Adjusting for intercurrent events using Bayesian joint models for longitudinal outcomes in clinical trials

Abstract

Background:

Intercurrent events in clinical trials can disrupt the interpretation and/or measurement of clinical endpoints. This article focuses on terminal intercurrent events that preclude complete measurement of a longitudinal outcome. When such events are related to the underlying outcome, particularly for physical signs, analyses based only on the available measurements can yield biased estimates. Consequently, a principled methodology is needed to effectively handle these intercurrent events.

Methods:

We propose a Bayesian joint modeling approach to account for terminal intercurrent events. Our model jointly analyzes longitudinal outcomes and terminal events using shared random effects. We employ multiple discrete-time survival submodels to accommodate different event types and evaluate operating characteristics through extensive simulations that resemble a clinical trial with recovery and death as competing events.

Results:

The proposed Bayesian joint modeling strategy demonstrates higher power than models that do not account for intercurrent events. Specifically, power increases by approximately 15% when bias due to intercurrent events is substantial and can be reduced by joint modeling.

Conclusion:

Our Bayesian joint modeling approach effectively addresses terminal intercurrent events in both the design and analysis phases of clinical trials. By explicitly accounting for event-related truncation of longitudinal follow-up, it improves the precision and reliability of treatment effect estimation when outcome measurement is incomplete.

1. Introduction

Intercurrent events are events that occur after the initiation of treatment that impact the interpretation or existence of the outcome interest in clinical trials [1]. Intercurrent events have recently emerged as a critical element in defining trial estimands, highlighting the need for methods to adjust for them in clinical trials [2].

In many clinical trials, intercurrent events can end follow-up early and preclude further measurement of the outcome. This issue is especially relevant when the outcome is a physical sign, such as blood pressure or diaphragm thickness, because such measures may be associated with the occurrence of terminal events. For example, follow-up may stop because of recovery or death, and these clinically distinct terminal events can occur in competition with each other. Ignoring the association between the longitudinal outcome process and the terminal events can bias inference for treatment effects on the longitudinal outcome. To address this setting, we propose a Bayesian joint modeling approach that links the longitudinal outcome with a competing terminal-event process to support valid inference for treatment comparisons on the longitudinal outcome while accounting for their association.

The proposed approach is motivated by the LANDMARK II randomized trial in patients with acute hypoxemic respiratory failure receiving mechanical ventilation, which compares a lung- and diaphragm-protective ventilation and sedation (LDPVS) strategy with standard care. The primary endpoint is the daily change in diaphragm thickness from baseline over the first 7 days; follow-up may end early due to liberation from mechanical ventilation or death, after which diaphragm thickness is no longer measurable.

Based on the motivating example, this article focuses on studies where the longitudinal outcome serves as the primary endpoint and intercurrent events hinder its measurement. The longitudinal outcome is modeled using a longitudinal model, whereas terminal events, represented by time-to-event data, are modeled using a discrete-time survival model [3], [4], [5]. This survival model can bypass the need to specify a parametric distribution for event times and can naturally accommodate multiple competing events [5], [6]. We integrate these two models through a joint modeling strategy to adjust for intercurrent events.

Bayesian joint models for longitudinal and time-to-event data have been developed and are becoming increasingly common in the analysis of clinical trials.[7] Recently, there has also been emerging research on Bayesian joint models in clinical trial design, with a focus on time-to-event endpoints and the integration of longitudinal processes in the modeling to enhance the power of treatment effect detection [8], [9], [10]. In this study, the primary endpoint is the longitudinal outcome, and a submodel is incorporated to account for intercurrent events.

We assess the performance of our proposed joint modeling approach through numerical simulations designed to resemble the conditions of our motivating trial. The results suggest improved operating characteristics for treatment effect detection when intercurrent events induce substantial bias in standard analyses, because this bias can be reduced by employing joint models.

The remainder of the article is organized as follows. We describe the trial design, statistical model, and simulation settings for evaluating the proposed approach. We then present simulation results motivated by the LANDMARK II trial. Finally, we conclude with a discussion of implications for the design and analysis of clinical trials.

2. Methods

2.1. Trial design

In this article, we consider a two-arm randomized clinical trial that aims to compare treatment against control. The primary outcome is measured repeatedly throughout the trial, but intercurrent events may prevent its measurement. Importantly, the occurrence of intercurrent events is associated with the primary outcome, which could introduce bias into the estimation of treatment effects if overlooked.

The primary objective of this study is to develop and evaluate a Bayesian joint modeling strategy for longitudinal primary endpoints whose measurement may be truncated by multiple terminal intercurrent events (e.g., recovery or death), and to support valid treatment effect inference by explicitly modeling the association between the longitudinal process and competing terminal events. To assess the performance of this strategy, we use simulation to characterize the frequentist operating characteristics of a Bayesian decision rule for detecting a clinically meaningful treatment effect under varying rates and types of intercurrent events. Accordingly, we assess whether the treatment yields a clinically meaningful improvement in the longitudinal outcome using the following one-sided test:

$$H_0:E\le c\text{vs}{H}_1:E>c$$

with significance level $1-\alpha$, where $E$ represents the treatment effect and $c$ indicates a minimum clinically meaningful improvement in the outcome, as specified by the clinical team. To implement this test in a Bayesian framework, we compute the posterior distribution of the treatment effect being greater than $c$,

$$p=\text{Pr}(E>c\mid \text{Data}).$$

Based on this posterior probability, the treatment is declared superior to the control if $p$ falls within the rejection region:

$$R=\left\{\text{Data}:p\ge T_{\alpha }\right\}.$$ (1)

We evaluate and calibrate this Bayesian decision rule using its operating characteristics under repeated sampling; thus, throughout the paper, “type I error” and “power” refer to the frequentist operating characteristics of the rule $p\ge T_{\alpha }$. The critical value $T_{\alpha }$ is determined through simulation to achieve a type I error of $1-\alpha$. Specifically, for data simulated under $H_0$, $T_{\alpha }$ corresponds to the $\alpha$ quantile of the posterior probabilities ${\left\{p_i\right\}}_{i=1}^m$ computed from each simulation run. Importantly, $T_{\alpha }$ is determined based on simulating the complete data without intercurrent events, as our knowledge about the exact distribution of these events in the trial is limited.

Because of the complexity of the proposed joint model, the operating characteristics of the trial are estimated through simulation. We suggest assessing these operating characteristics across various scenarios of intercurrent events to comprehensively understand their impact. The type I error and power for each scenario are approximated by examining the frequency of $\{p\ge T_{\alpha }\}$ when data is simulated under $H_0$ and $H_1$ with different simulated intercurrent events, respectively.

2.2. Statistical model

To implement the proposed Bayesian framework for detecting treatment effect significance, we now present the statistical model for analyzing the data. Firstly, we introduce the notation. We assume that in a trial with $I$ patients, the primary outcome is measured at $K$ different time points, denoted as $t_1,t_2,\dots ,t_K$. To facilitate the modeling, let $t_k=k-1$ for $k=1,2,\dots ,K$. Each patient’s treatment status is indicated by a binary variable $x_i$, with values 0 and 1 indicating the control and treatment arm, respectively. The outcome for patient $i$ at time $t$ is denoted as $Z_{it}$.

Assuming each patient has at least one observed outcome, we define $D_i$ as the last observed measurement time for patient $i$ before either the occurrence of intercurrent event or the end of the observation period. Additionally, We define ${\delta }_i$ as the type of intercurrent event for patient $i$, e.g., recovery or death for the LANDMARK II trial. For simplicity, we only consider two types of intercurrent events, where $\delta =0,1,2$ represents no event occurrence, type 1, and type 2 events, respectively. Hence, if patient $i$ experiences the type $h$ event between times $t$ and $t+1$, we denote this as $D_i=t$ and ${\delta }_i=h$, where $t$ ranges from $t_1=0$ to $t_{K-1}=K-2$, and $h$ is either 1 or 2. Conversely, if patient $i$ completes follow-up without experiencing intercurrent events, $D_i=t_K$ and ${\delta }_i=0$. Finally, we denote the observed responses for patient $i$ as ${\mathbf{Z}}_i^o={\left(Z_{it}\right)}_{t=0}^{D_i}$.

Our proposed joint modeling approach simultaneously models the observed longitudinal outcomes ${\mathbf{Z}}_i^o$ and intercurrent events using shared random effects to mitigate the bias caused by the intercurrent events. Fig. 1 provides a conceptual overview of how the joint model links the longitudinal trajectory and competing terminal intercurrent events through shared latent effects.

Fig. 1.

Schematic overview of the proposed Bayesian joint model. The longitudinal submodel and the discrete-time competing-risks terminal-event submodel are linked through shared latent effects, supporting inference for the treatment effect on the longitudinal endpoint while accounting for truncation by terminal intercurrent events.

Specifically, within the Bayesian framework, the modeling is based on the density of the observed data ${\mathbf{Z}}_i^o$, $D_i$, ${\delta }_i$, and random effect ${\mathbf{b}}_i$ for patient $i$, conditional on the treatment indicator $x_i$ and model parameters $\mathit{\theta }$, denoted as $f({\mathbf{Z}}_i^o,D_i,{\delta }_i,{\mathbf{b}}_i\mid x_i,\mathit{\theta })$. This joint distribution is factorized as follows:

$$f({\mathbf{Z}}_i^o,D_i,{\delta }_i,{\mathbf{b}}_i\mid x_i,\mathit{\theta })=f({\mathbf{b}}_i\mid x_i,{\mathit{\theta }}_1)f({\mathbf{Z}}_i^o\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_2)f(D_i,{\delta }_i\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_3),$$

where $\mathit{\theta }=({\mathit{\theta }}_1,{\mathit{\theta }}_2,{\mathit{\theta }}_3)$ where ${\mathit{\theta }}_1$, ${\mathit{\theta }}_2$ and ${\mathit{\theta }}_3$ are the parameters for the density of random effects, the conditional density of observed responses given random effects, and the conditional density of the intercurrent events given random effects, respectively. This model is commonly referred to as a shared parameter model [11], [12], [13] as we assume that $Z_i^o$ and $(D_i,{\delta }_i)$ are independent given the random effects ${\mathbf{b}}_i$.

Terminal intercurrent events such as recovery and death stop follow-up and make subsequent measurements of the longitudinal endpoint unavailable. When the risk of these events is related to the underlying outcome trajectory, analyses that ignore the event process can be biased. Our shared parameter model addresses this by linking the longitudinal trajectory and the competing terminal-event risks through shared random effects, so that inference for the treatment effect on the longitudinal endpoint is informed by both the observed longitudinal measurements and the timing/type of terminal events (Fig. 1).

In a general setting, we suggest employing Bayesian mixed-effects regression for the corresponding outcome to model the product $f({\mathbf{b}}_i\mid x_i,{\mathit{\theta }}_1)f({\mathbf{Z}}_i^o\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_2)$. Typically, the treatment effect is characterized by the coefficients of specific covariates within the model. To model the intercurrent events, $f(D_i,{\delta }_i\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_3)$, we suggest a discrete-time survival model based on the individuals’ hazards at each time point. Specifically, for patient $i$ at time $t$ where $t_1\le t<t_K$, we define the hazards of type 1 and type 2 intercurrent events, respectively, as

$${\lambda }_{it}=\text{Pr}(D_i=t,{\delta }_i=1\mid D_i\ge t,{\mathbf{b}}_i,x_i),{\kappa }_{it}=\text{Pr}(D_i=t,{\delta }_i=2\mid D_i\ge t,{\mathbf{b}}_i,x_i).$$

Then the submodel for the intercurrent events is represented by the likelihood:

$$\text{Pr}(D_i=t,{\delta }_i=h\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_3)={\lambda }_{it}^{I_1\left(h\right)}{\kappa }_{it}^{I_2\left(h\right)}\prod _{j=0}^{t-1}(1-{\lambda }_{ij}-{\kappa }_{ij}),0\le t<K-1,h=1,2,$$

$$\text{Pr}(D_i=K-1,{\delta }_i=0\mid {\mathbf{b}}_i,x_i,{\mathit{\theta }}_3)=\prod _{j=0}^{K-2}(1-{\lambda }_{ij}-{\kappa }_{ij}),$$

where $I_n\left(t\right)$ symbolizes the indicator function, taking the value 1 when $t=n$ and 0 otherwise. We then link the hazards to the random effects using a generalized logit link:

$$log\left(\frac{{\lambda }_{it}}{1-{\lambda }_{it}-{\kappa }_{it}}\right)=\left\{\begin{array}{cc}{q}_0\left(t\right)+p_{0}g({\mathbf{b}}_i,t),\hfill & x_i=0\hfill \\ q_1\left(t\right)+p_{1}g({\mathbf{b}}_i,t),\hfill & x_i=1\hfill \end{array}\right)log\left(\frac{{\kappa }_{it}}{1-{\lambda }_{it}-{\kappa }_{it}}\right)=\left\{\begin{array}{cc}{v}_0\left(t\right)+u_{0}g({\mathbf{b}}_i,t),\hfill & x_i=0\hfill \\ v_1\left(t\right)+u_{1}g({\mathbf{b}}_i,t),\hfill & x_i=1\hfill \end{array}\right)$$

where $g({\mathbf{b}}_i,t)$ represents a function of the latent random effects and, potentially, measurement time; its common specifications include the linear combination of the random effects, the current value of the longitudinal outcome, and so on. Parameters $p_0$, $p_1$, $u_0$ and $u_1$ quantify the association between $g({\mathbf{b}}_i,t)$ and the event hazards across treatment arms and event types. Meanwhile, $q_0\left(t\right)$, $q_1\left(t\right)$, $v_0\left(t\right)$ and $v_1\left(t\right)$ are functions that characterize the relationship between measurement time and these hazards. In the most general setting, the measurement time $t$ can be treated as a categorical variable in $q_0\left(t\right)$, $q_1\left(t\right)$, $v_0\left(t\right)$ and $v_1\left(t\right)$. However, this can result in an overparameterized model and a significant computational burden when the number of longitudinal measurements $K$ is high. This computational burden can be reduced either by collapsing measurement times into a single indicator or by treating measurement time $t$ as a continuous variable and specifying a linear form for $q_0\left(t\right)$, $q_1\left(t\right)$, $v_0\left(t\right)$ and $v_1\left(t\right)$. This proposed model should be implemented with minimally informative priors to mitigate their impact on the estimation of the treatment effect.

In the LANDMARK II trial, patients’ diaphragm thickness is recorded daily until they are either liberated from mechanical ventilation, die, or complete a week’s follow-up, resulting in $K=7$ measurements. The trial’s primary outcome is the magnitude of change in diaphragm thickness relative to the baseline value. We analyze this outcome using the Bayesian folded normal mixed-effects model [14]. The treatment effect of LDPVS is quantified as the average distance between the two regression lines for the two arms across all measurement times, denoted as $AD$ [14]. The trial aims to determine whether $AD$ significantly exceeds 0.1, i.e., $c=0.1$. The subsequent section delineates a variety of simulated scenarios for the intercurrent events.

2.3. Simulation settings

In this section, we outline the simulation plan used to assess our proposed method in the context of the LANDMARK II trial. All simulations are carried out using Markov Chain Monte Carlo method with JAGS 4.3.0 within the R 4.0.2 environment. The simulation plan is presented according to the framework described in Morris et al. [15].

Aims.

We aim to evaluate the operating characteristics of the trial analyzed using the proposed joint models in the presence of intercurrent events. We compare our joint models to a model that analyzes only the outcome without accounting for the intercurrent events, serving as a reference model.

Data-generating mechanisms.

We start by generating a complete dataset, randomly assigning $I=100$ patients to either the treatment or control arms with a probability of 0.5. The magnitude outcome is simulated at 7 time points. To simulate magnitude data consistent with the LANDMARK II trial, a binary sign indicator $\gamma$ is employed. This indicator takes the value −1 with a probability of 0.6 and the value 1 with a probability of 0.4, defining the sign of expected measurement trajectories for each patient, The values −1 and 1 correspond to the decrease and increase of diaphragm thickness, respectively. In the control arm, the measurement $Y_{it}$ for patient $i$ at time $t$ is generated from a normal distribution $N({\gamma }_i{\mu }_{it},{\sigma }^2)$, where ${\mu }_{it}=c_0+{\alpha }_{0i}+\left(c_1+{\alpha }_{1i}\right)t$, with $c_0=0.15$, $c_1=0.015$ and ${\alpha }_{0i}$, ${\alpha }_{1i}$ denoting the random effects. Similarly, in the treatment group, $Y_{it}$ is generated from $N({\gamma }_i{\mu }_{it},{\sigma }^2)$ where ${\mu }_{it}=d_0+{\beta }_{0i}+\left(d_1+{\beta }_{1i}\right)t$, with $d_0\in \{0.08,0.06,0.05,0.04\}$, $d_1=0.005$, and ${\beta }_{0i}$, ${\beta }_{1i}$ representing the random effects. The four values of $d_0$ correspond to increasing values of the average distance $AD$, 0.1, 0.12, 0.13 and 0.14, respectively. The standard deviation of the measurement error is fixed at $\sigma =0.06$. The individual random effects ${\alpha }_{0i}$, ${\alpha }_{1i}$, ${\beta }_{0i}$, and ${\beta }_{1i}$, are generated from normal distributions $N(0,{(c_0/2)}^2)$, $N(0,{(c_1/2)}^2)$, $N(0,{(d_0/2)}^2)$, and $N(0,{(d_1/2)}^2)$, respectively. Finally, $Z_{it}$, defined as the magnitude of the measurement $Y_{it}$, is the simulated outcome.

Once we have simulated the complete dataset, we generate two type of intercurrent events, attributed to recovery and death, then combining them into a competing risk model. For each patient $i$, we define a “risk score” $R_i={\sum }_{t=t_1}^{t_7}{\mu }_{it}/7$ as the mean expected trajectory of outcomes across times. We denote the time to recovery for patient $i$ as $T_i^r$ and the time to death as $T_i^d$. The submodels for $T_i^r$ and $T_i^d$ are specified conditional on $R_i$:

$$T_i^r-0.75\sim \text{Gamma}(1+r_1{R}_i,r_2{R}_i),$$

$$T_i^d\sim \text{Gamma}(1+d_1/R_i,d_2/R_i),$$

where $\text{Gamma}(k,\theta )$ is the Gamma distribution with shape parameter $k$, scale parameter $\theta$ and density function ${\left[\Gamma \left(k\right){\theta }^k\right]}^{-1}{t}^{k-1}{e}^{-t/\theta }$; parameter $r_1$, $r_2$, $d_1$ and $d_2$ are used to determine the distributions of the intercurrent events. We consider three scenarios for the intercurrent events; (1) events dominated by neither death or recovery, (2) events dominated by death and (3) events dominated by recovery. In scenario (1), parameters $r_1$, $r_2$, $d_1$ and $d_2$ are set to 10, 50, 0.5, and 0.3, respectively; in scenario (2), they are set to 50, 100, 0.5, and 0.3; and in scenario (3), they are set to 10, 50, 1, and 0.6. In these submodels, individuals with lower values for $R_i$ experience recovery sooner and individuals with higher values for $R_i$ experience death sooner.

For each patient $i$, we define the time to first event as $T_i=min(T_i^r,T_i^d)$. When $T_i\le t_7$, an intercurrent event is considered to have occurred and the last observed measurement time is set to $D_i=\lceil T_i\rceil -1$, the type of event ${\delta }_i$ is set to 1 if $T_i=T_i^r$ and 2 otherwise, where $\lceil t\rceil$ denotes the ceiling function of a real number t. If $T_i>t_7$, it indicates that event did not occur during the observation period for subject $i$ so we set $D_i=t_7$ and ${\delta }_i=0$.

Estimands.

We aim to estimate the operating characteristics (frequentist type I error and power) of the Bayesian decision rule in each setting. In the simulation study, we define type I error and power based on the posterior probability of observing a significant treatment effect (i.e., $AD>0.1$). Specifically, the type I error and power are the probability of this posterior probability exceeding the critical value $T_{0.95}$ under $H_0$ ($AD=0.1$) and $H_1$ ($AD=0.12,0.13,0.14$), respectively. The critical value $T_{0.95}$ is referred to in Eq. (1).

Methods.

We compare two joint models against a reference model. All models employ a folded normal mixed-effects regression for the observed magnitude outcomes. Model A, the reference model, does not include an additional submodel for the intercurrent events. Models B and C are joint models where the argument $t$ in $q_0\left(t\right)$, $q_1\left(t\right)$, $v_0\left(t\right)$ and $v_1\left(t\right)$ is treated as a continuous variable in Model B and as a categorical variable in Model C [14]. For both joint models, $g({\mathbf{b}}_i,t)$ corresponds to the true magnitude outcome at time $t$ [14]. For each setting and model, we conduct 1000 simulation runs. All models undergo a burn-in period of 2000 samples to ensure that Markov chains reach a stationary distribution. Following this, four chains are run, each with 2000 samples.

Performance measures.

The type I error and power are estimated as the proportion of simulated runs where the posterior probability of a significant treatment effect surpasses the critical value $T_{0.95}$ under $H_0$ and $H_1$, respectively. The treatment effect is estimated using the equation $AD=c_0-d_0+3(c_1-d_1)$. Additionally, we assess the impact of intercurrent events on the treatment effect estimate by examining the (posterior) mean difference, representing the change in posterior mean estimated by models A-C relative to the complete data scenario (using model A), and the standard deviation of the $AD$ estimate.

3. Results

We initially employed a complete data scenario, i.e., with no intercurrent events, to determine the critical value $T_{0.95}=0.951$. Our simulation results are summarized in tables Table 1, Table 2 for the 3 scenarios of intercurrent events. The proportion of missing primary outcome measurements due to both events (R+D), recovery (R), and death (D) is reported as the “Event Rate” in Table 1, providing detailed information for each scenario of intercurrent events.

Table 1.

The simulation results of operating characteristics (frequentist type I error and power) and event rate for settings with average distance (treatment effect) of 0.10, 0.12, 0.13 and 0.14. There are 3 scenarios, each with model A (reference), B (joint model with continuous time) and C (joint model with categorical time) evaluated for type I error and power, alongside event rates for event recovery (R), death (D), and both (R+D). Each simulated result represents the average value of 1000 simulated samples. The critical value of the rejection region is 0.951 for a level 0.05 hypothesis testing.

		Power				Event rate
Model	Type I error	0.12	0.13	0.14	Event type	0.10	0.12	0.13	0.14
Complete
	0.050	0.541	0.856	0.979		–	–	–	–
(1) Two Sources
A	0.001	0.058	0.273	0.606	R+D	0.6352	0.6665	0.7013	0.7419
B	0.012	0.204	0.464	0.762	R	0.2778	0.3495	0.3916	0.4371
C	0.009	0.205	0.465	0.772	D	0.3574	0.3170	0.3097	0.3048
(2) Death Dominated
A	0.000	0.122	0.407	0.769	R+D	0.4044	0.3789	0.3843	0.3990
B	0.015	0.314	0.648	0.899	R	0.0420	0.0564	0.0694	0.0917
C	0.014	0.318	0.641	0.903	D	0.3624	0.3226	0.3149	0.3073
(3) Recovery Dominated
A	0.019	0.365	0.713	0.926	R+D	0.3433	0.4135	0.4569	0.4973
B	0.024	0.379	0.696	0.902	R	0.2862	0.3562	0.3983	0.4410
C	0.024	0.374	0.694	0.901	D	0.0571	0.0573	0.0586	0.0562

Table 2.

The simulation results of change of posterior mean relative to the complete data scenario and standard deviation for settings with average distance (treatment effect) of 0.10, 0.12, 0.13 and 0.14. There are 3 scenarios, each with model A (reference), B (joint model with continuous time) and C (joint model with categorical time). Each simulated result represents the average value of 1000 simulated samples. The critical value of the rejection region is 0.951 for a level 0.05 hypothesis testing.

	Mean difference					Standard deviation
Scenario	0.10	0.12	0.13	0.14		0.10	0.12	0.13	0.14
Complete
	–	–	–	–		0.0129	0.0125	0.0123	0.0124
(1) Two Sources
A	0.01762	0.01904	0.01887	0.01773		0.0133	0.0131	0.0132	0.0139
B	0.00687	0.00890	0.00974	0.00916		0.0149	0.0146	0.0147	0.0154
C	0.00676	0.00896	0.00980	0.00932		0.0149	0.0146	0.0148	0.0154
(2) Death Dominated
A	0.01471	0.01519	0.01511	0.01469		0.0131	0.0127	0.0127	0.0130
B	0.00509	0.00523	0.00510	0.00480		0.0145	0.0141	0.0141	0.0144
C	0.00497	0.00523	0.00517	0.00489		0.0145	0.0141	0.0141	0.0144
(3) Recovery Dominated
A	0.00521	0.00568	0.00560	0.00529		0.0125	0.0119	0.0124	0.0129
B	0.00310	0.00455	0.00537	0.00558		0.0132	0.0129	0.0130	0.0134
C	0.00307	0.00451	0.00536	0.00563		0.0132	0.0130	0.0130	0.0134

The operating characteristics in each simulation setting depend on the location and shape of the posterior distribution of $AD$ compared to the complete data. This can be explored using the two key measures in Table 2: the mean difference and the standard deviation. In Table 2, the joint models B and C present very comparable results across all simulation settings, indicating similar performance in terms of estimating $AD$. Compared to Model A, the joint models mitigate the bias due to the intercurrent events as they have significantly smaller mean differences. However, they also exhibit slightly larger standard deviations due to the additional submodel required to account for intercurrent events. Across all three scenarios of intercurrent events, those dominated by recovery exhibit the smallest mean differences and the smallest differences in standard deviations. This suggests that, in this setting, recovery events have a lower impact on the results. The other two scenarios result in significantly larger effects, with greater mean differences and standard deviations, especially when the intercurrent events for recovery and death are common.

The type I error and power in Table 1 demonstrate that we typically underestimate the average distance $AD$ due to the bias introduced by intercurrent events. This results in reduced type I errors across all scenarios and models, particularly in Model A, which cannot mitigate this bias. Models B and C demonstrate similar power in all three settings, as they have comparable estimation performance. In the scenario where events are dominated by recovery, all models demonstrate comparable power as the estimation of $AD$ is similar, reflected in the mean differences and standard deviations. Furthermore, the power in this scenario aligns more closely with that of the complete data analysis. In the other two scenarios, Models B and C outperform Model A with an average increase in power of approximately 15%. This improvement is attributed to the joint models’ effective mitigation of bias.

From the simulation results, we can conclude that our proposed joint models significantly improve the trial’s power when substantial bias arises from intercurrent events. This improvement stems from the joint models’ effectiveness in mitigating bias in treatment effect estimation. Conversely, when the bias is insignificant or cannot be effectively addressed by joint models, their power resembles that of the reference model. Additionally, in this simulation study, the simpler joint model (Model B) yields comparable results to the more complex one (Model C).

4. Conclusion

Intercurrent events have recently been defined as a critical element of defining trial estimands and, as such, are an emerging research topic with significant interest and activity in the field [16]. These events occur after treatment initiation and can impact the interpretation, measurement, or existence of clinical endpoints, thereby influencing the estimation of treatment effects. This article focuses on intercurrent events that prevent the measurement of a longitudinal outcome in a clinical trial. In particular, when terminal events related to the outcome truncate follow-up, ignoring the association between the longitudinal outcome process and the terminal-event process can bias inference for treatment effects on the longitudinal outcome. To address this setting, we propose a Bayesian joint modeling approach that links the longitudinal outcome with a competing terminal-event process, enabling valid treatment effect inference while accounting for their association.

We introduced our joint modeling approach, which combines longitudinal clinical outcomes with time-to-event models for intercurrent events. We then explored the impact of these intercurrent events on the trial analysis. In our study, we employ a one-sided decision rule to assess the treatment effect, where efficacy is determined by the posterior probability of a clinically meaningful effect. Extensive simulations evaluated the joint modeling approach by examining the operating characteristics (frequentist type I error and power) of the Bayesian decision rule, using settings that closely resemble an ongoing pilot and feasibility trial. Our simulations demonstrate a substantial increase in power with the joint models in scenarios where bias due to intercurrent events is substantial and can be mitigated by the joint model. When the bias is minimal or the joint model is unable to effectively address it, we observe similar performance between the joint models and the longitudinal outcome model without considering intercurrent events.

When terminal events truncate follow-up for a longitudinal primary endpoint, standard longitudinal analyses (e.g., linear mixed models) relying on a missing-at-random assumption may be biased if early termination is informative [17]. Competing risks regression methods (e.g., Fine–Gray models) primarily address time-to-event estimands such as cumulative incidence and therefore do not directly estimate treatment effects on a longitudinal trajectory when the longitudinal variable is the primary endpoint [18]. Joint models address this challenge by linking the longitudinal process and the event process through shared latent effects, enabling inference that accounts for informative truncation [19], [20].

A key feature of our approach is the discrete-time competing-risks submodel for terminal intercurrent events, which avoids specifying a parametric continuous-time event-time distribution and extends naturally to multiple competing event types, making it well suited to settings with scheduled assessments [3], [5]. A trade-off is that this flexibility can be parameter-rich (indexed by event type and time interval) and may require adequate sample size and sufficient numbers of events in each category for stable estimation. Related discrete-time event formulations have also been used to address nonignorable dropout [21], whereas our focus is on multiple competing terminal intercurrent events within a Bayesian joint modeling strategy for a longitudinal primary endpoint.

One limitation of our study is that our proposed method is only applicable to intercurrent events that can be modeled as time-to-event variables. However, there are other intercurrent events, such as the dose of the drug taken by the patient, that cannot be adequately represented by time-to-event data. Hence, future research should investigate alternative modeling approaches capable of incorporating different types of data to capture a wider range of intercurrent events.

Another limitation of our current study is the absence of a proposed scheme for determining the optimal sample size in the presence of intercurrent events. If the event rate for intercurrent events is expected to be substantial, then estimating the sample size without considering them can significantly underestimate the required sample size. However, appropriately accounting for intercurrent events in the sample size determination procedure is challenging when limited knowledge about their distribution is available. Therefore, future research should prioritize the development of methods for determining the optimal sample size while appropriately accounting for intercurrent events, thereby ensuring sufficient power for the trial.

Overall, this article presents an initial exploration of a Bayesian joint modeling approach to address intercurrent events in the design and analysis of clinical trials. Moving forward, further research is warranted to refine and expand upon our proposed methods, particularly in incorporating various types of intercurrent events and determining optimal sample sizes. By addressing these challenges, we can advance the field of clinical trial methodology and improve the reliability and validity of clinical trial results in the face of intercurrent events.

CRediT authorship contribution statement

Wen Teng: Writing – review & editing, Writing – original draft, Methodology, Formal analysis, Conceptualization. Yongdong Ouyang: Writing – review & editing. Jose Dianti: Resources. Niall D. Ferguson: Resources. Ewan C. Goligher: Resources. Anna Heath: Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Conceptualization.

Funding

This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant RGPIN-2021-03366, and by the Canada Research Chairs program in Statistical Trial Design.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.