Abstract
This paper describes a targeted maximum likelihood estimator (TMLE) for the parameters of longitudinal static and dynamic marginal structural models. We consider a longitudinal data structure consisting of baseline covariates, time-dependent intervention nodes, intermediate time-dependent covariates, and a possibly time-dependent outcome. The intervention nodes at each time point can include a binary treatment as well as a right-censoring indicator. Given a class of dynamic or static interventions, a marginal structural model is used to model the mean of the intervention-specific counterfactual outcome as a function of the intervention, time point, and possibly a subset of baseline covariates. Because the true shape of this function is rarely known, the marginal structural model is used as a working model. The causal quantity of interest is defined as the projection of the true function onto this working model. Iterated conditional expectation double robust estimators for marginal structural model parameters were previously proposed by Robins (2000, 2002) and Bang and Robins (2005). Here we build on this work and present a pooled TMLE for the parameters of marginal structural working models. We compare this pooled estimator to a stratified TMLE (Schnitzer et al. 2014) that is based on estimating the intervention-specific mean separately for each intervention of interest. The performance of the pooled TMLE is compared to the performance of the stratified TMLE and the performance of inverse probability weighted (IPW) estimators using simulations. Concepts are illustrated using an example in which the aim is to estimate the causal effect of delayed switch following immunological failure of first line antiretroviral therapy among HIV-infected patients. Data from the International Epidemiological Databases to Evaluate AIDS, Southern Africa are analyzed to investigate this question using both TML and IPW estimators. Our results demonstrate practical advantages of the pooled TMLE over an IPW estimator for working marginal structural models for survival, as well as cases in which the pooled TMLE is superior to its stratified counterpart.
1 Introduction
Many studies aim to learn about the causal effects of longitudinal exposures or interventions using data in which these exposures are not randomly assigned. Specifically, consider a study in which baseline covariates, time-varying exposures or treatments, time-varying covariates, and an outcome of interest, such as death, are observed on a sample of subjects followed over time. The exposures of interest can both depend on past covariates and affect future covariates, as well as the outcome. Censoring may also occur, possibly in response to past treatment and covariates. Such data structures are ubiquitous in observational cohort studies. For example, a sample of HIV-infected patients might be followed longitudinally in clinic and data collected on antiretroviral prescriptions, determinants of prescription decisions including CD4+ T cell counts and plasma HIV RNA levels (viral loads), and vital status. Such data structures also occur in randomized trials when the exposure of interest is (non-random) compliance with a randomized exposure or includes non-randomized mediators of an exposure’s effect.
The causal effects of longitudinal exposures (as well as the effects of single time point exposures when the outcome is subject to censoring) can be formally defined by contrasting the distribution of a counterfactual outcome under different “interventions” to set the values of the exposure and censoring variables. For example, the counterfactual survival curve of HIV-infected subjects following immunological failure of antiretroviral therapy might be contrasted under a hypothetical intervention in which all subjects were switched immediately to a new antiretroviral regimen versus an intervention in which all subjects remained on their failing therapy [1]. In the presence of censoring due to losses to follow up, these counterfactuals of interest might be defined under a further intervention to prevent censoring. Interventions such as these, under which all subjects in a population are deterministically assigned the same vector of exposure and censoring decisions (for example, do not switch and remain under follow up) are referred to as “static regimes.”
More generally, counterfactuals can be defined under interventions that assign a treatment or exposure level to each subject at each time point based on that subject’s observed past. For example, counterfactual survival might be compared under interventions to switch all patients to second line antiretroviral therapy the first time their CD4+T cell count crosses a certain threshold, for some specified set of thresholds [2]. Such subject-responsive treatment strategies have been referred to as individualized treatment rules, adaptive treatment strategies, or “dynamic regimes” (see, for example, Robins [3]; Murphy et al. [4]; Hernan et al. [5]). Additional examples include strategies for deciding when to start antiretroviral therapy [6, 7] and strategies for modifying dose or drug choice based on prior response and adverse effects. Investigation of the effects of such dynamic regimes makes it possible to learn effective strategies for assigning an intervention based on a subject’s past and is thus relevant to any discipline that seeks to learn how best to use past information to make decisions that will optimize future outcomes.
The static and dynamic regimes described above are longitudinal – they involve interventions to set the value of multiple treatment and censoring variables over time. For example, counterfactual survival under no switch to second line therapy corresponds to a subject’s survival under an intervention to prevent a patient from switching at each time point from immunologic failure until death or the end of the study. A time-dependent causal dose–response curve, which plots the mean of the intervention-specific counterfactual outcome at time t as a function of the interventions through time t, can be used to summarize the effects of these longitudinal interventions. For example, a plot of the counterfactual survival probability as a function of time since immunologic failure, for a range of alternative CD4+ T cell count thresholds used to initiate a switch captures the effect of alternative switching strategies on survival.
Formal causal frameworks provide a tool to establish the conditions under which such causal dose–response curves can be identified from the observed data. Longitudinal static and dynamic regimes are often subject to time-dependent confounding – time-varying variables may confound the effect of future treatments while being affected by past treatment [8]. Traditional approaches to the identification of point treatment effects, which are based on selection of a single set of covariates for regression or stratification-based adjustment, break down when such time-dependent confounding is present. However, the mean counterfactual outcome under a longitudinal static or dynamic regime may still be identified under the appropriate sequential randomization and positivity assumptions (reviewed in Robins and Hernan [9]).
Under these assumptions, causal dose–response curves can be estimated by generating separate estimates of the mean counterfactual outcome for each time point and intervention (or regime) of interest. For example, one could generate separate estimates of the counterfactual survival curve for each CD4-based threshold for switching to second line therapy. In this manner, one obtains fits of the time-dependent causal dose–response curve for each of a range of possible thresholds, which together summarize how the mean counterfactual outcome at time t depends on the choice of threshold.
A number of estimators can be used to estimate intervention-specific mean counterfactual outcomes. These include inverse probability weighted (IPW) estimators (for example, [3, 5, 10]), “G-computation” estimators (typically based on parametric maximum likelihood estimation of the non-intervention components of the data generating process) (for example, [7, 11, 12]), augmented-IPW estimators (for example, [13–16, 31]), and targeted maximum likelihood (or minimum loss) estimators (TMLEs) (for example, [17, 18]). In particular, van der Laan and Gruber [19] combine the targeted maximum likelihood framework [20, 21] with important insights and the iterated conditional expectation estimators established in Robins [3, 29] and Bang and Robins [22].
Both the theoretical validity and the practical utility of these estimators rely, however, on reasonable support for each of the interventions of interest, both in the true data generating distribution and in the sample available for analysis. For example, in order to estimate how survival is affected by the threshold CD4 count used to initiate an antiretroviral treatment switch, a reasonable number of subjects must in fact switch at the time indicated by each threshold of interest. Without such support, estimators of the intervention-specific outcome will be ill-defined or extremely variable. Although one might respond to this challenge by creating coarsened versions of the desired regimes, so that sufficient subjects follow each coarsened version, such a method introduces bias and leaves open the question of how to choose an optimal degree of coarsening.
Since adequate support for every intervention of interest is often not available, Robins [23] introduced marginal structural models (MSMs) that pose parametric or small semiparametric models for the counterfactual conditional mean outcome as a function of the choice of intervention and time. For example, static MSMs have been used to summarize how the counterfactual hazard of death varies as a function of when antiretroviral therapy is initiated [24] and when an antiretroviral regimen is switched [25]. The extrapolation assumptions implicitly defined by non-saturated MSMs make it possible to estimate the coefficients of the model, and thereby the causal dose–response curve, even when few or no subjects follow some interventions of interest.
While MSMs were originally developed for static interventions [8, 10, 23, 24] they naturally generalize to classes of dynamic (or even more generally, stochastic) interventions as shown in van der Laan and Petersen [2] and Robins et al. [26]. Dynamic MSMs have been used, for example, to investigate how counterfactual hazard of death varies as a function of CD4+ T cell count threshold used to initiate antiretroviral therapy [6] or to switch antiretroviral therapy regimens [2]. Because the true shape of the causal dose–response curve is typically unknown, we have suggested that MSMs be used as working models. The target causal coefficients can then be defined by projecting the true causal dose–response curve onto this working model [20, 27].
The coefficients of both static and dynamic MSMs are frequently estimated using IPW estimators [2, 8, 10, 26]. These estimators have a number of attractive qualities: they can be intuitively understood, they are easy to implement, and they provide an influence curve-based approach to standard error estimation. However, IPW estimators also have substantial shortcomings. In particular, they are biased if the treatment mechanism used to construct the weights is estimated poorly (for example, using a misspecified parametric model). Further, IPW estimators are unstable in settings of strong confounding (near or partial positivity violations) and the resulting bias in both point and standard error estimates can result in poor inference (for a review of this issue see Petersen et al. [28]). Dynamic MSMs can exacerbate this problem, as the options for effective weight stabilization are limited [6, 26].
Asymptotically efficient and double robust augmented-IPW estimators of the estimand corresponding to longitudinal static MSM parameters were developed by Robins and Rotnitzky [14], Robins [13], Robins et al. [16]. These estimators are defined as a solution of an estimating equation, and as a result may be unstable due to failure to respect the global constraints implied by the model and the parameter. Robins [13, 29] and Bang and Robins [22] introduced an alternative double robust estimating equation-based estimator of longitudinal MSM parameters based on the key insight that both the statistical target parameter and the corresponding augmented-IPW estimating function (efficient influence curve) for MSMs on the intervention-specific mean can be represented as a series of iterated conditional expectations. In addition, they proposed a targeted sequential regression method to estimate the nuisance parameters of the augmented-IPW estimating equation. This innovative idea allowed construction of a double robust estimator that relies only on estimation of minimal nuisance parameters beyond the treatment mechanism.
In this paper, we describe a double robust substitution estimator of the parameters of a longitudinal marginal structural working model. The estimator presented incorporates the key insights and prior estimator of Robins [13, 29] and Bang and Robins [22] into the TMLE framework. Specifically, we expand on this prior work in several ways. We propose a TMLE for marginal structural working models for longitudinal dynamic regimes, possibly conditional on pre-treatment covariates. The TMLE described is defined as a substitution estimator rather than as solution to an estimating equation and incorporates data-adaptive/machine learning methods in generating initial fits of the sequential regressions. Finally, we further generalize the TMLE to apply to a larger class of parameters defined as arbitrary functions of intervention-specific means across a user-supplied class of interventions.
TMLE for the parameters of a MSM for “point treatment” problems, in which adjustment for a single set of covariates known not to be affected by the intervention of interest is sufficient to control for confounding, including history-adjusted MSMs, have been previously described [30, 31]. However, the parameter of a longitudinal MSM on the intervention-specific mean under sequential interventions subject to time-dependent confounding is identified as a distinct, and substantially more complex, estimand than the estimand corresponding to a point treatment MSM, and thus requires distinct estimators. An alternative TMLE for longitudinal static MSMs, which we refer to as a stratified TMLE, was described by Schnitzer et al. [32]. The stratified TMLE uses the longitudinal TMLE of van der Laan and Gruber [19] for the intervention-specific mean to estimate each of a set of static treatments and combines these estimates into a fit of the coefficients of a static longitudinal MSM on both survival and hazard functions. The stratified TMLE [32] resulted in substantially lower standard error estimates than an IPW estimator in an applied data analysis and naturally generalizes to dynamic MSMs. However, it remains vulnerable when there is insufficient support for some interventions of interest. In contrast, the TMLE we describe here pools over the set of dynamic or static interventions of interest as well as optionally over time when updating initial fits of the likelihood. It thus substantially relaxes the degree of data support required to remain an efficient double robust substitution estimator.
In summary, a large class of causal questions can be formally defined using static and dynamic longitudinal MSMs, and the parameters of these models can be identified from non-randomized data under well-studied assumptions. This article describes a TMLE that builds on the work of Robins [13, 29] and Bang and Robins [22] in order to directly target the coefficients of a marginal structural (working) model for a user-supplied class of longitudinal static or dynamic interventions. The theoretical properties of the pooled TMLE are presented, its implementation is reviewed, and its practical performance is compared to alternatives using both simulated and real data. R code [33] implementing the estimator and evaluating it in simulations is provided in online supplementary materials and as an open source R library ltmle [34].
1.1 Organization of paper
In Section 2, we define the observed data and a statistical model for its distribution. We then specify a non-parametric structural equation model for the process assumed to generate the observed data. We define counterfactual outcomes over time based on static or dynamic interventions on multiple treatment and censoring nodes in this system of structural equations. Our target causal quantity is defined using a marginal structural working model on the mean of these intervention-specific counterfactual outcomes at time t. The general case we present includes marginal structural working models on both the counterfactual survival and the hazard. We briefly review the assumptions under which this causal quantity is identified as a parameter of the observed data distribution. The statistical estimation problem is thus defined in terms of the statistical model and statistical target parameter.
Section 3 presents the TMLE defined by (a) representation of the statistical target parameter in terms of an iteratively defined set of conditional mean outcomes, (b) an initial estimator for the intervention mechanism and for these conditional means, (c) a submodel through this initial estimator and a loss function chosen so that the generalized score of the submodel with respect to this loss spans the efficient influence curve, (d) a corresponding updating algorithm that updates the initial estimator and iterates the updating till convergence, and (e) final evaluation of the TMLE as a plug-in estimator. We also present corresponding influence curve-based confidence intervals for our target parameter.
Section 4 illustrates the results presented in Section 3 using a simple three time point example and focusing on a marginal structural working model for counterfactual survival probability over time. This example is used to clarify understanding of notation and to provide a step-by-step overview of implementation of the pooled TMLE.
Section 5 compares the pooled TMLE described in this paper with alternative estimators for the parameters of longitudinal dynamic MSMs for survival. We provide a brief overview of the stratified TMLE [32], discuss scenarios in which each estimator may be expected to offer superior performance, and illustrate the breakdown of the stratified TMLE in a finite sample setting in which some interventions of interest have no support. As IPW estimators are currently the most common approach used to fit longitudinal dynamic MSMs, we also discuss two IPW estimators for these parameters.
Section 6 presents a simulation study in which the pooled TMLE is implemented for a marginal structural working model for survival at time t. Its performance is compared to IPW estimators and to the stratified TMLE for a simple data generating process and in a simulation designed to be similar to the data analysis presented in the following section, which includes time-dependent confounding and right censoring.
Section 7 presents the results of a data analysis investigating the effect of switching to second line therapy following immunologic failure of first line therapy using data from HIV-infected patients in the International Epidemiological Databases to Evaluate AIDS (IeDEA), Southern Africa. Throughout the paper, we illustrate notation and concepts using a simplified data structure based on this example.
Appendices contain a derivation of the efficient influence curve, further simulation details, an alternative TMLE, and reference table for notation. In online supplementary files, we present R code that implements the pooled TMLE, the stratified TMLE, and two IPW estimators for a marginal structural working model of survival. A corresponding publicly available R-package, ltmle, was released in May 2013 (http://cran.r-project.org/web/packages/ltmle/).
2 Definition of statistical estimation problem
Consider a longitudinal study in which the observed data structure O on a randomly sampled subject is coded as
where
Running example. Here and in subsequent sections, we illustrate notation using an example in which n i.i.d. HIV-infected subjects with immunological failure on first line therapy are sampled from some target population. Here
2.1 Statistical model
We use the notation
We use
The probability distribution
We consider a statistical model
2.2 Causal model and counterfactuals of interest
By specifying a structural causal model [35, 36] or equivalently, a system of non-parametric structural equations, it is assumed that each component of the observed longitudinal data structure (e.g.
and
in terms of a set of deterministic functions
To continue our HIV example, we might specify a causal model in which both time-varying CD4 count, death, and the decision to switch potentially depended on a subject’s entire observed past, as well as unmeasured factors. Alternatively, if we knew that switching decisions were made only in response to a subject’s most recent CD4 count and baseline covariates, the parent set of
This causal model represents a model
The intervention of interest might be static, with
Given a rule d, the counterfactual random variable
2.3 Marginal structural working model
Our causal quantity of interest is defined using a marginal structural working model to summarize how the mean counterfactual outcome at time t varies as a function of the intervention rule d, time point t, and possibly some baseline covariate V that is a function of the collection of all baseline covariates
We specify a working model
where
Such a
This equation can be replaced by
which corresponds with
In this case we have that
To be completely general, we will define our causal quantity of interest as a function f of
In addition to including the above example, this general formulation allows us to include marginal structural working models on continuous outcomes and on the intervention-specific hazard.
2.3.1 Choice of a weight function h ( d , t , V )
Unless one is willing to assume that the MSM
2.3.2 Running example
Continuing our HIV example, recall that static regimes are a special case of dynamic regimes and define the set of treatment rules of interest
The true time-dependent causal dose–response curve
where
2.4 Identifiability and definition of statistical target parameter
We assume the sequential randomization assumption [11]
(noting that weaker identifiability assumptions are also possible; see, for example, Robins and Hernan [9]). In our HIV example, the plausibility of this assumption would be strengthened by measuring all determinants of the decision to switch to second line therapy that also affect mortality via pathways other than switch time.
We further assume positivity, informally an assumption of support for each rule of interest across covariate histories compatible with that rule of interest. Specifically, for each
In our HIV example, in which
where
Let
The estimand
The causal identifiability assumptions put no restrictions on the probability distribution
The statistical estimation problem is now defined: We observe n i.i.d. copies
3 Pooled TMLE of working MSM for dynamic treatments and time-dependent outcome process
The TMLE algorithm starts out with defining the target parameter as a
An estimator of
Robins [13, 29] and Bang and Robins [22] reformulate the statistical target parameter and corresponding efficient influence curve for longitudinal MSMs on the intervention-specific mean as a series of iterated conditional expectations. For completeness, and to generalize to dynamic marginal structural working models possibly conditional on baseline covariates, as well as to general functions of the intervention-specific mean across a user-supplied class of interventions, we present this reformulation of the statistical target parameter below. The corresponding efficient influence curve is given in Appendix B. We will use the common notation
3.1 Reformulation of the statistical target parameter in terms of iteratively defined conditional means
For the case
where
where we define
To obtain
where we defined
The TMLE for the linear working model using the squared error loss function is obtained by simply redefining
In general, the above shows that we can represent
3.2 Estimation of intervention mechanism g 0
The log-likelihood loss function for
where
If there are certain variables in the
3.3 Loss functions and initial estimator of Q 0
We will alternate notation
This is an application of the log-likelihood loss function for the conditional mean of
In this loss function, the outcome
By summing over
for the whole
We will use the log-likelihood loss
3.4 Non-targeted substitution estimator
These loss functions imply a sequential regression methodology for fitting each of the required components of
Given the regression fit
The pooled TMLE presented below utilizes this same sequential regression algorithm and makes use of these initial fits of
3.5 Loss function and least favorable submodel that span the efficient influence curve
Recall that we use the notation
This parameter
This shows that
where
giving
In other words, the sum-loss function
and submodel
Consider also a submodel
3.6 Pooled TMLE
We now describe the TMLE algorithm based on the above choices of (1) the representation of
Recall that we define
The updating steps are implemented as follows: for each
and compute the corresponding update
Thus
using a data set pooled across
This defines the TMLE
The TMLE of
This plug-in estimator
An alternative pooled TMLE that only fits a single
3.7 Statistical inference for pooled TMLE
By construction, the TMLE solves the efficient influence curve equation
Specifically, if
As a consequence,
In the case that
4 Implementation of the pooled TMLE
The previous section reformulated the statistical parameter in terms of iteratively defined conditional means and described a pooled TMLE for this representation. In this section, we illustrate notation and implementation of this TMLE to estimate the parameters of a marginal structural working model on counterfactual survival over time.
4.1 The statistical estimation problem
We continue our motivating example, in which the goal is to learn the effect of switch time on survival. For illustration, focus on the two time point case where
Define the set of treatment rules of interest
The target causal parameter is defined as the projection of
Under the sequential randomization (2) and positivity (3) assumptions,
4.1.1 Reformulation of the statistical target parameter
Note that
while
4.2 Estimator implementation
We begin by describing implementation of a simple plug-in estimator of
4.2.1 Non-targeted substitution estimator
For each rule of interest
Fit a logistic regression of
Fit a logistic regression of the predicted values generated in the previous step on
For each rule of interest
The previous steps generated
We now describe how the pooled TMLE modifies this algorithm to update the initial estimator
4.2.2 Pooled TMLE
Estimate
Generate a vector
Fit a logistic regression of
For each subject,
For each subject
In our example,
Stack these matrices to create a matrix with
Among those subjects still alive at the previous time point (
Generate
For subjects with
Generate a vector
Fit a logistic regression of
For each subject
The following
Stack these matrices to create a matrix with
Fit a pooled logistic regression of
Generate
This gives an updated vector
Generate a vector
Fit a logistic regression of
For each subject,
For each subject
The following
Stack these matrices to create a matrix with
Fit a pooled logistic regression of
Generate
This gives an updated vector
The previous steps generated
5 Comparison with alternative estimators
In this section we compare the TMLE described with several alternative estimators available for dynamic MSMs for survival: non-targeted substitution estimators, IPW estimators, and the stratified TMLE of Schnitzer et al. [32].
5.1 Non-targeted substitution estimator
The consistency of non-targeted substitution estimators of
Correct a priori specification of parametric models for
5.2 Inverse probability weighted estimators
The IPW estimator described in van der Laan and Petersen [2], Robins et al. [26] is commonly used to estimate the parameters of a dynamic MSM. In brief, this estimator is implemented by creating one data line for each subject i, for each t, and for each d for which
The parameter mapping presented here for the TMLE suggests an alternative IPW estimator for dynamic MSM – namely, implement an IPW estimator for
or its bounded counterpart:
Robins and Rotnitzky [14].
The consistency of both IPW estimators relies on having a consistent estimator
5.3 Stratified targeted maximum likelihood estimator
Similar to the pooled TMLE, the stratified TMLE [32] also relies on reformulating the statistical target parameter in terms of iteratively defined conditional means and updating initial fits of these conditional means using covariates that are functions of an estimator
Let
The pooled and stratified estimators may nonetheless differ in both their asymptotic and finite sample performance. The stratified TMLE uses a more saturated model when updating
On the other hand, in some cases, it is no longer clear how to implement the stratified TMLE. For example, the target parameter may be defined using a MSM
In cases where V is discrete and there is support for some but not all rules
5.3.1 Numerical illustration
We use a simple simulation to illustrate a setting in which the positivity assumption holds, but many rules of interest have no support in a given finite sample. In this setting, the stratified estimator will be biased if the initial estimator
We implemented a simulation with observed data consisting of n i.i.d. copies of
The target parameter was defined as the projection of
Stratified and pooled TMLEs for
Breakdown of stratified TMLE when some rules
| Pooled TMLE | Stratified TMLE | |
| Bias | ||
| 0.1674 | −0.9346 | |
| −0.0563 | 0.2039 | |
| Bias/SE | ||
| 0.1821 | −2.3037 | |
| −0.2326 | 2.0955 | |
| Variance | ||
| 0.8451 | 0.1646 | |
| 0.0586 | 0.0095 | |
| MSE | ||
| 0.8714 | 1.0377 | |
| 0.0616 | 0.0510 | |
| 95% confidence interval coverage | ||
| 88% | 88% | |
| 22% | 33% |
This breakdown of the stratified TMLE in settings with no support for some rules of interest will not occur if the function h is chosen to give a weight of 0 to any rule (or more generally, any
6 Simulation study
6.1 Overview
In this section, we investigate the relative performance of the pooled TMLE, stratified TMLE, and IPW estimators for the parameters of a marginal structural working model. For each candidate estimator, we report bias, variance, MSE, and 95% confidence interval coverage estimates based on influence curve variance estimators. We note that our influence curve-based estimators assume the weight function
6.2 Simulation 1: baseline confounding only
6.2.1 Data generating process
We implemented a simulation with observed data consisting of n i.i.d. copies of
In order to investigate the impact of decreasing support in the data (practical violations or near violations of the positivity assumption), we considered two versions of this data generating process, with
6.2.2 Target parameter
The target parameter was defined as the projection of
6.2.4 Estimators
This is a static point treatment problem, and thus a number of additional estimators are available. However, we use this as a special case of longitudinal dynamic MSMs and investigate the relative performance of three estimators described in Section 5: the pooled TMLE, the stratified TMLE, and the standard IPW estimator. All estimators were implemented using two estimators of
6.2.5 Results
Results for Simulation 1a are shown in Table 2. When both
Results for Simulation 1b, with both
Simulation 1a
| g and Q correct | g correct and Q incorrect | g incorrect and Q correct | |||||||
| Pl TMLE | Str TMLE | IPW | Pl TMLE | Str TMLE | IPW | Pl TMLE | Str TMLE | IPW | |
| Bias | |||||||||
| −0.0047 | − | −0.0260 | − | −0.0083 | − | −0.4578 | |||
| −0.0038 | −0.0038 | ||||||||
| Bias/SE | |||||||||
| −0.0258 | − | −0.1450 | − | −0.0485 | − | −2.4870 | |||
| −0.0336 | −0.0336 | ||||||||
| Variance | |||||||||
| MSE | |||||||||
| 95% Cl Coverage | |||||||||
Simulation 1b
| Pooled TMLE | Stratified TMLE | IPW | |
| Bias | |||
| −0.0025 | − | −0.0486 | |
| −0.0007 | |||
| Bias/SE | |||
| −0.0092 | − | −0.1618 | |
| −0.0103 | |||
| Variance | |||
| MSE | |||
| 95% Cl coverage | |||
6.3 Simulation 2: resembling data analysis
In this simulation, we used a data generating process designed to resemble the data analysis presented in the following section, in which the goal was to investigate the effect of delayed switch to a new antiretroviral regimen on mortality among HIV-infected patients who have failed first line therapy. The data generating process thus contains both baseline and time-dependent confounders of a longitudinal binary treatment (time to switch), a repeated measures binary outcome (survival over time), and informative right censoring due to two causes (database closure and loss to follow up).
6.3.1 Data generating process
We implemented a simulation with observed data consisting of n i.i.d. copies of
for
Drawing a time updated CD4 count
Determining censoring due to database closure
If still uncensored, determining censoring due to loss to follow up
If still uncensored and not yet switched, determining switching using a Bernoulli trial with probability dependent on W and prior CD4 count.
Determining death using a Bernoulli trial with probability dependent on W, prior CD4 counts and regimen status
6.3.2 Target parameter
The target parameter was defined as the projection of the counterfactual survival curve for each switch time,
where
6.3.3 Estimators
We implemented the pooled TMLE, the stratified TMLE, the IPW estimator based on estimating
The intervention rules
All estimators were implemented using an estimator
6.3.4 Results
Results for Simulation 2 are shown in Table 4. In this simulation, both the pooled and the stratified TMLEs were essentially unbiased for all coefficients, and the two TMLEs had comparable MSEs. Both TMLEs exhibited less than nominal 95% confidence interval coverage when using influence curve-based variance estimators. The anti-conservative performance of the influence curve-based variance estimator is likely due to the presence of practical positivity violations and relatively rare outcomes; the fact that the weight function was treated as known may also make a small contribution. Further work is needed to develop improved diagnostics and variance estimators in these settings.
Simulation 2: resembling data analysis
| Pl TMLE | Str TMLE | Stan IPW | Str IPW | |
| Bias | ||||
| −0.0239 | − | |||
| −0.0041 | − | |||
| −0.0006 | ||||
| Bias/SE | ||||
| −0.0867 | − | |||
| −0.0849 | − | |||
| −0.0096 | ||||
| Variance | ||||
| MSE | ||||
| 95% CI coverage | ||||
In contrast, both IPW estimators were substantially biased for
7 Data analysis
We analyzed data from the International Epidemiological Databases – Southern Africa in order to investigate the effect of switching to second line therapy on mortality among HIV-infected patients with immunological failure on first line antiretroviral therapy. The data set and clinical question are described in detail in Gsponer et al. [1]. In brief, data were drawn from clinical care facilities in Zambia and Malawi, in which HIV-infected patients were followed longitudinally in clinic and data were collected on baseline demographic and clinical variables (sex, age, and baseline disease stage), time-varying CD4 count, and time-varying treatment, summarized here as switch to second line therapy. Death was independently reported. The 2,627 subjects meeting WHO immunological failure criteria were included in the current analysis beginning at time of immunologic failure. Following common practice and prior analysis, time was discretized into 3-month intervals; time updated CD4 count was coded such that CD4 count for an interval preceded switching decisions in that interval. Data on a subject were censored at time of database closure or after four consecutive intervals without clinical contact.
The data structure and target parameter were identical to those described in Simulation 2, with
Results are given in Table 5. The IPW estimates for the effect of switching on mortality (
The poor coverage observed in Simulation 2, despite absence of bias for the TMLEs, suggests that the influence curve-based variance estimators may be systematically underestimating the true variance in this analysis. While the non-parametric bootstrap offers an alternative approach, it is not expected to resolve the challenge of anti-conservative variance estimation in the setting of practical positivity violations. Intuitively, rare treatment/covariate combinations, despite being theoretically possible, may simply not occur in a given finite sample and as a result, the corresponding extreme weights implied by these combinations will not occur. Because the non-parametric bootstrap resamples from the same finite sample, it fails to address the underlying problem. Indeed, the bootstrap-based confidence intervals in the data analysis were slightly smaller than confidence intervals based on the influence curve. Thus in this realistic setting of rare outcomes and moderately strong confounding, our results caution against reliance on either approach to variance estimation, for either IPW or TML estimators, and suggest that additional work developing robust variance estimators in this setting is urgently needed.
Data analysis
| Estimate | |||||
| Pooled TMLE | |||||
| –4.8291 | – | –4.4348 | – | –4.5436 | |
| –0.1246 | – | –0.0217 | – | –0.0496 | |
| Stratified TMLE | |||||
| –4.8382 | – | –4.4407 | – | –4.5790 | |
| –0.1262 | – | –0.0221 | – | ||
| Standard IPTW | |||||
| –4.9166 | – | –4.5196 | – | –4.6413 | |
| –0.0696 | –0.0575 | ||||
| Stratified IPTW | |||||
| –5.0811 | – | –4.6601 | – | –4.8252 | |
| –0.0539 | –0.0374 |
In addition to the issues raised above, limitations of the analysis include the potential for unmeasured confounding by factors such as unmeasured health status and adherence, as well as bias due to incomplete health reporting, resulting in censoring due to loss to follow up that directly depends on death [43, 44]. These results also do not contradict the previous published results of Gsponer et al. [1], which found a protective effect of switching using a static IPW estimator for a hazard MSM; such an IPW estimator might perform substantially better than the dynamic IPW estimator for a survival MSM implemented here.
8 Discussion
In summary, we have presented a pooled TMLE for the parameters of static and dynamic longitudinal MSMs that builds on prior work by Robins [13, 29] and Bang and Robins [22]. We evaluated the performance of this estimator using simulated data and applied it, together with alternatives, in a data analysis. Both theory and simulations suggest settings in which the pooled TMLE offers advantages over alternative estimators. Software implementing this estimator, together with competitors, is included in supplementary files and is publicly available as part of R library ltmle (http://cran.r-project.org/web/packages/ltmle/).
The pooled TMLE presented in this paper, together with corresponding open source software, provides a new tool for estimation of the parameters of static or dynamic MSMs. It has clear theoretical advantages over available alternatives. Unlike IPW and augmented-IPW estimators, it is a substitution estimator. Unlike IPW estimators, it is double robust and asymptotically efficient depending on the initial estimators of g and Q. Unlike the previously proposed stratified TMLE, it does not require support in the data for every rule of interest and remains defined in the case that the target parameter is defined using a marginal structural working model conditional on a continuous baseline covariate.
In settings where some subset of discrete intervention rules has adequate support in a given sample, an alternative approach is to compare only this subset of rules [2]. However, in many settings smoothing over a large number of poorly supported rules is appropriate. For example, for a set of rules indexed by a continuous or multiple level ordered variable, smoothing over this variable provides a way to define a causal effect of interest despite inadequate support to estimate the counterfactual outcome under any of these rules individually.
The TMLE presented in this paper was developed for a causal model in which the non-intervention variables may be a function of the entire observed past (
Our simulations suggest that both stratified and pooled TMLEs may outperform both the IPW estimator typically used for dynamic MSMs, as well as an alternative “stratified” IPW estimator, in some settings with sparse data/near positivity violations. However, further work is needed to confirm this preliminary observation. Although the theory in this paper was developed for the general case of dynamic MSMs, including models of the time-specific hazard or survival functions, we focused our software implementation, examples, and simulations on MSMs for survival. The practical performance of the pooled TMLE relative to alternative estimators also remains to be investigated for the case of static and dynamic MSMs on the hazard. It also remains to implement and evaluate the relative performance of the alternative pooled TMLE described in Appendix C, in which the updating step pools not only over all rules of interest but also over all time points. Finally, the relative performance of the TMLE compared to double robust efficient estimating equation-based estimators for longitudinal MSM parameters, including those of Robins [13, 29] and Bang and Robins [22], remains to be evaluated.
Importantly, our simulations and data analysis illustrate the need for improved variance estimators for both TMLE and IPW in settings with moderately strong confounding, multiple time points, and relatively rare outcomes. Improved approaches to variance estimation and valid inference, as well as appropriate diagnostics to warn applied practitioners of settings in which coverage is likely to be poor, are crucial research priorities.
Acknowledgments
This work was supported by NIH Grant #U01AI069924 (NIAID, NICHD, and NCI) (PIs: Egger and Davies), the Doris Duke Charitable Foundation Grant #2011042 (PI: Petersen), and NIH Grant #R01AI074345-06 (NIAID) (PI: van der Laan).
Appendix A
Notation
Partial list of notation
| Variable | Description |
| Observed data | |
| covariates at time t | |
| outcome at time t | |
| intervention at time t | |
| treatment at time t | |
| right censoring at time t | |
| history of intervention nodes | |
| history of non-intervention nodes | |
| observed data structure | |
| parents of non-intervention nodes | |
| parents of intervention nodes | |
| Counterfactuals | |
| set of dynamic intervention rules | |
| intervention rule | |
| counterfactual L under rule d | |
| counterfactual Y under rule d | |
| Statistical counterparts | |
| random variable with distribution | |
| equal to | |
| outcome-component of | |
| Marginal structural working model | |
| working model for | |
| index set of time points (often | |
| V | “effect modifier” of interest, a function of |
| Distributions | |
| distribution of | |
| distribution of O | |
| distribution of counterfactual | |
| G-computation formula for post-intervention distribution of L under rule d equal to | |
| distribution of | |
| non-intervention factor of | |
| distribution of | |
| intervention mechanism factor of | |
| empirical distribution of | |
| Models | |
| SCM for | |
| statistical model for | |
| Functions and parameters | |
| causal target parameter mapping | |
| causal target parameter value (causal quantity) | |
| statistical target parameter mapping | |
| target parameter value equals | |
| structural equation | |
| structural equation | |
| dynamic intervention | |
| h | weight function |
| Efficient influence curve | |
| efficient influence curve of | |
| Least favorable submodels | |
| submodel through | |
| Loss functions | |
| loss function for | |
| Others | |
| Q | redefined as |
| Estimators | |
| estimator of | |
| initial estimator of | |
| initial estimator of | |
| updated estimator of | |
| TMLE of |
Appendix B
The efficient influence curve of the statistical target parameter
We have the following theorem for a general parameter
Theorem 1Consider a parameter
Then, the efficient influence curve of
(appendix A3, van der Laan and Rose [20]). By the delta-method, the efficient influence curve of
(appendix A3, van der Laan and Rose [20]). This completes the proof. ⃞
In order to determine the partial derivative of the function f and
In particular,
and
We can now apply our general Theorem 1 to the example with
The same equation applies for the linear working MSM but with the other definition of
Note that
and
This proves the following corollary [13, 22, 29].
Corollary 1Consider the target parameter
The efficient influence curve is double robust. In other words we have that
Appendix C
An alternative pooled TMLE that only fits a single ∈
The TMLE described in the main text relies on a separate
where the nuisance parameters of the loss function are estimated with the initial estimator
and the resulting update
Appendix D
Supplementary material, Simulation 2
Overview
Below we describe the true data generating process used in Simulation 2 in greater detail. We also provide a summary table comparing the support and number of events over time in the simulated data and in the real data set it was designed to resemble.
In our presentation of the simulation below we have altered our notation slightly from that presented in the paper to match notation in the accompanying R code. Specifically,
Further, the data generating process included a non-monotone monitoring process (denoted
Data generating process
Data were generated for a given individual according to the following process, where
Tables 9 and 10 compare the number of patients in the simulated data and actual data who are uncensored and following a given regime. In the data analysis, for example, in the first 3-month interval following immunologic failure (time = 1) there were no patient deaths among the 137 uncensored patients who switched immediately (switch time = 0) and 13 deaths among the 2,285 uncensored patients who did not switch immediately (switch time = 1). In the second 3-month interval following immunologic failure (time = 2), there was one patient death among the 120 uncensored patients who switched immediately (switch time = 0), no patient deaths among the 88 uncensored patients who switched during the first 3-month interval (switch time = 1) and 8 deaths among the 1,962 uncensored patients who did not switch immediately or during the first 3-month interval (switch time = 2).
Events in data analysis
| Time | Switch time | ||||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 0 | 13 | |||||||||
| 2 | 1 | 0 | 8 | ||||||||
| 3 | 0 | 1 | 0 | 11 | |||||||
| 4 | 0 | 0 | 0 | 0 | 5 | ||||||
| 5 | 1 | 0 | 0 | 0 | 0 | 3 | |||||
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | ||||
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |||
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | ||
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
Events in simulated data
| Time | Switch time | ||||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 0 | 9 | |||||||||
| 2 | 0 | 1 | 10 | ||||||||
| 3 | 0 | 0 | 1 | 9 | |||||||
| 4 | 0 | 0 | 0 | 1 | 7 | ||||||
| 5 | 0 | 0 | 0 | 0 | 1 | 5 | |||||
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | ||||
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | |||
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | ||
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Support (# uncensored and following rule) in data analysis
| Time | Switch time | ||||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 137 | 2,285 | |||||||||
| 2 | 120 | 88 | 1,962 | ||||||||
| 3 | 104 | 80 | 71 | 1,703 | |||||||
| 4 | 88 | 68 | 67 | 54 | 1,453 | ||||||
| 5 | 77 | 59 | 57 | 51 | 61 | 1,184 | |||||
| 6 | 54 | 44 | 50 | 48 | 57 | 57 | 951 | ||||
| 7 | 40 | 37 | 43 | 48 | 54 | 55 | 51 | 760 | |||
| 8 | 29 | 33 | 40 | 46 | 50 | 51 | 51 | 49 | 602 | ||
| 9 | 17 | 29 | 34 | 41 | 44 | 48 | 51 | 49 | 50 | 508 | |
| 10 | 10 | 28 | 34 | 39 | 44 | 46 | 51 | 48 | 49 | 53 | 432 |
Support (# uncensored and following rule) in simulated data
| Time | Switch time | ||||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 121 | 2,229 | |||||||||
| 2 | 107 | 118 | 1,890 | ||||||||
| 3 | 95 | 107 | 131 | 1,582 | |||||||
| 4 | 82 | 94 | 120 | 127 | 1,237 | ||||||
| 5 | 70 | 83 | 110 | 117 | 112 | 992 | |||||
| 6 | 61 | 74 | 97 | 108 | 102 | 101 | 798 | ||||
| 7 | 53 | 65 | 88 | 98 | 96 | 95 | 89 | 645 | |||
| 8 | 47 | 58 | 80 | 91 | 90 | 90 | 85 | 82 | 527 | ||
| 9 | 40 | 51 | 72 | 83 | 83 | 84 | 82 | 79 | 76 | 433 | |
| 10 | 35 | 47 | 65 | 77 | 77 | 79 | 76 | 75 | 74 | 72 | 358 |
References
1. GsponerT, PetersenM, EggerM, PhiridS, MaathuiseM, BoulleA, et al., and O. Keiser for IeDEA Southern Africa. The causal effect of switching to second line ART in programmes without access to routine viral load monitoring. AIDS2012;26:57–65.10.1097/QAD.0b013e32834e1b5fSearch in Google Scholar
2. van der LaanMJ, PetersenML. Causal effect models for realistic individualized treatment and intention to treat rules. Int J Biostat2007;3(1):Article 3.10.2202/1557-4679.1022Search in Google Scholar
3. RobinsJM.Analytic methods for estimating HIV treatment and cofactor effects. In: OstrowDG, KesslerR, editors. Methodological issues of AIDS Mental Health Research. New York: Plenum Publishing, 1993;213–90.Search in Google Scholar
4. MurphySA, van der LaanMJ, RobinsJM. Marginal mean models for dynamic regimes. J Am Stat Assoc2001;960:1410–23.10.1198/016214501753382327Search in Google Scholar
5. HernanMA, LanoyE, CostagliolaD, RobinsJM. Comparison of dynamic treatment regimes via inverse probability weighting. Basic Clin Pharmacol2006;98:237–42.10.1111/j.1742-7843.2006.pto_329.xSearch in Google Scholar
6. CainLE, RobinsJM, LanoyE, LoganR, CostagliolaD, HernnMA. When to start treatment? A systematic approach to the comparison of dynamic regimes using observational data. Int J Biostat2010;6(2):Article 18.10.2202/1557-4679.1212Search in Google Scholar
7. SchomakerM, EggerM, NdiranguJ, PhiriS, MoultrieH, TechnauK, et al.. When to start antiretroviral therapy in children aged 2–5 years: A collaborative causal modelling analysis of cohort studies from Southern Africa. PLoS Med2013;100:e1001555.Search in Google Scholar
8. RobinsJM, HernanMA, BrumbackB. Marginal structural models and causal inference in epidemiology. Epidemiology2000;110:550–60.10.1097/00001648-200009000-00011Search in Google Scholar
9. RobinsJM, HernanMA. Estimation of the causal effects of time-varying exposures. In: FitzmauriceG, DavidianMVerbekeMG, MolenberghsG, editors. Advances in longitudinal data analysis. New York: Chapman and Hall/CRC Press, 2009:553–599.Search in Google Scholar
10. RobinsJM. Marginal structural models versus structural nested models as tools for causal inference. In: HalloranME, BerryD, editors. Statistical models in epidemiology, the environment, and clinical trials. New York: Springer, 2000:95–133.Search in Google Scholar
11. RobinsJM. A new approach to causal inference in mortality studies with sustained exposure periods – application to control of the healthy worker survivor effect. Math Model1986;7:1393–512.10.1016/0270-0255(86)90088-6Search in Google Scholar
12. TaubmanSL, RobinsJM, MittlemanMA, HernanMA. Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol2009;380:1599–611.10.1093/ije/dyp192Search in Google Scholar PubMed PubMed Central
13. RobinsJM. Robust estimation in sequentially ignorable missing data and causal inference models. In: Proceedings of the American Statistical Association on Bayesian Statistical Science, 1999, 2000:6–10.Search in Google Scholar
14. RobinsJM, RotnitzkyA. Recovery of information and adjustment for dependent censoring using surrogate markers. In: Nicholas P.Jewell, KlausDietz, Vernon T.Farewell, editors, AIDS epidemiology. Boston: Birkhäuser, 1992:297–331.Search in Google Scholar
15. RobinsJM, RotnitzkyA. Comment on the Bickel and Kwon article, “Inference for semiparametric models: some questions and an answer”. Stat Sin2001;110:920–36.Search in Google Scholar
16. RobinsJM, RotnitzkyA, van der LaanMJ. Comment on “On profile likelihood”. J Am Stat Assoc2000;450:431–5.10.2307/2669391Search in Google Scholar
17. RosenblumM, van der LaanMJ. Simple examples of estimating causal effects using targeted maximum likelihood estimation. UC Berkeley Division of Biostatistics Working Paper Series, Working Paper 209. Available at: http://biostats.bepress.com/jhubiostat/paper209, 2011.Search in Google Scholar
18. StitelmanOM, De GruttolaV, van der LaanMJ. A general implementation of TMLE for longitudinal data applied to causal inference in survival analysis. UC Berkeley Division of Biostatistics Working Paper Series, Working Paper 281. Available at: http://biostats.bepress.com/ucbbiostat/paper281, 2011.Search in Google Scholar
19. van der LaanMJ, GruberS. Targeted minimum loss based estimation of causal effects of multiple time point interventions. Int J Biostat2012;8(1):Article 8.10.1515/1557-4679.1370Search in Google Scholar PubMed
20. van der LaanMJ, RoseS. Targeted learning: causal inference for observational and experimental data. Berlin/Heidelberg/New York: Springer, 2011.10.1007/978-1-4419-9782-1Search in Google Scholar
21. van der LaanMJ, RubinD. Targeted maximum likelihood learning. Int J Biostat2006;6(2):Article 2.10.2202/1557-4679.1043Search in Google Scholar
22 BangH, RobinsJM. Doubly robust estimation in missing data and causal inference models. Biometrics2005;61:962–72.10.1111/j.1541-0420.2005.00377.xSearch in Google Scholar PubMed
23. RobinsJM. Marginal structural models. In: 1997 Proceedings of the American Statistical Association, Section on Bayesian Statistical Science, 1998:1–10.Search in Google Scholar
24. HernanMA, BrumbackB, RobinsJM. Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology2000;110:561–70.10.1097/00001648-200009000-00012Search in Google Scholar PubMed
25. PetersenML, van der LaanMJ, NapravnikS, EronJ, MooreR, DeeksS. Long term consequences of the delay between virologic failure of highly active antiretroviral therapy and regimen modification: a prospective cohort study. AIDS2008;22:2097–106.10.1097/QAD.0b013e32830f97e2Search in Google Scholar PubMed
26. RobinsJM, OrellanaL, RotnitzkyA. Estimation and extrapolation of optimal treatment and testing strategies. Stat Med2008;27:4678–721.10.1002/sim.3301Search in Google Scholar PubMed
27. NeugebauerR, van der LaanM. Nonparametric causal effects based on marginal structural models. J Stat Plann Inference2007;137:419–34.10.1016/j.jspi.2005.12.008Search in Google Scholar
28. PetersenML, PorterKE, GruberS, WangY, van der LaanMJ. Diagnosing and responding to violations in the positivity assumption. Stat Methods Med Res2012;21:31–54.10.1177/0962280210386207Search in Google Scholar PubMed PubMed Central
29. RobinsJM. Commentary on “using inverse weighting and predictive inference to estimate the effects of time-varying treatments on the discrete-time hazard. Stat Med2002;210:1663–80.10.1002/sim.1110Search in Google Scholar
30. RosenblumM, van der LaanMJ. Targeted maximum likelihood estimation of the parameter of a marginal structural model. Int J Biostat2010;6(2):Article 19.10.2202/1557-4679.1238Search in Google Scholar PubMed PubMed Central
31. ScharfsteinDO, RotnitzkyA, RobinsJM. Adjusting for nonignorable drop-out using semiparametric nonresponse models (with discussion and rejoinder). J Am Stat Assoc1999;940:1096–1120 (1121–1146).Search in Google Scholar
32. SchnitzerME, MoodieEM, van der LaanMJ, PlattRW, KleinMB. Modeling the impact of hepatitis C viral clearance on end-stage liver disease in an HIV co-infected cohort with targeted maximum likelihood estimation. Biometrics2014;70(1):144–52.10.1111/biom.12105Search in Google Scholar PubMed PubMed Central
33. R Core Team. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2013. Available at: http://www.R-project.org/. ISBN 3-900051-07-0.Search in Google Scholar
34. SchwabJ, LendleS, PetersenM, van der LaanM. LTMLE: longitudinal targeted maximum likelihood estimation, 2013. Available at: http://cran.r-project.org/web/packages/ltmle/. R package version 0.9.3.Search in Google Scholar
35. PearlJ. Causal diagrams for empirical research. Biometrika1995;82:669–710.10.1093/biomet/82.4.702Search in Google Scholar
36. PearlJ. Causality: models, reasoning, and inference. Cambridge: Cambridge University Press, 2000.Search in Google Scholar
37. BickelPJ, KlaassenCA, RitovY, WellnerJA. Efficient and adaptive estimation for semiparametric models. Baltimore, MD: Johns Hopkins University Press, 1993.Search in Google Scholar
38. van der LaanMJ, RobinsJM. Unified methods for censored longitudinal data and causality. Berlin/Heidelberg/New York: Springer, 2003.10.1007/978-0-387-21700-0Search in Google Scholar
39. van der LaanMJ, PolleyE, HubbardA. Super learner. Stat Appl Genet Mol Biol2007;6:Article 25.10.2202/1544-6115.1309Search in Google Scholar PubMed
40. van der LaanMJ. Statistical inference when using data adaptive estimators of nuisance parameters. UC Berkeley Division of Biostatistics Working Paper Series, Working Paper 302, 2012. Available at: http://www.bepress.com/ucbbiostat/paper302.Search in Google Scholar
41. YoungJG, CainLE, RobinsJM, OÕReillyEJ, HernanMA. Comparative effectiveness of dynamic treatment regimes: an application of the parametric g-formula. Stat Biosci2011;30:119–43.10.1007/s12561-011-9040-7Search in Google Scholar PubMed PubMed Central
42. PicciottoS, HernanM, PageJ, YoungJ, RobinsJ. Structural nested cumulative failure time models to estimate the effects of interventions. J Am Stat Assoc2012;1070:866–900.10.1080/01621459.2012.682532Search in Google Scholar PubMed PubMed Central
43. GengEH, GliddenD, BangsbergDR, BwanaMB, MusinguziN, MetcalfeJ, et al. Causal framework for understanding the effect of losses to follow-up on epidemiologic analyses in clinic based cohorts: the case of HIV-infected patients on antiretroviral therapy in Africa. Am J Epidemiol2012;175:1080–7.10.1093/aje/kwr444Search in Google Scholar PubMed PubMed Central
44. SchomakerM, GsponerT, EstillJ, FoxM, BoulleA. Non-ignorable loss to follow-up: correcting mortality estimates based on additional outcome ascertainment. Stat Med2014;330:129–42.10.1002/sim.5912Search in Google Scholar PubMed PubMed Central
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