Adjustment for Variable Adherence under Hierarchical Structure: Instrumental Variable Modeling via Compound Residual Inclusion

Abstract

Background

Variable adherence to assigned conditions is common in randomized clinical trials.

Objectives

A generalized modeling framework under longitudinal data structures is proposed for regression estimation of the causal effect of variable adherence on outcome, with emphasis upon adjustment for unobserved confounders.

Research Design

A nonlinear, nonparametric random-coefficients modeling approach is described. Estimates of local average treatment effects among compliers can be obtained simultaneously for all assigned conditions to which participants are randomly assigned within the trial. Two techniques are combined to address time-varying and time-invariant unobserved confounding—residual inclusion and nonparametric random-coefficients modeling. Together these yield a compound, two-stage-residual inclusion, instrumental variables model.

Subjects

The proposed method is illustrated through a set of simulation studies to examine samll-sample bias and in application to neurocognitive outcome data from a large, multicenter, randomized clinical trial in sleep medicine for continuous positive airway pressure treatment of obstructive sleep apnea.

Results

Results of simulation studies indicate that, relative to a standard comparator, the proposed estimator reduces bias in estimates of the causal effect of variable adherence. Bias reductions were greatest at higher levels of residual variance and when confounders were time-varying.

Conclusions

The proposed modeling framework is flexible in the distributions of outcomes that can be modeled, applicable to repeated measures longitudinal structures, and provides effective reduction of bias due to unobserved confounders.

2. Introduction

2.1 Overview of problem and approach

Variable adherence to assigned condition is commonplace in randomized clinical trials (RCTs). Oftentimes, variable adherence is endogenous in nature, being driven by patient self-selection. A variety of statistical methods have been developed for the purpose of estimating consistently the cause-and-effect (causal) relationship between observed adherence and the clinical outcome of interest in the presence of unobserved confounding factors induced by endogeneity [1, 2]. Here we propose a generalized modeling framework that, for the first time, combines two-stage residual inclusion and nonparametric random-coefficients modeling to provide comprehensive adjustment for time-varying and time-invariant unobserved confounding within estimates of the causal effect of variable adherence for longitudinal sampling structures. This extension to longitudinal sampling structures is important because many RCTs monitor longitudinal outcomes on participants. By definition, RCTs assign participants to two or more conditions. The proposed modeling framework provides adjustment for time-varying and time-invariant unobserved confounding within each assigned condition of the RCT (Section 2.6).

2.2 Instrumental Variables

Under variable adherence, an intention-to-treat (ITT [3]) analysis gives estimates of treatment effectiveness (i.e., effect of treatment “as assigned” sensu [4]). However, the ITT estimate has limited clinical utility because it does not address response to the amount of treatment as actually received [5]. ITT analysis forgoes the opportunity to capitalize on a possible opportunity to estimate dose response from outcome data collected under variable adherence. Consistent estimation of dose response is complicated by possible confounders—factors which cause variation in adherence and cause variation in outcome. An example is “confounding by indication” [6] wherein a participant adjusts adherence based on current medical status per medical advice. Various methods to address confounding have been proposed, including propensity scores [7, 8, 9], G-estimation [10], and instrumental variables (IV [11]). IV methods are among those that offer a distinct advantage of permitting consistent estimation of dose-response in the presence of unobserved confounders, which is an important property because few if any situations may exist where investigators can be certain that all possible confounders have been measured. For this reason, we have incorporated IVs as a central component of our proposed modeling framework.

IV methods need to satisfy certain assumptions to generate consistent estimates of the causal effect of an intervention under variable adherence (i.e., the “local average treatment effect” among compliers, LATEC [12]). These assumptions need to be considered in each specific application of IV methods. The first assumption is that the IV must cause exposure to intervention and the second is that IVs affect outcome only indirectly through exposure to intervention (“exclusion restriction”) [13]. A third assumption is that IV and outcome have no common causes. As Hernan and Robins [13] explain, the random assignment mechanism satisfies all three of these conditions especially, regarding exclusion restriction, under double-blind RCTs. True random assignment is unquestionably exogenous and often associated strongly with degree of adherence to assigned condition. Unusual circumstances may arise where this may not be true, such as in the presence of “defiers” [14], participants predisposed to adhering to whichever assigned condition they are not assigned, which would violate the fourth requisite IV assumption of “monotonicity” [13]. A fifth assumption is that randomized units (e.g., patients or clusters) have mutually independent potential outcomes (“stable unit treatment value assumption,” SUTVA [14, 15]). For present purposes, SUTVA is assumed to apply.

2.3 Longitudinal structure

The current literature on the development and application of IV methods to variable adherence is active and broad (e.g., [16 - 20]). Slower to develop have been models, methods of estimation and example IV applications to hierarchical sampling structures, including longitudinal, despite the fact that that many RCTs collect longitudinal measurements on major outcomes within participants across study visits. Much existing work on longitudinal IV methods, especially earlier studies, focused on the limited circumstance in which adherence is coded as a binary presence-absence variable (e.g., [21]), which may be due to the fact that, in general, early literature on adjustment for variable adherence emphasized binary measures of adherence. In reality, adherence often varies across a gradation of levels. Bond et al. [11] allow for multi-level adherence in a longitudinal setting. They discuss application via generalized least squares (GLS) and employ generalized estimating equations (GEE [22]) in their worked example. Lamiraud et al. [23] also employed GEE.

GEE and GLS are marginal models, providing estimates of causal effects of treatment as received on average across the population [24]. Despite recent interest with “population health” [25], population average estimates are off-purpose for most RCTs, where the goal is to understand how individual patients respond to treatment (e.g., [26]). In contrast to GEE and GLS, random-coefficients models [24] are well-suited for characterizing individual patient response, because they offer estimates of longitudinal response trajectories of individual patients and of the average patient. Only under a linear model do estimates of population-average effects and average-patient effects coincide [24]; but many settings require a nonlinear modeling framework (e.g., binary outcome), which generalized linear random-coefficients models and nonlinear random-coefficients models [27] accommodate. To improve upon existing marginal modeling and/or linear modeling approaches, the method presented here employs random-coefficients models with allowance for nonlinear functional structure (Sections 2.4 and 3.1) and participant-level random coefficients.

2.4 Random coefficients

To date, random coefficients (“random effects”) have been employed only rarely to aid estimation of the LATEC. Sitlani et al. [28] assumed that random coefficients capture latent factors that are explanatory of endogenous selection to an assigned surgical intervention. Small et al. [15] adjusted for variable adherence in the setting of a binary outcome. Lamiraud et al. [23] employeded a correlated random-coefficients [29] structure. Correlated random coefficients are a very flexible structure because separate participant-level random coefficients could be employed for adherence and for outcome with these two random coefficients allowed to covary. A covariance structure is also specified at the level of individual observations. A shared random coefficients (SRC [29]) specification is another possibility (e.g., [28]). Within our proposed modeling framework we assume, as in previous literature [28], that random coefficients capture latent factors that are explanatory of endogenous selection. We go even further and recognize that the presence of unobserved confounders can, probably often, result in complex distributions for random coefficients. Specifically, unobserved confounders can generate a mixture distribution. We model that complexity in a first-stage equation with adherence as the outcome regressed on randomized condition as an instrument with a nonparametric distribution for random intercepts [30, 31]. Rather than shared or correlated random coefficient modeling, we employ the predicted values of these first-stage random intercepts as a fixed (latent effects) covariate in the second stage equations for the efficacy outcome. Using first-stage random coefficients as a second-stage fixed covariate conditions the second stage on the first and thereby eliminates the need for specification of a between-equation covariance structure, thereby simplifying model formulation. Second-stage equations regress the efficacy outcome on these predicted first-stage random intercepts and observed adherence. This facilitates adjustment for unobserved confounders (captured by the first-stage random coefficient) that are invariant across repeated measures as a form of two-stage residual inclusion (TSRI, [32]). Including the same predicted random coefficient values in both stages should avoid unnecessary introduction of inconsistency into the estimate of the causal effect of variable adherence [11].

2.5 Two-stage residual inclusion

TSRI has been shown to provide a consistent estimate of the causal effect of interest under linear and nonlinear model structures [30]. Indeed, TSRI addresses the concern raised by Lamiraud et al. [23] about application of IV methods to nonlinear settings. In addition to adjustment for time-invariant unobserved confounding, our method also employs the first-stage residuals as an additional fixed covariate in second-stage efficacy equations to offer adjustment for time-varying unobserved confounders thereby providing compound, TSRI, instrumental variable estimation (cTSRI-IV). Details are provided in Section 3.

2.6 Simultaneous adjustment

RCTs by definition consist of two or more conditions to which participants are assigned. The dose-response relationship between treatment as received and efficacy outcome may be potentially obscured by endogenous selection (and unobserved confounding) in some or all assigned conditions; and the precise nature of endogenous selection can easily vary among these conditions. Our system of equations (Section 3.1) permits simultaneous and condition-specific estimation of the LATEC for each and every of the study conditions to which participants are randomly assigned in the RCT. This efficient use of data from a RCT may be particularly helpful in application to comparative effectiveness studies, which are a class of RCT designs that are seeing widening use in efforts to enhance equipoise.

3. Methods

3.1 Model specification

Parameters, random variables, and observed values will be denoted by uppercase Greek letters, uppercase Roman letters, and lowercase Roman letters, respectively. Let random variable A_kit denote adherence to the randomly assigned study condition k for person i, i = {1, ..., n}, at time t; and let random variable O_kit denote outcome for condition k, person i at time t. Randomly assigned conditions are indexed by k, k = {1, ..., κ}, 2 ≤ κ. To simplify exposition here, we assume randomization is to two assigned conditions, κ ≡ 2, which allows illustration of the basic method and simplification of some notation. (Extension to 2 < κ is straightforward.) For 1 ≤ γ₁ < γ₂, models for the expectations of A_kit and O_kit are given by the following system of equations (i.e., k ∈ {1, 2} for adherence and for outcome):

$${\mu }_{ki}\stackrel{\mathrm{def}}{=}E[A_{kit}\mid R_{ki}]=g\left({\alpha }_0+{\alpha }_1(k-1)+{\sum }_{p=2}^{{\gamma }_1}{\alpha }_{pk}{x}_{pkit}+R_{ki}\right){\theta }_{ki}\stackrel{\mathrm{def}}{=}E[O_{kit}\mid A_{kit}=a_{kit},R_{ki}=r_{ki},{\acute{R}}_{ki}]={\acute{g}}_k\left({\beta }_0+{\beta }_{k1}{a}_{kit}+{\sum }_{p=2}^{{\gamma }_1}{\beta }_{pk}{x}_{pkit}+{\sum }_{p={\gamma }_1+1}^{{\gamma }_2}{\beta }_{pk}{x}_{pkit}+{\epsilon }_k(a_{kit}-{\mu }_{ki})+\varsigma k{g}^{-1}\left(r_{ki}\right)+{\acute{R}}_{ki}\right)$$ [1a]

where g and ǵ_k are some continuous real-valued mappings from $ℝ$ to $ℝ$ (e.g., link functions). The respective first-stage and second-stage participant level random coefficient are R_ki and Ŕ_ki. The term a_kit – μ_ki provides residual inclusion to adjust for time-varying endogeneity effect ε_k for the k^th assigned condition; ζ_k provides residual inclusion to adjust for time-invariant endogeneity effect ζ_k for the k^th assigned condition, with requisite inverse transformation g⁻¹; and x_pkitt is the p^th covariate measured on the i^th person at time t in the k^th condition. Covariates may include interactions with assigned condition and with time. Typically μ_ki will be a single equation, while the θ_ki can be formulated as a separate equation for each assigned condition. All first-stage covariates are included in the second stage as main effects to avoid introduction of bias [11]. Defining $Z_{kit}\stackrel{\mathrm{def}}{=}{A}_{kit}-{\mu }_{ki}$ and ${\acute{Z}}_{kit}\stackrel{\mathrm{def}}{=}{O}_{kit}-{\theta }_{ki}$ take

$$E\left[R_{ki}\right]=E\left[Z_{kit}\right]=E\left[{\acute{Z}}_{kit}\right]\equiv 0$$ [1b]

and

$$Var\left[\{R_{1i},R_{2i},{\acute{R}}_{1i},{\acute{R}}_{2i},Z_{1it},Z_{2it},{\acute{Z}}_{1it},{\acute{Z}}_{2it}\}\right]\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccccccc}\hfill {\sigma }_{11}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\sigma }_{21}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{11}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{21}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\sigma }_{12}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\sigma }_{12}^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{12}^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{22}^2\hfill \end{array}\right].$$ [1c]

Random coefficients and residuals are independent between assigned conditions due to randomized assignment; and random coefficients and residuals are independent between stages due to residual inclusion (conditioning). Since 1c does not account for longitudinal structure, we specify

$$Var\left[\{Z_{1i1}\cdots Z_{1i\tau },Z_{2i1}\cdots Z_{2i\tau },{\acute{Z}}_{1i1}\cdots {\acute{Z}}_{1i\tau },{\acute{Z}}_{2i1}\cdots {\acute{Z}}_{2i\tau }\}\right]\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}\hfill {\sigma }_{12}^2{\mathrm{P}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\sigma }_{22}^2{\mathrm{P}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{12}^2{\acute{P}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\acute{\sigma }}_{22}^2{\acute{\mathrm{P}}}_2\hfill \end{array}\right]$$ [1d]

with

$${\mathrm{P}}_k\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {\rho }_{k12}\hfill & \hfill \cdots \hfill & \hfill {\rho }_{k1\tau }\hfill \\ \hfill {\rho }_{k12}\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill {\rho }_{k2\tau }\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\rho }_{k1\tau }\hfill & \hfill {\rho }_{k2\tau }\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right],{\acute{\mathrm{P}}}_k\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {\acute{\rho }}_{k12}\hfill & \hfill \cdots \hfill & \hfill {\acute{\rho }}_{k1\tau }\hfill \\ \hfill {\acute{\rho }}_{k12}\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill {\acute{\rho }}_{k2\tau }\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\acute{\rho }}_{k1\tau }\hfill & \hfill {\acute{\rho }}_{k2\tau }\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right].$$ [1e]

Together, 1a through 1e constitute a completely general formulation to permit cTSRI-IV estimation for κ ≡ 2. The choice of mappings, covariates, any further restrictions on the covariance structure, as well as statistical distributions for adherence, outcomes, and random coefficients will be specific to each application; and, as indicated above, models can be specified for 2 < κ. Specific model structures are illustrated in Sections 3.3 and 4.

3.2 Parameter estimation

Various approaches to cTSRI-IV estimation of the parameters of model 1 are possible. Here maximum likelihood estimation is employed for its efficiency and availability. Let ψ_k denote the first-stage (adherence outcome) expectation parameters and ${\acute{\psi }}_k$ denote the second-stage (efficacy outcome) expectation parameters. Respective observed adherence and efficacy outcome vectors are a_k and O_k for assigned condition k. We appropriately condition on the random coefficients such that the resultant generalized sequence of likelihood specifications are

$$\mathcal{L}[a\mid {\psi }_1,{\psi }_2,{\sigma }_{11}^2,{\sigma }_{21}^2]=\prod _{k=1}^2\prod _{i=1}^n\mathcal{L}[a_{kit}\mid {\psi }_k,R_{ki}]\mathcal{L}[R_{ki}\mid {\sigma }_{k1}^2],$$ [2a]

$$\mathcal{L}[{\mathrm{o}}_k\mid {\acute{\psi }}_k]=\prod _{i=1}^n\mathcal{L}[o_{kit}\mid \{{\psi }_k,r_{ki},a_{kit}\}]\mathcal{L}[{\acute{R}}_{ki}\mid {\sigma }_{k2}^2].$$ [2b]

Because we wish to place no further restrictions on the distribution of the R_ki, estimation of 2a is via an expectation-maximization (EM) algorithm [31, 32]. The EM algorithm is initiated by drawing initial values for the random coefficients from a mixture of normal distributions, although final predictions r_ki resulting from application of the EM algorithm need not have a distribution that is a mixture of normal distributions. Use of an EM algorithm is intuitively appealing because it is designed to recover latent structures, in this case latent confounders. Likelihoods 2a and 2b can be combined as a product (due to independence between assigned conditions and conditioning in the second stage) for estimation purposes but that only offers an efficiency advantage in terms of reduced variance of parameter estimates if the second stage model for the efficacy outcome shares a common parameter with the first stage model for the adherence outcome—a restriction that may often not be warranted on scientific grounds. Nor do we wish to necessarily have common parameters across assigned conditions in the second stage, although this option could explored. A sequential estimation procedure is illustrated in Sections 3.3 and 4.

The structure of model 1 can induce strong correlation among A_kit, Z_kit, and R_ki. This multicollinearity complicates second-stage (efficacy outcome) parameter estimation. In addition to second-stage predictors a_kit, z_kit, and r_ki, this multicollinearity can also encompass the second-stage intercept. Regularization approaches such as a ridge penalty are one possibility (Section 5.1), although, without care, these may introduce bias. In Sections 3.3 through 4.5, we apply an alternative strategy that combines orthogonalization and white-noise data augmentation [33] to mitigate multicollinearity's impact on finite-sample bias and variance.

3.3 Simulation studies

A set of ninety-six simulation studies were conducted to assess small-sample bias properties of the proposed cTSRI-IV estimator. All studies assumed a RCT with 1:1 random assignment of participants to two different conditions and participants observed at six visits that were regularly-spaced in time. Each participant's dose was simulated as a linear function of a fixed intercept term α₀ = 100, a random participant-specific intercept term $Z_i\stackrel{iid}{\sim }N(0,25)$, participant's age ${\alpha }_1=\frac{1}{5}$ (time-invariant), assigned condition α₂ ∈ {10, 50} (instrument), and a time-varying random variable $I_{it}=B_{it}-1,B_{it}\stackrel{iid}{\sim }Bernoulli\left(\frac{1}{2}\right)$ of effect α₃ = 5. Parameter α₂ was varied in value to represent weak (α₂ = 10) and strong (α₂ = 50) instruments. In the first assigned condition, efficacy response was simulated as a linear function of a fixed intercept term , β₀ = 0, visit ${\beta }_1\in \{\frac{1}{2},1\}$, participant's dose β₂ ∈ {1, 2, 3, 4, 5, 6} of that condition, participant's age ${\beta }_3=\frac{1}{5}$, which served as a time-invariant confounder, and two random terms—a random participant-specific deviation Z_i from the fixed intercept $(Z_i\stackrel{iid}{\sim }N(0,c_{1}25),c_1\in \{0,5,10,15,20\})$ and a time-varying residual $E_{it}\stackrel{iid}{\sim }N(0,c_2{\sigma }^2),c_2\in \{1,20\}$, where, for example, c₂ ∈ {1, 20} indicates that c₂ was varied over the set of valuse across simulation studies (Table 1). The efficacy response in the second condition differed from the first due to the addition of I_it, β₄ = 5, which served as a time-varying confounder, and the removal of the age effect β₃ = 0 (time-invariant confounder) in the linear predictor. For computational efficiency, simulation studies were nested such that c₁ and β₂ varied over their each pseudo-random (Mersenne-Twister) draw of Z_i. Five hundred data sets were randomly generated at each of two sample sizes (n ∈ {50, 100}) for each combination of parameter values. The full set of simulations was repeated setting ǵ_k(·) = exp(·) to examine finite sample properties of the cTSRI-IV estimator under nonlinear structure.

Table 1.

Results of 96 simulation studies as described in section 3.3.

							Naïve		cTSRI-IV
							Time-invariant confounder	Time-varying confounder	Time-invariant confounder	Time-varying confounder
Type	n	α 2	β 1	c 1	c 2	β 2	${\widehat{\beta }}_2$	${\widehat{\beta }}_2$	${\widehat{\beta }}_2$	${\widehat{\beta }}_2$
L	50	10	1	10	1	1	1.41	1.49	1.20	1.09
L	50	10	1/2	0	1	2	2.25	2.37	2.19	2.08
L	50	10	1	5	1	3	3.46	3.60	3.20	3.09
L	50	10	1/2	15	1	4	4.73	4.14	4.18	4.07
L	50	10	1	20	1	5	5.77	5.09	5.20	5.08
L	50	10	1/2	10	1	6	6.42	6.50	6.18	6.07
L	50	10	1	10	20	1	2.24	2.45	1.18	1.07
L	50	10	1/2	0	20	2	2.13	2.25	2.19	2.15
L	50	10	1	5	20	3	3.70	3.87	3.28	3.16
L	50	10	1/2	15	20	4	5.71	5.96	4.30	4.06
L	50	10	1	20	20	5	7.06	7.34	5.31	5.18
L	50	10	1/2	10	20	6	7.25	7.46	6.29	6.01
L	50	50	1	10	1	1	1.34	1.50	1.15	1.05
L	50	50	1/2	0	1	2	2.21	2.35	2.14	2.04
L	50	50	1	5	1	3	3.40	3.60	3.15	3.05
L	50	50	1/2	15	1	4	4.35	4.44	4.14	4.05
L	50	50	1	20	1	5	5.67	5.16	5.15	5.05
L	50	50	1/2	10	1	6	6.34	6.50	6.14	6.04
L	50	50	1	10	20	1	2.11	2.54	1.22	1.09
L	50	50	1/2	0	20	2	2.12	2.24	2.16	2.01
L	50	50	1	5	20	3	3.65	3.93	3.16	3.03
L	50	50	1/2	15	20	4	5.53	6.09	4.08	4.05
L	50	50	1	20	20	5	6.81	7.46	5.26	5.07
L	50	50	1/2	10	20	6	7.12	7.55	6.01	6.05
L	100	10	1	10	1	1	1.42	1.50	1.20	1.09
L	100	10	1/2	0	1	2	2.25	2.37	2.18	2.07
L	100	10	1	5	1	3	3.47	3.61	3.20	3.08
L	100	10	1/2	15	1	4	4.76	4.10	4.19	4.07
L	100	10	1	20	1	5	5.77	5.09	5.20	5.09
L	100	10	1/2	10	1	6	6.41	6.50	6.19	6.07
L	100	10	1	10	20	1	2.26	2.47	1.17	1.01
L	100	10	1/2	0	20	2	2.13	2.24	2.21	2.07
L	100	10	1	5	20	3	3.70	3.86	3.16	3.14
L	100	10	1/2	15	20	4	5.73	5.98	4.12	4.03
L	100	10	1	20	20	5	7.07	7.35	5.17	5.05
L	100	10	1/2	10	20	6	7.26	7.47	6.20	6.11
L	100	50	1	10	1	1	1.34	1.50	1.15	1.05
L	100	50	1/2	0	1	2	2.20	2.34	2.14	2.04
L	100	50	1	5	1	3	3.40	3.60	3.15	3.05
L	100	50	1/2	15	1	4	4.34	4.44	4.14	4.04
L	100	50	1	20	1	5	5.71	5.12	5.15	5.05
L	100	50	1/2	10	1	6	6.34	6.50	6.14	6.04
L	100	50	1	10	20	1	2.11	2.54	1.13	1.07
L	100	50	1/2	0	20	2	2.12	2.24	2.13	2.08
L	100	50	1	5	20	3	3.63	3.90	3.12	3.04
L	100	50	1/2	15	20	4	5.53	6.09	4.10	4.10
L	100	50	1	20	20	5	6.82	7.47	5.14	5.06
L	100	50	1/2	10	20	6	7.12	7.55	6.17	6.07
NL	50	10	1	10	1	1	1.43	1.50	1.21	1.10
NL	50	10	1/2	0	1	2	2.26	2.38	2.21	2.09
NL	50	10	1	5	1	3	3.50	3.64	3.22	3.15
NL	50	10	1/2	15	1	4	4.71	4.19	4.18	4.08
NL	50	10	1	20	1	5	5.78	5.11	5.20	5.09
NL	50	10	1/2	10	1	6	6.47	6.56	6.19	6.09
NL	50	10	1	10	20	1	2.25	2.46	1.18	1.06
NL	50	10	1/2	0	20	2	2.14	2.26	2.15	2.03
NL	50	10	1	5	20	3	3.71	3.87	3.16	3.04
NL	50	10	1/2	15	20	4	5.73	5.98	4.17	4.08
NL	50	10	1	20	20	5	7.06	7.34	5.20	5.08
NL	50	10	1/2	10	20	6	7.25	7.45	6.17	6.03
NL	50	50	1	10	1	1	1.38	1.55	1.15	1.05
NL	50	50	1/2	0	1	2	2.25	2.41	2.14	2.05
NL	50	50	1	5	1	3	3.52	3.76	3.15	3.05
NL	50	50	1/2	15	1	4	4.51	4.68	4.14	4.05
NL	50	50	1	20	1	5	5.62	5.56	5.15	5.05
NL	50	50	1/2	10	1	6	−616.18*	−299.39*	6.14	6.05
NL	50	50	1	10	20	1	2.12	2.56	1.15	1.05
NL	50	50	1/2	0	20	2	2.12	2.23	2.14	2.04
NL	50	50	1	5	20	3	3.66	3.95	3.15	3.05
NL	50	50	1/2	15	20	4	5.59	6.17	4.14	4.05
NL	50	50	1	20	20	5	6.93	7.61	5.15	5.05
NL	50	50	1/2	10	20	6	−2905.40*	−3469.35*	6.14	6.04
NL	100	10	1	10	1	1	1.42	1.49	1.20	1.09
NL	100	10	1/2	0	1	2	2.25	2.37	2.19	2.08
NL	100	10	1	5	1	3	3.49	3.63	3.20	3.10
NL	100	10	1/2	15	1	4	4.74	4.13	4.19	4.07
NL	100	10	1	20	1	5	5.77	5.10	5.20	5.09
NL	100	10	1/2	10	1	6	6.46	6.55	6.19	6.08
NL	100	10	1	10	20	1	2.26	2.47	1.18	1.08
NL	100	10	1/2	0	20	2	2.12	2.24	2.19	2.05
NL	100	10	1	5	20	3	3.71	3.88	3.19	3.08
NL	100	10	1/2	15	20	4	5.73	5.98	4.18	4.07
NL	100	10	1	20	20	5	7.16	7.43	5.21	5.10
NL	100	10	1/2	10	20	6	7.26	7.47	6.18	6.05
NL	100	50	1	10	1	1	1.35	1.55	1.15	1.05
NL	100	50	1/2	0	1	2	2.25	2.40	2.14	2.05
NL	100	50	1	5	1	3	3.50	3.75	3.15	3.05
NL	100	50	1/2	15	1	4	4.50	4.70	4.14	4.05
NL	100	50	1	20	1	5	5.60	5.55	5.15	5.06
NL	100	50	1/2	10	1	6	−8196.30*	−8040.75*	6.14	6.05
NL	100	50	1	10	20	1	2.15	2.60	1.15	1.05
NL	100	50	1/2	0	20	2	2.15	2.25	2.14	2.04
NL	100	50	1	5	20	3	3.65	3.95	3.15	3.05
NL	100	50	1/2	15	20	4	5.60	6.15	4.14	4.05
NL	100	50	1	20	20	5	6.95	7.65	5.15	5.06
NL	100	50	1/2	10	20	6	−5196.85*	−5543.85*	6.14	6.04

Types of models are linear = L or nonlinear = N. n = total sample size, α₂ = strength of instrument, β₁ = effect on efficacy outcome of each visit, β₂ = effect on efficacy outcome of each dose unit, c₁ = scaling factor for the variance of the time-invariant random deviation Z_i from the fixed intercept, c₂ = scaling factor for the variance of the time-varying residual E_it, and ${\widehat{\beta }}_2$ is the estimate of effect on efficacy outcome of each dose unit for the indicated estimator and condition. Cells in gray marks instances where bias of CTSRI-IV estimator exceeded that for the naïve estimator.

^*failure of algorithm convergence.

For each simulated data set, contrasting approaches were used for fitting. The “naïve” approach entailed a single-stage regression of the efficacy outcome on dose, the dose by condition interaction, visit, and the visit by condition interaction. Fitting employed a either a linear mixed-coefficients model formulation with fixed coefficients for the four covariates, normally distributed random intercepts and residuals, and estimation via restricted maximum likelihood for ǵ_k(·) = 1 or, in the simulations under nonlinear structure, a generalized linear mixed-coefficients model formulation with fixed coefficients for the four covariates, normally distributed random intercepts, logarithm link, and estimation via adaptive Gaussian quadrature. The cTSRI-IV approach was in two stages. The first stage was a linear mixed-coefficients model formulation with fixed coefficients for condition, visit and their interaction; nonparametrically distributed random intercepts; normally distributed residuals; and estimation via the EM algorithm, initiated with a mixture of two normal distributions for the random intercepts. (In practice, different mixture quantities are applied and goodness of fit assessed, with the quantity of components dependent upon the structure of the time-invariant latent confounders). The second stage entailed fitting a separate fixed-coefficients generalized linear mixed model for each assigned condition. For each of these two models, the efficacy outcome was regressed on dose, visit, the participant's first-stage predicted value for the random intercept, and the participant's first-stage predicted residual values. Second-stage residuals were assumed to be normally (or log-normally) distributed, as designed. Multicollinearity was addressed through a combination of Gram-Schmidt orthogonalization of r_ki with respect to a_kit and white-noise augmentation of the intercept vector as 1 + λw, for tuning constant 0 < λ, where w is a 6 × n element vector of pseudo-random independent draws from a standard normal distribution. Constant λ will need to be optimized in each application. Settings too close to zero cause failure in convergence of the optimization algorithm while settings too far from zero increase bias and variance of the estimate of the LATEC. In general, λ should be employed as an algorithm convergence tuning parameter and set to a value just large enough to provide reliable algorithm convergence. We present results for settings of λ = 0.01 for the linear model and λ ∈ [0.01, 0.25] for the nonlinear model. Nonparametric fixed effects modeling was performed using the nplmreg package in R version 3.2 [34]. We found that the trust-region optimization algorithm, with central finite difference derivative approximation and a small initial trust region radius (e.g., < 0.01), available in the nlmixed procedure in SAS^® version 9.4 (SAS^® Institute, Cary, NC, USA), gave algorithm convergence failures below 4.7% and 13.5% during simulations for linear and nonlinear models, respectively. Second-stage random coefficients were predicted via Laplace approximation. Because neither the naïve approach nor cTSRI-IV estimation explicitly included age nor I_it as covariates, these served as unobserved time-invariant and time-varying confounders, respectively. Code for simulation studies is available upon request (THH).

4. Results

4.1 Simulation results

Results of the 96 simulation studies are summarized in Table 1. Compared to the naïve estimator, cTSRI-IV estimation reduced bias in estimates of adherence dosage in all but a few rare cases (gray cells in Table 1). Bias reduction was greatest for higher levels of residual variance (c₂ = 20). Among cTSRI-IV estimates, bias reduction was consistently greater for a time-varying confounder than the time-invariant confounder. Bias reduction was similar across linear and nonlinear models.

4.2 The APPLES trial

Application of the cTSRI-IV estimator is demonstrated using data from a RCT in sleep medicine. The Apnea Positive Pressure Long-term Efficacy Study (APPLES) was a multicenter, clinical trial sponsored by the National Heart, Lung and Blood Institute, in which 1,105 participants were randomized at a 1:1 ratio to either continuous positive airway pressure (CPAP) therapy for obstructive sleep apnea (OSA) or to a sham device [35]. Per the study's protocol, each participant was to complete baseline screening and diagnostic testing, be randomized to sham or active, and followed for 6 months. All participants provided informed consent. The study protocol was approved by the Institutional Review Board at each of five participating centers. Three primary neurocognitive outcomes, corresponding to each of three major domains of neurocognitive function (attention and vigilance, learning and memory, and executive/frontal lobe function), were assessed pre-randomization (baseline) and at 2-month and 6-month visits during the active intervention period: the Buschke Selective Reminding Test - Sum Recall (BSRT-SR), the Pathfinder Number Test - Total Time (Trails A of Trail Making Test; PFNT-TT), and the primary mid-day score from the Sustained Working Memory Test – Overall Mid-Day Index (SWMT-OMI [36]). Greater neurocognitive ability is indicated by higher scores for BSRT-SR and SWMT-OMI and lower scores for PFNT-TT Here we restrict analysis to the PFNT-TT as the efficacy outcome. A total of 849 participants had data at the 2-month and 6-month visits for the analysis presented here, n₁ = 412 for sham and n₂ = 437 for active. Information about the impact of missing data due to dropout is detailed in Kushida et al. [35]. All analyses described in the remainder of Section 4 were performed in SAS® version 9.4 (SAS® Institute, Cary, NC, USA) and R [34].

4.3 cTSRI-IV estimation for APPLES

Participants in APPLES were randomized to two assigned conditions, so κ = 2 as in Section 3.1. Separately for each of these three outcomes, we fit model 1 with the following tailoring to APPLES:

$$\begin{array}{cc}\hfill {\mu }_{ki}\stackrel{\mathrm{def}}{=}E[A_{kit}\mid R_{ki}]=& {\alpha }_0+{\alpha }_1(k-1)+{\sum }_{p=2}^6{\alpha }_p{x}_{pi}+R_{ki}\hfill \\ \hfill {\theta }_{ki}\stackrel{\mathrm{def}}{=}E[O_{kit}\mid A_{kit}=a_{kit},R_{ki}=r_{ki},{\acute{R}}_{ki}]=& {\beta }_{k0}+{\beta }_{k1}{a}_{kit}+{\sum }_{p=2}^6{\beta }_{pk}{x}_{pki}+{\epsilon }_k(a_{kit}-{\mu }_{ki})+{\varsigma }_k{r}_{ki}+{\acute{R}}_{ki}\hfill \end{array}$$ [3a]

and

$${\mathrm{P}}_k\stackrel{\mathrm{def}}{=}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right],{\acute{\mathrm{P}}}_k\stackrel{\mathrm{def}}{=}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right].$$ [3b]

Here k = 1 for sham and k = 2 for active. Covariates {x₂, x₃, x₄, x₅} are those used to construct randomization strata: race (non-Caucasian x₂ = 0, Caucasian x₂ = 1), gender (female x₃ = 0, male x₃ = 1), moderately severe OSA at baseline (x₄ = 1, x₄ = 0 otherwise) and severe OSA at baseline (x₅ = 1, x₅ = 0 otherwise). Baseline age in years is also a covariate (x₆). Age is a potentially strong confounder. In our experience, adherence often tends to increase with participants’ age in RCTs. Also, performance on neurocognitive measures can decline with age. For purposes of exposition, this model assumes a common LATEC across the subpopulations defined by the covariates. The common LATEC assumption can be relaxed [11] and indeed this possibility should be examined in a complete analysis, including for the APPLES trial. Random coefficients Ŕ_ki~N(0,σ²) are also included in the second stage to account for possible correlation between the two repeated measurements. All parameter estimates are time-averages across the 2-month and 6-month visits. Alternative formulations that allow for interaction with time are also possible. Specification 3b assumes that repeated measurements on the same outcome (adherence or a neurocognitive measure) within a person are uncorrelated conditional upon the random coefficient R_ki. We assume $Z_{kit}\sim N(0,{\sigma }_{k2}^2)$ and ${\acute{Z}}_{kit}\sim N(0,{\sigma }_{k3}^2)$ and ǵ_k = 1, k ∈ {1,2}. Likelihood formulation is

$$\mathcal{L}[a\mid {\psi }_1,{\psi }_2,{\sigma }_{11}^2,{\sigma }_{21}^2]=\prod _{k=1}^2\prod _{i=1}^n\varphi [a_{kit}\mid {\psi }_k,R_{ki}]f_2[R_{ki}\mid {\sigma }_1^2],$$ [4a]

$$\mathcal{L}[{\mathrm{o}}_k\mid {\acute{\psi }}_k]=\prod _{i=1}^n\varphi [o_{kit}\mid \{{\acute{\psi }}_k,r_{ki},a_{kit}\}]\varphi [{\acute{R}}_{ki}\mid {\sigma }_{k2}^2],$$ [4b]

where ϕ[·] denotes the normal density, f₂ is the density function for an unspecified distribution where the subscript indicates a two-component mixture density, and k ∈ {1,2}.

4.4 Model fitting and hypothesis testing

Adherence was measured nightly as hours of device usage per night via a monitor built into the sham and active CPAP devices (Encore ® Pro SmartCard ©, Phillips Respironics ® Inc., Andover, MA, USA) and formulated as the average nightly value over the prior two months. Approximately 21% and 29% of nightly adherence data were missing at 2 months and 6 months, respectively, yielding the sample size of 849 participants for fitting model 3. For present analysis, missing status was assumed to be non-informative (i.e., due to monitor malfunction) so that 2-month averages were calculated using only observed adherence values. Alternative approaches could entail some form of imputation, including setting missing values to zero assuming the participant was not using the device.

Parameters were estimated by separately optimizing each of the three likelihood equations (4a and 4b) using the trust-region optimization algorithm, as described in Section 3.3, with an initial trust region radius setting of 0.01; and we found λ = 0.01 provided reliable optimization algorithm convergence. Second-stage random coefficients were predicted via Laplace approximation. To propagate sampling error properly from the first stage into the second, we employed bootstrap resampling stratified on assigned condition. From 1,000 bootstrap resamples we calculated simple percentile 95% confidence bounds.

4.5 Findings

Table 2 reports estimates of bootstrap 95% confidence intervals from the fit of model 3 to the PFNT-TT efficacy outcome data from APPLES. Results suggest that an increasing dose of active adherence decreases (improves) average PFNT-TT scores, a finding not detected in the absence of cTSRI-IV estimation in the original report [35]. The distribution of second-stage efficacy residuals were somewhat skewed (sample skewness = 1.46) and strongly leptokurtic (sample kurtosis = 7.89), indicating a violation of the model's assumption that these residuals are drawn from a Gaussian distribution. We suggest possible remedies in Section 5.1.

Table 2.

Ninety-five percent confidence bounds on parameter estimates from fit of model 3 to the PFNT-TT efficacy outcome from APPLES. RE = random effect.

Parameter	2.5%	97.5%
Sham Intercept	21.03	24.18
Active Intercept	22.12	25.01
Sham RE	8.98	11.20
Active RE	−5.01	−2.21
Sham Residual	−5.40	−1.73
Active Residual	−0.02	1.79
Sham Age	0.49	2.78
Active Age	0.68	3.51
Sham White	−0.45	−0.08
Active White	−0.68	−0.08
Sham Male	−0.22	0.00
Active Male	−0.10	0.06
Sham Moderate	0.00	0.40
Active Moderate	0.03	0.31
Sham Severe	−0.19	0.13
Active Severe	0.15	0.69
Sham Adherence	−0.07	0.10
Active Adherence	−0.66	−0.35

5. Discussion

5.1 Statistical implications

The proposed method addresses estimation of the causal effect of treatment as received (LATEC) under possibly nonlinear and longitudinal model structures. Reduction in bias due to unobserved confounding is provided through isolation and adjustment for exogenous and endogenous components of adherence via time-invariant and time-variant residual inclusion (TSRI). Adjustment for observed confounders, such as age and known comorbidities, is also essential. Age may be useful because adherence and health outcomes often vary among RCT participants of differing ages. Altogether, this especially comprehensive approach to adjustment for unobserved confounders should effectively mitigate inconsistency in estimates of the LATEC in RCTs of longitudinal sampling structures, as our simulation studies suggest.

A potential drawback to comprehensive adjustment for unobserved confounders is that the cTSRI- IV model is enriched in fixed parameters, especially with incorporation of interaction terms and/or spline bases for covariates. When sample sizes are modest, identification of model parameters may oftentimes require further restrictions on those parameters. One general approach would be to add a L_q-penalty term (e.g., q = 1 for lasso or q = 2 for ridge regression) to the likelihood (equation 2) or, even better, apply these penalties to derive inputs such as principal components [37] as an additional means for addressing multicollinearity. This approach comes with additional computational complexity as penalty parameters may need be estimated via sample reuse procedures [37]. Further, additional sample reuse procedures, such as bootstrapping [38], would be needed to generate confidence intervals on parameter estimates that propagate sampling error in the penalty parameter estimate into LATEC estimates. To minimize introduction of bias into the estimate of the LATEC, the LATEC parameters could be excluded from the penalty term. Penalized estimation of the LATEC under the cTSRI-IV model merits study.

The cTSRI-IV model also contains random components (random effects and residuals). The standard approach to modeling random coefficients is to assume that these coefficients follow a Gaussian distribution. We propose a nonparametric finite mixture alternative to account for the overdispersion induced by the latent confounder(s). Another nonparametric approach is generalized maximum entropy estimation [39]. Generalized maximum entropy estimation can be useful for small sample sizes, in the presence of multicollinearity among covariates, and can also accommodate residual distributions that are skewed or strongly leptokurtic (Section 4.5) [40]. Bayesian estimation is yet another alternative and an attractive one in a parameter-rich setting because, like generalized maximum entropy estimation, Bayesian estimation permits borrowing information about parameters from prior published studies via empirically informed priors [41]; and Bayesian estimation can employ finite mixtures to model complex (e.g. leptokurtic) residual distributions.

We employ a nonparametric bootstrap procedure for constructing confidence intervals on model parameters (Section 4.4) to accommodate the possibly nonlinear structure of the model and two-stage estimation procedure. Various standard diagnostics, bias correctives, and efficiency gains are available for the nonparametric bootstrap [42], including those that allow for residual heteroscedasticity [43], so that, altogether the nonparametric bootstrap allows proper inference without sacrificing flexibility of model structure.

5.2 Limitations

The list of assumptions for consistent IV estimates of dose response is not short (Section 2.2). Among those assumptions is that the IV impacts outcome only indirectly through exposure to intervention (“exclusion restriction”). This should result if the RCT is effectively double-blinded [13]. APPLES was designed as a double-blind study; yet, analysis of exit interviews suggested that some participants, especially in the sham condition, may have been aware of their assigned condition, although overall degree of agreement between assigned and blinded guesses was poor [34]. Another assumption, SUTVA, will not hold if some participants interfere with other participants’ outcomes through social interactions [44].

5.3 Conclusions

We introduce a method for estimating the causal effects of treatment as received simultaneously for each assigned condition of a RCT within a framework that permits nonlinear modeling of longitudinal data structures. A comprehensive approach to adjustment for unobserved confounders is included. Future work could explore asymptotic properties of the proposed estimator and the implications of stochastic adherence [45].