Joint mixed-effects models for causal inference with longitudinal data

Abstract

Causal inference with observational longitudinal data and time-varying exposures is complicated due to the potential for time-dependent confounding and unmeasured confounding. Most causal inference methods that handle time-dependent confounding rely on either the assumption of no unmeasured confounders or the availability of an unconfounded variable that is associated with the exposure (e.g., an instrumental variable). Furthermore, when data are incomplete, validity of many methods often depends on the assumption of missing at random. We propose an approach that combines a parametric joint mixed-effects model for the study outcome and the exposure with g-computation to identify and estimate causal effects in the presence of time-dependent confounding and unmeasured confounding. G-computation can estimate participant-specific or population-average causal effects using parameters of the joint model. The joint model is a type of shared parameter model where the outcome and exposure-selection models share common random effect(s). We also extend the joint model to handle missing data and truncation by death when missingness is possibly not at random. We evaluate the performance of the proposed method using simulation studies and compare the method to both linear mixed-effects models and fixed-effects models combined with g-computation as well as to targeted maximum likelihood estimation. We apply the method to an epidemiologic study of vitamin D and depressive symptoms in older adults and include code using SAS PROC NLMIXED software to enhance the accessibility of the method to applied researchers.

1. Introduction

Estimating causal effects from observational longitudinal studies with time-varying exposures is rife with challenges, including time-dependent confounding and unmeasured confounding. Appropriately addressing these challenges is essential to unbiased estimation of causal effects.

The available methodology for performing causal inference with longitudinal data has largely focused on semiparametric modeling of population-average causal effects. Most methods require either the assumption of no unmeasured confounding (e.g., marginal structural models, targeted maximum likelihood estimation) or availability of an unconfounded variable that is associated with the exposure (e.g., instrumental variable methods or g-estimation of structural nested models).^1–4

There has been relatively little work on causal inference with longitudinal data using mixed-effects models⁵ in which causal effects are defined using potential outcomes. Sitlani et al⁶ considered linear structural mixed models for randomized trials with non-compliance. Although the authors mentioned a joint model for outcome and exposure to handle unmeasured confounding and an outcome model that includes serial correlation as a source of time-dependent confounding, the authors focused evaluation and discussion on outcome models without serial correlation conditioned on the random effects and exposure models that were independent of random effects. Bind et al⁷ considered causal mediation analysis with separate mixed-effects models for the mediator and the outcome assuming no unmeasured confounding and no time-dependent confounding. The models were linked through (possibly) correlated random effects. Kennedy et al⁸ used mixed-effects models as the first of a two-stage approach to appropriately overcome time-dependent confounding in time-to-event analysis. Small et al⁹ and Holmes et al¹⁰ proposed mixed-effects models to handle non-adherence in randomized trials. Zhang et al¹¹ addressed heterogeneity of treatment effects in randomized trials by sensitivity analysis of a mixed-effects model in which potential outcomes are presumed conditionally independent given observed and unobserved factors.

In this paper, we focus on joint mixed-effects models for longitudinal outcomes and time-varying exposures in settings with time-dependent confounding. Joint models were originally introduced as a method for handling nonignorably missing data^{12, 13} when estimating the outcome data-generating model. We adapt the method to handle unmeasured confounding and to identify causal effects by performing g-computation¹⁴ using the joint-model parameters. We additionally consider these issues in conjunction with selective attrition in the form of data missing not at random in the sense of Rubin;¹⁵ that is, nonignorably missing data, and truncation by death. A simulation study compares and contrasts the performance of g-computation using parameters from the joint mixed-effects model with that of g-computation using parameters from conventional linear mixed models and fixed-effects models. We also compare the joint mixed-effects model to targeted maximum likelihood estimation, a semiparametric method that depends on the assumption of no unmeasured confounding and is doubly robust in that correct estimation of only one of the outcome-generating model or exposure-selection model is required for unbiased estimation.²

We consider epidemiological studies of older adults to motivate the methods. When estimating causal effects of potential risk factors for aging-related adverse outcomes, important sources of outcome heterogeneity and unmeasured confounding include lifecourse factors; that is, exposures that occurred during gestation, childhood, adolescence, young adulthood, and adult life that may affect health in old age.¹⁶ Thus, mixed-effects models may be particularly attractive when studying the older adult population. To be concrete, we focus on a study examining the causal effects of serum vitamin D on depressive symptoms over time in older adults. It is plausible that unmeasured antecedent factors may affect severity of depressive symptoms in later life, and that some of these same factors may influence later lifesyle choices and physical health with implications on serum vitamin D concentrations.

2. Data, Definitions, Models, and Estimation

2.1. Data and Definitions

Consider a longitudinal study with n participants where participant i has J_i scheduled follow-up visits. Let Y_ij denote the study outcome for participant i at visit j(j = 0, …, J_i) such that Y_i0 is the baseline value of the outcome. Similarly, A_ij is the study exposure for participant i at visit j. Let X_i be a vector of observed baseline exposure-outcome confounders.

The analytical objective is to estimate causal effects of exposure history through visit j − 1, denoted Ā_i,j−1 = {A_i0, A_i1, …, A_i,j−2, A_i,j−1}, on Y_ij. Let Y_ij(ā_j−1) denote the potential outcome for participant i at visit j that would have been observed if exposure history Ā_i,j−1 were set to ā_j−1 = {a₀, …, a_j−1}. The causal effect of Ā_i,j−1 on Y_ij is a contrast of the average of Y_ij(ā_j−1) versus $Y_{ij}({\overline{a}}_{j-1}^{\prime })$, where ${\overline{a}}_{j-1}^{\prime }$ is an alternative exposure history. We make the consistency assumption to link potential outcomes with observed outcomes. That is, we assume that Y_ij = Y_ij(ā_j−1) if Ā_i,j−1 = ā_j−1. We make the stable unit treatment value assumption,¹⁷ which means that the exposure history of a participant does not influence the outcomes of other participants and that there are not multiple versions of the exposure. This assumption allows us to denote potential outcomes with Y_ij(Ā_i,j−1) rather than Y_ij(Ā_j−1), where Ā_j−1 = {Ā_1,j−1, …, Ā_n,j−1}.

2.2. Models

We posit a linear mixed-effects model for the data-generating mechanism of outcome Y. In general, let

$$Y_{ij}={\beta }_0+{\mathit{X}}_i^t{\mathit{\beta }}_1+{\overline{Y}}_{i,j-1}^t{\mathit{\beta }}_2+{\overline{A}}_{i,j-1}^t{\mathit{\beta }}_3+{({\mathit{X}}_i\otimes {\overline{A}}_{i,j-1})}^t{\mathit{\beta }}_4+{\beta }_{5}j+{\mathit{Z}}_{ij}^t{\mathit{b}}_i+{\epsilon }_{ij}\text{for}j=0,\dots ,J_i,$$ (1)

where b_i is a vector of random effects, Z_ij is a design matrix for the random effects, ε_ij is a measurement error, and ${\overline{A}}_{i,j-1}^t{\mathit{\beta }}_3={\beta }_{30}{A}_{i0}+\cdots +{\beta }_{3,j-1}{A}_{i,j-1}$ with other terms similarly defined. Note that serial correlation (and hence time-dependent confounding) is considered by allowing Y_ij to depend on past values of Y, Ȳ_i,j−1. Thus, coefficients of A may not have a causal interpretation.

Throughout, we assume that

$${\mathit{\epsilon }}_i\text{iid}N(0,{\sigma }^2)$$ (2)

and

$${\mathit{b}}_i~N(\mathbf{0},\mathit{D}).$$

To be more specific and parsimonious, we will focus on binary exposures using the following outcome model:

$$Y_{ij}={\beta }_0+{\mathit{X}}_i^t{\mathit{\beta }}_1+{\beta }_2{Y}_{i,j-1}+{\overline{A}}_{i,j-1}^t{\mathit{\beta }}_3+{(X_{i1}{\overline{A}}_{i,j-1})}^t{\mathit{\beta }}_4+{\beta }_{5}j+b_{i0}+b_{i1}(X_{i1}+X_{i1}\sum _{k=0}^{j-1}{A}_{ik})+{\epsilon }_{ij},$$ (3)

where Ȳ_i,k = Ā_i,k = Ø for k < 0.

Thus, we consider a random intercept (b_i0) as well as a random slope (b_i1) for a baseline covariate X_i1 that also depends on cumulative exposure history. These random effects can be interpreted as unmeasured sources of heterogeneity that influence the value of the outcome and the effect of observed factors on the outcome. For example, in an epidemiologic study of vitamin D and depressive symptoms in older adults, X_i1 may be age at study baseline; thus b_i1 accounts for the possibility that heterogeneity of age and age-specific vitamin D effects on depressive symptoms may be due, for example, to unmeasured lifecourse factors or subclinical biological factors. As noted by Sitlani et al,⁶ a special case of Equation (3) that does not include heterogeneity of exposure effects (i.e., b_i1 = 0 for all i) is an example of a rank-preserving model,¹⁸ because the participants’ outcome ranks are preserved across all possible exposure histories. In contrast, Equation (3) in general is an example of a non-rank-preserving model and may be more realistic in some applications.

The same unmeasured factors that influence the study outcome may also influence selection of exposure over time. The selection process for binary exposures may follow a model such as

$$\text{logit}(p_{i,j-1}^A)={\alpha }_0+{\mathit{X}}_i^t{\mathit{\alpha }}_1+{\alpha }_2{Y}_{i,j-1}+{\overline{A}}_{i,j-2}^t{\mathit{\alpha }}_3+{({\mathit{X}}_i{Y}_{i,j-1})}^t{\mathit{\alpha }}_4+{\alpha }_{5}j+{\alpha }_6{b}_{i0}+{\alpha }_7{b}_{i1}\text{for}j=1,\dots ,J_i,$$ (4)

where $p_{i,j-1}^A$ denotes E[A_i,j−1 |, X_i, Ȳ_i,j−1, Ā_i,j−2, b_i].

If the outcome and exposure-selection models are correctly specified by Equations (3) and (4), respectively, then unbiased parameter estimation can proceed by maximizing the joint likelihood for the outcome and exposure selection,

$$\prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{A}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i=\prod _i\int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)\times f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{b}}_i;\mathit{\alpha })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i$$ (5)

where f(·) denotes a probabilty density (or mass) function. As per Equation (2), f(Y_ij | X_i, Ȳ_i,j−1, Ā_i,j−1, b_i; β, σ²) is the normal density. As per Equation (4), The joint mixed-effects model Equation (5) is an example of a shared parameter model,^{12, 13, 19} because the outcome model and exposure-selection model have common (i.e, shared) random effects. In Equation (5), the outcome model and exposure-selection model are simultaneously fit to derive unbiased estimates of β. If, however, b_i does not influence exposure selection, then Equation (5) can be factored as

$$\prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{A}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i=\prod _{i}f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2};\mathit{\alpha })\times \int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i.$$ (6)

In model (6), the exposure-selection model factors from the outcome model. Thus, parameters β can be estimated by maximizing the outcome likelihood using standard methods for linear mixed-effects models.

We have hitherto focused on estimating parameters that describe the data-generating mechanism, but that may not have a causal interpretation. We now turn our attention to estimating causal effects using these parameters. First, we consider a directed acyclic graph (Figure 1) to convey the assumptions of unmeasured confounding via arrows from b_i to both the exposure A and the outcome Y. Figure 1 also conveys time-dependent confounding via the pathways A_i,j−1 → Y_ij → A_ij → Y_i,j+1 and A_i,j−1 → Y_i,j+1. Maximizing Equation (5) addresses unmeasured confounding when estimating the data-generating process. The parameters β do not, in general, have a causal interpretation owing to the time-dependent confounding. However, under certain assumptions, functions of β can identify causal effects. Namely, the following assumption, sequential randomization conditional on random effects, must hold to identify causal effects from β:

$${\overline{Y}}_i(\overline{a})\perp A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2}={\overline{a}}_{j-2},{\mathit{b}}_i\text{for}\text{all}\overline{a}\in \overline{\mathcal{A}}\text{and}j=1,\dots ,J_i,$$ (7)

where Ȳ_i(ā) = {Y_i1(a₀), Y_i2(ā₁), …, Y_i,Ji(ā_Ji−1)} and 𝒜̄ is the support of Ā; that is, 𝒜̄ is the collection of all possible exposure histories. Assumption (7) holds if there are no unmeasured exposure-outcome confounders at time j after conditioning on baseline covariates, outcome and exposure history, and random effects. For example, causal effects can be identified by maximizing Equation (5) under (correct specification of) the exposure-selection model (4) and outcome model (3). If α₆ = 0 and α₇ = 0, then causal effects can be identified by fitting the outcome-model using standard methods for linear mixed-effects models; that is, the exposure-selection model need not be estimated owing to the factoring in Equation (6). Alternatively, methods that can produce unbiased estimates by correctly modeling the exposure-selection mechanism can be used in this case including marginal structural models or doubly robust methods such as longitudinal targeted maximum likelihood.^{1, 2}

Figure 1.

Directed acyclic graph for the joint mixed-effects model conveying unmeasured confounding and time-varying confounding.

We use g-computation¹⁴ to identify causal effects as functions of β. G-computation generalizes the idea of covariate standardization to complex longitudinal settings. The idea is to integrate outcome expected values over covariate histories while treating the exposure as fixed (see Daniel at al²⁰ for an accessible illustration). Thus, we define

$$g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)=\int E[Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1}={\overline{a}}_{j-1},{\mathit{b}}_i]\prod _{k=0}^{j-1}f(Y_{i,k}\mid {\mathit{X}}_i,{\overline{Y}}_{i,k-1},{\overline{A}}_{i,k-1}={\overline{a}}_{k-1},{\mathit{b}}_i)d{Y}_{i,k}.$$ (8)

We refer to (8) as conditional g-computation, because contrasts of the form $g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)-g_j({\overline{a}}_{j-1}^{\prime }\mid {\mathit{X}}_i,{\mathit{b}}_i)$ are causal contrasts conditioned on baseline covariates and random effects. Depending on study objectives, researchers may prefer to additionally integrate over baseline covariates and/or random effects. Computing

$$g_j({\overline{a}}_{j-1})=\int g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)f({\mathit{X}}_i)d{\mathit{X}}_{i}f({\mathit{b}}_i)d{\mathit{b}}_i$$ (9)

and $g_j({\overline{a}}_{j-1})-g_j({\overline{a}}_{j-1}^{\prime })$ results in population-average causal contrasts. For example, consider the model in Equation (3) for j = 1, 2. At j = 1,

$$E[Y_{i1}\mid {\mathit{X}}_i,Y_{i0},A_{i0}=a_0,{\mathit{b}}_i]=({\beta }_0+{\beta }_5)+{\mathit{X}}_i^t{\mathit{\beta }}_1+{\beta }_2{Y}_{i0}+{\beta }_{30}{a}_0+{\beta }_{40}{X}_{i1}{a}_0+b_{i0}+b_{i1}(X_{i1}+X_{i1}{a}_0).$$

Integrating over f(Y_i0 | X_i, b_i) involves replacing Y_i0 with its expected value conditioned on X_i and b_i,

$$E[Y_{i0}\mid {\mathit{X}}_i,{\mathit{b}}_i]={\beta }_0+{\mathit{X}}_i^t{\mathit{\beta }}_1+b_{i0}+b_{i1}{X}_{i1},$$

resulting in

$$\begin{gathered} g_1(a_0\mid {\mathit{X}}_i,{\mathit{b}}_i)=({\beta }_0+{\beta }_5)+{\mathit{X}}_i^t{\mathit{\beta }}_1+{\beta }_2({\beta }_0+{\mathit{X}}_i^t{\mathit{\beta }}_1+b_{i0}+b_{i1}{X}_{i1})+{\beta }_{30}{a}_0+{\beta }_{40}{X}_{i1}{a}_0+b_{i0}+b_{i1}(X_{i1}+X_{i1}{a}_0)= \\ ({\beta }_0+{\beta }_5+b_{i0}+{\beta }_2{\beta }_0+{\beta }_2{b}_{i0})+{\mathit{X}}_i^t({\mathit{\beta }}_1+{\beta }_2{\mathit{\beta }}_1)+X_{i1}(b_{i1}+{\beta }_2{b}_{i1})+a_0({\beta }_{30}+{\beta }_{40}{X}_{i1}+b_{i1}{X}_{i1}). \end{gathered}$$ (10)

Therefore, g₁(a₀ = 1 | X_i, b_i) − g₁(a₀ = 0 | X_i, b_i) = β₃₀ + β₄₀X_i1 + b_i1X_i1. Furthermore, g₁(a₀ = 1) − g₁(a₀ = 0) = β₃₀ + β₄₀E[X_i1], because random effects are assumed to be independent of X_i and to have mean 0.

To perform g-computation at j = 2, note that

$$E[Y_{i2}\mid {\mathit{X}}_i,{\overline{Y}}_{i1},{\overline{A}}_{i1}={\overline{a}}_i,{\mathit{b}}_i]=({\beta }_0+2{\beta }_5)+{\mathit{X}}_i^t{\mathit{\beta }}_1+{\beta }_2{Y}_{i1}+{\beta }_{30}{a}_0+{\beta }_{31}{a}_1+{\beta }_{40}{X}_{i1}{a}_0+{\beta }_{41}{X}_{i1}{a}_1+b_{i0}+b_{i1}[X_{i1}+X_{i1}(a_0+a_1)].$$

Integrating over f(Y_i1 | X_i, Y_i0, A_i0 = a₀, b_i) and f(Y_i0 | X_i, b_i) results in

$$g_2({\overline{a}}_1\mid {\mathit{X}}_i,{\mathit{b}}_i)-g_2(\{0,0\}\mid {\mathit{X}}_i,{\mathit{b}}_i)=a_0({\beta }_{30}+{\beta }_2{\beta }_{30}+{\beta }_{40}{X}_{i1}+{\beta }_2{\beta }_{40}{X}_{i1}+b_{i1}{X}_{i1}+{\beta }_2{b}_{i1}{X}_{i1})+a_1({\beta }_{31}+{\beta }_{41}{X}_{i1}+b_{i1}{X}_{i1}).$$

Further integration over X_i and b_i results in

$$g_2({\overline{a}}_1)-g_2(\{0,0\})=a_0({\beta }_{30}+{\beta }_2{\beta }_{30}+{\beta }_{40}E[X_{i1}]+{\beta }_2{\beta }_{40}E[X_{i1}])+a_1({\beta }_{31}+{\beta }_{41}E[X_{i1}]).$$

2.3. Some Extensions

2.3.1. Missing Data and Truncation by Death

Joint mixed-effects models were originally proposed to address nonignorable attrition.^{12, 13} These models have evolved to handle joint longitudinal and time-to-event analysis.²¹ The previously proposed methods have largely focused on point exposure studies (e.g., intent-to-treat analysis of randomized trials) for estimating the data-generating process; but they have not been motivated by the objective of estimating causal effects of time-varying exposures using the potential outcomes framework.

To tackle this problem in which we treat the longitudinal outcome as the focus and death as a nuisance, let S_ij = 1 indicate that participant i was alive at the time of scheduled visit j, and S_ij = 0 indicate that participant i was dead at the time of scheduled visit j. Therefore, S_ij = 0 implies that S_ik = 0 for k > j, and S_ij = 1 implies that S_ik = 1 for k < j. Similarly, let R_ij = 1 indicate that participant i responded (i.e., provided data) at scheduled visit j, and R_ij = 0 indicate that participant i did not respond at scheduled visit j. If missingness is monotone; e.g., where participants are permanently lost to follow-up if they miss a visit, then R_ij = 0 implies that R_ik = 0 for k > j, and R_ij = 1 implies that R_ik = 1 for k < j.

Let $p_{ij}^S$ denote E[S_ij | S_i,j−1 = 1, X_i, Ȳ_ij, Ā_ij, b_i] = E[S_ij | S_i,j−1 = 1, X_i, Ȳ_i,j−1, Ā_i,j−1, b_i]. Consider the following model for the death process:

$$\text{logit}(p_{ij}^S)={\gamma }_0+{\mathit{X}}_i^t{\mathit{\gamma }}_1+{\gamma }_2{Y}_{i,j-1}+{\overline{A}}_{i,j-1}^t{\mathit{\gamma }}_3+{\gamma }_4{X}_{i1}{Y}_{i,j-1}+{\gamma }_{5}j+{\gamma }_6{b}_{i0}+{\gamma }_7{b}_{i1},$$ (11)

where S_ij follows a Bernoulli distribution,

$$f(S_{ij}\mid S_{i,j-1}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\gamma })={(p_{ij}^S)}^{S_{ij}}{(1-p_{ij}^S)}^{(1-S_{ij})}.$$

Model (11) implies that among participants who survived to visit j − 1, survival to visit j is statistically independent of A_ij and Y_ij given the observed histories and random effects.

Similarly, let $p_{ij}^R$ denote E[R_ij | S_i,j = 1, R_i,j−1 = 1, X_i, Ȳ_ij, Ā_ij, b_i] = E[R_ij | S_i,j = 1, R_i,j−1 = 1, X_i, Ȳ_i,j−1, Ā_i,j−1, b_i]. Consider the following model for a monotone response process among survivors

$$\text{logit}(p_{ij}^R)={\delta }_0+{\mathit{X}}_i^t{\mathit{\delta }}_1+{\delta }_2{Y}_{i,j-1}+{\overline{A}}_{i,j-1}^t{\mathit{\delta }}_3+{\delta }_4{X}_{i1}{Y}_{i,j-1}+{\delta }_{5}j+{\delta }_6{b}_{i0}+{\delta }_7{b}_{i1},$$ (12)

where R_ij follows a Bernoulli distribution,

$$f(R_{ij}\mid S_{i,j}=1,R_{i,j-1}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\delta })={(p_{ij}^R)}^{R_{ij}}{(1-R_{ij}^S)}^{(1-R_{ij})}.$$

Model (12) implies that, among survivors who have thus far remained in the study, response at visit j is statistically independent of A_ij and Y_ij given the observed histories and random effects. Model (12) includes a missing not at random mechanism, because R_ij may depend on possibly missing Y_ij after integrating over b_i;¹⁵ that is, in general missingness is nonignorable. If δ₆ = δ₇ = 0, then the likelihood factors and the missingness mechanism simplifies to missing at random;¹⁵ that is, missingness is ignorable.

We can estimate the parameters of the data-generating process by extending model (5) to include the death and response processes in the joint likelihood:

$$\begin{gathered} \prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{R}}_{i{J}_i},{\overline{S}}_{i{J}_i},{\overline{A}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha },\mathit{\gamma },\mathit{\delta })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i \\ =\prod _i\int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)\times f(R_{ij}\mid S_{ij}=1,R_{i,j-1}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\delta }) \\ \times f(S_{ij}\mid S_{i,j-1}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\gamma })f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{b}}_i;\mathit{\alpha })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i. \end{gathered}$$ (13)

Models (11) and (12) imply

$$f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\mathit{\sigma }}^2)=f(Y_{ij}\mid S_{ij}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)=f(Y_{ij}\mid R_{ij}=1,S_{ij}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2).$$

Maximizing the likelihood (13) estimates β; however, once again, Equation (3) does not have a causal interpretation. Furthermore, defining causal effects in the presence of truncation by death is particularly challenging, because death renders missing outcomes to be undefined, whereas nonresponse simply renders them unobserved. Tchetgen Tchetgen et al²² and Shardell et al²³ showed that, in general, g-computation in the presence of truncation by death can identify a causal effect that is a weighted sum of principal strata (PS) direct effects and indirect effects.²⁴ That is, it identifies a causal contrast of exposure history ā_j−1 versus ${\overline{a}}_{j-1}^{\prime }$ on the outcome at visit j within the latent subgroup who would have survived to visit j whether they followed exposure history ā_j−1 or ${\overline{a}}_{j-1}^{\prime }$. Unlike in previous work on causal inference with truncation by death,^{22, 23} we consider unmeasured common causes of the outcome, missingness, and death in conjunction with possible unmeasured confounding. G-computation is performed as in Equation (8) by conditioning on survival,

$$g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)=\int E[Y_{ij}\mid R_{ij}=1,S_{ij}=1,{\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1}={\overline{a}}_{j-1},{\mathit{b}}_i]\times \prod _{k=0}^{j-1}f(Y_{ik}\mid R_{ik}=1,S_{ik}=1,{\mathit{X}}_i,{\overline{Y}}_{i,k-1},{\overline{A}}_{i,k-1}={\overline{a}}_{k-1},{\mathit{b}}_i)d{Y}_{ik}.$$ (14)

Whether contrasts $g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)-g_j({\overline{a}}_{j-1}^{\prime }\mid {\mathit{X}}_i,{\mathit{b}}_i)$ computed from Equation (14) have a causal interpretation depends on whether an unidentifiable assumption is satisfied. Namely,

$$Y_{ij}({\overline{a}}_{j-1},{\overline{y}}_{j-1}^{\ast })\perp S_{ij}({\overline{a}}_{j-1}^{\prime },{\overline{y}}_{j-1}^{\ast \prime })\mid S_{ij}({\overline{a}}_{j-1},{\overline{y}}_{j-1}^{\ast }),{\mathit{X}}_i,Y_{i0},$$ (15)

where ${\overline{y}}_{j-1}^{\ast }=\{y_1,\dots ,y_{j-1}\}$, i.e., post-exposure outcome history through visit j − 1, which are the outcomes at all past visits except at j = 0; and $Y_{ij}({\overline{a}}_{j-1},{\overline{y}}_{j-1}^{\ast })$ and $S_{ij}({\overline{a}}_{j-1},{\overline{y}}_{j-1}^{\ast })$ are counterfactual outcome and survival, respectively, at visit j of exposure history and post-exposure outcome history through j − 1. Assumption (15) is a “cross-world” independence assumption that depends on counterfactuals of Ā_j−1 and variables that mediate the effect of Ā_j−1 on Y_ij and S_ij. The assumption implies that the counterfactual outcome set to one joint exposure and outcome history is conditionally independent of survival set to another joint exposure and outcome history.

Using the arguments of Tchetgen Tchetgen et al²² and Shardell et al,²³ the causal contrast at j = 1, $g_1(a_0\mid {\mathit{X}}_i,{\mathit{b}}_i)-g_1(a_0^{\prime }\mid {\mathit{X}}_i,{\mathit{b}}_i)$, equals

$$\int E[Y_{i1}(a_0)-Y_{i1}(a_0^{\prime })\mid S_{i1}(a_0)=S_{i1}(a_0^{\prime })=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i]f(Y_{i0}\mid {\mathit{X}}_i,{\mathit{b}}_i)d{Y}_{i0}.$$ (16)

The causal contrast $E[Y_{i1}(a_0)-Y_{i1}(a_0^{\prime })\mid S_{i1}(a_0)=S_{i1}(a_0^{\prime })=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i]$ is a PS effect of the exposure within the latent subgroup who would have survived if A_i0 were set to a₀ or $a_0^{\prime }$, conditioned on {Y_i0, X_i, b_i}.²⁴ Thus, the causal contrast in Equation (16) is a weighted average of PS effects.

The causal contrast at j = 2, $g_2({\overline{a}}_1\mid {\mathit{X}}_i,{\mathit{b}}_i)-g_2({\overline{a}}_1^{\prime }\mid {\mathit{X}}_i,{\mathit{b}}_i)$ is more complicated owing to a possible A₀ → Y₁ → A₁ → Y₂ pathway. Specifically, it equals^{22, 23}

$$\begin{gathered} \int E[Y_{i2}({\overline{a}}_1,y_1)-Y_{i2}({\overline{a}}_1^{\prime },y_1)\mid S_{i2}({\overline{a}}_1,y_1)=S_{i2}({\overline{a}}_1^{\prime },y_1)=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i] \\ \times f_{Y_{i1}(a_0^{\prime })}(Y_{i1}=y_1\mid S_{i1}(a_0)=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i)d{y}_1\times f(Y_{i0}\mid {\mathit{X}}_i,{\mathit{b}}_i)d{Y}_{i0} \\ +\int E[Y_{i2}({\overline{a}}_1,y_1)-Y_{i2}({\overline{a}}_1,y_1^{\prime })\mid S_{i2}({\overline{a}}_1,y_1)=S_{i2}({\overline{a}}_1,y_1^{\prime })=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i] \\ \times [f_{Y_{i1}(a_0)}(Y_{i1}=y_1\mid S_{i1}(a_0)=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i)-f_{Y_{i1}(a_0^{\prime })}(Y_{i1}=y_1\mid S_{i1}(a_0^{\prime })=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i)]d{y}_1\times f(Y_{i0}\mid {\mathit{X}}_i,{\mathit{b}}_i)d{Y}_{i0}. \\ =\text{Direct}\text{Effect}+\text{Indirect}\text{Effect} \end{gathered}$$ (17)

The Direct Effect in Equation (17) is a weighted average of the PS-controlled direct effect $E[Y_{i2}({\overline{a}}_1,y_1)-Y_{i2}({\overline{a}}_1^{\prime },y_1)\mid S_{i2}({\overline{a}}_1,y_1)=S_{i2}({\overline{a}}_1^{\prime },y_1)=1,Y_{i0},{\mathit{X}}_i,{\mathit{b}}_i]$ averaged over the distribution of { $Y_{i1}(a_0^{\prime })$, Y_i0}. The Indirect Effect is the difference in average PS effects of Y_i1 on Y_i2 setting Ā_i1 to ā₁ versus ${\overline{a}}_1^{\prime }$, comparing the weighted average over the distributions of Y_i1(a₀) versus $Y_{i1}(a_0^{\prime })$.

Equation (17) differs from the target of estimation for most methods that jointly model longitudinal and mortality data. Most methods that focus on the longitudinal outcome aim to estimate the data-generating process of the longitudinal outcome if there were no deaths. In contrast, we aim to estimate causal effects, and Equation (17) is the causal effect identified by g-computation in the presence of truncation by death.

Slight variations are needed in studies with non-monotone missingness. In particular, Equation (12) would not require conditioning on R_i,j−1 = 1, but would involve conditioning on the vector (R_i0, R_i1, …, R_i,j−1) as a way to model the joint distribution of R_i = (R_i0, R_i1, …, R_i,J). Similarly, the probability for R_ij in Equation (13) would not condition on R_i,j−1 = 1 for the same reason. The use of JMM is advantageous here because it allows integration over missing data with relative ease.^{25, 26}

2.3.2. Additional Time-Dependent Confounding

We have heretofore only considered time-dependent confounding due to serial correlation. However, other endogenous time-varying covariates may also confound the effect of Ā on Y, resulting in time-dependent confounding. That is, some factors may affect the exposure and the outcome at one time point, but may be affected by the exposure, directly or indirectly, at an earlier time point.^{1, 20}

Let L_ik be a set of time-dependent confounders at visit k with density f(L_ik | X_i, Ȳ_i,k−1, L̄_i,k−1, Ā_i,k−1, b_i; ϕ), k = 0, …, J_i − 1, indexed by parameters ϕ. if L_ik is independent of b_i for all k, then the likelihood for L_i = {L_i0, …, L_i,Ji−1} factors from that of {Ā_i, Ȳ_i} and can be maximized separately. However, if L_i shares random effects with {Ā_i, Ȳ_i}, then the likelihood (5) needs to be extended to include ϕ:

$$\begin{gathered} \prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{A}}_{i{J}_i-1},{\overline{\mathit{L}}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha },\mathit{\varphi })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i \\ =\prod _i\int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,j-1},{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,j-1},{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{b}}_i;\mathit{\alpha }) \\ \times f({\mathit{L}}_{i,j-1}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,j-2},{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{b}}_i;\mathit{\varphi })f({\mathit{b}}_i;\mathit{D})d{\mathit{b}}_i. \end{gathered}$$ (18)

Equation (18) shows that L̄_i,j−1 may have an effect on Y_ij and A_i,j−1. This phenomenon can be accommodated by including components of L_i on the right-hand side of Equations (3) and (4). To identify causal effects, g-computation involves integrating over L̄_i (e.g., see Daniel et al²⁰),

$$\begin{gathered} g_j({\overline{a}}_{j-1}\mid {\mathit{X}}_i,{\mathit{b}}_i)=\int E[Y_{ij}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,j-1},{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1}={\overline{a}}_{j-1},{\mathit{b}}_i] \\ \times \prod _{k=0}^{j-1}f({\mathit{L}}_{ik}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,k-1},{\overline{Y}}_{ik},{\overline{A}}_{i,k-1}={\overline{a}}_{k-1},{\mathit{b}}_i)d{\mathit{L}}_{ik}f(Y_{ik}\mid {\mathit{X}}_i,{\overline{\mathit{L}}}_{i,k-1},{\overline{Y}}_{i,k-1},{\overline{A}}_{i,k-1}={\overline{a}}_{k-1},{\mathit{b}}_i)d{Y}_{ik}. \end{gathered}$$ (19)

Equation (18) exemplifies the case where variables assessed at some visit k are assumed to be realized in the order Y_k, then L_k, then A_k. Slight variations of Equation (18) are needed for alternative ordering. Unbiased estimation of causal effects requires the model for L to be correctly specified. Exogenous time-varying covariates; that is, covariates that are not influenced (directly or indirectly) by past exposures; need not be modeled.

2.3.3. Process-Specific Random Effects

Thus far, the only random effects that we have considered for the exposure-selection process are those that are shared with the outcome process. Thus, in this model, the assumption of no unmeasured confounders implies the assumption of no exposure-selection heterogeneity. We therefore extend the model to allow separate random effects for the exposure-selection and outcome processes, some of which may be shared between the two processes. For example, let

$$\text{logit}(p_{i,j-1}^A)={\alpha }_0+{\mathit{X}}_i^t{\mathit{\alpha }}_1+{\alpha }_2{Y}_{i,j-1}+{\overline{A}}_{i,j-2}^t{\mathit{\alpha }}_3+{\mathit{\alpha }}_4{X}_{i1}{Y}_{i,j-1}+{\alpha }_5(j-1)+{\alpha }_6{b}_{i0}+{\alpha }_7{b}_{i1}+{\mathit{V}}_i^t{\mathit{g}}_i,$$

where g_i is a vector of random effects for the exposure-selection process and V_i is the design matrix for the exposure-selection process random effects. Parameter estimation proceeds by maximizing the joint likelihood

$$\begin{gathered} \prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{A}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i,{\mathit{g}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha })f({\mathit{b}}_i,{\mathit{g}}_i;{\mathit{D}}_{\mathit{b},\mathit{g}})d({\mathit{b}}_i,{\mathit{g}}_i) \\ =\prod _i\int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{b}}_i,{\mathit{g}}_i;\mathit{\alpha })f({\mathit{b}}_i,{\mathit{g}}_i;{\mathit{D}}_{\mathit{b},\mathit{g}})d({\mathit{b}}_i,{\mathit{g}}_i), \end{gathered}$$ (20)

where D_b,g is the variance-covariance matrix of {b_i, g_i}. A likelihood analogous to Equation (20) has been proposed to handle nonignorably missing data or time-to-event data and has been referred to as a generalized shared-parameter model.^{27, 28} Likelihood (20) has the benefit of accounting for potential heterogeneity in the exposure-selection process, even in the absence of unmeasured confounding. If α₆ = α₇ = 0, then Equation (20) factors as

$$\begin{gathered} \prod _i\int f({\overline{Y}}_{i{J}_i},{\overline{A}}_{i{J}_i-1}\mid {\mathit{X}}_i,{\mathit{b}}_i,{\mathit{g}}_i;\mathit{\beta },{\sigma }^2,\mathit{\alpha })f({\mathit{b}}_i,{\mathit{g}}_i;{\mathit{D}}_{\mathit{b},\mathit{g}})d({\mathit{b}}_i,{\mathit{g}}_i) \\ =\prod _i\int \prod _{j}f(A_{i,j-1}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-2},{\mathit{g}}_i;\mathit{\alpha })d({\mathit{g}}_i;{\mathit{D}}_{\mathit{g}})d{\mathit{g}}_i\int \prod _{j}f(Y_{ij}\mid {\mathit{X}}_i,{\overline{Y}}_{i,j-1},{\overline{A}}_{i,j-1},{\mathit{b}}_i;\mathit{\beta },{\sigma }^2)f({\mathit{b}}_i;{\mathit{D}}_{\mathit{b}})d{\mathit{b}}_i, \end{gathered}$$ (21)

where D_g is the variance-covariance matrix for g_i and D_b is the variance-covariance matrix for b_i, since the outcome and exposure-selection processes have no shared random effects (i.e., no unmeasured confounding). Therefore, α and β can be estimated separately. A variation of model (21) has been proposed for mediation analysis in which the mediator and outcome have separate, yet possibly correlated, random effects.⁷

3. Simulation Studies

We performed a series of simulation studies to evaluate the performance of the joint mixed-effects model combined with g-computation to estimate causal effects. We aimed to compare the joint mixed effects model (JMM) with conventional linear mixed-effects models (LMM) combined with g-computation, fixed-effects models (FEM) combined with g-computation, and longitudinal targeted maximum likelihood estimation (LTMLE). Like JMM, LTMLE involves fitting a model for the outcome and a model for the exposure. Unlike JMM, LTMLE is semiparametric, relies on the assumption of no unmeasured confounders, and is doubly robust, which means that it is unbiased if at least one of the models is correctly specified.

We simulated 1,000 datasets of size n = 1,000. We fit JMM, LMM, and FEM using SAS version 9.3 with PROC NLMIXED. Numerical integration was performed by Gaussian adaptive quadrature with five quadrature points; optimization was performed via the dual quasi-Newton algorithm using the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm²⁹ to update the Cholesky factor of the Hessian matrix. LTMLE was fit using the ltmle package in R Statistical Software.³⁰

We performed three sets of simulation studies. The first set involved unmeasured confounding only; the second set involved unmeasured confounding and a misspecified random effects distribution (JMM and LMM); and the third set involved unmeasured confounding and nonignorable dropout. JMM was implemented using SAS PROC NLMIXED by fitting likelihoods for Y, A, and (for the third set of simulations) R and including a RANDOM statement for random effects. LMM was implemented in PROC NLMIXED by fitting a likelihood for Y (deleting likelihoods for A and R) and including a RANDOM statement for random effects. Lastly, FEM was implemented in PROC NLMIXED by fitting a likelihood for Y without random effects and deleting the corresponding RANDOM statement (and deleting likelihoods for A and R).

For all simulations, we considered J_i = 2 follow-up visits, a single baseline covariate, X_i, and binary time-varying exposure, A_ij. X_i was generated as a standard normal random variable. The outcome Y_ij was generated to have error variance σ² = 1 and mean as in Equation (3) with β₀ = 1, β₁ = −1, β₂ = 1, β₃₀ = 1, β₃₁ = 2, β₄₀ = 0, β₄₁ = 0, and β₅ = −1. Thus, at j = 1, g(a₀ = 1) − g(a₀ = 0) = β₃₀ = 1, and at j = 2, g(ā₁) − g({0, 0}) = a₀(β₃₀ + β₂β₃₀) + a₁β₃₁ = 2a₀ + 2a₁.

We performed studies with random intercept only (i.e, b_i1 = 0 for all participants, hence Var(b_i1) = 0.0) and studies with both random intercept and slope. In the exposure-selection model, A_i,j−1 was generated as a Bernoulli random variable to have mean as in Equation (4) with α₀ = 0, α₁ = 0.5, α₂ = −0.75, α₃ = 1.0, α₄ = 0.25, and α₅ = 0. For studies with random intercept only, we set α₆ = 0 or 1 and α₇ = 0. For studies with a random slope and intercept, we set α₆ = 0 and α₇ = 0 or 1.

In the third set of simulations, we additionally simulated R_ij, conditional on R_i,j−1 = 1 (where R_i0 = 1 for all) as a Bernoulli random variable to have mean as in Equation (12) with δ₀ = 1.25, δ₁ = 0.5, δ₂ = 0.5, δ₃₀ = 0.5, δ₃₁ = 0.5, δ₄ = 0.25, and δ₅ = −0.25. For studies with random intercept only, we set δ₆ = 0 or 0.5 and δ₇ = 0. For studies with a random slope and intercept, we set δ₆ = 0 and δ₇ = 0 or 0.5. When R_ij = 0, Y_ik and A_ik were set to missing for k ≥ j.

We set Var(b_i0) = 1. For studies that additionally had a random slope, we set Var(b_i1) = 0.3 or 1 and Corr(b_i0, b_i1) = 0.5. In the first and third set of simulations, b_i0 was simulated from a normal distribution, and (b_i0, b_i1) were simulated from a bivariate normal distribution. In the second set of simulations, b_i0 was simulated from an exponential distribution and centered to have mean 0; (b_i0, b_i1) were simulated from a bivariate exponential distribution with correlation 0.5 and centered to have mean 0.³¹

Table 1 displays the simulation results in scenarios with only unmeasured confounding and normally distributed random effects. In the absence of random slope (Var(b_i1)=0) and unmeasured confounding (α₆ = 0); JMM, LMM, and LTMLE produced unbiased estimates; but LMM was the most efficient and LTMLE was the least efficient. In contrast, using FEM and failing to account for heterogeneity resulted in non-negligible bias. These findings are consistent with expectations given that JMM, LMM, and LTMLE account for within-person clustering whereas FEM does not. In the absence of random slope (Var(b_i1)=0) and the presence of unmeasured confounding (α₆ = 1), JMM remained unbiased whereas LMM, FEM, and LTMLE all demonstrated bias exceeding 10% for all causal effects. These findings are also consistent with expectations because JMM was the only method that does not rely on the assumption of no unmeasured confounding.

Table 1.

Simulation study with unmeasured confounding and normally distributed random effects. Simulations include either random intercept only (Var(b_i1) = 0.0) or both random slope and intercept (Var(b_i1) ∈ {0.3, 1.0}).

Var(bi1)	α6	α7	Method	ĝ1(a0 = 1) − ĝ1(a0 = 0)			ĝ2(a0 = 1, a1) − ĝ2(a0 = 0, a1)			ĝ2(a0, a1 = 1) − ĝ2(a0, a1 = 0)			% Non. Conv.
				Estimate	SE	ESD	Estimate	SE	ESD	Estimate	SE	ESD
0.0	0	0	JMM	0.986	0.119	0.120	1.973	0.253	0.257	1.996	0.108	0.109	0.0
			LMM	0.998	0.068	0.066	1.996	0.140	0.136	2.003	0.082	0.083	0.0
			FEM	0.858	0.065	0.060	1.998	0.155	0.143	2.303	0.093	0.087	0.0
			LTMLE	0.975	0.123	0.130	1.942	0.258	0.253	1.928	0.182	0.197	0.0
	1		JMM	1.001	0.095	0.094	2.004	0.195	0.194	2.002	0.088	0.088	0.1
			LMM	1.351	0.065	0.062	2.756	0.137	0.131	2.236	0.078	0.077	0.0
			FEM	1.258	0.063	0.058	2.943	0.148	0.135	2.533	0.088	0.079	0.0
			LTMLE	1.443	0.115	0.121	2.862	0.217	0.215	2.242	0.150	0.167	0.0
0.3	0	0	JMM	0.997	0.082	0.081	1.996	0.169	0.168	1.996	0.090	0.091	0.0
			LMM	1.000	0.075	0.076	2.003	0.155	0.157	1.998	0.087	0.089	0.0
			FEM	0.903	0.072	0.076	2.229	0.181	0.194	2.576	0.103	0.106	0.0
			LTMLE	0.984	0.142	0.158	2.010	0.303	0.228	1.999	0.204	0.228	0.0
		1	JMM	0.997	0.081	0.082	1.994	0.165	0.167	2.001	0.086	0.086	0.0
			LMM	1.102	0.074	0.075	2.213	0.153	0.155	2.068	0.084	0.085	0.0
			FEM	1.027	0.070	0.074	2.549	0.178	0.188	2.637	0.101	0.105	0.0
			LTMLE	1.005	0.133	0.149	2.054	0.286	0.328	1.992	0.186	0.201	0.0
1.0		0	JMM	0.998	0.083	0.083	1.996	0.171	0.171	1.999	0.091	0.091	0.1
			LMM	0.996	0.082	0.083	1.990	0.167	0.169	1.997	0.090	0.097	0.1
			FEM	0.885	0.082	0.091	2.328	0.220	0.246	2.843	0.118	0.132	0.1
			LTMLE	0.926	0.167	0.179	1.960	0.388	0.397	2.057	0.246	0.271	0.0
		1	JMM	1.000	0.080	0.083	2.001	0.162	0.168	1.996	0.085	0.082	0.3
			LMM	1.137	0.079	0.081	2.281	0.160	0.166	2.071	0.085	0.083	0.1
			FEM	1.005	0.080	0.091	2.647	0.212	0.241	2.765	0.114	0.118	0.1
			LTMLE	0.900	0.140	0.151	1.837	0.327	0.356	1.923	0.220	0.240	0.0

Abbreviations: SE: arithmetic mean of asymptotic standard errors, ESD: empirical standard deviation of estimates, % Non. Conv: % non-convergence (no solution found), JMM: joint mixed-effects model, LMM: linear mixed-effects model, FEM: fixed-effects model, LTMLE: longitudinal targeted maximum likelihood. True g₁(a₀ = 1) − g₂(a₀ = 0) = 1.0, g₂(a₀ = 1, a₁) − g₂(a₀ = 0, a₁) = 2.0, g₂(a₀, a₁ = 1) − g₂(a₀, a₁ = 0) = 2.0.

In the presence of a random slope (Var(b_i1)∈ {0.3, 1.0}), Table 1 shows that JMM produced unbiased estimates regardless of the values of α₇ and Var(b_i1), and results from LMM were unbiased in the absence of unmeasured confounding (α₇ = 0). However, LMM demonstrated bias exceeding 10% for at least one causal effect in the presence of unmeasured confounding (α₇ = 1), where larger biases occurred when Var(b_i1) = 1.0 than when Var(b_i1) = 0.3, which reflects the strength of unmeasured confounding. FEM exhibited bias exceeding 10% for at least one causal effect for all scenarios. LTMLE was unbiased when Var(b_i1) = 0.3 regardless of α₇; however, LTMLE exhibited some (2.0% to 10%) bias when Var(b_i1) = 1.0 regardless of α₇. JMM was expected to be unbiased irrespective of the presence of unmeasured confounding because it was the only method that does not rely on the assumption of no unmeasured confounding. However, LTMLE was more robust to unmeasured confounding than was LMM, which may reflect the robustness of empirical models of exposure over parametric assumptions of outcome models in this case. Once again, LTMLE was the least efficient method, and LMM tended to be somewhat more efficient than JMM. Non-convergence was rare (0% to 0.3%) across all methods.

Table 2 displays the simulation results in scenarios with unmeasured confounding and exponentially distributed random effects. In this case, JMM and LMM were fit by misspecifying the random effects distribution as normal (random intercept only) or bivariate normal (random intercept and slope). In the absence of random slope (Var(b_i1)=0) and unmeasured confounding (α₆ = 0), LTMLE was unbiased and LMM showed little bias whereas FEM had 7.3% to 19.1% bias and JMM estimates all had bias exceeding 10%. In contrast, in the presence of unmeasured confounding (α₆ = 1), JMM was unbiased whereas all other methods’ estimates had bias exceeding 10%.

Table 2.

Simulation study with unmeasured confounding and exponentially distributed random effects. JMM and LMM misspecified the the random effect distribution as normal. Simulations include either random intercept only (Var(b_i1) = 0.0) or both random slope and intercept (Var(b_i1) ∈ {0.3, 1.0}).

Var(bi1)	α6	α7	Method	ĝ1(a0 = 1) − ĝ1(a0 = 0)			ĝ2(a0 = 1, a1) − ĝ2(a0 = 0, a1)			ĝ2(a0, a1 = 1) − ĝ2(a0, a1 = 0)			% Non. Conv.
				Estimate	SE	ESD	Estimate	SE	ESD	Estimate	SE	ESD
0.0	0	0	JMM	1.241	0.108	0.126	2.538	0.240	0.281	2.208	0.103	0.114	0.0
			LMM	1.064	0.067	0.065	2.140	0.140	0.134	2.077	0.080	0.086	0.0
			FEM	0.927	0.064	0.059	2.167	0.154	0.142	2.383	0.091	0.094	0.0
			LTMLE	0.997	0.134	0.149	1.959	0.268	0.226	2.058	0.198	0.226	0.0
	1		JMM	1.037	0.102	0.111	2.080	0.212	0.232	2.046	0.091	0.086	0.0
			LMM	1.346	0.065	0.064	2.749	0.137	0.134	2.255	0.078	0.076	0.0
			FEM	1.250	0.063	0.058	2.924	0.148	0.142	2.558	0.087	0.082	0.0
			LTMLE	1.485	0.130	0.160	2.877	0.238	0.276	2.258	0.159	0.192	0.0
0.3	0	0	JMM	1.070	0.080	0.068	2.159	0.167	0.141	2.070	0.087	0.089	0.6
			LMM	1.055	0.074	0.063	2.125	0.154	0.129	2.059	0.085	0.086	0.0
			FEM	0.896	0.071	0.065	2.201	0.178	0.166	2.526	0.101	0.105	0.0
			LTMLE	1.004	0.148	0.160	2.002	0.302	0.319	2.048	0.193	0.223	0.0
		1	JMM	1.029	0.079	0.076	2.065	0.162	0.158	2.037	0.084	0.088	0.7
			LMM	1.122	0.074	0.076	2.258	0.153	0.157	2.094	0.083	0.085	0.0
			FEM	0.999	0.071	0.081	2.481	0.179	0.210	2.587	0.101	0.112	0.0
			LTMLE	1.034	0.152	0.158	2.086	0.290	0.313	2.056	0.191	0.233	0.0
1.0		0	JMM	1.042	0.083	0.078	2.092	0.171	0.160	2.031	0.089	0.084	13.3
			LMM	1.037	0.081	0.075	2.081	0.167	0.154	2.022	0.088	0.089	8.8
			FEM	0.814	0.083	0.094	2.125	0.219	0.252	2.715	0.118	0.135	0.0
			LTMLE	0.973	0.167	0.184	1.994	0.368	0.396	2.054	0.238	0.279	0.0
		1	JMM	0.984	0.082	0.083	1.972	0.165	0.171	1.995	0.086	0.094	23.4
			LMM	1.122	0.080	0.084	2.256	0.163	0.174	2.072	0.086	0.095	14.0
			FEM	0.902	0.083	0.102	2.387	0.221	0.262	2.731	0.118	0.128	0.0
			LTMLE	0.904	0.174	0.201	1.884	0.359	0.382	2.073	0.197	0.240	0.0

Abbreviations: SE: arithmetic mean of asymptotic standard errors, ESD: empirical standard deviation of estimates, % Non. Conv: % non-convergence (no solution found), JMM: joint mixed-effects model, LMM: linear mixed-effects model, FEM: fixed-effects model, LTMLE: longitudinal targeted maximum likelihood. True g₁(a₀ = 1) − g₂(a₀ = 0) = 1.0, g₂(a₀ = 1, a₁) − g₂(a₀ = 0, a₁) = 2.0, g₂(a₀, a₁ = 1) − g₂(a₀, a₁ = 0) = 2.0.

In the presence of a random slope (Var(b_i1) ∈ {0.3, 1.0}), Table 2 shows that JMM produced unbiased estimates in the presence of unmeasured confounding (α₇ = 1) and small (1.6% to 7.9%) bias in the absence of unmeasured confounding (α₇ = 0). LMM produced estimates with small bias when Var(b_i1) = 1.0 in the absence of unmeasured confounding (α₇ = 0), estimates with small (2.9% to 6.2%) bias when Var(b_i1) = 0.3 in the absence of unmeasured confounding (α₇ = 0), and at least one estimate exceeding 10% bias in the presence of unmeasured confounding (α₇ = 1). FEM exhibited bias exceeding 10% for at least one causal effect for all scenarios. LTMLE displayed small to no bias in all scenarios with its largest (5.6% to 9.6%) bias when Var(b_i1) = 1.0 in the presence of unmeasured confounding (α₇ = 1). These findings reflect the robustness of LTMLE and the sensitivity of parametric methods to the outcome distribution in the absence of unmeasured confounding, and that the benefit of accommodating unmeasured confounding may overcome the limitation of misspecifying the random effect distribution. Once again, LTMLE was the least efficient estimation method, and LMM tended to be somewhat more efficient than JMM. Non-convergence was frequent (8.8% to 23.4%) when Var(b_i1) = 1.0 in JMM or LMM.

Table 3 displays the simulation results in scenarios with unmeasured confounding, nonignorable dropout, and normally distributed random effects. In the absence of random slope (Var(b_i1)=0), and when dropout was ignorable (δ₆ = 0), JMM performed well whether unmeasured confounding was present (α₆ = 1) or absent (α₆ = 0); LMM performed well when unmeasured confounding was absent (α₆ = 0), but not present (α₆ = 1); and FEM and LTMLE were biased whether unmeasured confounding was absent (α₆ = 0) or present (α₆ = 1). This pattern was largely repeated when dropout was nonignorable (δ₆ = 0.5). LTMLE tended to be the least efficient method across all specifications, and LMM was more efficient than JMM.

Table 3.

Simulation study with unmeasured confounding, nonignorable dropout, and normally distributed random effects. Simulations include either random intercept only (Var(b_i1) = 0.0) or both random slope and intercept (Var(b_i1) ∈ {0.3, 1.0}).

Var (bi1)	δ6	δ7	α6	α7	Method	ĝ1(a0 = 1) − ĝ1(a0 = 0)			ĝ2(a0 = 1, a1) − ĝ2(a0 = 0, a1)			ĝ2(a0, a1 = 1) − ĝ2(a0, a1 = 0)			% Non. Conv.
						Estimate	SE	ESD	Estimate	SE	ESD	Estimate	SE	ESD
0.0	0	0	0	0	JMM	1.015	0.148	0.153	2.035	0.323	0.326	1.999	0.130	0.126	14.7
					LMM	0.990	0.079	0.077	1.981	0.166	0.162	1.990	0.101	0.098	0.0
					FEM	0.907	0.078	0.077	2.125	0.189	0.187	2.293	0.116	0.102	0.0
					LTMLE	0.964	0.142	0.154	1.828	0.281	0.325	1.781	0.213	0.238	0.0
			1		JMM	1.001	0.114	0.127	2.005	0.239	0.269	2.002	0.103	0.118	10.8
					LMM	1.345	0.076	0.066	2.749	0.163	0.141	2.226	0.096	0.102	0.0
					FEM	1.316	0.075	0.066	3.101	0.181	0.158	2.524	0.110	0.100	0.0
					LTMLE	1.437	0.130	0.139	2.746	0.282	0.286	2.154	0.230	0.249	0.0
	0.5		0		JMM	0.981	0.142	0.164	1.960	0.311	0.347	1.972	0.128	0.139	16.5
					LMM	0.980	0.081	0.084	1.954	0.170	0.169	1.977	0.104	0.099	0.0
					FEM	0.904	0.080	0.080	2.095	0.191	0.188	2.253	0.120	0.106	0.0
					LTMLE	0.950	0.141	0.148	1.776	0.280	0.287	1.770	0.220	0.255	0.0
			1		JMM	0.965	0.117	0.136	1.929	0.245	0.286	1.966	0.107	0.129	28.7
					LMM	1.337	0.077	0.078	2.725	0.166	0.164	2.195	0.099	0.112	0.0
					FEM	1.310	0.076	0.068	3.059	0.182	0.160	2.479	0.112	0.102	0.0
					LTMLE	1.415	0.131	0.147	2.636	0.278	0.303	2.082	0.233	0.256	0.0
0.3	0	0	0	0	JMM	1.001	0.097	0.099	2.006	0.203	0.207	1.996	0.113	0.104	4.1
					LMM	1.002	0.088	0.087	2.008	0.184	0.178	1.999	0.109	0.101	0.0
					FEM	1.014	0.086	0.092	2.494	0.218	0.236	2.570	0.128	0.138	0.0
					LTMLE	0.948	0.170	0.200	1.929	0.312	0.376	1.852	0.220	0.270	0.0
				1	JMM	0.995	0.094	0.104	1.994	0.194	0.223	2.003	0.107	0.101	2.5
					LMM	1.104	0.086	0.088	2.217	0.179	0.189	2.080	0.105	0.097	0.0
					FEM	1.164	0.084	0.088	2.889	0.215	0.235	2.688	0.125	0.108	0.0
					LTMLE	1.024	0.156	0.182	1.998	0.294	0.318	1.878	0.232	0.265	0.0
		0.5		0	JMM	1.008	0.096	0.103	2.014	0.200	0.216	1.991	0.113	0.117	4.0
					LMM	1.009	0.088	0.087	2.011	0.183	0.181	1.995	0.109	0.111	0.0
					FEM	1.022	0.085	0.086	2.505	0.218	0.224	2.562	0.129	0.128	0.0
					LTMLE	0.948	0.160	0.184	1.850	0.282	0.304	1.848	0.217	0.271	0.0
				1	JMM	1.012	0.094	0.102	2.014	0.195	0.210	1.991	0.108	0.097	7.9
					LMM	1.120	0.086	0.092	2.236	0.179	0.192	2.080	0.105	0.093	0.0
					FEM	1.173	0.084	0.089	2.899	0.217	0.228	2.683	0.126	0.109	0.0
					LTMLE	1.025	0.148	0.173	1.965	0.288	0.359	1.863	0.224	0.256	0.0
1.0		0		0	JMM	0.987	0.097	0.088	1.971	0.201	0.181	1.978	0.116	0.120	1.7
					LMM	0.986	0.095	0.084	1.968	0.196	0.173	1.978	0.115	0.124	0.0
					FEM	1.063	0.097	0.105	2.756	0.259	0.292	2.788	0.147	0.159	0.0
					LTMLE	0.927	0.197	0.231	1.883	0.369	0.454	1.880	0.242	0.328	0.0
				1	JMM	0.982	0.094	0.094	1.966	0.191	0.194	1.995	0.107	0.107	3.1
					LMM	1.127	0.091	0.096	2.262	0.188	0.200	2.092	0.107	0.109	0.0
					FEM	1.264	0.095	0.101	3.326	0.256	0.274	2.864	0.141	0.139	0.0
					LTMLE	0.895	0.182	0.208	1.905	0.331	0.392	1.827	0.278	0.362	0.0
		0.5		0	JMM	1.002	0.098	0.106	1.993	0.202	0.217	1.997	0.116	0.119	8.2
					LMM	1.007	0.095	0.102	1.998	0.197	0.208	2.016	0.115	0.116	0.0
					FEM	1.081	0.098	0.106	2.807	0.264	0.285	2.770	0.150	0.161	0.0
					LTMLE	0.858	0.198	0.223	1.832	0.348	0.422	1.882	0.234	0.322	0.0
				1	JMM	1.014	0.093	0.076	2.012	0.187	0.158	2.001	0.106	0.104	10.4
					LMM	1.154	0.091	0.092	2.291	0.186	0.191	2.106	0.106	0.110	0.0
					FEM	1.269	0.096	0.104	3.350	0.260	0.292	2.821	0.142	0.149	0.0
					LTMLE	0.890	0.202	0.239	1.843	0.328	0.366	1.883	0.247	0.289	0.0

Abbreviations: SE: arithmetic mean of asymptotic standard errors, ESD: empirical standard deviation of estimates, % Non. Conv: % non-convergence (no solution found), JMM: joint mixed-effects model, LMM: linear mixed-effects model, FEM: fixed-effects model, LTMLE: longitudinal targeted maximum likelihood. True g₁(a₀ = 1) − g₂(a₀ = 0) = 1.0, g₂(a₀ = 1, a₁) − g₂(a₀ = 0, a₁) = 2.0, g₂(a₀, a₁ = 1) − g₂(a₀, a₁ = 0) = 2.0.

In the presence of a random slope, Table 3 shows that JMM was unbiased regardless of whether unmeasured confounding was present (α₇ = 1) or absent (α₇ = 0), or whether dropout was nonignorable (δ₇ = 0.5) or not (δ₇ = 0). LMM was unbiased in the absence of unmeasured confounding (α₇ = 0) regardless of the dropout mechanism (δ₇), but LMM produced at least one estimate with bias exceeding 10% in the presence of unmeasured confounding (α₇ = 1). LTMLE produced small bias when Var(b_i1) = 0.3 regardless of unmeasured confounding (α₇) and the dropout mechanism (δ₇); however, LTMLE produced some estimates that had at least 10% bias when Var(b_i1) = 1.0 in the presence of unmeasured confounding (α₇ = 1) or nonignorable dropout (δ₇ = 0.5). FEM exhibited bias exceeding 10% for at least one causal effect for all scenarios. LTMLE was the least efficient method across all specifications, and LMM was more efficient than JMM. Non-convergence ranged from 2.5% to 28.7% in JMM and was higher in random intercept models than in models with random both a random intercept and slope.

In summary, no single method outperformed all other methods in every scenario studied; however, all methods outperformed FEM. JMM performed well except when estimating random intercept models with a misspecified random effect distribution in the absence of unmeasured confounding, which reflects the sensitivity to parametric assumptions. In contrast, LTMLE performed particularly well in this scenario, which reflects the fact that it does not rely on parametric assumptions and the assumption of no unmeasured confounders was satisfied. Not surprisngly, LTMLE uniformly outperformed FEM, although both methods omitted the random effects thereby misspecifying the outcome model. The difference was that LTMLE was doubly robust; it estimated the exposure-selection mechanism and, where relevant, the dropout models. Therefore, in cases where the random effect did not affect the exposure mechanism and where dropout was ignorable, LTMLE correctly specified these models. LTMLE also demonstrated small bias when the random slope explained a small portion of the outcome variance (Var(b_i1) = 0.3), which reflected a moderate misspecification of the outcome model. The limitation of LTMLE is that it was the least efficient method studied. Interestingly, LMM performed as well as JMM, and with greater efficiency, in the absence of unmeasured confounding even when drop-out was nonignorable. In general, dropout may induce a type of bias referred to as collider-stratification bias.³² This bias occurs when the exposure (A) and random effect (b) both affect the dropout mechanism (R) due to restricting analysis on observations that are not lost to follow-up. That is, arrows in a DAG from A and b collide at R: A → R ← b. This restriction induces an association between the exposure and random effect resulting in unmeasured confounding. Hernań et al³² demonstrated that this bias does not occur when A and b have a multiplicitive effect on the cumulative probability that R = 1. In the simulation herein, A and b had a multiplicative effect on the conditional odds that R = 1; thus, except in the presence of very high dropout rates, bias from nonignorable dropout in LMM was expected to be small. This phenomenon was also empirically demonstrated by a recently published set of simulation studies.³³ Lastly, an additional caveat of JMM is its potential for non-convergence.

4. Data Application

We applied the proposed joint mixed-effects model to a study of circulating vitamin D and depressive symptoms among participants enrolled in “Aging in Chianti” (InCHIANTI), a study of older adults residing in the Chianti region of Italy (near Tuscany).³⁴ Specifically, we estimated the effect of serum concentrations of 25-hydroxyvitamin D [25(OH)D], the gold-standard measure of vitamin D body stores, on severity of depressive symptoms in community-dwelling older adults.³⁵

The analysis included 1,203 participants from the InCHIANTI study. Study visits in InCHIANTI were scheduled to be approximately three years apart. In this analysis, we included as exposure 25(OH)D measured at baseline and at the first and second follow-up visits; we included as outcome depressive symptoms assessed at the first, second, and third follow-up visits. Depressive symptoms were measured using the Center for Epidemiologic Studies Depression (CES-D) Scale,³⁶ a measurement scale comprising 20 items resulting in a score ranging from 0 to 60, with higher scores indicating more severe depressive symptoms. We dichotomized 25(OH)D at 20 ng/mL. We regressed CES-D at time j (j=1, 2, or 3) on CES-D at time j − 1, 25(OH)D at time 0 through j − 1, random slope (models with an additional random intercept did not converge), and covariates. Covariates included participant demographics, lifestyle factors, season of blood collection, body mass index, renal function (creatinine clearance), and comorbid conditions. We jointly logistically regressed 25(OH)D (≥ 20 ng/mL versus <20 ng/mL) at time j − 1 on 25(OH)D at time 0 through j − 2, CES-D at time j − 1, the random slope, and covariates. Lastly, we logistically regressed missingness (yes versus no) at time j on CES-D at time j − 1, baseline 25(OH)D, covariates, and the random slope. Missingness was not monotone in this study, so CES-D at time j − 1 was multiplied by R_i,j−1 in the missing-data model.

We estimated causal mean differences using JMM, LMM, and FEM; all combined with g-computation; and using LTMLE. We computed structural distributed lags by the joint causal effects of 25(OH)D at times j − 1, j − 2, and j − 3 on depressive symptoms at time j. We refer to the effects of 25(OH)D at times j − 1, j − 2, and j − 3 as the lag 1, lag 2, and lag 3 effects, respectively. The estimated distributed lags are interpreted as the causal effect of setting 25(OH)D ≥ 20 ng/mL at all three time points versus setting 25(OH)D <20 ng/mL at all three time points on CES-D.

Table 4 displays participant characteristics by time (study visit). The mean (standard deviation) age was 74.9 (7.9) years, and 56.7% of participants were women. By the last study visit, over half of the participants were missing or dead.

Table 4.

Descriptive Statistics in the InCHIANTI Study (n=1,203)

Characteristic	Visit 0	Visit 1	Visit 2	Visit 3
	Mean (SD) or Number (%)
CES-D score	12.1 (9.0)	15.7 (8.7)	14.2 (7.7)	14.8 (8.4)
25(OH)D ≥ 20 ng/mL	368 (35.0)	615 (74.7)	371 (50.1)
Female Sex	682 (56.7)	–	–
Age (y)	74.9 (7.9)	–	–
Education (y)	5.3 (3.4)	–	–
Smoker	104 (8.6)	–	–
Alcohol Consumption(drinks week)	7.2 (10.2)	6.5 (8.8)	5.9 (8.3)
MMSE score	24.2 (5.3)	23.7 (6.2)	22.8 (6.8)
Creatinine Clearance (mL/min/1.73 m2)	71.0 (13.6)	66.9 (15.4)	69.4 (15.9)
Body Mass Index (kg m2)	27.5 (3.8)	26.8 (3.8)	27.1 (3.9)
Calcium Intake (mg/day)	815 (299)	822 (304)	812 (286)
Renal Disease	629 (52.2)	541 (57.6)	410 (48.9)
Congestive Heart Failure	59 (4.9)	57 (6.1)	86 (10.2)
Hypertension	525 (43.6)	283 (30.1)	257 (30.6)
Missing Alive	0	259 (21.5)	261 (21.7)	244 (20.2)
Dead	0	143 (11.9)	298 (24.8)	435 (36.2)

Abbreviations: SD, standard deviation; ; CES-D, Center for Epidemiologic Studies Depression Scale; 25(OH)D, 25-hydroxyvitamin D; y, years; MMSE, Mini-Mental State Examination score. Missingness at visit j defined as missing CES-D at visit j or missing 25(OH)D or covariates at visit j − 1, for j = 1, 2, 3. Dashes (−) indicate time-invariant covariates assessed at visit 0.

Table 5 shows the estimated causal mean differences of 25(OH)D on depressive symptoms. All methods (JMM, LMM, FEM, and LTMLE) led to the same qualitative conclusion that the joint effect of 25(OH)D ≥ 20 ng/mL at all three time points causes less severe depressive symptoms than 25(OH)D < 20 ng/mL; although JMM and FEM were the only methods for which the 95% confidence interval excluded 0. JMM estimated the largest magnitude structural distributed lag (−1.80; 95% confidence interval, −3.42, −0.18); LTMLE estimated the smallest magnitude structural distributed lag (−1.01; 95% confidence interval, −3.00, 0.98) and had the widest confidence intervals. JMM, LMM, and FEM all estimated that lag 1 25(OH)D had the strongest causal effect and lag 3 25(OH)D had the weakest causal effect. Surprisingly, LTMLE estimated that lag 3 25(OH)D ≥ 20 ng/mL caused more severe depressive symptoms.

Table 5.

Estimated Causal Effect of Vitamin D on Depressive Symptoms in the InCHIANTI Study (n=1,203)

Method	Parameter	Estimate	95% CI
JMM	Lag 1	−1.37	(−2.06, −0.67)
	Lag 2	−0.31	(−1.27, 0.65)
	Lag 3	−0.12	(−0.50, 0.26)
	Lag 1,2, and 3	−1.80	(−3.42, −0.18)
LMM	Lag 1	−1.35	(−2.04, −0.66)
	Lag 2	−0.16	(−1.11, 0.80)
	Lag 3	−0.06	(−0.44, 0.31)
	Lag 1,2, and 3	−1.57	(−3.18, 0.04)
FEM	Lag 1	−1.19	(−1.88, −0.50)
	Lag 2	−0.40	(−1.35, 0.55)
	Lag 3	−0.16	(−0.54, 0.22)
	Lag 1,2, and 3	−1.75	(−3.35, −0.14)
LTMLE	Lag 1	−1.68	(−4.99, 1.64)
	Lag 2	−1.43	(−3.71, 0.84)
	Lag 3	1.29	(−0.78, 3.37)
	Lag 1,2, and 3	−1.01	(−3.00, 0.98)

Abbreviations: CI, confidence interval; JMM: joint mixed-effects model; LMM: linear mixed-effects model; FEM: fixed-effects model, LTMLE: longitudinal targeted maximum likelihood.

JMM estimated the coefficient of the random slope as 2.51 (95% confidence interval, −0.73, 5.75) for 25(OH)D and −11.88 (95% confidence interval, −27.31, 3.55) for missingness. Furthermore, the estimated variance of the random slope was 0.046 (95% confidence interval, 0.003, 0.613). After accounting for measured factors, including lagged depressive symptoms, unmeasured sources of heterogeneity that impact the effect of 25(OH)D on depressive symptoms may be relatively small (i.e., very little unmeasured confounding); thus the generally concordant results between JMM, LMM, and FEM. the difference between LTMLE and the other three methods may be explained, in part, by the fact that LTMLE tends to have greater variability than the other methods that may reflect sensitivity to large inverse probability of exposure weights, and it is less dependent on correct specification of the outcome model than the other methods. Lastly, unlike the other three methods, LTMLE constrains the estimated potential outcomes to be in the observed range of the observed outcomes.

5. Discussion

Causal effects are a common target of estimation in observational longitudinal studies. However, such studies have the potential for unmeasured confounding and time-dependent confounding. Combining joint mixed-effects models with g-computation is an attractive approach for simultaneously addressing these challenges.

A particular advantage of the joint mixed-effects model is that it performs well in most situations with either the presence or absence of unmeasured confounding. The method’s accessibility is enhanced because the model can be estimated using off-the-shelf statistical packages, such as SAS PROC NLMIXED (see Appendix for sample code). Since the joint mixed-effects model relies on neither the assumption of no unmeasured confounding, in contrast with conventional g-computation and marginal structural models, nor the availability of an unconfounded variable, in contrast with instrumental variable methods, the approach can expand the scenarios within which unbiased estimation of causal effects can occur.

A caveat of the joint mixed-effects model is that unbiased estimation depends on correct specification of the outcome data-generating process, the exposure-selection process, and the random-effects distribution.^37–39 Robustness to misspecification in the random-effects distribution in the context of joint models has been reported.⁴⁰ In our simulation study, JMM performmed well with a misspecified random effect distribution except with random intercept only models in the absence of unmeasured confounding. In this case, LTMLE performed the best of all four methods assessed. Furthermore, as with any g-computation, estimation of causal effects can be computationally intensive in the presence of many time-varying confounders. The joint-mixed effects model addresses unmeasured confounding via random effects, which serve as proxies for the unmeasured confounders. However, the random effects may not capture unmeasured time-varying confounders. Despite these caveats, the proposed method may be particularly beneficial when a known strong confounder is not measured and there is sufficient scientific background knowledge to inform the outcome and exposure-selection models. This scenario is plausible when analyzing data from large cohort studies, because typically not all hypotheses are specified at the study design stage and not all information is collected in order to reduce participant burden and to satisfy budget constraints.

A practical way to balance the strengths and limitations of JMM with those of other methods is to perform multiple methods as part of a sensitivity analysis. In particular, JMM was motivated by studies where important strong confounders were not measured, as exemplified by the analysis of vitamin D and depressive symptoms in InCHIANTI, where sunlight exposure was not measured. However, measured confounders may be omitted resulting in misspecification. For example, if confounder X is modeled, but transformed X should have also been modeled, then this omission is a type of unmeasured (more accurately, omitted) confounding. Although the random effects may capture this omission, we recommend researchers perform extensive exploratory analysis, informed by scientific background knowledge, using the measured covariates to identify strong confounders and appropriate functional forms to help build the model. Prioritizing modeling known strong measured confounders may help reduce bias without a major sacrifice in efficiency. Bias from omitted or mis-modeled weak confounders may be partially overcome by the JMM, and such bias may be small relative to the gains in efficiency from omitting some functions of weak confounders. Given the strong parametric assumptions of JMM, we recommend that researchers perform sensitivity analysis using flexible methods that rely on different assumptions. For example, LTMLE is a flexible method that can help researchers relax parametric assumptions and consider more complex functional forms. However, this type of approach relies on the assumption of no unmeasured confounders; therefore, if results differ between the methods, it is not clear if this is due to violating the assumption of no unmeasured confounders in LTMLE or due to misspecifying the random effect distribution or functional form in JMM. Thus, the results of LTMLE can be used to motivate changes to the functional form of JMM or variable transformations to better satisfy JMM assumptions.

The proposed method contributes to a growing body of literature aiming to account for participant heterogeneity while estimating causal effects.^{6, 7} This and other work has mainly focused on heterogeneity in the context of longitudinal studies,^{6, 7} but others have noted that heterogeneity may be due to clustering.⁴¹ Lastly, future extensions of the joint mixed-effects model should consider categorical and count outcomes.

6. Data Accessibility

Computer code used to generate and analyze simulated data is available for readers. Data from the InCHIANTI study is available with permission from the InCHIANTI Publications Committee at www.inchiantistudy.net.

Appendix: SAS Code and Example Data

We include SAS code using PROC NLMIXED to fit the joint mixed-effects model and perform g-computation. The variable ‘theoutcome’ is a placeholder. Below are example data from four participants. In InCHIANTI, missingness was not necessarily monotone. For example, the participant with ID = 4 has Y missing at time 1 and observed at times 2 and 3. Thus whether this participant contributes data to the likelihood for Y at time 2 or 3 depends on whether Y at time 2 or 3 is regressed on data collected at time 1. Lagged variables were set to 0 for times prior to data collection (time = 0); X1 was standardized.

  ID  time   Y     Alag1   Ylag1     RYlag1  Alag2   Alag3   Atime0     X1      X2
1      1    20       0       31        31      0       0        0     1.3875     1
1      2     .       0       20        20      0       0        0     1.3875     1
1      3     .       .        .         0      0       0        0     1.3875     1
2      1    36       0       34        34      0       0        0    −0.1125     1
2      2    33       1       36        36      0       0        0    −0.1125     1
2      3    20       1       33        33      1       0        0    −0.1125     1
3      1    24       0        7         7      0       0        0    −0.9875     0
3      2     9       1       24        24      0       0        0    −0.9875     0
3      3    19       1        9         9      1       0        0    −0.9875     0
4      1     .       1        3         3      0       0        1    −0.7375     0
4      2    10       .        .         0      1       0        1    −0.7375     0
4      3    13       0       10        10      .       1        1    −0.7375     0
proc nlmixed data=jmmdata qpoints=5;
parms beta0=14 beta11=0 beta12=1 beta2=0 beta30=0 beta31=0 beta32=0
           beta40=0 beta41=0 beta5=0 logsigmasq=2 /*f(Y) parms */
           alpha0=0 alpha11=0 alpha12=0 alpha2=0 alpha30=0 alpha31=0
           alpha41=0 alpha5=0 alpha7=0 /* f(A) parms */
           delta0=0 delta11=0 delta12=0 delta2=0 delta3=0 delta4=0
           delta5=0 /* f(R) parms */
           logvarb1=0; /*f(b) parms */
/*normal likelihood for Y */
mu = beta0 + beta11*X1 + beta12*X2 + beta2*Ylag1 + beta30*Alag1 +
         beta31*Alag2 + beta32*Alag3 + beta40*X1*Alag1 + beta41*X1*Alag2 +
  beta42*X1*Alag3 + beta5*time + b1*(1 + Alag1 + Alag2 + Alag3)*X1;
py = pdf(’NORMAL’,Y,mu,exp(logsigmasq));
llhY = log(py);
if llhY=. then llhY=0;
/* Bernoulli likelihood for A */
logitA=alpha0 + alpha11*X1 + alpha12*X2 + alpha2*Ylag1 + alpha30*Alag2 +
             alpha31*Alag3 + alpha41*X1*Ylag1 + alpha5*time + alpha7*b1;
pA=exp(logitA)/(1 + exp(logitA));
llhA = Alag1*log(PA) + (1-Alag1)*log(1-pA);
if llhA=. then llhA=0;
/* Bernoulli likelihood for (non-monotone) R */
logitR = delta0 + delta11*X1 + delta12*X2 + delta2*Atime0 + delta3*RYlag1 +
              delta4*time + delta5*b1;
pR = exp(logitR)/(1 + exp(logitR));
llhR= R*log(pR) + (1-R)*log(1-pR);
llik= llhY + llhA +llhR;
model theoutcome~general(llik);
/*normal likelihood for b1 */
random b1 ~normal(0,exp(logvarb1)) subject=id;
/* g-computation (X1 centered at 0) */
estimate ’g-computation: lag 1’ beta30;
estimate ’g-computation: lag 2’ beta2*beta30 + beta31;
estimate ’g-computation: lag 3’ beta2*beta2*beta30 + beta2*beta31 + beta32;
estimate ’g-computation: lag 1-3’ beta30 + beta2*beta30 + beta31 + beta2*beta2*beta30 +
                                 beta2*beta31 + beta32;
ods output ParameterEstimates=ParameterEstimates
                  AdditionalEstimates=AdditionalEstimates;
run;