effect of community ART coverage on HIV acquisition
| sra | iv | |
|---|---|---|
| Hazard ratio | 0.45 | 0.19 |
95 confidence interval |
(0.20, 0.92) | (0.06, 0.83) |
sra, standard inverse probability of treatment weighted estimation of the marginal structural Cox model; iv, the proposed instrumental variable estimator, with weights truncated at the 
th and 
th percentiles.
6. Discussion
The proposed method can be improved or extended in multiple ways. It would be of interest to look at truncation of the weights in simulations. Formal justification of truncating weights as a means of stabilizing inverse probability weighting analyses is currently lacking, and represents a fruitful avenue of future research. Another potential extension is in the direction of semiparametric efficiency and enhanced robustness against partial model misspecification of nuisance parameters. The efficient influence function for the proposed marginal structural model is significantly more complicated than that in the marginal structural mean model (Tchetgen Tchetgen et al., 2018), and its study is beyond the scope of this paper.
Supplementary Material
Acknowledgement
We thank the two reviewers, associate editor and editor for many useful comments which have led to an improved manuscript. Cui was supported in part by the National University of Singapore and Singapore Ministry of Education. Tchetgen Tchetgen was supported in part by the U.S. National Institutes of Health.
Contributor Information
Y Cui, Department of Statistics and Data Science, National University of Singapore, 6 Science Drive 2, 117546 Singapore.
H Michael, Department of Mathematics and Statistics, University of Massachusetts, 710 N. Pleasant Street, Amherst, Massachusetts 01003, U.S.A.
F Tanser, Lincoln Institute for Health, University of Lincoln, Brayford Way, Brayford Pool, Lincoln LN6 7TS, U.K.
E Tchetgen Tchetgen, Department of Statistics and Data Science, The Wharton School, University of Pennsylvania, 265 South 37th Street, Philadelphia, Pennsylvania 19104, U.S.A.
Supplementary material
The Supplementary Material includes proofs of the theoretical results and additional simulation scenarios.
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