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Int J Biostat. 2011 January 1; 7(1): Article 33.
Published online 2011 September 2. doi:  10.2202/1557-4679.1351
PMCID: PMC3204669

On Causal Mediation Analysis with a Survival Outcome

Abstract

Suppose that having established a marginal total effect of a point exposure on a time-to-event outcome, an investigator wishes to decompose this effect into its direct and indirect pathways, also known as natural direct and indirect effects, mediated by a variable known to occur after the exposure and prior to the outcome. This paper proposes a theory of estimation of natural direct and indirect effects in two important semiparametric models for a failure time outcome. The underlying survival model for the marginal total effect and thus for the direct and indirect effects, can either be a marginal structural Cox proportional hazards model, or a marginal structural additive hazards model. The proposed theory delivers new estimators for mediation analysis in each of these models, with appealing robustness properties. Specifically, in order to guarantee ignorability with respect to the exposure and mediator variables, the approach, which is multiply robust, allows the investigator to use several flexible working models to adjust for confounding by a large number of pre-exposure variables. Multiple robustness is appealing because it only requires a subset of working models to be correct for consistency; furthermore, the analyst need not know which subset of working models is in fact correct to report valid inferences. Finally, a novel semiparametric sensitivity analysis technique is developed for each of these models, to assess the impact on inference, of a violation of the assumption of ignorability of the mediator.

Keywords: natural direct effect, natural indirect effect, Cox proportional hazards model, additive hazards model, multiple robustness

1. Introduction

Suppose that, upon establishing a marginal total effect of a point exposure on an outcome of interest, an investigator wishes to decompose this effect into its direct and indirect pathways, also known as natural or pure direct and indirect effects, mediated by a variable known to occur after the exposure and prior to the outcome (Robins and Greenland, 1992, Pearl, 2001). The literature on statistical methods for causal mediation analysis has blossomed in recent years with new results on identification of direct and indirect effects, and a number of novel techniques for obtaining statistical inferences about these effects (van der Laan and Petersen, 2005, VandeWeele, 2009, Imai 2010a,b, Lange and Hansen, 2011, VanderWeele, 2011, Tchetgen Tchetgen and Shpitser, 2011a,b). With the exception of Tein and Mackinnon (2003), and the recent paper by Lange and Hansen (2011) and the accompanying commentary by VanderWeele (2011), who consider a survival context, the existing literature on causal mediation analysis has largely focused on structural models for a mean effect. The current paper aims to further develop methodology for mediation analysis for survival data. In fact, we propose a general theory of estimation of natural direct and indirect effects for two important semiparametric models of a failure time outcome. We assume that the underlying survival model for the marginal total effect and thus for the direct and indirect effects, can either be a marginal structural Cox proportional hazards model as in Robins (1998), or a marginal structural additive hazards model. Lange and Hansen (2011) were the first to consider the use of the additive hazards model for causal mediation analysis in a survival context; whereas Tein and Mackinnon (2003) and VanderWeele (2011) also consider the use of a Cox proportional hazards model for mediation analysis.

The current paper aims to extend these existing results in several important ways. Thus, we develop some new semiparametric estimators of direct and indirect effects for each of these models, with appealing robustness properties. Specifically, the proposed approach which is so-called multiply robust, allows the investigator to use several flexible working models in order to adjust for a possibly large number of pre-exposure confounders for both exposure and mediating variables. Multiple robustness is appealing because it only requires a subset of these working models to be correct for unbiasedness (more precisely for consistency); furthermore, the analyst need not know which subset of working models is in fact correct to report valid inferences. Finally, in this paper, a novel semiparametric sensitivity analysis technique is also developed for each model, to assess the impact on mediation inferences, of a violation of the assumption of ignorability of the mediating variable. This is an important contribution in its own right, particularly because no methodology currently exist, for performing a sensitivity analysis for unmeasured confounding in the current survival context.

The theory developed in this paper closely parallels similar theory recently proposed by Tchetgen Tchetgen and Shpitser (2011a,b), who were the first to formalize the semipararametric theory for making multiply robust inferences about natural direct and indirect effect of the exposure on the mean of the outcome. In section 2, we adapt their previous results to obtain multiply robust inferences about natural direct and indirect effects of a binary exposure on the marginal survival curve in the presence of confounding and right censoring. Because the previous theory does not directly apply to semiparametric regression models for survival data, new methodology is developed in Section 3 for obtaining multiply robust inferences about natural direct and indirect effects under a Cox proportional hazards model and an additive hazards model. Then, we develop similar multiply robust estimators of natural indirect effects for each model. Finally, Section 4 gives new results on semiparametric sensitivity analysis in a survival context.

First we introduce some notation. Throughout, we suppose independent and identically distributed data on a vector (E, M, X, T*, Δ) is collected for n subjects. Here, E is the binary exposure variable, M is a mediator variable with support S, known to occur subsequently to E and prior to T*, and X is a vector of pre-exposure variables with support X that confound the association between (E, M) and the underlying failure time of interest T. Because of censoring, we observe Δ = I(TC) and T* = min(T, C) where C denotes an individual’s right censoring time. Unless stated otherwise, we assume that conditional on E, censoring is independent of (M, X, T). Although, we show in Section 4 how this latter assumption can be relaxed. To limit the amount of unmeasured confounding, we suppose that X contains several variables, and thus is likely of moderate to high dimension. We assume that for each level {E = e, M = m}, there exist a counterfactual variable Te, m corresponding to the outcome had possibly contrary to fact the exposure and mediator variables taken the value (e, m) and for {E = e}, there exist a counterfactual variable Me corresponding to the mediator variable had possibly contrary to fact the exposure variable taken the value e.

Although the paper focuses on a binary exposure, we note that the extension to a polytomous exposure is trivially deduced from the exposition.

2. Mediation analysis for a marginal survival probability

Let D*(t) denote I(T* ≥ t), D(t) denote I(Tt) and define the corresponding counterfactual at risk process Dem(t) = I(Temt). Also, let Sem(t) = E {Dem(t)} = E {I(Temt)} denote the survival probability at time t had possibly contrary to fact the exposure and mediator variables taken the value (e, m); and let ST|E, M, X (t|E, M, X) denote the conditional survival probability of T at t. Consider the following decomposition of the total effect of E on the survival probability at time t:

S1M1(t)S0M0(t)[horiz curly bracket]total effect=S1M1(t)S1M0(t)[horiz curly bracket]natural indirect effect+S1M0(t)S0M0(t)natural direct effect
(1)

As shown in the display above, the natural direct effect captures the effect of the exposure when one intervenes to set the mediator to the (random) level it would have been in the absence of exposure (Robins and Greenland, 1992, Pearl 2001). Such an effect generally differs from the controlled direct effect which refers to the exposure effect that arises upon intervening to set the mediator to a fixed level that may differ from its actual observed value (Robins and Greenland, 1992, Pearl, 2001, Robins, 2003). As noted by Pearl (2001), controlled direct and indirect effects are particularly relevant for policy making whereas natural direct and indirect effects are more useful for understanding the underlying mechanism by which the exposure operates.

Identification of natural direct and indirect effects requires additional assumptions. To proceed, we make the consistency assumption also known as the stable unit treatment value assumption (SUTVA):

if E=e, then Me=M w.p.1and if E=e and M=m then Te,m=T w.p.1

In addition, we adopt the sequential ignorability assumption of Imai et al (2010) which states that for e, e′ [set membership] {0, 1}:

{Te,m,Me}[coproduct operator]E|X
(2)

Tem[coproduct operator]M|E=e,X
(3)

paired with a standard positivity assumption:

fM|E,X (m|E,X) >0 w.p.1 for each m[set membership]Sand fE|X (e|X) >0 w.p.1 for each e[set membership]{0,1}

where fM|E,X is the density of [M|E, X] and fE|X is the density of [E|X]. Then, under the consistency assumption, the first part of the sequential ignorability assumption (2) and the positivity assumption, one can show that SeMe(t) is identified by the g-formula of Robins (1997); under the additional assumption given by the second part of the sequential ignorability assumption (3), one can further show as in Imai et al (2010a), that:

S1M0(t)=θt=[double integral operator]S×XST|E,M,X(t|E=1,M=m,X=x)fM|E,X(m|E=0,X)fX(x)dμ(m,x)
(4)

where fM|E,X and fX are respectively the conditional density of the mediator M given (E, X) and the density of X, and μ is a dominating measure for the distribution of [M, X]. Thus S1M0 (t) is identified from the observed data (See Pearl, 2011 and van der Laan and Petersen (2005) for related identification results). We note that the second part of the sequential ignorability assumption (3) is particularly strong and must be made with care. This is partly because, it is always possible that there might be unobserved variables that confound the relationship between the outcome and the mediator variables even upon conditioning on the observed exposure and covariates. Furthermore, the confounders X must all be pre-exposure variables, i.e. they must precede E. In fact, Avin et al (2005) proved that without additional assumptions, one cannot identify natural direct and indirect effects if there are confounding variables that are affected by the exposure even if such variables are observed by the investigator. This implies that similar to the ignorability assumption of the exposure in observational studies, ignorability of the mediator cannot be established with certainty even after collecting as many pre-exposure confounders as possible. Furthermore, as Robins and Richardson (2010) point out, whereas the first part of the sequential ignorability assumption (2) could in principle be enforced in a randomized study, by randomizing E within levels of X; the second part of the sequential ignorability assumption (3) cannot similarly be enforced experimentally, even by randomization. And thus for this latter assumption to hold, one must entirely rely on expert knowledge about the mechanism under study. For this reason, it will be crucial in practice to supplement mediation analyses with a sensitivity analysis that accurately quantifies the degree to which results are robust to a potential violation of the sequential ignorability assumption. For this purpose, later in the paper, we adapt and extend the sensitivity analysis technique of Tchetgen Tchetgen and Shpitser (2011a,b) to a survival analysis context.

Theorem 1 of Tchetgen Tchetgen and Shpitser (2011a) implies that in order to obtain a consistent and asymptotically normal (CAN) estimator of the functional displayed in equation (4) and thus a CAN estimator of S1M0 (t) under the three assumptions given above, one must consistently estimate a subset of the following quantities {ST|E,M,X, fM|E,X, fE|X}. Thus, let {ŜT|E,M,X, fM|E,X, fE|X} denote estimates of these required quantities, based on standard parametric or semiparametric working models for regression and density estimation. Because of the curse of dimensionality due to a high dimensional X, nonparametric methods for estimating these quantities are likely impractical for the sample sizes encountered in practice, and thus parametric/semiparametric models must be used. We emphasize that these three models are not of primary scientific interest but as later demonstrated, are needed for making inferences about mediation.

In principle, one could simply evaluate the functional under the estimated model to obtain the maximum likelihood estimator (MLE):

θ^ttm=PnSS^T|E,M,X (t|E=1,M=m,X) f^M|E,X (m|E=0,X) dμ(m)

where Pn [·] [equivalent] n−1i [·]i. However, one should then be concerned that model mis-specification of either ŜT|E,M,X or fM|E,X would likely lead to biased estimates of direct and indirect effects. Note that the MLE does not rely on a model for fE|X and thus is completely robust to a mis-specified and thus likely biased estimate fE|X. Two alternative estimators can be constructed, by essentially following the approach of Tchetgen Tchetgen and Shpitser (2011a) for mean effects, that respectively use {ŜT|E,M,X, fE|X} only and {fM|E,X, fE|X} only, and therefore, that are respectively robust to mis-specification of fM|E,X and ŜT|E,M,X. Indeed, in the first case, one could use:

θ^tte=Pn{I(E=0)f^E|X(0|X)S^T|E,M,X(t|E=1,M=m,X)}

and in the second case one could use :

θ^tem=Pn{ΔS^C|E(T*|E=1)I(E=1)I(T*t)f^E|X(E|X)f^M|E,X(M|E=0,X)f^M|E,X(M|E,X)}

where ŜC|E (T*|E = e) denotes the exposure arm specific Kaplan-Meier estimator of the survival curve of censoring. S^C|E1(T*|E=1) weights are needed here to correctly account for censoring (Robins and Rotnitzky, 1992, Satten and Datta, 2001) under the current assumption that censoring is ignorable conditional on E, and an additional standard positivity assumption for the censoring mechanism (Robins and Rotnitzky, 1992). Unfortunately, as was the case for the MLE, these alternative estimators are likely severely biased if any of the required working models is incorrect.

Theorem 2 of Tchetgen Tchetgen and Shpitser (2011a) allows us to partially resolve this potential difficulty, by providing a basic roadmap for constructing an estimator [theta w/ hat]t = [theta w/ hat]t (ŜT|E,M,X, fM|E,X, fE|X) that is partially robust to such model mis-specification, and that remains CAN in the union model 𝒨union in which at least one but not necessarily all of the following hold:

  1. the estimates of the conditional survival probability ŜT|E,M,X and of the conditional density of the mediator fM|E,X are consistent;
  2. the estimates of the conditional survival probability ŜT|E,M,X and of the conditional density of the exposure fE|X are consistent
  3. the estimates of the conditional densities of the exposure and mediator variables are consistent.

Clearly, such an estimator [theta w/ hat]t should generally be preferred to θ^ttm, θ^tte and θ^tem because an inference using [theta w/ hat]t is guaranteed to remain valid under many more data generating laws than an inference based on each of the other three estimators. [theta w/ hat]t is in fact so-called triply robust, as it delivers the correct inferences under the union of the three submodels (a), (b) and (c). By Theorem 2 of Tchetgen Tchetgen and Shpitser (2011a), the following estimator is in fact triply robust:

θ^t=Pn[ΔS^C,1(T*|E=1)I(E=1)f^E|X(E|X)f^M|E,X(M|E=0,X)f^M|E,X(M|E,X)×{I(T*t)S^T|E,M,X(t|E=1,M,X)}+I(E=0)f^E|X(0|X){S^T|E,M,X(t|E=1,M,X)η^t(1,0,X)}+η^t(1,0,X)]

where

η^t(1,0,X)=SS^T|E,M,X(t|E=1,M=m,X)f^M|E,X(m|E=0,X)dμ(m)

[theta w/ hat]t may in turn be combined as in Tchetgen Tchetgen and Shpitser (2011a) with an existing doubly robust estimator of the g-formula for SeMe (t) (van der Laan and Robins, 2003, Bang and Robins, 2005), to obtain a triply robust estimator of the natural direct and indirect effects given in equation (1). To report confidence intervals, the nonparametric bootstrap could be used although an analytic expression and a corresponding estimator for the asymptotic variance of [theta w/ hat]t is easily derived from a standard Taylor series argument (see for example Tchetgen Tchetgen and Shpitser, 2011a).

3. Mediation analysis for two survival models

In this section, we consider the estimation of natural direct effects under two alternative structural models for the total effect of exposure: a Cox proportional hazards model (Cox PH) and an additive hazards model.

3.1. Proportional hazards model

The first model posits a Cox PH regression for the average total effect of the exposure, that is

λTe(t)=λT0(t) exp (βce)

where λTe (t) denotes an individual’s average hazard of experiencing an event at time t, had possibly contrary to fact, the person been exposed to E = e, and βc encodes on the log-hazards scale, the total causal effect of exposure. As in VanderWeele (2011), one can decompose exp (βc) = λT1 (t) / λT0 (t) into natural direct and indirect components:

λT1(t)λT0(t)=λT1M1(t)[horiz curly bracket]total effectλT0M0(t)=λT1M1(t)[horiz curly bracket]natural indirect effectλT1M0(t)×λT1M0(t)λT0M0(t)natural direct effect
(5)

For estimation, whereas VanderWeele (2011) requires an additional assumption that the outcome is rare over the entire follow-up period, here we make no such rare disease assumption. However, it is assumed that the natural direct hazards ratio, and thus the indirect hazards ratio, both agree with the proportional hazards assumption of the total effect; specifically, we assume that

λTeM0(t)=λT0M0(t) exp (βcdire)

follows a Cox PH model where βcdir represents the direct effect of exposure, and similarly

λT1Me(t)=λT1M0(t) exp (βcinde)

where βcind represents the indirect effect of exposure. This is an additional assumption, since although unlikely in practice, in principle both direct and indirect effect could be functions of time in such a way that they combine to produce a time-constant total effect on the hazards ratio scale. Next, we describe some procedures for estimating the direct effect parameter βcdir.

Our first result generalizes the simple weighted strategy that previously gave θ^tem, and relies on the assumption that {fM|E,X, fE|X} converges to the truth, and it does not use ŜT|E,M,X.

Theorem 1: Under the consistency, sequential ignorability and positivity assumptions, Uw(βcdir) is an unbiased estimating function for βcdir, where

Uw(βcdir)=Uw(βcdir;fM|E,X,fE|X)=dN*(t)W[Eξ1(t;βcdir)ξ2(t;βcdir)],
(6)

with

ξ1(t;βcdir)=E{D*(t)W E exp (βcdirE)},ξ2(t;βcdir)=E{D*(t)W exp (βcdirE)},W=fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X)

and N* (t) = I(T* ≤ t, Δ = 1) is the counting process of an observed failure time. Thus, βcdir is the solution of the equation:

E{Uw(βcdir)}=0

The proof of Theorem 1 is provided in the appendix; the result motivates the estimator β˜cdir that solves:

Pn{U^w(β˜cdir)}=0

where Ûw (β) = Ûw(β; fM|E,X, fE|X) is an empirical version of Uw (β) defined as:

dN*(t)W^[EPn{D*(t)W^ E exp (βcdirE)}Pn{D*(t)W^ exp (βcdirE)}]
(7)

with

ξ^1(t)=Pn{D*(t)W^ E exp (βcdirE)},ξ^2(t)=Pn{D*(t)W^ exp (βcdirE)},

and Ŵ defined as W under {fM|E,X, fE|X}. Thus, under the key assumption that {fM|E,X, fE|X} is consistent (and converges in probability at rates faster than n−1/4, see Newey (1994)), and under further standard regularity conditions β˜cdir is CAN with asymptotic variance that can be obtained by a standard Taylor expansion, or more conveniently by the nonparametric bootstrap.

This simple weighing strategy for estimating βcdir holds appeal in that it is easy to implement in standard statistical software packages for survival analysis. This is because, equation (7) is a modified score equation for the partial likelihood of a marginal Cox proportional hazards model (which is recovered by setting the weights Ŵ [equivalent] 1), and thus the modification mainly entails setting non-unity weights, which can be done in most software packages for Cox regression analysis. For instance, β˜cdir can be obtained using PROC PHREG in SAS with the WEIGHT statement to incorporate the individual specific weight Ŵ. However, as we have mentioned before, in the event that either fM|E,X or fE|X fails to converge to the truth, β˜cdir will generally be biased. Thus we propose an alternative approach to estimate βcdir.

We proceed by first finding a modification to equation (6) that delivers the desired robustness property. Note that because both quantities ξ1 (t) and ξ2 (t) in equation (6) involve W, unbiased estimation (more precisely consistent estimation) of these two functions effectively requires correct models for {fM|E,X, fE|X}. Thus, a key step in developing a multiply robust estimator of βcdir involves first finding multiply robust estimators of these two functions. In this vein, to further allow for generality, for any function of E, say H = h(E), let

R(t,H;βcdir)=R(t,H;βcdir,ST|E,M,X,fM|E,X,fE|X,SC|E)={D*(t)SC|E(t|E)ST|E,M,X(t|E,M,X)}W h(E) exp (βcdirE)+{eSC|E)(t|E=e)ST|E,M,X(t|E=e,M=m,XfM|E,X(m|E=0,X)h(e) exp (βcdire)dμ(m)}+I(E=0)f(E|X)eSC|E(t|E=e)ST|E,M,X(t|E=e,M,X)h(e) exp (βcdire)I(E=0)f(E|X)[eSC|E(t|E=e)ST|E,M,X(t|E=e,M=m,X)fM|E,X(m|E=0,X)h(e) exp (βcdire)dμ(m)]

and let

ξjmr(t;βcdir)=ξjmr(t;βcdir,ST|E,M,X,fM|E,X,fE|X,SC|E)=E{R(t,Hj;βcdir)},j=1,2;

where H1 = E and H2 = 1.

Next, define R(t, Hj; βcdir) as R(t, Hj; βcdir) under the law { ST|E,M,X, fM|E,X, fE|X}. In the appendix, we establish that ξjmr(t;βcdir)=ξj(t;βcdir), j = 1, 2, and in fact, we prove that this alternative representation is multiply robust, in the sense that ξjmr,(t;βcdir)=E{R(t,Hj;βcdir)}=ξj(t;βcdir) provided that at least one of the following three conditions hold: either { ST|E,M,X, fM|E,X} = {ST|E,M,X, fM|E,X} or { ST|E,M,X, fE|X} = {ST|E,M,X, fE|X}, or { fM|E,X, fE|X} = {fM|E, X, fE|X}. In the appendix, we use this result to establish the following theorem:

Theorem 2: Under the consistency, sequential ignorability and positivity assumptions, Umr(βcdir)=Umr(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E) is an unbiased estimating function for βcdir, where

Umr(βcdir)={dN*(t)+SC|E(t|E)dST|E,M,X(t|E,M,X)}W{Eξ1mr(t;βcdir)ξ2mr(t)}e[set membership]{0,1}[dST|E,M,X(t|E=e,m,X)×SC|E(t|E=e)fM|E,X(m|E=0,X)×{eξ1mr(t;βcdir)ξ2mr(t)}]dμ(m)I(E=0)fE|X(E|X)e[set membership]{0,1}[{SC|E(t|E=e)×dST|E,M,X(t|E=e,M,X)}×{eξ1mr(t;βcdir)ξ2mr(t;βcdir)}]+I(E=0)fE|X(E|X)e[set membership]{0,1}[{SC|E(t|E=e)×dST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)}×{eξ1mr(t;βcdir)ξ2mr(t;βcdir)}]dμ(m)

Furthermore,

E{Umr(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0
(8)

if one but not necessarily all three of the following conditions holds: either { ST|E,M,X, fM|E,X} = {ST|E,M,X, fM|E,X} or { ST|E,M,X, fE|X} = {ST|E,M,X, fE|X}, or { fM|E,X, fE|X} = {fM|E,X, fE|X}.

According to Theorem 2, a multiply robust estimator β^cdir is obtained by solving the equation:

Pn{U^mr(β^cdir;S^T|E,M,X,f^M|E,X,f^E|X,S^C|E)}=0

where Ûmr (·; ·, ·, ·) is obtained by substituting Pn [·] for all marginal expectations. so that under standard regularity conditions it can be shown that β^cdir is CAN in model 𝒨union. An analytical expression for the asymptotic variance of β^cdir under 𝒨union can be obtained using large sample theory for martingales which is not pursued here. Alternatively, one could also use the nonparametric bootstrap for inference which is more convenient.

To estimate the indirect log hazards ratio βcind, we observe that by the decomposition given in equation (5), βcind=βcβcdir where βc is the total log hazards ratio, i.e. λT1 (t) / λT0 (t) = exp (βc). This gives a simple approach for estimating the indirect effect, by first estimating βc using standard inverse-probability-of-treatment weighting for total effects. That is, following Robins (1998), βc can be estimated by solving equation (7) upon substituting Ŵ with f^E|X1(E|X). Then, βcind can be estimated either by β˜cβ^cdir or alternatively by β˜cβ^cdir. Unfortunately, both of these estimators are likely biased if f^E|X1(E|X) is not consistent. As a remedy, the next theorem gives a multiply robust estimating function of βcind.

Theorem 3: Suppose that βcdir is known, then under the consistency, sequential ignorability and positivity assumptions, Vmr(βcdir,βcind)=Vmr(βcdir,βcind;ST|E,M,X,fM|E,X,fE|X,SC|E) is an unbiased estimating function for βcind, where

Vmr(βcdir,βcind)=[{dN*(t)+{SC|E(t|E)×dST|E,M,X(t|E,M=m,X)×fM|E,X(m|E,X)}dμ(m)}×fE|X1(E|X)×{E[theta]1mr(t;βcdir,βcind)[theta]2mr(t;βcdir,βcind)}][double integral operator]e[set membership]{0,1}[SC|E(t|E=e)×dST|E,M,X(t|E=e,M,X)×fM|E,X(m|E=e,X)×{e[theta]1mr(t;βcdir,βcind)[theta]2mr(t;βcdir,βcind)}]dμ(m),

with

[theta]jmr(t;βcdir,βcind)=[theta]jmr(t;βcdir,βcind,ST|E,M,X,fM|E,X,fE|X,SC|E)=E{G(t,Hj;βcdir,βcind)},j=1,2G(t,H;βcdir,βcind)={D*(t){SC|E(t|E)×ST|E,M,X(t|E,M=m,X)×fM|E,X(m|E,X)}dμ(m)}×fE|X1(E|X)h(E) exp{(βcdir+βcind)E}+e{SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=e,X)×h(e) exp {(βcdir+βcind)e}dμ(m)}

Furthermore, ( βcdir, βcind) solves

E{Vmr(βcind,βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0E{Umr(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0

if one but not necessarily all three of the following conditions holds: either { ST|E,M,X, fM|E,X} = {ST|E,M,X, fM|E, X} or { ST|E,M,X, fE|X} = {ST|E,M,X, fE|X}, or { fM|E,X, fE|X} = {fM|E, X, fE|X}.

According to theorem 3, a multiply robust estimator β^cind is obtained by solving the equation:

Pn{V^mr(β^cind,β^cdir;S^T|E,M,X,f^M|E,X,f^E|X,S^C|E)}=0

where Vmr (·, ·; ·, ·, ·) is obtained by substituting Pn [·] for all marginal expectations, so that under standard regularity conditions, β^cind is CAN in model 𝒨union. The nonparametric bootstrap can be used to obtain standard errors for inference.

3.2. Additive hazards model

In some situations, assuming proportional hazards may not fit the data well, in which case, an additive hazards model will often fit the data better (Lin and Ying, 2004). This alternative model assumes the average total effect of the exposure is additive on the hazards scale :

λTe(t)=λT0(t)+βae

where βa encodes the total causal effect of exposure. As in Lange and Hansen (2011), one can decompose βa = λT1 (t) − λT0 (t) into natural direct and indirect components:

=λT1(t)λT0(t)=λT1M1(t)λT0M0(t)[horiz curly bracket]total effect=λT1M1(t)λT1M0(t)[horiz curly bracket]natural indirect effect+λT1M0(t)λT0M0(t)natural direct effect
(9)

We further assume that the natural direct effect, and thus the indirect effect, agrees with the assumption of additive hazards, and thus

λTeM0(t)=λT0M0(t)+βadire

where βadir represents the direct effect of the exposure, and similarly

λT1Me(t)=λT1M0(t)+βainde

where βaind represents the indirect effect of the exposure. As was the case for the Cox PH model, this is an assumption, because although unlikely in practice, in principle both direct and indirect effects could be functions of time in such a way that they combine to produce a constant additive total effect. Next we discuss a variety of estimating approaches for the direct effect parameter βadir.

The next result gives a weighted approach analogous to that proposed for the Cox PH model

Theorem 4: Under the consistency, sequential ignorability and positivity assumptions, Zw(βadir) is an unbiased estimating function for βadir, where

Zw(βadir)={dN*(t)EβadirD*(t)dt}W[E[var pi]1(t)ω2(t)],
(10)

with

[var pi]1(t)=E{D*(t)WE},[var pi]2(t)=E{D*(t)W}

Thus, βcdir is the solution of the equation:

E{Zw(βadir)}=0

The theorem implies that β˜adir is CAN provided {fM|E,X, fE|X} is consistent, where β˜adir solves

Pn{Z^w(β˜adir)}=0

with

Z^w(β)={dN*(t)EβD*(t)dt}W^[E[var pi]^1(t)[var pi]^2(t)]
(11)

an empirical version of Zw. Note that β˜adir is available in closed form

β˜adir=PndN*(t)W^[E[var pi]^1(t)[var pi]^2(t)]PnED*(t)dtW^[E[var pi]^1(t)[var pi]^2(t)]

But β˜adir is not multiply robust. The next theorem provides a multiply robust estimating function of βadir. First, we introduce some additional notation and let

[var pi]jmr(t)=[var pi]jmr(t;ST|E,M,X,fM|E,X,fE|X,SC|E)=E[{D*(t)SC|E(t|E)ST|E,M,X(t|E,M,X)}W hj(E)+e[set membership]{0,1}{SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)hj(e)}dμ(m)+I(E=0)fE|X(E|X)e[set membership]{0,1}SC|E(t|E=e)ST|E,M,X(t|E=e,M,X)hj(e)I(E=0)fE|X(E|X)e[set membership]{0,1}{SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)hj(e)}dμ(m)]

Theorem 5: Under the consistency, sequential ignorability and positivity assumptions, Zmr(βadir)=Zmr(βadir;ST|E,M,X,fM|E,X,fE|X,SC|E) is an unbiased estimating function for βadir, where

Zmr(βadir)={dN*(t)EβadirD*(t)dt+SC|E(t|E)dST|E,M,X(t|E,M,X)+EβadirSC|E(t|E)ST|E,M,X(t|E,M,X)dt} W{E[var pi]1mr(t)[var pi]2mr(t)}+[double integral operator]e[set membership]{0,1}[{[SC|E(t|E=e)×dST|E,M,X(t|E=e,M=m,X)][eβadirSC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)dt]}×{eξ1mr,(t;βcdir)ξ2mr,(t)}fM|E,X(m|E=0,X)dμ(m)]+I(E=0)fE|X(E|X)e[set membership]{0,1}{{[SC|E(t|E=e)×dST|E,M,X(t|E=e,M,X)][eβadirSC|E(t|E=e)×ST|E,M,X(t|E=e,M,X)d(t)]}×{eξ1mr,(t;βcdir)ξ2mr,(t;βcdir)}}I(E=0)fE|X(E|X)[double integral operator]e[set membership]{0,1}{{[SC|E(t|E=e)×dST|E,M,X(t|E=e,M=m,X)][eβadirSC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)dt]}×{eξ1mr,(t;βcdir)ξ2mr,(t;βcdir)}fM|E,X(m|E=0,X)dμ(m)}

Furthermore,

E{Zmr(βadir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0
(12)

if one but not necessarily all three of the following conditions holds: either { ST|E,M,X, fM|E,X} = {ST|E,M,X, fM|E,X} or { ST|E,M,X, fE|X} = {ST|E,M,X, fE|X}, or { fM|E,X, fE|X} = {fM|E,X, fE|X}

By theorem 5, a multiply robust estimator β^adir is obtained by solving the equation:

Pn{Z^mr(β^adir;S^T|E,M,X,f^M|E,X,f^E|X,S^C|E)}=0

so that under standard regularity conditions, β^adir is CAN in model 𝒨union. The nonparametric bootstrap can be used to compute standard errors for inference.

Suppose now that one wishes to estimate the indirect hazards difference βaind. By the decomposition given in equation (9), βadir=βaβadir where βatotal is the total hazards difference, i.e. λT1 (t) − λT0 (t) = βa. This decomposition immediately gives a simple estimator of the indirect effect based on a weighting scheme. The approach entails first estimating βa by using inverse-probability-of-treatment weighting. Following Robins (1998), betaa is obtained by solving equation (11) upon replacing Ŵ by f^E|X1(E|X). Then, we can define an estimator of βaind by β˜aβ^adir or alternatively by β˜aβ˜adir. Unfortunately, just as in the Cox model, both of these estimators are likely biased if f^E|X1(E|X) is not consistent. As a remedy, the next theorem gives a multiply robust estimating function of βaind.

Theorem 6: Suppose βadir is known, then under the consistency, sequential ignorability and positivity assumptions, Pmr(βadir,βaind)=Pmr(βadir,βaind;ST|E,M,X,fM|E,X,fE|X,SC|E) is an unbiased estimating function for βaind, where

Pmr(βadir,βaind)={dN*(t)E(βadir+βaind)D*(t)dt+SC|E(t|E)dST|E,M,X(t|E,m,X)fM|E,X(m|E,X)dμ(m)+[E(βadir+βaind)SC|E(t|E)×ST|E,M,X(t|E,m,X)fM|E,X(m|E,X)d(μ(m),t)]}×fE|X1(E|X){E[var phi]1mr(t)[var phi]2mr(t)}+e[set membership]{0,1}[{SC|E(t|E=e)dST|E,M,X(t|E=e,M=m,X)e(βadir+βaind)SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)d(μ(m),t)}×{e[var phi]1mr(t)[var phi]2mr(t)}fM|E,X(m|E=e,X)]

with ϕjmr(t)=ϕjmr(t;ST|E,M,X,fM|E,X,fE|X,SC|E)

=E[{D*(t)[SC|E(t|E)×ST|E,M,X(t|E,m,X)fM|E,X(m|E,X)dμ(m)]}×fE|X1(E|X)hj(E)]+e[set membership]{0,1}{SC|E(t|E=e)[ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=e,X)hj(e)]}dμ(m)

Furthermore,

E{Pmr(βadir,βaind;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0E{Zmr(βadir,ST|E,M,X,fM|E,X,fE|X,SC|E)}=0
(13)

if one but not necessarily all three of the following conditions holds: either { ST|E,M,X, fM|E,X} = {ST|E,M,X, fM|E,X} or { ST|E,M,X, fE|X} = {ST|E,M,X, fE|X}, or { fM|E,X, fE|X} = {fM|E, X, fE|X}.

According to theorem 6, a multiply robust estimator β^aind is obtained by solving the equation:

Pn{P^mr(β^aind,β^adir;S^T|E,M,X,f^M|E,X,f^E|X,S^C|E)}=0

where Pmr (·, ·; ·, ·, ·, ·) is obtained by substituting Pn [·] for all marginal expectations, then under standard regularity conditions, β^aind is CAN in model 𝒨union; thus one can use the nonparametric bootstrap for inference.

4. Dependent censoring

We briefly consider how to modify the proposed methods if the previous assumption that censoring is independent of (M, X, T) conditional on E, is believed not to hold, but instead, censoring is known to be independent of T given (E, M, X). For brevity, we focus attention to the Cox proportional hazards model, but the approach is easily adapted to the additive hazards model. Consider the simple weighted estimating equation given in Theorem 1, the modification entails simply replacing the weight W in equation (6) with the time-dependent weight:

Wt*=W×1SC|E,M,X(t|E,M,X)

where SC|E,M,X is the conditional survival curve of censoring, so that equation (6) is replaced by

Uw*(βcdir)=Uw*(βcdir;SC|E,M,X,fM|E,X,fE|X)=dN*(t)Wt*[Eξ1*(t;βcdir)ξ2*(t;βcdir)],

with

ξ1*(t;βcdir)=E{D*(t)Wt*E exp(βcdirE)},

ξ2*(t;βcdir)=E{D*(t)Wt* exp(βcdirE)},

then, one can show that Uw*(βcdir) is an unbiased estimating equation under the assumption that censoring is independent of T given (E, M, X), provided a standard positivity assumption holds for the censoring mechanism (van der Laan and Robins, 2003). A feasible estimator is obtained by replacing all unknown quantities by empirical versions, using parametric or semiparametric models for (SC|E,M,X, fM|E,X, fE|X). The multiply robust estimating function given in Theorem 2 can similarly be modified to accommodate this particular form of dependent censoring. Details are relegated to the appendix.

5. A semiparametric sensitivity analysis

In this section, we extend the semiparametric sensitivity analysis technique proposed by Tchetgen Tchetgen and Shpitser (2011a,b), to assess the extent to which a violation of the ignorability assumption for the mediator might alter inferences about natural direct or indirect effects in the survival context. Let

γ(t,e,m,x)=λT1,m|E,M,X(t|E=e,M=m,X=x)λT1,m|E,M,X(t|E=e,Mm,X=x)

then

Te,m∐̸/M|E=e,X

i.e. a violation of the ignorability assumption for the mediator variable, generally implies that γ (t, e, m, x) ≠ 0 for some (t, e, m, x). Suppose M is binary and larger values of T are beneficial for health, then if γ (t, e, 1, x) < 0 but γ (t, e, 0, x) > 0 for all t, then on average, individuals with {E = e, X = x} and mediator value {M = 0} have a higher hazard function for each of the potential outcomes {T11, T10} than individuals with {E = e, X = x} but {M = 1} ; i.e. healthier individuals are more likely to receive the mediator. On the other hand, if γ (t, e, 0, x) < 0 but γ (t, e, 1, x) > 0 for all t, suggests confounding by indication for the mediator variable; i.e. unhealthier individuals are more likely to receive the mediating factor.

We proceed as in Robins et al (1999) who proposed using a selection bias function for the purposes of conducting a sensitivity analysis for total effects, and Tchetgen Tchetgen and Shpitser (2011a,b) who adapted the approach for assessing the impact of unmeasured confounding on the estimation of average natural direct and indirect effects. Here we propose to recover inferences about natural direct effects on the hazard function, under either an additive or a proportional hazards model, by assuming the selection bias function γ (t, e, m, x) is known, which encodes the magnitude and direction of the unmeasured confounding for the mediator. In the following, S is assumed to be finite. To motivate the proposed approach, suppose for the moment that fM|E,X is known, then under the assumption that the exposure is ignorable given X, we show in the appendix that the following lemma holds:

Lemma 1:Let

δ(t,e,m,x)=δ(t,e,m,x;fM|E,X)=fM|E,X(m|E=e,X=x)+{1fM|E,X(m|E=e,X=x)} exp{0tγ(u,e,m,x)du}fM|E,X(m|E=0,X=x)+{1fM|E,X(m|E=0,X=x)} exp{0tγ(u,0,m,x)du}

and

δ˙(t,1,m,x)=[partial differential] log δ(u,1,m,x)[partial differential]u|u=t

Under the consistency assumption and the first part of the sequential ignorability assumption (2)

ST1,M0|M0,X(t|M0=m,X=x)=ST1,m|E,M,X(t|E=0,M=m,X=x)=ST|E,M,X(t|E=1,M=m,X=x)×δ(t,1,m,x)

Furthermore,

λT1,M0|M0,X(t|M0=m,X=x)=λT1,m|E,M,X(t|E=0,M=m,X=x)=λT|E,M,X(t|E=1,M=m,X=x)δ˙(t,1,m,x)

Lemma 1 implies that ST1,M0 (t) is identified by:

E(m[set membership]SST|E,M,X(t|E=1,M=m,X=x)δ(t,1,m,x)fM|E,X(m|E=0,X))
(14)

Below, we use this result to obtain consistent estimators of { βjdir, βjind: j = a, c} assuming γ (·, ·, ·, ·) is known. A sensitivity analysis is then obtained as in Tchetgen Tchetgen and Shpitser (2011a,b) by repeating this process and by reporting inferences for each choice of γ (·, ·, ·, ·) in a finite set of user–specified functions Γ = {γα (·, ·, ·, ·) :α} indexed by a finite dimensional parameter α with γ0 (·, ·, ·, ·) [set membership] Γ corresponding to the ignorability assumption of M, i.e. γ0 (·, ·, ·, ·) [equivalent] 0. Throughout, models for the probability mass functions of [M|E, X] and [E|X] are assumed to be correct. Thus, to implement the sensitivity analysis technique, we develop a semiparametric estimator of { βjdir, βjind : j = a, c} in a model 𝒨1 that assumes the model for [M, E|X] is known up to a set of finite dimensional parameters, and in which the selection bias function is known, γ (·, ·, ·, ·) = γα* (·, ·, ·, ·) for α* fixed.

For the Cox PH model, we propose to use the following modified estimating function for estimating the direct effect under 𝒨1, which carefully incorporates the selection bias function:

Uw(βcdir,α*)=δα*(t,E,M,X){dN*(t)δ˙α*(t,M,X)D*(t)dt}W×{EE{D*(t)WEδα*(t,E,M,X) exp(βcdirE)}E{D*(t)Wδα*(t,E,M,X) exp (βcdirE)}}

where δα* (·, ·, ·, ·) is defined as δ (·, ·, ·, ·) under γα* (·, ·, ·, ·). For the additive model, one can use the following modified estimating function under 𝒨1:

Zw(βadir,α*)={dN*(t)δ˙α*(t,E,M,X)D*(t)dtEβadirD*(t) δ˙α*(t,E,M,X)dt}δα*(t,E,M,X)W×{EE{D*(t)W Eδα*(t,E,M,X)}E{D*(t)Wδα*(t,E,M,X)}}

In the appendix, we show the following result holds:

Theorem 7:Suppose γ (·, ·, ·, ·) = γα* (·, ·, ·, ·), then under the consistency and positivity assumptions, and the ignorability assumption for the exposure, and under the Cox PH model, βcdir=βcdir(α*) solves the equation

E{Uw(βcdir,α*)}=0
(15)

Similarly, under the additive hazards model, βadir=βadir(α*) solves the equation

E{Zw(βadir,α*)}=0

Thus, under model 𝒨1 and the Cox PH assumption, a sensitivity analysis then entails reporting the set {β^cdir(α):α} (and the associated confidence intervals) which summarizes how sensitive inferences are to a deviation from the ignorability assumption α = 0, where β˜cdir(α) solves an empirical version of equation (15) with unknown quantities estimated under the model. A sensitivity analysis is similarly obtained for the additive hazards model, and inferences about indirect effects are obtained as in Section 3, upon substituting {β^cdir(α),β^adir(α):α} for {β^cdir,β^adir}. In the appendix, we describe a doubly robust sensitivity analysis technique which further extends these results, by recovering correct sensitivity analyses under a union model in which, fM|E,X is assumed to be consistent, however, only one but not necessarily both fT|M,E,X and fE|X need to be consistently estimated.

It is helpful for practice, to briefly describe possible functional forms for the selection bias function γα(·, ·, ·, ·). In the simple case where M is binary, it may be convenient to specify a single parameter model such as one of the following:

γα,1(t,e,m,x)=αt(2m1)γα,2(t,e,m,x)=αtmγα,3(t,e,m,x)=αt(2m1)eγα,4(t,e,m,x)=αtmeγα,5(t,e,m,x)=αt(2m1)ex1γα,6(t,e,m,x)=αtmex1

where for each of the above functional forms, the scalar parameter α encodes the magnitude and direction of unmeasured confounding for the mediator.

The functions γα,3, γα,4, γα,5 and γα,6 model interactions with the exposure variable and a component X1 of X, thus allowing for heterogeneity in the selection bias function over time. Since the functional form of γα is not identified from the observed data, we generally recommend reporting results for a variety of functional forms.

It is important to note that the sensitivity analysis technique introduced above appears to be the first of its kind for survival data. While a variety of techniques have previously been proposed for conducting sensitivity analyses for unmeasured confounding in the context of mediation, for example, VanderWeele (2010), Imai et al (2010a), Tchetgen Tchetgen and Shpitser (2011a,b), none of the existing techniques apply to mediation in the survival context under either a Cox PH model or an additive hazards model. It is also important to note that by concisely encoding a possible violation of the ignorability assumption for the mediator through a selection bias function the proposed approach avoids having to spell out in detail, the possible nature of the unmeasured confounding; although in practice, as illustrated above, a parsimonious model must be used for the selection bias function. A further appeal of the approach is that it may be used to perform a sensitivity analysis, in settings where the ignorability violation arises due to a confounder of the mediator-outcome relationship that is also an effect of the exposure variable; in which case, as observed in Section 2, such a variable even when observed, cannot be used towards identification of natural direct and indirect effects without additional assumptions.

Finally, we note that while in this section, the support of M was finite, the proposed sensitivity analysis methodology can be extended to accommodate a continuous mediator by further adapting the approach of Robins et al (1999) to the present setting.

6. Discussion

The current paper makes a number of contributions to the study of statistical methods for causal mediation analysis. Focusing on survival data, we have proposed a number of new estimators of natural direct and indirect effects for the Cox PH and the additive hazards models. The weighted approach developed in section 3 is appealing for its simplicity and because it is easy to implement in existing software, provided individual-specific weights are accommodated. We should note that, whereas it is common practice when estimating total effects via inverse-probability-weights to report conservative standard errors based on the sandwich variance formula, this ignores the first stage estimation of the treatment weights. Results by Tchetgen Tchetgen and Shpitser (2011a) imply that such a practice gives the wrong answer for natural direct and indirect effects. However, a standard bootstrap may be used for inference. We also note that, in general, the more involved multiply robust approach of Section 3 should be preferred to the simpler weighted approach on theoretical grounds, because the former delivers valid inferences under weaker assumptions than the latter. However, implementing these improved methods for routine application presents a significant challenge that we plan to take on elsewhere. In addition, as pointed out by a referee, we note that in the setting of a randomized trial, the exposure mechanism is known by design and therefore, the multiply robust estimator described above becomes doubly robust in the sense that for correct inferences, one only needs either fM|E,X or ST|E,M,X to be correctly specified, but not necessarily both. Finally, we emphasize that the proposed multiply robust strategies should not be viewed as a substitute for sound model checking, and therefore, we encourage users of these methods to treat any multiply robust analysis they conduct with the same level of model scrutiny as they would apply to a non-robust approach.

APPENDIX. 

PROOF OF THEOREM 1:

Under the consistency, sequential ignorability and positivity assumptions,

E{D*(t)fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X)h(E) exp (βcdirE)}=E{ST|E,M,X(t|E,M,X)SC|E(t|E)×fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X)h(E)exp(βcdirE)}=eE{ST|E,M,X(t|E=e,M=m,X)SC|E(t|E=e)×fM|E,X(m|E=0,X)×h(e) exp (βcdire)dμ(m)}={eSC|E(t|E=e)h(e) exp (βcdire)×E[ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)dμ(m)]}={eSC|E(t|E=e)h(e) exp (βcdire)STe,M0(t)}={eSC|E(t|E=e)h(e) exp (βcdire)STe,M0(t)}

and E {dN* (t)W h (E)}

=E{λT|E,M,X(t|E,M,X)ST|E,M,X(t|e,M,X)×SC|E(t|E)Wh(E)dt}=E{fT|E,M,X(t|E,M,X)SC|E(t|E)Wh(E)}dt=e[SC|E(t|E=e)h(e)dt×E{fT|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)dμ(m)}]=efTe,M0(t)SC|E(t|E=e)h(e)dt=eSC|E(t|E=e)h(e)λT0(t) exp (βcdire)STe,M0(t)dt

Therefore E{dN*(t)W[Eξ1(t;βcdir)ξ2(t;βcdir)]}

=E{dN*(t)W[EE{D*(t)fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X)h1(E) exp (βcdirE)}E{D*(t)fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X)h2(E) exp (βcdirE)}]}=E{dN*(t)W[E{eSC|E(t|E=e)e exp (βcdire)STe,M0(t)}{eSC|E(t|E=e) exp (βcdire)STe,M0(t)}]}=[E{dN*(t)W h1(E)}E{dN*(t)W h2(E)}{eSC|E(t|E=e) exp (βcdire)STe,M0(t)}{eSC|E(t|E=e) exp (βcdire)STe,M0(t)}]=dt[eSC|E(t|E=e)eλT0(t) exp (βcdire)STe,M0(t)e{SC|E(t|E=e)λT0(t) exp (βcdire)STe,M0(t)}×e{SC|E(t|E=e)eexp(βcdire)STe,M0(t)}{eSC|E(t|E=e) exp (βcdire)STe,M0(t)}]=dt[eSC|E(t|E=e)eλT0(t) exp (βcdire)STe,M0(t)e{SC|E(t|E=e)λT0(t) exp (βcdire)STe,M0(t)}×e{SC|E(t|E=e)eλT0(t) exp (βcdire)STe,M0(t)}{eSC|E(t|E=e)λT0(t) exp  (βcdire)STe,M0(t)}]=0

The following lemma will be used repeatedly to establish multiple robustness of a given estimating function.

LEMMA A.1

Given i.i.d data (O, M, E, X), define the weighted functional κ (l) with weight L = l(E) as:

κ(l)=e=01L(e)E{E(O|M=m,E=e,X)×fM|E,X(m|E=0,X)dμ(m)}

Let B(m, e, x) = E (O|M = m, E = e, X = x). Then, the random variable J = J(B, fM|E,X, fE|X) satisfies the triply robust unbiasedness property

E{J(B,fM|E,X,fE|X)}=κ(l)

if at least one but not necessarily all of the following conditions hold: either {B, fM|E,X} = {B, fM|E,X} or {B, fE|X} = {B, fE|X}, or { fM|E,X, fE|X} = {fM|E,X, fE|X}; where

J(B,fM|E,X,fE|X)=fM|E,X(M|E=0,X)fE|X(E|X)fM|E,X(M|E,X){OB(M,E,X)}L(E)+e=01L(e)B(M,e,X)fM|E,X(m|E=0,X)dμ(m)+I(E=0)fE|X(0|X){e=01L(e)[B(M,e,X)B(M,e,X)fM|E,X(m|E=0,X)dμ(m)]}

PROOF OF LEMMA A.1:

By Theorem 1 of Tchetgen Tchetgen and Shpitser (2011a), the random variable J(B, fM|E,X, fE|X) – κ (l) is the efficient influence function of κ (l) and thus the result follows from Theorem 2 of their paper. For an alternative proof consider the bias of J(B, fM|E,X, fE|X):

E{J(B,fM|E,X,fE|X)}κ(l)=e=01E[fE|X(e|X)fM|E,X(m|E=e,X)fM|E,X(m|E=0,X)fE|X(e|X)fM|E,X(m|E=e,X){B(m,e,X)B(m,e,X)}L(e)dμ(m)+e=01L(e)B(M,e,X)fM|E,X(m|E=0,X)dμ(m)e=01L(e)B(M,e,X)fM|E,X(m|E=0,X)dμ(m)+fE|X(0|X)fE|X(0|X)e=01{L(e)×[B(m,e,X)fM|E,X(m|E=0,X)B(m,e,X)fM|E,X(m|E=0,X)]dμ(m)}]=E[e=01{fE|X(e|X)fM|E,X(m|E=e,X)fE|X(e|X)fM|E,X(m|E=e,X)1}×{B(m,e,X)B(m,e,X)}fM|E,X(m|E=0,X)L(e)e=01{fM|E,X(m|E=0,X)fM|E,X(m|E=0,X)}{B(m,e,X)B(m,e,X)}L(e)dμ(m)+e=01{fM|E,X(m|E=0,X)fM|E,X(m|E=0,X)}{fE|X(e|X)fE|X(e|X)1}B(m,e,X)L(e)dμ(m)]

= 0 if at least one of the three conditions of the Lemma holds, proving the result.

PROOF OF THEOREM 2:

Under the consistency, sequential ignorability and positivity assumptions, in the proof of Theorem 1 we showed

ξ1(t;βcdir)={eh(e) exp (βcdire)×E[SC|E(t|E=e)ST|E,M,X(t|E=e,M=m,X)fM|E,X(m|E=0,X)dμ(m)]}

which is of the form κ(l), with L(e)=h(e) exp (βcdire) and O = D* (t). Therefore, by Lemma 1, R(t, H; βcdir) has the desired triply robust unbiasedness property, such that L(e)=SC|E(t|E=e)h(e) exp βcdire, thus we have E{R(t,H;βcdir)}=ξ1(t;βcdir) under the conditions of the Theorem. Similarly, we have previously established in the proof of Theorem 1, that

  E{dN*(t)W h(E)}=e[h(e)×E{SC|E(t|E=e)×fT|E,M,X(t|E=e,M=m,X)dtfM|E,X(m|E=0,X)dμ(m)}]

which is of the form κ (l), with L(e) = h(e) and O = dN* (t).Therefore, by Lemma 1, the theorem holds upon setting h(E)={Eξ1mr,(t;βcdir)ξ2mr,(t)}.

The following Lemma is key to proving Theorem 3

LEMMA A.2:

Define the weighted functional σ (l) with weight L = l(E) as:

σ(l)=e=01L(e)E{E(O|E=e,X)}=e=01L(e)E{E(O|M=m, E=e,X)×fM|E,X(m|E=e,X)dμ(m)}

The random variable A = A(B, fM|E,X, fE|X) satisfies the double robust unbiasedness property which states that E{A(B,fM|E,X,fE|X)}=σ(l) if at least one but not necessarily both of the following conditions hold: either {B, fM|E,X} = {B, fM|E,X} or fE|X=fE|X; where

A(B,fM|E,X,fE|X)=1fE|X(E|X){OB(m,E,X)fM|E,X(m|E,X)dμ(m)}L(E)+e=01L(e)B(M,e,X)fM|E,X(m|E=e,X)dμ(m).

PROOF OF LEMMA A.2:

E{A(B,fM|E,X,fE|X)}σ(l)=e=01E{fE|X(e|X)fE|X(e|X){B(m,e,X)fM|E,X(m|E=e,X)B(m,e,X)fM|E,X(m|E=e,X)}dμ(m)L(e) +e=01L(e)B(M,e,X)fM|E,X(m|E=e,X)dμ(m)B(m,e,X)fM|E,X(m|E=e,X)dμ(m)}=e=01E{{fE|X(e|X)fE|X(e|X)1}×{B(m,e,X)fM|E,X(m|E=e,X)dμ(m)B(m,e,X)fM|E,X(m|e,X)dμ(m)}L(e)

= 0 under the assumptions of the theorem.

PROOF OF THEOREM 3:

We note that [theta]jmr(t;βcdir,βcind)

=[theta]j(t;βcdir,βcind)=[theta]j(t;βcdir,βcind,ST|E,M,X,fM|E,X,fE|X,SC|E)=E[e{SC|E(t|E=e)ST|E,M,X(t|E=e,M=m,X)fM|E,X(m|E=e,X)h(e) exp {(βcdir+βcind)e}dμ(m)}]

is of the form of the weighted functional σ (l) with L(e)=h(e) exp {(βcdir+βcind)e} and O = D*(t) thus by Lemma A.2,

E[G(t,Hj;βcdir,βcind,ST|E,M,X,fM|E,X,fE|X,SC|E)]=[theta]j(t;βcdir,βcind)

if either fE|X=fE|X or {ST|E,M,X,fM|E,X}={ST|E,M,X,fM|E,X} but not necessarily both. Furthermore, note that

=E{e[set membership]{0,1}[SC|E(t|E=e)×fT|E,M,X(t|E=e,M,X)×fM|E,X(m|E=e,X)×{e[theta]1mr(t;β1,β2)[theta]2mr(t;β1,β2)}dt]dμ(m)}

is of the form of the weighted functional σ (l) with L(e)={e[theta]1mr(t;β1,β2)[theta]2mr(t;β1,β2)}dt and O = dN* (t), therefore, by Lemma A.2, if either fE|X=fE|X or {ST|E,M,X,fM|E,X}={ST|E,M,X,fM|E,X}

E{Vmr(βcdir,βcind;ST|E,M,X,fM|E,X,fE|X,SC|E)}=E{e[set membership]{0,1}[SC|E(t|E=e)×fT|E,M,X(t|E=e,M,X)×fM|E,X(m|E=e,X)×{e[theta]1(t;βcdir,βcind)[theta]2(t;β1,β2)}dt]dμ(m)}=E{e[set membership]{0,1}[SC|E(t|E=e)STe(t)λT0(t) exp (βce)×{ee[set membership]{0,1}SC|E(t|E=e)eSTe(t)λT0(t) exp (βce)e[set membership]{0,1}SC|E(t|E=e)STe(t)λT0(t) exp (βce)}dt]}=0

The result then follows by noting that βcdir solves

E{Umr(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0

which is triply robust by Theorem 2.

PROOF OF THEOREM 4:

It is straightforward to verify that [var pi]j (t) is of the form of κ(l) with L(e) = hj(e) and O = D* (t). Thus,

[var pi]1(t)=e=01eE{SC|E(t|E=e)×ST|E,M,X(t|E=e,M,X)×fM|E,X(m|E=0,X)dμ(m)}[var pi]2(t)=e=01E{SC|E(t|E=e)×ST|E,M,X(t|E=e,M,X)×fM|E,X(m|E=0,X)dμ(m)}

Under the assumed structural model, and the consistency, sequential ignorability and positivity assumptions,

E[{dN*(t)EβadirD*(t)dt}W hj(E)]=e=01hj(e)λT0(t)dtE{SC|E(t|E=e)×ST|E,M,X(t|E=e,M,X)×fM|E,X(m|E=0,X)dμ(m)}

proving the result.

PROOF OF THEOREM 5:

The proof is similar to that of Theorem 2, by applying Lemma A.1 to the three functionals [var pi]1(t)=[var pi]1mr, [var pi]2(t)=[var pi]2mr(t) and E[{dN*(t)EβadirD*(t)dt}W hj(E)].

PROOF OF THEOREM 6:

The proof is similar to that of Theorem 3, by applying Lemma A.2 to the four key functionals:

ϕjmr(t)=e[set membership]{0,1}hj(e)E{SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=e,X)}dμ(m),j=1, 2

and thus L(e) = hj(e) respectively and O = D* (t) ; furthermore,

E[{dN*(t)E(βadir+βaind)D*(t)dt}fE|X1(E|X){Eϕ1mr(t)ϕ2mr(t)}]=E[{dN*(t)EβaD*(t)dt}fE|X1(E|X){Eϕ1mr(t)ϕ2mr(t)}]=e[set membership]{0,1}{eϕ1mr(t)ϕ2mr(t)}[{SC|E(t|E=e)×fT|E,M,X(t|E=e,M=m,X)}×fM|E,X(m|E=e,X)dμ(m)]e[set membership]{0,1}e(βadir+βaind){eϕ1mr(t)ϕ2mr(t)}[SC|E(t|E=e)×ST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=e,X)dμ(m)]

is a difference of two σ (l) – functionals with respectively L(e)={eϕ1mr(t)ϕ2mr(t)} and O = dN* (t) for the first functional, and L(e)

=e(βadir+βaind){eϕ1mr(t)ϕ2mr(t)}

and O = D*(t) for the second functional. This implies that if if either fE|X=fE|X or {ST|E,M,X,fM|E,X}={ST|E,M,X,fM|E,X}

E{Pmr(βadir,βaind;ST|E,M,X,fM|E,X,fE|X,SC|E)}=λT0(t)e=01SC|E(t|E=e)STe(t){eϕ1mr(t)ϕ2mr(t)}dt=λT0(t)e=01SC|E(t|E=e)STe(t){ee=01eSC|E(t|E=e)STe(t)e=01SC|E(t|E=e)STe(t)}dt

= 0 and therefore Pmr ( βadir, βaind; ST|E,M,X, fM|E,X, fE|X, SC|E) is a doubly robust estimating function for βa=βadir+βaind.

The result then follows by noting that βaind=βaβadir and βadir solves

E{Zmr(βadir;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0

which is triply robust by Theorem 5, and thus, βaind solves

E{Pmr(βadir,βaind;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0 provided one of the three conditions given in the theorem hold.

PROOF OF LEMMA 1:

We observe that

ST1,m|E,X(t|E=e,X=x)=ST1,m|E,M,X(t|E=e,M=m,X=x)fM|E,X(m|E=e,X=x)+ST1,m|E,M,X(t|E=e,Mm,X=x){1fM|E,X(m|E=e,X=x)}=exp{0t(λT1,m|E,M,X(u|E=e,M=m,X=x))du}×fM|E,X(m|E=e,X=x)+exp{0t(λT1,m|E,M,X(u|E=e,Mm,X=x)du)}×{1fM|E,X(m|E=e,X=x)}=exp{0t(λT1,m|E,M,X(u|E=e,M=m,X=x)du)}×[fM|E,X(m|E=e,X=x)+exp{0tγ(u,e,m,x)du}×{1fM|E,X(m|E=e,X=x)}]

Thus, by ignorability of E, we obtain

exp{0t(λT1,m|E,M,X(u|E=0,M=m,X=x)du)}=exp{0t(λT1,m|E,M,X(u|E=1,M=m,X=x)du)}×[fM|E,X(m|E=1,X=x)+exp{0tγ(u,1,m,x)du}{1fM|E,X(m|E=1,X=x)}][fM|E,X(m|E=0,X=x)+exp{0tγ(u,0,m,x)du}{1fM|E,X(m|E=0,X=x)}]= exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}×[fM|E,X(m|E=1,X=x)+exp{0tγ(u,1,m,x)du}{1fM|E,X(m|E=1,X=x)}][fM|E,X(m|E=0,X=x)+exp{0tγ(u,0,m,x)du}{1fM|E,X(m|E=0,X=x)}]

proving the first result by consistency.

Furthermore, by differentiating with respect to t :

λT1,m|E,M,X(t|E=0,M=m,X=x)× exp{0t(λT1,m|E,M,X(u|E=0,M=m,X=x)du)}=λT|E,M,X(t|E=1,M=m,X=x)× exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}×δ(t,e,m,x)+δ˙(t,e,m,x)×δ(t,e,m,x)× exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}[left and right double arrow ]λT1,m|E,M,X(t|E=0,M=m,X=x)× exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}×δ(t,e,m,x)=λT|E,M,X(t|E=1,M=m,X=x)× exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}×δ(t,e,m,x)+δ˙(t,e,m,x)×δ(t,e,m,x)× exp{0t(λT|E,M,X(u|E=1,M=m,X=x)du)}[left and right double arrow ]λT1,m|E,M,X(t|E=0,M=m,X=x)=λT|E,M,X(u|E=1,M=m,X=x)δ˙(t,e,m,x)

proving the second part of the Lemma.

PROOF OF THEOREM 7:

By Lemma 1 and the assumptions of the theorem,

E[δa*(t,E,M,X){dN*(t)δ˙α*(t,E,M,X)D*(t)dt}W hj(E)]=E[{λT|E,M,X(t|E,M,X)δ˙α*(t,E,M,X)}δα*(t,E,M,X)D*(t)W hj(E)dt]=E[{λT|E,M,X(t|E,M,X)δ˙a*(t,E,M,X)}×SC|E(t|E)ST|E,M,X(t|E,M,X)×δa*(t,E,M,X)W hj(E)dt]=e[set membership]{0,1}m[set membership]SSC|E(t|e)E[{λT|E,M,X(t|e,m,X)δ˙α*(t,e,M,X)}×{ST|E,M,X(t|e,m,X)×δα*(t,e,M,X)}fM|E,X(m|E=0,X)hj(e)dt]=m[set membership]SSC|E(t|e)E[{λT|E,M,X(t|1,m,X)δ˙a*(t,1,M,X)}×{ST|E,M,X(t|1,m,X)×δα*(t|1,m,X)}fM|E,X(m|E=0,X)dμ(m)hj(1)dt]+m[set membership]SSC|E(t|e)E[λT|E,M,X(t|0,m,X)×ST|E,M,X(t|0,m,X)×fM|E,X(m|E=0,X)hj(0)dt]=m[set membership]SSC|E(t|e)E[λT1,M0|M0,X(t|M0=m,X)×ST1,M0|M0,X(t|M0=m,X)×fM|E,X(m|E=0,X)hj(1)dt]+m[set membership]SSC|E(t|e)E[λT0,M0|M0,X(t|M0=m,X)×ST0,M0|M0,X(t|M0=m,X)×fM|E,X(m|E=0,X)hj(0)dt]=e[set membership]{0,1}SC|E(t|e)λTe,M0(t)STe,M0(t)hj(e)dt=e[set membership]{0,1}SC|E(t|e)λT0,M0(t) exp (βcdir)STe,M0(t)hj(e)dt

One can similarly show that

E{D*(t)Whj(E)δα*(t,E,M,X) exp (βcdirE)}=e[set membership]{0,1}SC|E(t|e) exp (eβcdir)STe,M0(t)hj(e)dt

which implies the result since

E{Uw(βcdir,α*)}=e[set membership]{0,1}SC|E(t|e)λT0,M0(t)exp(βcdir)STe,M0(t)×{eΣe[set membership]{0,1}SC|E(t|e)exp(eβcdir)STe,M0(t)eΣe[set membership]{0,1}SC|E(t|e)exp(eβcdir)STe,M0(t)dt}dt=0

For the case of an additive structural model

E[W{dN*(t)δ˙α*(t,E,M,X)D*(t)dtEβadirD*(t)dt}δα*(t,E,M,X)hj(E)]=E[{λT|E,M,X(t|E,M,X)δ˙α*(t,E,M,X)Eβadir}δα*(t,E,M,X)D*(t)Whj(E)dt]=E[{λT|E,M,X(t|E,M,X)δ˙α*(t,E,M,X)Eβadir}δα*(t,E,M,X)D*(t)Whj(E)dt]=E[{λT|E,M,X(t|E,M,X)δ˙α*(t,E,M,X)Eβadir}×SC|E(t|E)ST|E,M,X(t|E,M,X)δα*(t,E,M,X)Whj(E)dt]=e[set membership]{0,1}m[set membership]SSC|E(t|e)E[{λT|E,M,X(t|e,m,X)δ˙α*(t,e,M,X)eβadir}×{ST|E,M,X(t|e,m,X)×δα*(t,e,M,X)}×fM|E,X(m|E=0,X)hj(e)dt]=m[set membership]SSC|E(t|0)E[λT|E,M,X(t|0,m,X){ST|E,M,X(t|0,m,X)}×fM|E,X(m|E=0,X)hj(0)dt]+m[set membership]SSC|E(t|1)E[{λT|E,M,X(t|1,m,X)δ˙α*(t,1,M,X)βadir}×{ST|E,M,X(t|1,m,X)×δα*(t|1,M,X)}×fM|E,X(m|E=0,X)hj(1)dt]=m[set membership]SSC|E(t|0)E[λT0,M0|M0,X(t|M0=m,X)×ST0,M0|M0,X(t|M0=m,X)×fM|E,X(m|E=0,X)dμ(m)hj(0)dt]+m[set membership]SSC|E(t|1)E[λT1,M0|M0,X(t|M0=m,X)βadir×{ST1,M0|M0,X(t|M0=m,X)}×fM|E,X(m|E=0,X)hj(1)dt]=SC|E(t|0)[λT0,M0(t)ST0,M0(t)hj(0)dt]+SC|E(t|1){λT1,M0(t)βadir}ST1,M0(t)hj(1)dt=λT0,M0(t){SC|E(t|0)ST0,M0(t)hj(0)dt+SC|E(t|1)ST1,M0(t)hj(1)dt}=λT0,M0(t)e[set membership]{0,1}SC|E(t|e)STe,M0(t)hj(e)dt

One can similarly show that

E{D*(t)Whj(E)δα*(t,E,M,X)}=e[set membership]{0,1}SC|E(t|e)STe,M0(t)hj(e)dt

which gives the result.

MULTIPLY ROBUST ESTIMATING FUNCTION UNDER DEPENDENT CENSORING:

Let

R*(t,H;βcdir)=R*(t,H;βcdir,ST|E,M,X,fM|E,X,fE|X,SC|E,M,X)={D*(t)SC|E,M,X(t|E,M,X)ST|E,M,X(t|E,M,X)}W*h(E)exp(βcdirE)+{eST|E,M,X(t|E=e,M=m,X)fM|E,X(m|E=0,X)h(e)exp(βcdire)dμ(m)}+I(E=0)f(E|X)eST|E,M,X(t|E=e,M,X)h(e)exp(βcdire)I(E=0)f(E|X)[eST|E,M,X(t|E=e,M=m,X)fM|E,X(m|E=0,X)h(e)exp(βcdire)dμ(m)]

and let ξjmr(t;βcdir)=ξjmr,*(t;βcdir,ST|E,M,X,fM|E,X,fE|X,SC|E,M,X)=E{R*(t,Hj;βcdir)}, j = 1, 2;

Then, ξjmr*(t;βcdir)=ξj*(t;βcdir), j = 1, 2, and in fact, ξjmr*(t;βcdir)=E{R*(t,Hj;βcdir)}=ξj(t;βcdir) provided that SC|E,M,X=SC|E,M,X and at least one of the following three conditions hold: either {ST|E,M,X,fM|E,X}={ST|E,M,X,fM|E,X} or {ST|E,M,X,fE|X}={ST|E,M,X,fE|X}, or {fM|E,X,fE|X}={fM|E,X,fE|X}. One may use this result to establish the following theorem:

Theorem A.1: Under the consistency, sequential ignorability and positivity assumptions, Umr*(βcdir)=Umr*(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E,M,X) is an unbiased estimating function for βcdir, where

Umr*(βcdir)={dN*(t)+SC|E(t|E)×dST|E,M,X(t|E,M,X)}W*{Eξ1mr(t;βcdir)ξ2mr(t)}[double integral operator]e[set membership]{0,1}[dST|E,M,X(t|E=e,m,X)fM|E,X(m|E=0,X)×{eξ1mr(t;βcdir)ξ2mr(t)}]dμ(m)I(E=0)fE|X(E|X)e[set membership]{0,1}[{dST|E,M,X(t|E=e,M,X)}×{eξ1mr(t;βcdir)ξ2mr(t;βcdir)}]+I(E=0)fE|X(E|X)[double integral operator]Σe[set membership]{0,1}[{dST|E,M,X(t|E=e,M=m,X)×fM|E,X(m|E=0,X)}×{eξ1mr(t;βcdir)ξ2mr(t;βcdir)}]dμ(m)

Furthermore,

E{Umr*(βcdir;ST|E,M,X,fM|E,X,fE|X,SC|E,M,X)}=0
(16)

if SC|E,M,X=SC|E,M,X and one but not necessarily all three of the following conditions holds: either {ST|E,M,X,fM|E,X}={ST|E,M,X,fM|E,X} or {ST|E,M,X,fE|X}={ST|E,M,X,fE|X}, or {fM|E,X,fE|X}={fM|E,X,fE|X}.

The proof of this theorem is similar to that of Theorem 2.

DOUBLY ROBUST SENSITIVITY ANALYSIS FOR SEMI-PARAMETRIC REGRESSION MODELS FOR SURVIVAL DATA:

We propose a sensitivity analysis that is doubly robust under the Cox PH model. For each fixed α=α*, consider the following modified estimating function for βcdir:

Uw,dr(βcdir,α*)=Uw,dr(βcdir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E)={δα*(t,E,M,X){dN*(t)δ˙α*(t,E,M,X)D*(t)dtEβadirD*(t)δ˙α*(t,E,M,X)dt}W×{Eχ1(t;βcdir,α*)χ2(t;βcdir,α*)}×δα*(t,E,M,X)SC|E(t|E){fT|E,M,X(t|E,M,X)dtδ˙α*(t,E,M,X)ST|E,M,X(t|E,M,X)dtEβadirST|E,M,X(t|E,M,X)dtδ˙α*(t,E,M,X)dt}W}×{Eχ1(t;βcdir,α*)χ2(t;βcdir,α*)}×{+m[set membership]Se[set membership]{0,1}δα*(t,e,m,X)SC|E(t|e)fT|E,M,X(t|e,m,X)dtδ˙α*(t,e,m,X)ST|E,M,X(t|e,m,X)dteβadirδ˙α*(t,e,m,X)ST|E,M,X(t|e,m,X)dt×{eχ1(t;βcdir,α*)χ2(t;βcdir,α*)fM|E,X(m|E=0,X)}}

with

χj(t;βcdir,α*)=E{(D*(t)SC|E(t|E)×ST|E,M,X(t|E,M,X)dt)×Whj(E)δα*(t,E,M,X)exp(βcdirE)}+m[set membership]Se[set membership]{0,1}(SC|E(t|e)ST|E,M,X(t|e,m,X)dt)×fM|E,X(m|E=0,X)hj(e)δα*(t,e,m,X)exp(βcdire)

One can then easily verify that Uw,dr(βcdir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E) is doubly robust in the sense that

E{Uw,dr(βcdir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0 if either ST|E,M,X=ST|E,M,X or fE|X=fE|X.

For the additive hazards model, we propose to use the following modified estimating function of βadir:

Zw,dr(βadir,α*)=Zw,dr(βadir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E)={δα*(t,E,M,X)×{dN*(t)δ˙α*(t,E,M,X)D*(t)dt}W×{Eζ1(t,α*)ζ2(t;α*)}δα*(t,E,M,X)SC|E(t|E)×{fT|E,M,X(t|E,M,X)dtδ˙α*(t,E,M,X)ST|E,M,X(t|E,M,X)dt}W}×{Eζ1(t,α*)ζ2(t;α*)}       +m[set membership]Se[set membership]{0,1}δα*(t,e,m,X)SC|E(t|e)×{fT|E,M,X(t|e,m,X)dtδ˙α*(t,e,m,X)ST|E,M,X(t|e,m,X)dt×{eζ1(t,α*)ζ2(t;α*)}fM|E,X(m|E=0,X)}

with

ζj(t,α*)=E{(D*(t)SC|E(t|E)ST|E,M,X(t|E,M,X)dt)×Whj(E)δα*(t,E,M,X)}+m[set membership]Se[set membership]{0,1}(SC|E(t|e)ST|E,M,X(t|e,m,X)dt)×fM|E,X(m|E=0,X)hj(e)δα*(t,e,m,X)

One can easily verify that Zw,dr(βadir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E) is doubly robust in the sense that

E{Zw,dr(βadir,α*;ST|E,M,X,fM|E,X,fE|X,SC|E)}=0 if either ST|E,M,X=ST|E,M,X or fE|X=fE|X.

References

[1] Avin C, Shpitser I, Pearl J. Identifiability of path-specific effects. IJCAI-05, Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence; Edinburgh, Scotland, UK. July 30–August 5, 2005; 2005. pp. 357–363.
[2] Bang H, Robins J. Doubly robust estimation in Missing data and causal inference models. Biometrics. 2005;61:692–972. doi: 10.1111/j.1541-0420.2005.00377.x. [PubMed] [Cross Ref]
[3] van der Laan MJ, Robins JM. Unified Methods for Censored Longitudinal Data and Causality. Springer Verlag; New York: 2003.
[4] van der Laan M, Petersen M. Direct Effect Models. 2005. UC Berkeley Division of Biostatistics Working Paper Series Working Paper 187. http://www.bepress.com/ucbbiostat/paper187.
[5] Imai K, Keele L, Yamamoto T. Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science. 2010a;25:51–71. doi: 10.1214/10-STS321. [Cross Ref]
[6] Imai K, Keele L, Tingley D. A General Approach to Causal Mediation Analysis. Psychological Methods. 2010b Dec;15(4):309–334. doi: 10.1037/a0020761. [PubMed] [Cross Ref]
[7] Lin DY, Ying Z. Semiparametric analysis of the additive risk model. Biometrika. 1994;81:61–71. doi: 10.1093/biomet/81.1.61. [Cross Ref]
[8] Newey W. Semiparametric efficiency bounds. Journal of Applied Econometric. 1994;5(2):99–135. doi: 10.1002/jae.3950050202. [Cross Ref]
[9] Pearl J. Direct and indirect effects. Proceedings of the 17th Annual Conference on Uncertainty in Artificial Intelligence (UAI-01); San Francisco, CA. 2001. pp. 411–42. Morgan Kaufmann.
[10] Pearl J. The Mediation Formula: A guide to the assessment of causal pathways in nonlinear models. 2011. Technical report. http://ftp.cs.ucla.edu/pub/stat_ser/ [PubMed]
[11] Robins JM, Greenland S. Identifiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:143–155. doi: 10.1097/00001648-199203000-00013. [PubMed] [Cross Ref]
[12] Robins JM, Rotnitzky A. Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell N, Dietz K, Farewell V, editors. AIDS Epidemiology - Methodological Issues. Boston, MA: Birkhäuser; 1992. pp. 297–331.
[13] Robins JM, Rotnitzky A, Scharfstein D. Sensitivity Analysis for Selection Bias and Unmeasured Confounding in Missing Data and Causal Inference Models. In: Halloran ME, Berry D, editors. Statistical Models in Epidemiology: The Environment and Clinical Trials. Vol. 116. NY: Springer-Verlag; 1999. pp. 1–92. IMA. [Cross Ref]
[14] Robins JM. Marginal structural models 1997. Proceedings of the American Statistical Association; 1998. pp. 1–10. Section on Bayesian Statistical Science, Reproduced courtesy of the American Statistical Association.
[15] Robins J. Semantics of causal DAG models and the identification of direct and indirect effects. In: Green P, Hjort N, Richardson S, editors. Highly Structured Stochastic Systems. Oxford, UK: Oxford University Press; 2003. pp. 70–81.
[16] Robins JM, Richardson TS. Alternative graphical causal models and the identification of direct effects. In: Shrout P, editor. To appear in Causality and Psychopathology: Finding the Determinants of Disorders and Their Cures. Oxford University Press; 2010.
[17] Scharfstein DO, Rotnitzky A, Robins JM. Rejoinder to comments on “Adjusting for non-ignorable drop-out using semiparametric non-response models” Journal of the American Statistical Association. 1999;94:1096–1120. doi: 10.2307/2669923. Journal of the American Statistical Association, 94:1121–1146. [Cross Ref]
[18] Tchetgen Tchetgen EJ, Shpit I. Semiparametric Theory for Causal Mediation Analysis: efficiency bounds, multiple robustness and sensitivity analysis. 2011b. Jun 3rd, 2011. http://www.bepress.com/harvardbiostat/paper130/
[19] Tchetgen Tchetgen EJ, Shpit I. Semiparametric Estimation of Models for Natural Direct and Indirect Effects. 2011a. Jun 3rd, 2011. http://www.bepress.com/harvardbiostat/paper129/
[20] Tsiatis AA. Semiparametric Theory and Missing Data. Springer. Verlag; New York: 2006.
[21] VanderWeele TJ. Marginal structural models for the estimation of direct and indirect effects. Epidemiology. 2009;20:18–26. doi: 10.1097/EDE.0b013e31818f69ce. [PubMed] [Cross Ref]
[22] VanderWeele TJ. Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology. 2010;21:540–551. doi: 10.1097/EDE.0b013e3181df191c. [PubMed] [Cross Ref]

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