Identification of the Joint Effect of a Dynamic Treatment Intervention and a Stochastic Monitoring Intervention Under the No Direct Effect Assumption

1 Introduction

Effective management of chronic conditions such as diabetes involves frequent re-evaluation of treatment decisions over the course of the patient’s illness. Patients and their clinicians must choose not only adequate therapies among many options, but they must also decide throughout the course of the illness when to initiate, switch or intensify treatments and how to titrate medication to balance health benefits and harms. These clinical decisions are informed by periodic monitoring of laboratory test results (e.g., hemoglobin A1c for patients with diabetes), changes in symptoms, and more general considerations of patients’ preferences, concurrent conditions, adherence to previously prescribed treatments, and overall health. Evidence most suitable to guide the real-world management of chronic conditions must therefore involve the evaluation of complex treatment strategies that continuously adapt to new information about the patient’s changing circumstances collected during routine clinic visits. For instance, in diabetes research, contrasting (e.g., by randomization) adaptive treatment regimens of the type:

“Initiate insulin the first time the patient’s observed A1c drifts above X” with X=7%,7.5%, 8%, or 8.5%

will generate evidence that is clinically more relevant that contrasting static treatment regimens of the type

“Initiate insulin X months from now” with X =3, 6, 9, or 12

because the former regimens emulate (albeit simplistically) how treatment decisions are made in real-world clinical practice by personalizing treatment decisions over time based on the latest A1c information available for the patient. With the help of healthcare providers, the above adaptive treatment regimens can be made more complex so they better emulate how treatment decisions are informed in real-world clinical settings and thus ensure that the evidence generated is most useful by contrasting decision making procedures that are (or could be realistically) used by clinicians in practice.

Causal inference methods to estimate the effects of treatment regimens in the presence of time-dependent confounding have been developed specifically to generate such evidence using data from observational studies [1, 2, 3, 4, 5, 6] or randomized experiments [7, 8, 9, 10, 1112]. Until recently [13, 14] however, applications have ignored the role of clinical monitoring frequency in the definition of the effects of adaptive treatment strategies. As a result, conclusions about health benefits and harms remain specific [4, Section 6] to the particular clinical monitoring conditions of the study which may or not reflect a particular real-world practice of interest. For example, two perfect randomized trials (no non-compliance and no loss to follow-up) that are conducted in the same patient population but using two distinct clinical monitoring schedules could provide different conclusions (both valid) about the effectiveness of an adaptive treatment strategy compared to the same, for example, static (i.e., non-adaptive) treatment strategy (e.g., never treat).

Evaluating the joint effect of an adaptive treatment strategy and a particular clinical monitoring intervention is thus desirable to, not only, improve the generalizibility [4, 15, 16] of study results, but also, more importantly, to generate evidence on how clinical monitoring modify the health effects of adaptive treatment strategies. For instance, in diabetes research, contrasting joint treatment and monitoring regimens of the type:

“Monitor the patient’s A1c every X months and initiate insulin the first time her A1c drifts above 8%” with X = 3, 6, 9, 12

can generate evidence about the value of increased A1c monitoring (“value of information” [4]) that can be highly relevant for both patients living with chronic conditions and for the health systems that care for them: A patient’s quality of life is impacted by the burdens imposed by frequent laboratory and other clinical surveillance and this surveillance also contributes to the significant burdens that chronic conditions put on healthcare systems. Thus, not only is evidence on how to possibly minimize the intensity of clinical monitoring without significantly undermining health outcomes relevant to inform care, but also evidence that aim to optimize outcomes by adapting monitoring decisions over time based on a patient’s evolving condition. The latter type of evidence can, for example, be generated by contrasting joint treatment and monitoring regimens of the type:

“Monitor the patient’s A1c every X months as long as her latest A1c remain below 7% and increase monitoring to every 3 months otherwise. Initiate insulin the first time her A1c drifts above 8%” with X = 3, 6, 9, 12.

Standard estimation approaches for evaluating the effect of general time-varying exposures with observational data can be readily applied to the evaluation of the joint effect of an adaptive treatment strategy and a particular clinical monitoring intervention by simply defining the exposure variable of interest at each time point as a two-dimensional variable where the first and second element represents the treatment and monitoring decision, respectively. These methods include not only the most popular Inverse Probability Weighting (IPW) [17] and G-computation [18] estimation approaches but also the two doubly robust and locally efficient augmented-IPW [19, 20, 21, 22, 23] and Targeted Minimum Loss based Estimation (TMLE) [24, 25, 26] approaches. Effect identifiability with these various estimators relies on both a no unmeasured confounders assumption and a positivity assumption. The latter assumption requires that each patient in the study can possibly experience the time-varying interventions of interest at each time point whatever the levels of the time-dependent confounders. Upholding of this assumption when the time-varying interventions include rigid monitoring control (e.g., “patient should have her A1c monitored exactly every 3 months”) might often be unrealistic in observational studies conducted in real-world settings. Concerns over effect non-identifiability due to violation of the positivity assumption associated with monitoring decisions have motivated prior work [4, Section 6] and led to the development of new IPW estimators which remain consistent under a weaker positivity assumption for monitoring decisions as long as it can also be assumed that these monitoring decisions have no direct effects on covariates (including the outcome) once treatment decisions are made. This assumption was named the No Direct Effect (NDE) assumption and the resulting NDE-based IPW estimators were developed in problems with an end-of-study outcome and for static or more general (adaptive or not) stochastic monitoring interventions.

In this article, we extend this prior work in the context of time-to-event outcomes with the presentation of two counterfactual identifiability results. These two counterfactual equalities are developed based on the nonparametric structural equation model (NPSEM) framework [27, 28, 29, 30] and applicable to the evaluation of the joint effect of an adaptive treatment strategy and either a static or general stochastic clinical monitoring intervention under the NDE assumption. The practical implication of the results presented here is the straightforward derivation of new IPW, G-computation, augmented IPW and TMLE estimators that rely on a weaker positivity assumption for the monitoring process compared to their standard analogs (i.e., not based on the NDE assumption). We note, however, that while we illustrate the implication of the results in this paper for estimator construction using the IPW methodology with the simple example of static monitoring interventions, the detailed derivation and evaluation of alternate estimators and of the required assumptions for estimation consistency (e.g., sequential randomization assumption) are beyond the scope of this article. In separate work, we will further exploit the two identifiability results presented here to demonstrate that they lead, not only, to another innovative NDE-based IPW estimator for general stochastic monitoring interventions derived in a prior paper [4] (without formally evoking the results of this article), but also to the straightforward derivation of NDE-based TMLE estimators for either static or general stochastic monitoring interventions.

In the next section, we present the notation adopted in this paper to represent the general longitudinal observational data available in practice to evaluate the joint treatment and monitoring effects of interest. In Section 3, we introduce the causal model defined by a general NPSEM that is assumed to have generated the observed data and that defines the various classes of counterfactual variables considered in this work. We detail the definition of these counterfactuals in Section 4 before formalizing the NDE assumption. In Sections 6 and 7, we present the two identifiability results for evaluating joint effects defined by static and general stochastic monitoring interventions, respectively. We illustrate the practical implication of these results for estimator construction in Section 8 and conclude with a discussion of the contribution of this paper to the existing literature on the same topic in Section 9. For clarity, we continue to use examples in diabetes comparative effectiveness research (CER) throughout the manuscript to demonstrate the applicability of the formalism introduced in the next sections.

2 Observed data

We consider an observational study (e.g., retrospective study based on Electronic Health Records from a healthcare database) in which longitudinal data are collected on a random sample of n patients from a target population starting at the index date (i.e., cohort entry date denoted by t=0) and until end of follow-up defined by the earliest occurrence of a failure event of interest (e.g., onset of diabetic peripheral neuropathy) or a right-censoring event (e.g., health plan disenrollment). Right-censoring events include reaching the administrative end of the study which occurs for all patients at time t=K+1 at the latest. At each time point t during follow-up (expressed in an arbitrary unit of time, e.g., 30-day interval), measurements on the following variables are assembled. The sequence in which we describe these variables reflects the actual temporal ordering of the underlying information these variables encode. The binary outcome variable Y(t) represents whether the failure event occurred since time point t−1 (Y(0)=0 by convention). The vector of covariates Z(t) represents various patient attributes (e.g., demographic, vital signs, comorbidities, etc.) at time t. We note that some of these attributes may not be monitored at each time point in which case indicators of monitoring are included as part of the definition of Z(t) and the unmonitored attribute measurements are then equal to 0 by convention. The covariate I(t) represents the patient’s attribute at time t (e.g., A1c) that is involved in the definition of the adaptive treatment regimen of interest and for which a monitoring intervention is also of interest. By convention, I(0) is always monitored and I(t)≡0 if the attribute is not monitored at time t. The binary variable A2(t) represents whether the patient experiences a right-censoring event at time t. The variable A1(t) represents the treatment decision of interest at time t (e.g., insulin exposure). The binary variable N(t) represents the monitoring decision of interest at time t and indicates whether a measurement for I(t+1) will be collected (e.g., whether an A1c test will be conducted at the next time point). We thus have I(t+1)=0 if N(t)=0. In short, the observed data are realizations of n independent and identically distributed copies of the following temporally ordered sequence of random variables:

where L(t)≡(Y(t),Z(t),I(t)), A2(K+1)=1 if Y(K+1)=0 (i.e., right-censoring from reaching the administrative end of the study is deterministic at K+1), and where, by convention, all variables become degenerate (defined at their last observed value) after occurrence of a failure or right-censoring event. Time-independent covariates (e.g., sex) are included in the vector of baseline covariates L(0) alongside the baseline measurements of time-varying covariates. We denote the distribution of these observed data O by P0. To simplify notation below, we use the overbar symbol to denote the history of a variable. More specifically, Xˉ(t) and Xˉ(t,t′) represent the vectors of measurements of the time-varying attribute X from baseline to time point t: Xˉ(t)≡(X(0),…,X(t)) and from time point t to t′: Xˉ(t,t′)≡(X(t),…,X(t′)), respectively. By convention, Xˉ(t) and X(t) are nil for t<0 and Xˉ(t,t′) is nil for t′<t. We also use lower case letters to represent realizations of random variables. Finally, we denote the support of the treatment and monitoring processes Aˉ1(K) and Nˉ(K) by Aˉ and Nˉ, respectively.

3 Causal model

We assume the following nonparametric structural equation model (NPSEM) that links (Appendix A) the observed data distribution P0 to the probability distribution (denoted by PU) of a vector of random disturbances U≡((UY(t),UZ(t),UI0(t),UA2(t))t=0,…,K+1,(UA1(t),UN(t))t=0,…,K) and a fixed vector of functions f≡((fY(t),fZ(t),fI0(t),fA2(t))t=0,…,K+1,(fA1(t),fN(t))t=0,…,K):

1. Y(t)=fY(t)(PY(t)0,UY(t)) with parents PY(t)0≡(Yˉ(t−1),Zˉ(t−1),Iˉ0(t−1),Aˉ2(t−1),Aˉ1(t−1),Nˉ(t−1)) for t=0,…,K+1,

2. Z(t)=fZ(t)(PZ(t)0,UZ(t)) with parents PZ(t)0≡(Yˉ(t),Zˉ(t−1),Iˉ0(t−1),Aˉ2(t−1),Aˉ1(t−1),Nˉ(t−1)) for t=0,…,K+1,

3. I0(t)=fI0(t)(PI0(t)0,UI0(t)) with parents PI0(t)0≡(Yˉ(t),Zˉ(t),Iˉ0(t−1),Aˉ2(t−1),Aˉ1(t−1),Nˉ(t−1)) for t=0,…,K+1,

4. I(t)=Y(t)I(t−1)+(1−Y(t))N(t−1)I0(t) for t=0,…,K+1 where N(−1)≡1 by convention,

5. A2(t)=fA2(t)(PA2(t)0,UA2(t)) with parents PA2(t)0≡(Yˉ(t),Zˉ(t),Iˉ0(t),Aˉ2(t−1),Aˉ1(t−1),Nˉ(t−1)) for t=0,…,K+1,

6. A1(t)=fA1(t)(PA10,UA1(t)) with parents PA1(t)0≡(Yˉ(t),Zˉ(t),Iˉ0(t),Aˉ2(t),Aˉ1(t−1),Nˉ(t−1)) for t=0,…,K,

7. N(t)=fN(t)(PN(t)0,UN(t)) with parents PN(t)0≡(Yˉ(t),Zˉ(t),Iˉ0(t),Aˉ2(t),Aˉ1(t),Nˉ(t−1)) for t=0,…,K,

where the functions f above are left unspecified except for the following constraints:

1. fY(0)(UY(0))=0,

2. fY(t)(PY(t)0,UY(t))=Y(t−1) if A2(t−1)=1 or Y(t−1)=1 for t=1,…,K+1,

3. fZ(t)(PZ(t)0,UZ(t))=Z(t−1) if A2(t−1)=1 or Y(t)=1 for t=1,…,K+1,

4. fI0(t)(PI0(t)0,UI0(t))=I0(t−1) if A2(t−1)=1 or Y(t)=1 for t=1,…,K+1,

5. fA2(t)(PA2(t)0,UA2(t))=A2(t−1) if A2(t−1)=1 or Y(t)=1 for t=1,…,K+1,

6. fA1(t)(PA1(t)0,UA1(t))=A1(t−1) if A2(t)=1 or Y(t)=1 for t=1,…,K,

7. fN(t)(PN(t)0,UN(t))=N(t−1) if A2(t)=1 or Y(t)=1 for t=1,…,K,

8. fA2(K+1)(PA2(K+1)0,UA2(K+1))=1 if Y(K+1)=0.

In this NPSEM, the observed covariate I(t) is explicitly linked to a latent variable I0(t) which represents the patient’s underlying attribute measurement of interest at time t. More specifically, as long as the patient has not experienced failure yet, if a decision was made not to collect the measurement I0(t) at the previous time point (i.e., when N(t−1)=0) then I(t)=0 (which follows the convention described in the previous section to define I(t)) and otherwise I(t)=I0(t) (i.e., when N(t−1)=1).

The combination of the latent variable I0(t) and observed variables Y(t), Z(t), A2(t), A1(t), and N(t) are referred to as endogenous variables ^[1] and we note that the previous constraints on the functions f merely encode 1) the convention that the baseline outcome is 0, 2) the convention that the endogenous variables are defined as degenerate random variables after end of follow-up from occurrence of a failure or right-censoring event, and 3) deterministic right-censoring from administrative end of study at t=K+1.

We note also that the definition of the observed measurement I(t) above implies that, like all other endogenous variables, it becomes degenerate at its last observed value after the occurrence of, not only, the failure event, but also a right-censoring event, i.e., I(t)=I(t−1) if Y(t)=1 or A2(t−1)=1.

Finally, we introduce the notation L0(t)≡(Y(t),Z(t),I0(t)) for t=0,…,K+1 and bring attention to the distinction between L0(t) and the observed vector of covariates L(t)=(Y(t),Z(t),I(t)).

4 Counterfactual variables of interest

Whether conducted with a randomized experiment or an observational study, CER typically aims to evaluate causal estimands defined by the distribution of “potential outcomes”, i.e., outcomes if all patients in the target population experienced the “same” (possibly adaptive or stochastic) exposure intervention. In this section, we differentiate five classes of general counterfactual variables considered in this work. Each is defined by a particular modification of the previous NPSEM as described below.

The NPSEM is modified by changing the structural equations associated with a set (denoted by X) of endogenous variables referred to as the action variables. More specifically, for each action variable X(t)∈X, the original structural equation X(t)=fX(t)(PX(t)0,UX(t)) is replaced with a new structural equation X(t)=f∗X(t)(PX(t)0,UX(t),U∗X(t)) and a joint conditional distribution (denoted by PU∗∣U) is specified for the new random disturbances associated with the action variables, (UX(t)∗)X(t)∈X, given the original disturbances U. The endogenous variables resulting from such a modified NPSEM are referred to as counterfactual variables and their joint distribution is entirely defined by the joint distribution of the disturbances (U,(U∗X(t))X(t)∈X) which we denote by PU,U∗. To differentiate such counterfactual variables from the original endogenous variables in the unmodifed NPSEM, we use the generic superscript notation ∗, i.e., the endogenous variables in any given modifed NPSEM are denoted by Y∗(t), Z∗(t), I∗0(t), I∗(t), A2∗(t), A1∗(t), and N∗(t) and the modified NPSEM is then expressed as the following collection of equations:

1. X∗(t)=f∗X(t)(PX∗(t)0,UX(t),U∗X(t)) for each counterfactual variable X∗(t) corresponding with an action variable X(t)∈X, where PX∗(t)0 denotes the counterfactual analog of the parents PX(t)0 in the modified NPSEM (e.g., PN∗(t)0≡(Yˉ∗(t),Zˉ∗(t),Iˉ0∗(t),Aˉ2∗(t),Aˉ1∗(t),Nˉ∗(t−1))),

2. X∗(t)=fX(t)(PX∗(t)0,UX(t)) for each counterfactual variables X∗(t) corresponding with a non-action variable X(t)∉X.

5 No direct effect (NDE) assumption

We denote, respectively, by Aˉ∗ and Nˉ∗, the union of the supports of the counterfactual treatment processes Aˉ∗1(K) and counterfactual monitoring processes Nˉ∗(K) introduced in Section 4.2 for any choices of static treatment and monitoring interventions aˉ∗1 and nˉ∗:

We note that these sets contain the supports of, not only, the observed treatment and monitoring processes, but also the supports of the various counterfactual treatment and monitoring processes defined in Section 4 based on specific choices for aˉ∗1, nˉ∗, dx, dx∗, and h∗N(t), i.e.,

We define the NDE assumption as the following set of equalities: