Estimating mean potential outcome under adaptive treatment length strategies in continuous time

Abstract

An adaptive treatment length strategy is a sequential stage-wise treatment strategy where a subject’s treatment begins at baseline and one chooses to stop or continue treatment at each stage provided the subject has been continuously treated. The effects of treatment are assumed to be cumulative and, therefore, the effect of treatment length on clinical endpoint, measured at the end of the study, is of primary scientific interest. At the same time, adverse treatment-terminating events may occur during the course of treatment that require treatment be stopped immediately. Because the presence of a treatment-terminating event may be strongly associated with the study outcome, the treatment-terminating event is informative. In observational studies, decisions to stop or continue treatment depend on covariate history that confounds the relationship between treatment length on outcome. We propose a new risk-set weighted estimator of the mean potential outcome under the condition that time-dependent covariates update at a set of common landmarks. We show that our proposed estimator is asymptotically linear given mild assumptions and correctly specified working models. Specifically, we study the theoretical properties of our estimator when the nuisance parameters are modeled using either parametric or semiparametric methods. The finite sample performance and theoretical results of the proposed estimator are evaluated through simulation studies and demonstrated by application to the Enhanced Suppression of the Platelet Receptor IIb/IIIa with Integrilin Therapy (ESPRIT) infusion trial data.

1 |. INTRODUCTION

An adaptive treatment length strategy is a sequential dynamic treatment strategy to give treatment continuously from baseline to a point in time when a subject chooses to stop treatment or until a competing treatment-terminating event occurs, whichever comes first. In both the adaptive treatment length strategy and the observational study, treatment length is determined sequentially as decisions to stop or continue treatment are repeated over time, as long as the subject has been treated continuously up to that time when a decision to stop or continue is made. In the observational data, the decision to stop or continue treatment also depends on covariate history up to the current time. In short, the broad scientific objective is to describe how mean outcome changes as a function of treatment length when the decision to stop or continue throughout the treatment window depends on a history of tailoring information that induces confounding by indication in the observational study. Applications of adaptive treatment length strategies include informative treatment discontinuation in infusion studies (Johnson and Tsiatis, 2004, 2005; Zhang et al., 2011; Yang et al., 2018), switching antiretroviral therapies (Johnson et al., 2013), and palliative care for patients suffering from neuromuscular diseases (Lu and Johnson, 2017). By swapping treatment initiation for treatment discontinuation and the role of intermediate events, Hu et al. (2018) showed how to extend these ideas to evaluate strategies for initiating combination antiretroviral therapies in the presence of comorbidities, such as tuberculosis (see also, Robins et al., 2008).

Estimation of causal estimands for adaptive treatment strategies (ATSs), or dynamic treatment regimes, from longitudinal data with time-varying confounders has well-known challenges with, by now, equally well-known methods to overcome such challenges. The successful application of existing methods to adaptive treatment length strategies depends critically on the observed data and methodological assumptions. To fix ideas, a dynamic treatment regime ${\overline{d}}_K$ is a sequence of decision rules ${\overline{d}}_K=\left(d_t,t\in \mathcal{T}\right)$, where the decision rule d_t : dom(H_t) → dom(A_t), H_t is the combined treatment and covariate histories up to time t, A_t is treatment selection at time t, dom(·) is short-hand for the domain of the argument and the time domain is $\mathcal{T}$. Compared with static regimes that are fixed a priori at the beginning of the study, dynamic regimes are adaptive in that treatment selection at time t depends on a subject’s history of data up to that point. If ${\mathcal{A}}_K$ is the collection of all possible treatment vectors ${\overline{\mathbf{a}}}_K=\left(a_1,\dots ,a_K\right)$, then the potential outcome of interest under regime ${\overline{d}}_K$ is

$$Y\left({\overline{d}}_K\right)=\sum _{{\overline{\mathbf{a}}}_K\in {\mathcal{A}}_K}Y\left({\overline{\mathbf{a}}}_K\right)I\left\{d_1\left(H_1\right)=a_1\right\}\cdots I\left\{d_K\left(H_K\right)=a_K\right\},$$ (1)

where $Y\left({\overline{\mathbf{a}}}_K\right)$ is the potential response had a subject followed treatment regime ${\overline{\mathbf{a}}}_K$. More generally, the potential outcome $Y\left({\overline{d}}_K\right)$ is defined through the sequence of stochastic rules, say ${\overline{p}}_{{\overline{d}}_K}=(p_{1,{\overline{d}}_K},\dots ,p_{K,{\overline{d}}_K})$ with treatment assignment at the t-th stage in the dynamic regime given by $p_{t,{\overline{d}}_K}\left(a_t|H_t\right),t=1,\dots ,K$. Without loss of generality, we will refer to the potential outcome $Y\left({\overline{d}}_K\right)$ under both general stochastic treatment assignment mechanisms and the degenerate case when $p_{t,{\overline{d}}_K}\left(a_t|H_t\right)=I\left\{d_t\left(H_t\right)=a_t\right\}$. The objective is to estimate the value function, $\mathbb{E}\left[Y\left({\overline{d}}_K\right)\right]$. In our application to adaptive treatment length strategies in Section 5, treatment assignment A_t is a binary indicator of whether treatment has been stopped by time t, and C_t is the indicator that treatment has been stopped due to involuntary adverse treatment-terminating events by time t. Then our objective in discrete time may be summarized as estimating $\mathbb{E}\left[Y\left({\overline{d}}_K\right)\right]$ for ATS ${\overline{d}}_K$ defined by the rule, d_t(H_t) = 1 − I(t ≤ t^o)I(C_t = 0), that treats continuously until provider stops treatment at t^o or until a treatment-terminating event, whichever comes first.

In the observational treatment length study, decisions to stop treatment by choice depend on time-varying covariate information. Because time-dependent covariates occur downstream of earlier treatment choices but prior to the final outcome of interest, this leads to a classic case of confounding and one challenge to overcome in estimating the causal estimand. A second challenge that our proposed method attempts to overcome is to estimate the causal estimand when the time domain $\mathcal{T}$ is an interval. When the domain $\mathcal{T}$ is a finite number of time points, other authors (Johnson and Tsiatis, 2004; Lu and Johnson, 2017) have shown how to apply causal inference methods for dynamic treatment regimes (Robins, 1997; Robins et al., 2000; Murphy et al., 2001) to estimate the causal estimand from observational data. However, when the treatment assignment is a continuous random variable or, in our case, the domain $\mathcal{T}$ is an interval, the literature of relevant methodology is less robust. Johnson and Tsiatis (2005) proposed an inverse probability weighted estimator of $\mathbb{E}\left[Y\left({\overline{d}}_K\right)\right]$ under a continuous treatment assignment mechanism when the value function is assumed to be a known parametric function in treatment length. Lu and Johnson (2015) proposed a direct estimator of $\mathbb{E}\left[Y\left({\overline{d}}_K\right)\right]$ assuming baseline confounding but no time-varying confounders. Methods for continuous treatment assignment mechanisms in related problems have been proposed elsewhere in the literature (e.g., Lok, 2008; Zhang et al., 2011; Díaz and van der Laan, 2013; Kennedy et al., 2017). However, these methods are not directly applicable to our problem as they ignore the informative nature of the treatment-terminating event.

In this paper, we propose an estimator that is initially motivated by Robins’ (1986) G-computation formula for estimating the mean potential outcome of a static treatment regime from longitudinal data extended to the case of continuously varying covariates and treatments (Gill and Robins, 2001). One practical difficulty with G-estimation in continuous time is modeling general covariate histories that can change in infinitesimals of time. To avoid this challenge, our method uses a variation of risk-set weighting and, as motivated by our application, assumes that covariate histories are updated at a finite number of landmarks. Then, our estimator is constructed as stochastic integrals of weighted predictions from regression models for the set of potential outcomes as functions of covariate histories. Because our estimator does not model the treatment assignment mechanism, our method may be applied to data irrespective of the domain $\mathcal{T}$ for the treatment length random variable. We study the asymptotic behavior of the proposed estimator and show that, under regularity conditions, our estimator is asymptotically linear.

2 |. BACKGROUND

While the text in Section 1 introduced basic ideas of adaptive treatment length strategies using notational conventions for ATSs in discrete time, the methods here are tailored for causal inference in continuous time for which appropriate modifications are needed. Section 2.1 gives the notation used in the sequel followed by g-formula principles in Sections 2.2 and 2.3 used to motivate our proposed approach in Section 3.

2.1 |. Notation

In our setup, the presumption is that treatment decisions are made repeatedly in infinitesimals [t, t + dt) throughout the treatment window $\mathcal{T}=[0,\tau ]$, provided that the subject has been treated up to time t. Hence, let A_t represent the counting process that treatment is stopped by time t, ${\overline{\mathbf{A}}}_t=\left\{A_s,s\in [0,t]\right\}$ is treatment history up to and including time t, ${\overline{\mathbf{a}}}_t$ is a sample path of treatment assignment to time t, and let ${\mathcal{A}}_{\tau }$ be the collection of all sample paths. Similarly, define $C_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)$ as the random indicator that a treatment-terminating event has occurred by time t if a patient is following treatment sequence ${\overline{\mathbf{a}}}_{t^{-}}$; in particular, let $C_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)$ realize the value +1 if treatment has stopped by time t due to treatment-terminating event. Let ${\mathbf{X}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)$ be a potential auxiliary time-dependent covariate if following treatment sequence ${\overline{\mathbf{a}}}_{t^{-}}$ and ${\overline{\mathbf{X}}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)$ is the covariate history through time t. The potential outcome $Y\left({\overline{\mathbf{a}}}_{\tau }\right)$ is defined as the outcome if a patient follows the treatment sequence ${\overline{\mathbf{a}}}_{\tau }$. The complete set of potential treatment-terminating events and outcome for the adaptive treatment length study is $\left\{C_t\left({\overline{\mathbf{a}}}_{t^{-}}\right),Y\left({\overline{\mathbf{a}}}_{\tau }\right),{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }\right\}$ and the complete collection of auxiliary covariates is $\left\{{\mathbf{X}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right),{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }\right\}$. Thus, if we let ${\mathbf{L}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)=\left\{C_t\left({\overline{\mathbf{a}}}_{t^{-}}\right),{\mathbf{X}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right)\right\}$, then the complete set of potential random variables is $O_{po}=\left\{{\mathbf{L}}_t\left({\overline{\mathbf{a}}}_{t^{-}}\right),Y\left({\overline{\mathbf{a}}}_{\tau }\right),{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau },t\in [0,\tau ]\right\}$. By the stable unit treatment value assumption (SUTVA; Rubin, 1978), a subject’s treatment assignment is not affected by the treatment assignment of others. Hence, the observed data are $\mathbf{O}=\left\{{\mathbf{L}}_t\left({\overline{\mathbf{A}}}_{t^{-}}\right),A_t,Y\left({\overline{\mathbf{A}}}_{\tau }\right),t\in [0,\tau ]\right\}$ and written concisely as O = {L_t, A_t, Y, t ∈ [0, τ]}. The observed data for all n subjects are O₁, … , O_n and assumed to be independent random vectors from a common distribution.

In order to identify the causal estimand, we require that treatment assignment at time t be a function of covariate and treatment histories up to time t but not a function of information beyond time t. This is known as the sequential randomization assumption and is summarized in (2),

$$A_t\perp \left\{{\mathbf{L}}_{t+dt}\left({\overline{\mathbf{a}}}_t\right),\dots ,{\mathbf{L}}_{\tau }\left({\overline{\mathbf{a}}}_{{\tau }^{-}}\right),Y\left({\overline{\mathbf{a}}}_{\tau }\right)\right\}|H_t,$$ (2)

where $H_t=\left({\overline{\mathbf{L}}}_t,{\overline{\mathbf{A}}}_{t^{-}}\right)$. To describe the treatment assignment mechanism in the observational study, we define the instantaneous probability of stopping a subject’s treatment by choice at time t, provided the subject is still receiving treatment at time t, as

$${\lambda }_{t,obs}\left(H_t\right)=\underset{dt\downarrow 0}{\lim }P\left(d{A}_t=1|A_{t^{-}}=0,C_t=0,{\overline{\mathbf{X}}}_t\right)/dt.$$ (3)

Using our above notation where dA_t is a Bernoulli event for stopping treatment by choice in [t, t + dt), we heuristically define the intensity of treatment assignment in the observational study as $P\left(d{A}_t=1|H_t\right)=d{\tilde{\mathrm{\Lambda }}}_{t,obs}\left(H_t\right)$, where $d{\tilde{\mathrm{\Lambda }}}_{t,obs}\left(H_t\right)=I\left(A_{t^{-}}=0,C_t=0\right){\lambda }_{t,obs}\left(H_t\right)dt$. Using this notation, the treatment assignment mechanism in the interval [t, t + dt) from the observational study is written as $p_{t,obs}\left(d{A}_t|H_t\right)=d{\tilde{\mathrm{\Lambda }}}_{t,obs}{\left(H_t\right)}^{d{A}_t}\{1-d{\tilde{\mathrm{\Lambda }}}_{t,obs}\left(H_t\right)\}$, and under the conditional independence assumption in (2), $p_{t,obs}\left(\cdot |H_t\right)=p_{t,obs}\left(\cdot |H_{\tau },Y\left({\overline{\mathbf{a}}}_{\tau }\right)\right)$.

Analogous to treatment assignment in the observational study, we define treatment assignment in the ATS through integrated intensity processes, that is, $p_{t,{\overline{d}}_{\tau }}\left(d{A}_t|H_t\right)=d{\tilde{\mathrm{\Lambda }}}_{t,{\overline{d}}_{\tau }}{\left(H_t\right)}^{d{A}_t}\{1-d{\tilde{\mathrm{\Lambda }}}_{t,{\overline{d}}_{\tau }}\left(H_t\right)\}$, where $P_{{\overline{d}}_{\tau }}\left(d{A}_t=1|H_t\right)=d{\tilde{\mathrm{\Lambda }}}_{t,{\overline{d}}_{\tau }}\left(H_t\right)$ and $P_{{\overline{d}}_{\tau }}$ is probability measure in the ATS. Consequently, using the treatment assignment mechanism in the ATS, $\left\{p_{t,{\overline{d}}_{\tau }}\left(\cdot |H_t\right),t\in [0,\tau ]\right\}$, we may define the potential outcome

$$Y\left({\overline{d}}_{\tau }\right)=\sum _{{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }}Y\left({\overline{\mathbf{a}}}_{\tau }\right)\underset{[0,\tau ]}{\pi }{p}_{t,{\overline{d}}_{\tau }}\left(a_t(dt)|H_t\right),$$

where $\underset{[0,\tau ]}{\pi }$ is the product integral from 0 to τ and the target causal estimand in continuous time is the value function $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$.

2.2 |. G-computation in continuous time

Robins’ (1986) G-computation method is a powerful technique that is widely used to infer the causal effect from longitudinal observational data. In principle, Robins’ G-computation formula is used to express the distribution of potential outcome on a dynamic treatment regime as nested functions of the observed data. A plug-in estimator is derived by replacing unknown distributions with parametric families of statistical models, estimating unknown parameters from the observed data, and evaluating the causal estimand with fitted quantities. In our setting, where the time-varying covariates are measured only up to the treatment stopping time, the G-computation method is challenging because it requires predictions from correctly specified models of complex stochastic processes in continuous time.

To gain a better insight into the challenge of G-computation in our setting and the proposed remedy, we describe the causal estimand and likelihood for the adaptive treatment length problem. Now, for a specified ATS, ${\overline{d}}_{\tau }$, the causal estimand $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$ may be written as

$$\begin{gathered} \sum _{{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }}{\int }_{{\overline{\mathbf{l}}}_{\tau }}{\int }_{y}y{\mathcal{P}}_{{\overline{A}}_{\tau }\left({\overline{d}}_{\tau }\right)}\left(h_{{\tau }^{+}}\right) \\ \times \left\{f_Y\left(y|{\overline{\mathbf{a}}}_{\tau },{\overline{\mathbf{l}}}_{\tau }\right)f_{{\overline{L}}_{\tau }}\left({\overline{\mathbf{l}}}_{\tau }|{\overline{\mathbf{a}}}_{{\tau }^{-}}\right)\right\}dydv\left({\overline{\mathbf{l}}}_{\tau }\right), \end{gathered}$$ (4)

where ${\sum }_{{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }}$ indicates all treatment paths ${\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }$ consistent with ${\overline{d}}_{\tau }$, ${\mathcal{P}}_{{\overline{A}}_{\tau }\left({\overline{d}}_{\tau }\right)}\left(H_{{\tau }^{+}}\right)=\underset{{}_{[0,\tau ]}}{\pi }{p}_{t,{\overline{d}}_{\tau }}\left(d{A}_t\mid H_t\right)$ with $H_{{\tau }^{+}}\equiv \left(A_{\tau },H_{\tau }\right)$, f_Y is the conditional density of the endpoint Y given the entire covariate and treatment histories, and $f_{{\overline{L}}_{\tau }}$ is the conditional density of ${\overline{\mathbf{L}}}_{\tau }$ given treatment history ${\overline{\mathbf{A}}}_{{\tau }^{-}}={\overline{\mathbf{a}}}_{{\tau }^{-}}$, and ν is the dominating measure for ${\overline{\mathbf{L}}}_{\tau }$. Without loss of generality, we write $f_{{\overline{L}}_{\tau }}$ as an infinite product of joint conditional densities of dL_t in the infinitesimal [t, t + dt) given ${\overline{\mathbf{L}}}_{t^{-}}={\overline{\mathbf{l}}}_{t^{-}}$ and that a subject has been continuously treated up to time t, that is,

$$f_{{\overline{L}}_{\tau }}\left({\overline{\mathbf{l}}}_{\tau }|{\overline{\mathbf{a}}}_{{\tau }^{-}}\right)=\underset{[0,\tau ]}{\pi }P\left(d{\mathbf{L}}_t\in d{\mathbf{l}}_t|{\overline{\mathbf{A}}}_{t^{-}}={\overline{\mathbf{a}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}}={\overline{\mathbf{l}}}_{t^{-}}\right),$$ (5)

where $P\left(d{\mathbf{L}}_0\in d{\mathbf{l}}_0|A_{0^{-}},{\mathbf{L}}_{0^{-}}\right)=P\left(d{\mathbf{L}}_0\in d{\mathbf{l}}_0\right)$. The likelihood of the data is

$${\mathcal{L}}^{‡}={\mathcal{P}}_{{\overline{A}}_{\tau }(obs)}\left(H_{{\tau }^{+}}\right)\left\{f_Y\left(Y|{\overline{\mathbf{A}}}_{\tau },{\overline{\mathbf{L}}}_{\tau }\right)f_{{\overline{L}}_{\tau }}\left({\overline{\mathbf{L}}}_{\tau }|{\overline{\mathbf{A}}}_{{\tau }^{-}}\right)\right\},$$ (6)

where ${\mathcal{P}}_{{\overline{A}}_{\tau }(obs)}\left(H_{{\tau }^{+}}\right)=\underset{[0,\tau ]}{\pi }{p}_{t,obs}\left(d{A}_t|H_t\right)$. Because the estimand in (4) is a function of parameters in curly brackets in (6) and randomization probabilities $p_{t,{\overline{d}}_{\tau }}\left(\cdot |H_t\right)$ in ${\mathcal{P}}_{{\overline{A}}_{\tau }\left({\overline{d}}_{\tau }\right)}$ that are known a priori by design, the causal estimand can be estimated by replacing the unknown parameters with estimates informed by the likelihood ${\mathcal{L}}^{‡}$ in (6). This is, in principle, Robins’ (1986) G-computation method for the adaptive treatment length strategy ${\overline{d}}_{\tau }$ in continuous time.

When L_t is measured at a finite number of grid points, $f_{{\overline{L}}_{\tau }}$ in (5) reduces to a finite product of conditional densities (e.g., Murphy et al., 2001, (4.4)). But if any part of L_t is measured in continuous time, then $f_{{\overline{L}}_{\tau }}$ in (5) does not simplify and is difficult to model using familiar stochastic processes, in general. First, ${\overline{\mathbf{L}}}_{\tau }=\left\{{\mathbf{L}}_t,t\in [0,\tau ]\right\}$ is the combined histories of the terminating event ${\overline{\mathbf{C}}}_{\tau }$, that is, a point process, and covariate history ${\overline{\mathbf{X}}}_{\tau }$. Second, even if we could develop a method using a convenient Gaussian process, for example, to model covariate history ${\overline{\mathbf{X}}}_{\tau }$, it is rarely the case that the covariate history is a vector of continuous variables only and such a method would be of limited practical value. Positing a flexible model for the event history ${\overline{\mathbf{L}}}_{\tau }$ and estimating the parameters of such a model from a selective biased sample of subjects are two practical challenges for any direct plug-in estimator of (4) via G-computation formula. In Section 2.3, we show how a common set of landmark updates to covariate history simplifies the expression in (5).

2.3 |. $f_{{\overline{\mathbf{L}}}_{\mathbf{\tau }}}$ under common landmark updates to covariate histories

Now we assume that while treatment could be stopped at any time in the interval (0, τ], either by choice or treatment-terminating event, covariate information updates only at a finite, common number of landmarks, 0 < t₁ < t₂ < ⋯ < t_K < τ. To avail ourselves of this special simplifying structure on the covariate process, we split ${\overline{\mathbf{L}}}_t=\left({\overline{\mathbf{C}}}_t,{\overline{\mathbf{X}}}_t\right)$ into covariate history ${\overline{\mathbf{X}}}_t$ and history of terminating events ${\overline{\mathbf{C}}}_t$. We write $P\left(d{\mathbf{L}}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}}\right)$ as the product $P\left(d{\mathbf{X}}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}}\right)P\left(d{C}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}},{\mathbf{X}}_t\right)$, where $P\left(d{C}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}},{\mathbf{X}}_t\right)$ is the probabilistic mechanism for treatment-terminating event in the interval [t, t + dt). By assuming that the covariate history is left-continuous, we have that $P\left(d{C}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{C}}}_{t^{-}},{\overline{\mathbf{X}}}_t\right)=d{\tilde{\mathrm{\Lambda }}}_t^{\xi }{\left(H_t\right)}^{d{C}_t}\{1-d{\tilde{\mathrm{\Lambda }}}_t^{\xi }\left(H_t\right)\}$, where $P\left(d{C}_t=1|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}},{\mathbf{X}}_t\right)=d{\tilde{\mathrm{\Lambda }}}_t^{\xi }\left(H_t\right)$, $d{\tilde{\mathrm{\Lambda }}}_t^{\xi }\left(H_t\right)=I\left(A_{t^{-}}=0,C_{t^{-}}=0\right){\lambda }_t^{\xi }\left(H_t\right)dt$, and ${\lambda }_t^{\xi }\left(H_t\right)$ is the hazard of a treatment-terminating event at time t given covariate history and that the subject has been continuously treated to time t. Furthermore, a piece-wise constant assumption on covariate history implies that $P\left(d{\mathbf{X}}_t|{\overline{\mathbf{A}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}}\right)$ is a density function at the landmark t_j, given covariate history up to t_j and that a subject is continuously treated to time t_j for j = 1, … , K, and is degenerate otherwise. Define $P\left(d{\mathbf{X}}_t\in d{\mathbf{x}}_t|{\overline{\mathbf{X}}}_{t^{-}}={\overline{\mathbf{x}}}_{t^{-}}\right)=q_t\left({\mathbf{x}}_t|{\overline{\mathbf{x}}}_{t^{-}}\right)$ and $P\left(d{\mathbf{X}}_t\in d{\mathbf{x}}_t|{\overline{\mathbf{A}}}_{t^{-}}={\overline{\mathbf{a}}}_{t^{-}},{\overline{\mathbf{L}}}_{t^{-}}={\overline{\mathbf{l}}}_{t^{-}}\right)={\tilde{q}}_t\left({\mathbf{x}}_t|{\overline{\mathbf{a}}}_{t^{-}},{\overline{\mathbf{l}}}_{t^{-}}\right)$ with ${\tilde{q}}_t\left({\mathbf{x}}_t|{\overline{\mathbf{a}}}_{t^{-}},{\overline{\mathbf{l}}}_{t^{-}}\right)=q_t{\left({\mathbf{x}}_t|{\overline{\mathbf{x}}}_{t^{-}}\right)}^{I\left(a_{t^{-}}=0,c_{t^{-}}=0\right)}$. Then, the density $f_{{\overline{L}}_{\tau }}\left({\overline{\mathbf{l}}}_{\tau }(t)|{\overline{\mathbf{a}}}_{{\tau }^{-}}\right)$ simplifies to

$$\begin{gathered} f_{{\overline{L}}_{\tau }}^{*}\left({\overline{\mathbf{l}}}_{\tau }|{\overline{\mathbf{a}}}_{{\tau }^{-}}\right)=\underset{[0,\tau ]}{\pi }{\left\{d{\tilde{\mathrm{\Lambda }}}_t^{\xi }\left(H_t\right)\right\}}^{d{c}_t}\left\{1-d{\tilde{\mathrm{\Lambda }}}_t^{\xi }\left(H_t\right)\right\} \\ \times \prod _{j=1}^K{\tilde{q}}_{t_j}\left({\mathbf{x}}_{t_j}|{\overline{\mathbf{a}}}_{t_j^{-}},{\overline{\mathbf{l}}}_{t_j^{-}}\right). \end{gathered}$$ (7)

Therefore, under a piece-wise constant assumption on covariate history, we rewrite the likelihood ${\mathcal{L}}^{‡}$ in (6) as:

$$\mathcal{L}={\mathcal{P}}_{{\overline{A}}_{\tau }(obs)}\left(H_{{\tau }^{+}}\right)f_Y\left(Y|{\overline{\mathbf{A}}}_{\tau },{\overline{\mathbf{L}}}_{\tau }\right)f_{{\overline{L}}_{\tau }}^{*}\left({\overline{\mathbf{L}}}_{\tau }|{\overline{\mathbf{A}}}_{{\tau }^{-}}\right),$$ (8)

where $f_{{\overline{L}}_{\tau }}^{*}$ is given in (7). Our landmark condition on covariate history similarly affects our expression (4) for the causal estimand $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$, that is,

$$\begin{gathered} \sum _{{\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }}{\int }_{{\overline{\mathbf{l}}}_{\tau }}{\int }_{y}y{\mathcal{P}}_{{\overline{A}}_{\tau }\left({\overline{d}}_{\tau }\right)}\left(H_{{\tau }^{+}}\right)f_Y\left(y|{\overline{\mathbf{a}}}_{\tau },{\overline{\mathbf{l}}}_{\tau }\right) \\ \times f_{{\overline{L}}_{\tau }^{*}}^{*}\left({\overline{\mathbf{l}}}_{\tau }|{\overline{\mathbf{a}}}_{{\tau }^{-}}\right)dyd{\nu }^{*}\left({\overline{\mathbf{l}}}_{\tau }\right), \end{gathered}$$ (9)

where ν* is the dominating measure for ${\overline{\mathbf{L}}}_{\tau }$ under a piece-wise constant assumption on covariate history. Expression (9) leads directly to a plug-in G-computation estimator for general dynamic treatment regime ${\overline{d}}_{\tau }$ given by the randomization probabilities $p_{t,{\overline{d}}_{\tau }}\left(\cdot |H_t\right)$, where the treatment-terminating event can occur at any time in the interval (0, τ] and the covariate history is piece-wise constant with updates at landmarks t₁, … , t_K. Namely, if the functions in (9) can be estimated consistently, then replacing the unknown functions with their estimated counterparts leads to a plug-in estimator of $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$.

3 |. METHODS

In order to derive a plug-in estimator from expression (9), each conditional density $q_t\left({\mathbf{x}}_t|{\overline{\mathbf{x}}}_{t^{-}}\right)$ must be modeled for each landmark t_j. Such models are arguably easier in discrete time than in continuous time but can still be a formidable practical challenge because of the arbitrary nature of covariate history ${\overline{\mathbf{X}}}_t$. Also, because the covariate process fails to update at landmarks after treatment has stopped, we have missing data for all subjects with observed treatment length less than than target treatment length. Therefore, we have incomplete data to estimate the parameters of $q_t\left({\mathbf{x}}_t|{\overline{\mathbf{x}}}_{t^{-}}\right)$ and, because of informative treatment discontinuation, data get both sparser and more selective as the landmark t_j increases toward τ, the right-hand endpoint of the treatment window. Applying a simple degenerate data structure—for example, projection via last observation carried forward (Petersen et al., 2014) or projection through eligible subjects only (Lu and Johnson, 2017)—eases the computational burden but can lead to inconsistent estimation of $q_t\left({\mathbf{x}}_t|{\overline{\mathbf{x}}}_{t^{-}}\right)$, and hence $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$, when such assumptions are not true. The proposed estimator circumvents some challenges in the modeling of covariate histories through the idea of risk-set weighting.

Let the decision rule d_t(H_t) = 1 − I(t ≤ t^o)I(C_t = 0) define the adaptive treatment length strategy ${\overline{d}}_{\tau }$. We evaluate the causal estimand in (9) by considering each ${\overline{\mathbf{a}}}_{\tau }\in {\mathcal{A}}_{\tau }$ on a case-by-case basis. Write $m_Y\left({\overline{\mathbf{a}}}_{\tau },{\overline{\mathbf{c}}}_{\tau },{\overline{\mathbf{x}}}_{\tau }\right)=\mathbb{E}(Y|{\overline{\mathbf{A}}}_{\tau }={\overline{\mathbf{a}}}_{\tau },{\overline{\mathbf{C}}}_{\tau }={\overline{\mathbf{c}}}_{\tau },{\overline{\mathbf{X}}}_{\tau }={\overline{\mathbf{x}}}_{\tau })$, $G_t\left(H_t\right)=\underset{[0,t]}{\pi }\{1-d{\mathrm{\Lambda }}_S^{\xi }\left(H_S\right)\}$.

• (Case 1) Subject was continuously treated until the provider stopped treatment by choice at time t^o, that is, ${\overline{\mathbf{a}}}_{\tau }^{(1)}=\left\{a_s:=I\left\{s>t^{°}\right\},s\in [0,\tau ]\right\}$. The estimand in (9) evaluates to $\mathbb{E}[m_Y({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)},{\overline{\mathbf{X}}}_{t^o})G_{t^o}({\tilde{H}}_{t^o}^{(1)})]$, where ${\overline{\mathbf{c}}}_{\tau }^{(1)}=\left\{c_s:=0,s\in [0,\tau ]\right\}$, ${\tilde{H}}_t^{(1)}=({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)},{\overline{\mathbf{X}}}_t)$, the constrained treatment and covariate history H_t under $({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)})$.

• (Case 2) Subject was continuously treated until they experienced a random treatment-terminating event at some t < t^o. For each infinitesimal [t, t + dt), t < t^o, let ${\overline{\mathbf{a}}}_{\tau }^{(2).t}=\{a_s:=I\{s>t\},s\in [0,\tau ]\}$, ${\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t}=\{c_s:=I\{s>t\},s\in [0,\tau ]\}$. Then, at time t, t < t^o, (9) is $\mathbb{E}[m_Y({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{X}}}_t)d{\mathrm{\Lambda }}_t^{\xi }({\tilde{H}}_t^{(2)\cdot t})G_t({\tilde{H}}_t^{(2)\cdot t})]$, where ${\tilde{H}}_t^{(2)\cdot t}=({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{X}}}_t)$ is the constrained treatment and covariate history H_t under $({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t})$. The full expression under Case 2 is the sum over all infinitesimals [t, t + dt) from 0 to t^o.

Then, the target estimand (9) is the sum of the marginal expectations in Case 1 and Case 2. Writing the contribution to the summand for Case 2 as a stochastic integral with integrator $d{G}_t\left(H_t\right)=-G_t\left(H_t\right)d{\mathrm{\Lambda }}_t^{\xi }\left(H_t\right)$, the target estimand is

$$\begin{gathered} \mathbb{E}[m_Y\left({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)},{\overline{\mathbf{X}}}_{t^o}\right)G_{t^o}\left({\tilde{H}}_{t^o}^{(1)}\right) \\ -{\int }_0^{t^o}{m}_Y\left({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{X}}}_t\right)d{G}_t\left({\tilde{H}}_t^{(2)\cdot t}\right)], \end{gathered}$$ (10)

where the expectation in (10) is taken with respect to the distribution of covariate history ${\overline{\mathbf{X}}}_{\tau }$. Importantly, the conditional mean m_Y in expression (10) assumes that covariate history affects the outcome Y when the subject is being treated but not after treatment is stopped. This assumption is consistent with available data for the motivating example in Section 5 and would be true, for example, if the reason treatment stopped was due to death. However, if covariate data were collected after treatment was stopped and did affect outcome, the methods proposed here could be extended by replacing ${\overline{\mathbf{X}}}_{\tau }$ with ${\overline{\mathbf{X}}}_{\tau }$ in m_Y in (10). We also note that covariate information collected after treatment has stopped cannot affect treatment assignment and cannot, therefore, be a confounding variable regardless of whether it affects the outcome or not.

Unfortunately, the covariate history ${\overline{\mathbf{X}}}_{\tau }$ is only observed for those subjects who are still receiving treatment at the most recent landmark, by the landmark assumption in Section 2.3, and so (10) cannot be estimated with an ordinary sample average. For this purpose, let $R_t=1-A_{\lfloor t\rfloor }=\underset{s\le \lfloor t\rfloor }{\pi }\left(1-A_S\right)$ be the indicator that a subject is still being treated at time ⌊t⌋ = max{t_j|t_j ≤ t, j = 1, … , K} and let $W_t\left(H_t\right)=\underset{s\le \lfloor t\rfloor }{\pi }\left\{1-{\lambda }_s^U\left(H_s\right)\right\}$, where ${\lambda }_t^U\left(H_t\right)$ is the all-cause hazard of treatment discontinuation at time t given treatment and covariate history H_t. Then, under the sequential randomization assumption, a standard argument in the analysis of observational data using Horwitz–Thompson weighting applied to the “risk-set” indicator R_t leads us to conclude that (10) is equal to the expression

$$\begin{gathered} \mathbb{E}[\frac{R_{t^o}{m}_Y\left({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)},{\overline{\mathbf{X}}}_{t^o}\right)G_{t^o}\left({\tilde{H}}_{t^o}^{(1)}\right)}{W_{t^o}\left(H_{t^o}\right)} \\ -{\int }_0^{t^o}\frac{R_t{m}_Y\left({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{X}}}_t\right)d{G}_t\left({\tilde{H}}_t^{(2)\cdot t}\right)}{W_t\left(H_t\right)}]. \end{gathered}$$ (11)

Substituting consistent estimators ${\widehat{m}}_Y$, $\widehat{G}$, and $\widehat{W}$ for m_Y, G, and W, respectively, and taking a sample average of the expression in square brackets in (11) leads to our estimator

$$\begin{gathered} {\mathbb{P}}_n[\frac{R_{t^o}{\widehat{m}}_Y\left({\overline{\mathbf{a}}}_{\tau }^{(1)},{\overline{\mathbf{c}}}_{\tau }^{(1)},{\overline{\mathbf{X}}}_{t^o}\right){\widehat{G}}_{t^o}\left({\tilde{H}}_{t^o}^{(1)}\right)}{{\widehat{W}}_{t^o}\left(H_{t^o}\right)} \\ -{\int }_0^{t^o}\frac{R_t{\widehat{m}}_Y\left({\overline{\mathbf{a}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{c}}}_{\tau }^{(2)\cdot t},{\overline{\mathbf{X}}}_t\right)d{\widehat{G}}_t\left({\tilde{H}}_t^{(2).t}\right)}{{\widehat{W}}_t\left(H_t\right)}]. \end{gathered}$$ (12)

Naturally, the performance and behavior of the statistic in (12) is tied closely to the flexibility of the regression functions used to model m_Y, G, and W. We adopt Cox’s (1972) proportional hazards model for ${\lambda }_t^{\xi }\left(H_t\right)$ and ${\lambda }_t^U\left(H_t\right)$ and consider different strategies for modeling m_Y with varying amounts of flexibility. The large sample behavior of the proposed estimator in (12) is outlined in the Appendix with details given in the Supporting Information. We use bootstrap resampling to estimate the asymptotic variance of our estimator in (12) for its ease in practical implementation by data analysts. Resampling can be justified formally using techniques described in Kosorok (2008, Theorem 10.2), for example. The finite sample behavior of the point and interval estimator are evaluated in Section 4.

Remark 1.

When there is confounding through baseline factors only and those factors are observed on the entire study sample, the risk-set weighting is superfluous and the estimand in (10) can be estimated directly with a sample average as shown by Lu and Johnson (2015). Our method generalizes the estimator by Lu and Johnson (2015) to the case of time-dependent confounding, when covariate history updates at landmark time points t₁, … , t_K throughout the study follow-up period.

4 |. SIMULATION STUDIES

We conducted simulation studies to evaluate the proposed estimator $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$ in finite samples under the ATS ${\overline{d}}_{\tau }$ given by d_t(H_t) = 1 − I(t ≤ t^o)I(C_t = 0). The performance of our proposed estimator depends on the choice of modeling and estimation strategies leading to estimators of the regression functions m_Y, ${\lambda }_t^{\xi }$, and ${\lambda }_t^U$. To facilitate connections and comparison to earlier works, we write the observed data as $O=\left(Y,U,\mathrm{\Delta },{\overline{\mathbf{X}}}_U\right)$, where $U={\int }_0^{\tau }(1-A_t)dt$ is the observed treatment length, $\mathrm{\Delta }=1-{\int }_0^{\tau }d{C}_t$ is a binary indicator that realizes the value 0 if treatment was stopped due to treatment-terminating event and 1 otherwise, and ${\overline{\mathbf{X}}}_U$ is covariate history through time U (e.g., Johnson and Tsiatis, 2004, 2005; Lu and Johnson, 2015). Through this parameterization, the conditional mean $m_Y\left({\overline{\mathbf{A}}}_{\tau },{\overline{\mathbf{C}}}_{\tau },{\overline{\mathbf{X}}}_{\tau }\right)=m_Y\left(U,\mathrm{\Delta },{\overline{\mathbf{X}}}_U\right),I(U\le t,\mathrm{\Delta }=0)$ is the counting process for the treatment-terminating event, the at-risk indicator is R_t = I(U ≥ ⌊t⌋), and ${\lambda }_t^U\left(H_t\right)$ is the hazard function for the at-risk process given history H_t.

Our simulation scenarios are motivated by the data in Section 5. There are a total of five prespecified landmarks are t_j ∈ {6, 12, 18, 24, 30} and t₀ = 0. We consider two covariates ${\mathbf{X}}_t=(X_0^{(1)},X_t^{(2)})$, where $X_0^{(1)}$ is a time-independent standard normal random variable and $X_t^{(2)}$ is a time-dependent covariate. The time-dependent covariate $X_t^{(2)}$ follows a first-order Markov model: $X_{t_k}^{(2)}$ is normal with mean $X_{t_{k-1}}^{(2)}$ and unit variance for k = 1, … , 5, and $X_{t_0}^{(2)}$ is standard normal. The cumulative covariate history is ${\overline{\mathbf{X}}}_t=\left({\mathbf{X}}_0,\dots ,{\mathbf{X}}_t\right)$ and the upper limit of the treatment length window is τ = 30.

Let λ_t(H_t) be the cause-specific hazard function for treatment discontinuation time with Δ = 1, and ${\lambda }_t^{\xi }\left(H_t\right)$ be the cause-specific hazard function for treatment discontinuation time with Δ = 0. We assume that both hazard of treatment length choice and λ_t(H_t) and ${\lambda }_t^{\xi }\left(H_t\right)$ follow proportional hazard models, ${\lambda }_t\left(H_t;\gamma \right)={\lambda }_{0,t}\exp \left({\gamma }^{\top }{\overline{\mathbf{X}}}_t\right)$, and ${\lambda }_t^{\xi }\left(H_t;\eta \right)={\lambda }_{0,t}^{\xi }\exp \left({\eta }^{\top }{\overline{\mathbf{X}}}_t\right)$. We use piecewise constant baseline hazard functions for λ₀,_t and ${\lambda }_{0,t}^{\xi }$ such that ${\lambda }_{0,t}={\sum }_{j=1}^K{\kappa }_{j}I\left\{t\in \left(t_{j-1},t_j\right]\right\}$ and ${\lambda }_{0,t}^{\xi }={\sum }_{j=1}^K{\kappa }_j^{\xi }I\left\{t\in \left(t_{j-1},t_j\right]\right\}$, where κ_j and ${\kappa }_j^{\xi }$ are the baseline rates in the jth interval for T and ξ, respectively. We assume the outcome models follow the general semiparametric models, $m_Y\left(U,\mathrm{\Delta },{\overline{\mathbf{X}}}_t\right)=\mathrm{\Delta }{m}_1\left(U,{\overline{\mathbf{X}}}_t\right)+(1-\mathrm{\Delta })m_0\left(U,{\overline{\mathbf{X}}}_t\right)$, $m_0\left(t,{\overline{\mathbf{X}}}_t\right)=\mathrm{\Phi }\left\{{\varphi }_0(t)+{\beta }^{\top }{\overline{\mathbf{X}}}_t\right\}$ and $m_1\left(t,{\overline{\mathbf{X}}}_t\right)=\mathrm{\Phi }\left\{{\varphi }_1(t)+{\alpha }^{\top }{\overline{\mathbf{X}}}_t\right\}$, where ${\varphi }_1(t)=\log (t)$, ${\varphi }_0(t)=1+t^{1/2}$. The observed data are thus $\left(Y,U,\mathrm{\Delta },{\overline{\mathbf{X}}}_U\right)$. Note, under this setup, the induced hazard function for U is ${\lambda }_t^U\left(H_t\right)={\lambda }_t^{\xi }\left(H_t\right)+{\lambda }_t\left(H_t\right)$. In the interest of space, the results for continuous endpoint with identity link Φ(·) are presented here while the the results for binary endpoint with logit link function are relegated to Web Appendix D. Specific parameters for the simulation results are presented below:

• Scenario 1 (time-dependent confounding): We set α = β = (0.5, −0.5)^⊤, and:

  • for a roughly 45% adverse event rate: γ = (−0.3, 0.5)^⊤, η = (0.3, −0.5)^⊤, (κ₁, … , κ₅) = (1.55, 2, 2.8, 4.4, 14)/100, $({\kappa }_1^{\xi },\dots ,{\kappa }_5^{\xi })=(1.35,1.6,2.4,3.8,11)/100$;
  • for a roughly 20% adverse event rate: γ = (−0.1, 0.5)^⊤, (κ₁, … , κ₅) = (2.2, 3, 4, 6.2, 23), $({\kappa }_1^{\xi },\dots ,{\kappa }_5^{\xi })=(0.6,0.75,1,1,4)/100$.
• Scenario 2 (baseline confounding only): We set α = β = (0, −0.5)^⊤, and:

  • for a roughly 45% adverse event rate: γ = (0, 0.5)^⊤, η = (0, −0.5)^⊤, (κ₁, … , κ₅) = (1.75, 2.1, 3.3, 4.7, 14)/100, $({\kappa }_1^{\xi },\dots ,{\kappa }_5^{\xi })=(1.4,1.6,2.5,4,11)/100$;
  • for a roughly 20% adverse event rate: (κ₁, … , κ₅) = (2.5, 3, 4, 6.2, 23)/100, and $({\kappa }_1^{\xi },\dots ,{\kappa }_5^{\xi })=(0.6,0.75,1,1,4)/100$.

In all scenarios, we generate 500 Monte Carlo data sets of sizes n = 250 and n = 500. The chosen parameters lead to observed treatment length distribution with approximate quantiles of 20%, 40%, 60%, and 80% at the landmarks 6, 12, 18, and 24, respectively. True values of the estimand were computed via numerical integration.

Figure 1 summarizes simulation results for estimating $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$ with different t^o in the presence of time-dependent confounding. For both the proposed estimator and Lu and Johnson (LJ, 2015), m_Y is fit through parametric models with the correct function of time (see also Web Appendix D for the simulation results where m_Y is fit by partly linear models). In the presence of time-dependent confounding, LJ is not consistent and the bias does not shrink to zero as the sample size increases as seen in the top row of Figure 1. The bias is especially severe in the middle part of the investigated time interval due to the misspecification of the outcome model m_Y. On the other hand, the proposed estimator performs well across a range of adverse events rates, 45% or 20%. When the sample size is modest, for example, n = 250, there is some bias in the tails of the treatment window. However, this extraneous finite sample bias diminishes substantially as the sample size increases to n = 500.

FIGURE 1.

Simulation results based on 500 Monte Carlo data set. The empirical bias, standard deviation, and coverage rates for two different adverse events rates, 45% and 20%, under time-dependent confounding. The outcome is Gaussian. MCSD is the Monte Carlo standard deviation and SEE is the standard error estimate

The panels in the second row describe the empirical standard error (MCSD, gray line) and the bootstrap-based standard error estimates (SEE, black line) for our proposed estimator (i.e., we do not present the variance estimation results for LJ method, but see Lu and Johnson, 2015). These two curves agree closely over the entire treatment window for both sample sizes suggesting that bootstrap resampling performs well. As exemplified in Figure 1, realizations of the estimator over treatment length can appear as nonsmooth functions in time. But this is a direct consequence of how the proposed estimator adjusts for time-dependent confounding, with informative updates at the landmarks. The last row shows that the empirical coverage rate of LJ may be much lower than nominal level when important time-dependent confounders are ignored. The empirical coverage rate curve of our proposed estimator is close to nominal level, which justifies our asymptotic arguments.

When there is only baseline confounding, LJ showed that their estimator was consistent and asymptotically normal. It is important to compare estimators in this setting to evaluate how much precision is lost due to our estimation scheme to accommodate time-dependent confounders. In simulation scenario 2, only the baseline covariate $X_0^{(1)}$ has a nontrivial effect on the outcome. As predicted, Figure 2 shows that both the proposed estimator and LJ are relatively unbiased when time-dependent confounding plays no role. Finite sample bias in the tails disappears as the sample size increases. The second row in Figure 2 shows that the MCSD curve of LJ is 25% smaller than the proposed estimator for treatment duration greater than or equal to 25 with continuous outcomes. This result suggests LJ is more efficient than the proposed estimator when there is confounding through baseline factors only, and one explanation for this result is because there is additional uncertainty in risk-set weighting or any method that attempts to adjust for time-dependent confounding. As seen in Figure 1, the bootstrap-based variance estimation performs well here and the empirical coverage probability is close to the nominal level, which support our theoretical results in Web Appendix A.

FIGURE 2.

Simulation results based on 500 Monte Carlo data set. The empirical bias, standard deviation, and coverage rates for two different adverse events rates, 45% and 20%, without time-dependent confounding. The outcome is Gaussian. MCSD is the Monte Carlo standard deviation

Web Appendix D provides additional simulation results to show the performance of the proposed estimator when m_Y is modeled through the general semiparametric model and estimated via profile least squares. We also give results for the case where m_Y is misspecified by both parametric and semiparametric models but can be learned by flexible machine learning methods. In particular, we show that the point-wise bias tends to zero as the sample size increases and thus supports our claim of consistency using nonregular methods.

5 |. ANALYSIS OF THE INFUSION TRIAL DATA

The proposed methods are motivated by the Enhanced Suppression of the Platelet Receptor IIb/IIIa with Integrilin Therapy (ESPRIT), a double-blind, randomized, placebo-controlled clinical trial of a novel dosing regimen (O’Shea et al., 2001). Briefly, patients undergoing elective stent percutaneous coronary intervention were given two bolus doses of eptifibatide followed by eptifibatide or placebo infusion just prior to coronary intervention for patients on the intervention or standard of care arm, respectively. If an adverse event occurred, the protocol advised that infusion be stopped immediately; in the absence of such adverse events, providers decided when to stop the infusion. Then, the adaptive treatment length policy in this context is to infuse for a fixed amount of time t or until an infusion-terminating event occurs, whichever comes first.

To facilitate a comparison to earlier work, we considered similar clinical definitions for adverse event and clinical endpoint. We define the outcome Y as the composite binary endpoint of death, myocardial infarction, or urgent target vessel revascularization within 30 days. Furthermore, an adverse event necessitating infusion termination included any combination of abrupt closure, no reflow, or coronary thrombosis. The baseline covariate data include indicators of diabetes, percutaneous transluminal coronary angioplasty, angina, heparin use, and the continuous variable weight. Enzyme level is the one time-dependent covariate and was collected at landmarks of 6, 12, 18, and 24 h. There were 1040 patients on the integrilin arm but 18 patients (1.73%) are missing enzyme levels. We analyze the complete-case data and thus the effective sample size is n = 1022.

Table 1 gives estimates of $\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$ at five time points using variations of the newly proposed estimator and the estimator by Lu and Johnson (2015), where the latter method does not adjust for time-dependent confounding through enzyme level. Both methods use proportional hazards models to model the hazard of treatment-terminating event and allow for flexible definitions of regression functions m_Y. We note that point estimates for LJ in Table 1 and in Lu and Johnson (2015, Section 4.2) are not identical because the sample size is different; that is, if time-dependent enzyme level is not used, one can retain data for 18 patients with missing enzyme levels. Nevertheless, the point estimates are rather similar.

TABLE 1.

Thirty-day event proportions (%) from the ESPRIT infusion study. Proportional hazards models for ${\lambda }_t^{\xi }$ and ${\lambda }_t^U$ functions while different statistical models for m_Y include linear models (LM), partially linear models (PLM), and super learner (SL) ensembles with gam, glm, MARS, and random forest. The estimator that adjusts for baseline confounders only is the direct estimator proposed by Lu and Johnson (LJ, 2015) whereas the new proposed estimator adjusts for time-varying confounders. Table entries include point estimates (Est.) and bootstrap standard errors (SE) at five time points

Time (h)	Potential confounding
	Time-dependent						Baseline only
	SL		PLM		LM		SL		PLM		LM
	Est.	SE	Est.	SE	Est.	SE	Est.	SE	Est.	SE	Est.	SE
16	5.98	0.92	5.67	1.43	5.88	1.11	6.92	1.01	5.00	1.38	5.43	1.03
18	6.56	0.97	6.60	0.91	6.53	0.82	7.18	1.21	6.44	0.94	6.50	0.83
20	7.11	1.07	7.71	1.24	7.45	0.78	7.40	1.14	7.88	1.25	7.56	0.79
22	7.20	0.92	8.25	1.34	7.99	1.02	7.70	1.16	8.46	1.37	8.69	1.05
24	8.32	1.27	8.81	2.31	8.53	1.48	8.14	2.26	10.64	2.28	9.89	1.58

Table 1 shows the numerical results of the proposed method and LJ method when outcome parts are modeled by super learner (SL), partially linear model (PLM), and linear model (LM). All variations of the two methods return similar patterns of estimates of the expected event proportion across different treatment strategies. In general, the expected event proportion is gently increasing for infusion length strategies ranging from 16 to 24 h. Hence, there is some evidence to suggest that infusing with integrilin for shorter durations is better than infusing for longer duration (Johnson and Tsiatis, 2004). Point estimates of the proposed method are relatively comparable under different working outcome models. When adjusting for baseline confounding only, the results from PLM and LM are similar but they differ from the SL results. We interpret this as mild evidence, though not statistically significant, that a parametric outcome working model may be misspecified when time-dependent enzyme level is not included among the set of potential confounders in the method. When time-dependent enzyme level is included in the analysis, the agreement among the point estimates is very good using different outcome working models. Standard error estimates using a parametric working outcome model are uniformly smaller compared to those using a general semiparametric working model, as expected, and the same pattern is observed regardless of whether one adjusts for time-dependent confounders or not. The bootstrap-based standard error estimates for SL may be invalid but are shown for completeness.

6 |. DISCUSSION

Motivated by the ESPRIT trial, we developed a method to evaluate the effect of continuous adaptive treatment length subject to eligibility criteria in the presence of potential time-dependent covariates. The proposed approach combines a likelihood-based method and risk-set weighting method to address the incomplete covariate history problem and flexible working models can be utilized to construct the estimation. Additionally, simulation studies show that the adjustment of time-dependent covariates is necessary to understand the effect of a nonrandomized treatment on outcome in the presence of time-dependent confounders.

The consistency of the proposed estimator relies on correctly specified models and assumptions for the nuisance parameters, including but not limited to the piece-wise constant assumption on the time-dependent covariate process. One future research topic is to construct a doubly robust estimator for adaptive treatment length policies so that the consistency follows if at least one of the nuisance parameters is modeled correctly (Kennedy et al., 2017). We have proposed to use machine learning methods (e.g., superlearner) to reduce the chance of obtaining a biased estimate caused by some model misspecification. This, however, may lead to estimators that are not asymptotically linear, which can complicate deriving a variance formula for the corresponding estimator. One can overcome this problem by using a targeted minimum loss–based estimator where the nuisance parameters are targeted (van der Laan, 2014).

In our framework, covariate history is not collected beyond the observed treatment length U. This is an artifact regarding the data made available to our team. In the clinic, it is reasonable to believe that a subset of time-varying covariates X_t might be collected beyond treatment termination, especially for a patient admitted for heart surgery as in our data application in Section 5. Under a different set of circumstances, say, if we were investigating a dynamic treatment regime where treatment could be restarted, covariate information collected after the first stoppage would be important for the data analysis.

APPENDIX: OUTLINE OF OPERATING CHARACTERISTICS

We outline the operating characteristics of the proposed estimator and describe the theoretical results formally in Web Appendix A. We make the following assumptions to derive a new expression for the causal estimand that leads to our proposed estimator.

A1. Consistency of the observed outcome (SUTVA);

A2. Sequential randomization assumption in (2);

A3. Positivity (or overlap) of the treatment assignment and treatment-terminating events:

A4. The probability that treatment is stopped by choice and also stopped by treatment-terminating event in the same infinitesimal [t, t + dt) has zero probability;

A5. ${\overline{\mathbf{X}}}_t$ is left-continuous, piece-wise constant with jumps at the landmarks such that ${\overline{L}}_{\tau }$ has density (7).

Assumptions A1 and A2 are the consistency and identifiability assumptions used in causal inference to estimate causal estimands. Assumption A3 ensures that statistical inference of the treatment effect is possible. Assumption A4 prevents ties between competing events while Assumption A5 makes clear our landmark condition on covariate history.

Let ${\mu }_{t^o}=\mathbb{E}\left[Y\left({\overline{d}}_{\tau }\right)\right]$ under ${\overline{d}}_{\tau }$ given by d_t(H_t) = 1 − I(t ≤ t^o)I(C_t = 0) when $\mathcal{G}$ is the entire population. Assume conditions A1–A5 above, conditions C1–C3 in Web Appendix C at fixed target time t^o, and that ${\lambda }_t^{\xi }\left(H_t\right)$ and ${\lambda }_t^U\left(H_t\right)$ are correctly modeled through a proportional hazards assumption. If ${\widehat{m}}_Y$ is a parametric estimator of m_Y and the model m_Y is correct, then ${\widehat{\mu }}_{t^o}$ is consistent, asymptotically linear, and $n^{1/2}\left({\widehat{\mu }}_{t^o}-{\mu }_{t^o}\right)$ converges in distribution to a mean-zero normal random variable with variance $\mathbb{E}[\phi {\left({\overline{\mathbf{X}}}_{t^o},t^o\right)}^{\otimes 2}]$, where $\phi \left({\overline{\mathbf{X}}}_t,t\right)$ is the influence curve at time t and detailed in Web Appendix C. We prove a similar result where we allow m_Y to follow a general semiparametric model, estimate the nonparametric component using local polynomials and the parametric components via profile least squares. Those details are provided in Web Appendix D.

Additionally, we provide conditions on the regression function m_Y under which the proposed estimator ${\widehat{\mu }}_{t^o}$ will be consistent but the Donsker condition used in Theorem 10.2 in Kosorok (2008) for inference procedures may be violated. Assume assumptions A1–A5, ${\lambda }_t^{\xi }\left(H_t\right)$ and ${\lambda }_t^U\left(H_t\right)$ are correctly modeled via PH, and conditions C1–C3 in Web Appendix C at fixed target time t^o. At fixed target time t^o, if m_Y is fit by models such that ${\Vert {\widehat{m}}_Y-m_Y\Vert }_{\infty }=o_p(1)$, then $\left|{\widehat{\mu }}_{t^o}-{\mu }_{t^o}\right|=o_p(1)$ for t^o ∈ (0, τ].

DATA AVAILABILITY STATEMENT

The data that support the findings in this paper are available upon request from Dr. James Tcheng, principal investigator of the Enhanced Suppression of the Platelet Receptor IIb/IIIa with Integrilin Therapy (ESPRIT) trial. Restrictions apply to the availability of these data, which were used in this paper with the explicit written permission of the principal investigator. Data are available from the authors (brent_johnson@urmc.rochester.edu) with the permission of Dr. James Tcheng (tchen001@mc.duke.edu).