Abstract
Background
Dementia often has an insidious onset with considerable individual differences in disease manifestation. Disease progression models incorporating latent time shifts have been proposed to investigate long-term disease progression and individual disease stages. The latent time shift is a horizontal shift in time that aligns patients along a global timeline for disease progression. However, these models ignore informative dropout due to dementia or death, which may lead to bias in estimating longitudinal trajectories.
Methods
To account for informative dropout, we propose a nonlinear joint model with latent time shifts. This joint model uses a multivariate nonlinear disease progression model with latent time shifts to model the correlated longitudinal cognitive measures, and simultaneously, a proportional hazards model to incorporate time to dementia or death. We evaluate and compare different association structures between the longitudinal and dropout processes within the joint models via a simulation study. In addition, we compare the joint models with the disease progression model that ignores informative dropout. We further applied the proposed joint model to a cohort study to analyze longitudinal cognitive measures from neuropsychological tests, together with time to dementia or death.
Results
In the simulation study, the proposed joint model with the correct association structure yields optimal performance in estimating both longitudinal trajectories and latent time shifts. Even the joint model with a misspecified association structure outperforms the disease progression model that neglects informative dropout. In the real data application, we estimated both trajectories and latent time shifts from longitudinal cognitive measures and demonstrated their associations with the hazard of dementia or death, confirming the presence of informative dropout.
Conclusions
Our proposed joint model offers more accurate and robust estimates than the latent time disease progression model that does not consider informative dropout. Our method provides insights into cognitive health in the presence of dementia and death.
Supplementary Information
The online version contains supplementary material available at https://doi.org/10.1186/s12874-026-02942-w.
Keywords: Joint model, Disease progression, Dementia, Cognitive decline
Introduction
Dementia is characterized by a decline in mental ability that impairs memory, thinking, and decision-making abilities, which significantly affects people’s daily activities [1]. Modeling disease progression using cognitive decline preceding dementia onset is challenging since dementia often has an insidious onset with considerable individual differences in disease manifestation [2]. Linear and non-linear mixed models have been proposed to describe the progression of dementia [3–6]. However, traditional timescales used in these models face challenges in accurately depicting the disease progression [7]. Specifically, time to dementia diagnosis can be subjective, confining the analysis to the participants with dementia. Time since inclusion lacks biological meaning and exhibits heterogeneity due to varying clinical stages at enrollment. Chronological age introduces considerable individual heterogeneity as disease manifests at different ages for different people. The true longitudinal trajectory for disease progression may be obscured using these timescales.
To address the limitations of traditional timescales, disease progression models that utilize a latent disease time have been proposed, primarily in the area of neurodegenerative diseases, such as Alzheimer’s Disease, which is the leading cause of dementia [2, 8–10]. The latent disease time is retrieved from the data by estimating a subject-level latent time shift for individual heterogeneity in disease stages. Compared to traditional timescales, the latent disease time serves as a continuous representation of the disease stage by re-ordering the participants, providing a more accurate staging of participants [2]. In addition, these models aim to reconstruct the true long-term disease progression trajectory from short-term observations, while noting that limited follow-up may potentially bias progression estimates [8, 11]. The disease progression models differ in how they parameterize latent time shifts, the shapes of mean disease progression trajectories, and the number of cognitive measures they can model simultaneously. Donohue et al. proposed a semiparametric model and iterative estimation procedure to estimate the long-term disease trends based on short-term data [9]. Li et al. developed a latent time joint mixed model using Bayesian estimation, where the nonlinear disease progression trends were modeled by a nonlinear transformation prior to trajectory modeling [8]. Based on Li et al., Lespinasse et al. further used the partially observed clinical diagnosis information to help anchor the estimation of latent time shifts [7]. Lorenzi et al. proposed a nonparametric Gaussian process model that allows uncertainty to vary along the trajectory [12]. Raket et al. directly modeled a nonlinear longitudinal cognitive trajectory on its original scale by a parametrized family of exponential functions and included covariate effects on disease stage, rate of decline and deviation from the mean [10]. Building on this work, Kühnel et al. simultaneously modeled multivariate longitudinal trajectories [2]. Disease progression models have great potential in analyzing a variety of chronic diseases, providing meaningful insights into their complex and heterogeneous pathophysiology [13]. In addition to multiple neurodegenerative diseases, disease progression models have been expanded to other disease areas, including respiratory, autoimmune diseases, and ophthalmological diseases [13, 14].
In longitudinal studies to track progression, dropout is often caused by the occurrence of dementia or death [15]. This type of dropout process is identified as informative, or missing not at random (MNAR), as the probability of the dropout depends on the unobserved data [16]. However, the existing disease progression models usually assume missing data as missing at random (MAR). To account for informative dropout, the longitudinal and dropout processes should be jointly modeled. Joint models combine a linear mixed model for longitudinal outcomes with a survival model for time-to-event outcomes (e.g., dropout due to dementia or death) [17]. Joint models employing the shared random effect association structure have been used, where dropouts are assumed to be directly related to the random effects from the longitudinal process [18–21]. Another commonly used association structure is the current value structure, where dropouts are linked to the true unobserved values of the longitudinal outcomes [22–28]. Studies have shown that joint models outperform linear mixed models or separate models of longitudinal and missingness (e.g., dropout) processes in terms of bias and fit, offering more accurate longitudinal estimates, whereas linear mixed models often yield overly optimistic estimates [20–22].
In this paper, we aim to describe a multivariate nonlinear joint model with latent time shifts. The joint model is based on two longitudinal cognitive measures in the memory and language domains derived from neuropsychological (NP) tests, along with time to dementia or death from the Framingham Heart Study (FHS) Offspring cohort. Detailed descriptions of this cohort study are provided in Sect. " Real Data Application". To demonstrate the ability of our proposed model to account for informative dropout, we compare the proposed joint models with varying association structures to a disease progression model with latent time shifts that ignores informative dropout. We then present the proposed method, its estimation procedures, and the evaluation of the performance of our proposed method via a simulation study. Lastly, we apply our proposed approach to the FHS Offspring cohort dataset, followed by summarizing our main findings and future directions.
Methods
Our proposed joint model extends the multivariate continuous-time disease progression (MCDP) model proposed by Kühnel et al. (2020) by combining the MCDP model with a proportional hazard model [2]. The MCDP model is a nonlinear mixed model that aligns patients according to their predicted disease progression along a global latent timescale. Compared to other disease progression models, such as the MCDP model demonstrates superior performance by more accurately predicting postbaseline trajectories and individual patient stages via subject-level latent time shifts [2]. However, the MCDP model ignores informative missingness, such as dropout due to death or dementia. To address informative missingness, our proposed joint model is composed of two sub-models: (i) the MCDP model for longitudinal cognitive measures (longitudinal sub-model); (ii) a proportional hazards model for the dropout process, i.e., the time to dementia or death (survival sub-model). We use Fig. 1 to illustrate our proposed nonlinear joint model assuming K longitudinal cognitive measures.
Fig. 1.
The joint model framework. 
is a longitudinal measure, 
represents the true unobserved values for the longitudinal measure. 
is the random latent time shift and 
is a vector of covariates associated with 
. 
represents the random intercept for the longitudinal measure. 
denotes the hazard function for the survival process. 
represents the covariates associated with the hazard 
. Both longitudinal and survival processes are linked by either random effects (the paths from 
and 
to 
) or the true unobserved values of longitudinal measures (the path from 
to 
). The red arrows represent the longitudinal process, and the blue arrows represent the survival process
Longitudinal sub-model: MCDP
Let 
be the value of the k th longitudinal cognitive measure at time 
for subjecti, wherei=1, 2, ..., N,k=1, 2, … , K,
, and 
is the number of longitudinal measurements for subjecti. For participanti, let 
be a vector of length p for baseline covariates and 
be a vector of length q for covariates that have an influence on disease stages. The MCDP model can be formulated as
 |
1 |
where 
denotes the true unobserved value of 
. The random error has the following distribution 
. The function 
represents the mean disease progression in the population. An individual’s deviation from the mean disease progression can be divided into two parts: the horizontal shift, 
, and the vertical shift, 
.
Horizontal shift
The horizontal shift refers to the latent time shift that moves subjects backward or forward in time according to their true disease stages, so subjects are realigned along a global timeline. The latent time shift is composed of a fixed effect, 
, for individual differences in disease stage that can be ascribed to covariates, 
, and a random effect, 
, that represents the unobserved random variation in disease stage with 
. The timescale formed by the shifted time points, 
, also referred to as the ‘disease time’ or ‘disease age’, constitutes a global timescale for disease progression in the population, which uncovers the participants’ true disease stages. A smaller disease age indicates an earlier true disease stage and better health status. The subject-specific latent time shift is assumed to be consistent across all K outcomes for each participant. The horizontal latent time shift is visualized in Fig. 2.
Fig. 2.
Simulated trajectories with latent time shifts. The three panels illustrate individual trajectories for one longitudinal measure. The left panel plots trajectories against the original years since baseline (
, showing all participant data. The middle panel adjusts the x-axis to disease time (
) and realigned all trajectories from the first panel based on this adjusted timeline. Blue and red lines/dots depict participants with 
and 
, respectively. The solid black curve in the middle panel marks the predicted mean trajectory using the adjusted disease time. The right panel shows trajectories for two specific individuals against disease time. For one individual, the blue solid trajectory shifts to a dashed trajectory after adjusting for a fixed time shift (
), then to a dotted trajectory with a further random time shift (
), 
. Similarly, the red solid trajectory for another individual shifts left to a dotted line after accounting for a random time shift (
) only, 
. Vertical shifts are the deviations of each individual trajectories from the predicted mean trajectory, which is not shown in the figures
Vertical shift
The vertical shift represents a deviation from the mean disease progression. It is composed of three parts: a fixed effect, 
which accounts for the deviation that can be attributed to covariates 
; a random intercept, 
, which captures the subject-level unobserved random deviation; and a random error term, 
. The random intercept, 
, resembles the random intercept in the traditional linear mixed model. We assume the vector of the random effects across K longitudinal measures, denoted as 
, follows a multivariate normal distribution: 
, and 
is the standard deviation of the random intercept for the k th longitudinal measure. We further assume independence among the random latent time shift 
, the random intercept 
, and the random error term 
.
Mean disease progression
To model the mean disease progression trajectory for the k th longitudinal outcome, we use a nonlinear form for 
 |
2 |
where 
is an intercept parameter describing the left asymptote; the scaling parameter 
for the exponential function can be interpreted as the mean deviation from 
at time t=0; and 
is a scaling parameter for time. The use of an exponential mean curve is based on the observation that cognitive decline is generally characterized by an escalating rate of deterioration over the course of disease progression, which is consistent with the shape of an exponential curve [2].
Survival sub-model: proportional hazards model
We use a parametric proportional hazards model for time to dropout due to dementia or death. The model is as follows
 |
3 |
where 
is the baseline hazard. It has been shown that an unspecified baseline hazard in the joint model is likely to underestimate the standard errors of the parameter estimates [29]. To handle this issue, we use a parametric Weibull baseline hazard, which introduces a shape parameter allowing the baseline hazard to increase or decrease over time, while maintaining a parsimonious model structure. It can be extended to other parametric forms such as exponential or piecewise constant. The vector of parameters 
is the vector of coefficients for covariates 
. 
is a function of random effects from the longitudinal sub-model (i.e., 
and 
) at time t, which represents the association structure between the longitudinal and survival (e.g., missingness or dropout) processes, and 
denotes the coefficients for 
. Specifically, we consider two commonly used forms for 
: the current value structure and the shared random effect structure. In addition, to facilitate model comparison we include the separate models that ignore the association between the longitudinal and survival processes, where 
=0.
Current value association structure (CV)
This association structure assumes that the true unobserved value for longitudinal measures at time t, denoted as 
, is predictive of the event risk at t [30]. Therefore, we have 
. Equation (3) can be rewritten as
 |
4 |
where 
measures the association between 
and the risk for the event of our interest (e.g., dropout due to dementia or death), indicating that each one-unit increase in the current value of 
at time t is associated with an 
-fold increase in the hazard of the event at the same time t. This association structure is often employed in scenarios where the primary focus is on survival time [29].
Shared random effect association structure (RE)
The shared random effect association structure postulates that the risk for the event is directly associated with the random effects from the longitudinal sub-model, i.e., the random time shift 
and the random intercepts 
. We have 
. Equation (3) can be rewritten as
 |
5 |
where 
and 
measure the strength of the association between the event risk and the random effects. Typically, this structure is often utilized in scenarios where the primary focus is the longitudinal process with informative dropout, or in situations where both the longitudinal and survival processes are of equal interest [29]. In real-world scenarios, the choice of association structure depends on the applications and underlying assumptions. When prior knowledge is limited, both association structures can be compared using model fit criteria to guide selection.
Separate models (independent association structure)
If we set 
in both association structures, the survival sub-model is reduced to a standard proportional hazards model, i.e.,
 |
6 |
Since this model fails to accommodate the association between the longitudinal and survival processes, it is reduced to the separate models of the MCDP model and the proportional hazards model with an independent association structure. That is, the longitudinal and survival processes are modeled independently. The estimates from the longitudinal sub-model are equivalent to those obtained from the standard MCDP model that ignores informative dropout.
Model estimation
Likelihood
Let 
be the true time to dropout due to dementia or death for subject i, and 
be the censoring time, which corresponds to dropout due to other reasons. Let 
denote the observed time and 
be the event indicator, where 
indicates that the event is observed and 
indicates censoring. Let 
denote the full parameter vector. 
and 
are the random latent time shift and the vector of random intercepts. The conditional likelihood is constructed under the commonly used assumption that both the longitudinal and survival processes are independent given the random effects. Therefore, the likelihood conditional on the random effects is as follows:
 |
For the longitudinal part, we have
 |
For the survival part, we have
 |
where 
is the hazard function described in Eqs. (4) to (6). 
, which represents the survival function. Note that we use Gauss–Legendre quadrature with 8 nodes to approximate the integral in the cumulative hazard function when the survival sub-model has a current value association structure. Specifically, 
, where 
is the standardized weight for quadrature node q, with 
, and 
is the location for quadrature node q.
Bayesian inference
We use a Bayesian approach based on Markov Chain Monte Carlo (MCMC) methods for the parameter estimation of the joint model. The joint posterior probability distribution is analogous to:
 |
where 
is the joint prior distribution for the full parameter vector 
.
and 
are the distributions for the random latent time shift and random intercepts, respectively.
We use weakly informative priors 
on all model parameters. For the fixed effects in the longitudinal sub-model, i.e., 
,
, we assign normal priors with large variance, i.e., N(0, 100). Note that for the scaling factor 
, we use a uniform prior with an upper limit of 0, denoted as Uniform(-1, 0), to ensure that all cognitive trajectories are oriented to be decreasing (
), consistent with real-world scenarios. The fixed parameters in the survival sub-models, denoted as 
and 
, are also assigned weakly informative priors N(0, 100). The variance–covariance matrix for the random intercepts (
) is decomposed into a correlation matrix (
) and a diagonal matrix (
), with standard deviations on the diagonal using Cholesky decomposition, i.e., 
, 
We employ the “Lewandowski-Kurowicka-Joe (LKJ) priors” [31] with a shape parameter equal to 1, i.e., the parameterization of the LKJ correlation matrix density in terms of its Cholesky factor, for estimating 
. The shape parameter of 1 corresponds to a uniform prior over the correlation matrix. We use a Half-Cauchy (0, 5) prior for the standard deviation of 
for the latent time shift, which allows substantial variability in the latent time, and Half-Cauchy (0, 0.5) for the standard deviation of random errors 
and random intercepts 
, with most prior values below 1.5.
The model parameters are estimated using the R interface of Stan with the R package “rstan” [32]. Stan utilizes a No-U-Turn Sampler version of Hamiltonian Monte Carlo (HMC) algorithm which optimizes the exploration of the target distribution based on Hamiltonian dynamics [33]. It has been shown that HMC has the capacity to handle complex nonlinear estimation tasks that might be challenging for Gibbs sampling [34]. To assess model convergence, we use the potential scale reduction statistic 
[35]. For all parameters, 
values < 1.1 indicate successful model convergence. Model comparison is evaluated using the Widely Applicable Information Criterion (WAIC) [36]. WAIC is a useful tool for estimating pointwise out-of-sample prediction accuracy within the Bayesian framework. It is calculated based on the log-likelihood evaluated at the posterior simulations of the parameter values [37].
Simulation study
In this section, we performed simulation studies to evaluate and compare the joint models based on the shared random effect and current value structures. We also compared the joint models with the separate models.
Data generation
We generated two correlated longitudinal measures with n = 1000 subjects in each simulated dataset. The baseline time variable 
was generated from a uniform distribution U(-5, 5) to be the baseline age subtracted by 60. Each participant was examined every four years until the 20th year during follow-up. The longitudinal measures were observed until the occurrence of an event or at the 20th year. We employed the MCDP model to generate these two longitudinal measures, 
based on the following equation:
 |
7 |
We considered one binary covariate 
that was generated from Bernoulli (0.5), which had an impact on the latent time shift. We also included a continuous covariate, 
, generated from Uniform (-5, 5). The correlation between two longitudinal measures were captured by both the random latent time shift and the random intercepts. We assume that the random latent time shift followed a normal distribution, i.e., 
, and the random intercepts were generated with 
.
We considered the dropout times, due to dementia or death, under the three association structures described in the Methods section: the shared random effects structure, the current value structure, and the separate models. When the current value structure was implemented, the event times were generated based on the following equation
 |
8 |
In addition to the binary covariate 
and the continuous covariate 
shared with the longitudinal sub-model, we further incorporated the true unobserved values of both longitudinal measures at time t, 
and 
, to capture the association between two processes.
After specifying the proportional hazards model, the time to event data were generated using an inverse cumulative hazard function transformation [38]. Specifically, we first simulated a variable 
for participant i. Next, we obtained the true event time 
by solving for the equation 
, where 
was survival function for the hazard function in Eq. (8). We then defined the time to censoring, 
as the time to the maximum follow-up duration for participant i. Lastly, the observed event time was 
. For participant i, 
and 
that occurred after 
were dropped.
When the shared random effect structure was our primary interest, the event times were generated based on the following equation
 |
9 |
We incorporated the random latent time shift 
and the random intercepts (
and 
from both longitudinal measures to capture the association between two processes. The time to event data were generated following the procedures outlined in the current value structure or directly from a Weibull distribution based on the hazard function as specified in Eq. (9).
In contrast to the two association structures, we also considered the separate models, where the MCDP model was used to model the longitudinal measures and the dropout times were not related to the longitudinal measures, reflecting noninformative missingness. The dropout times were generated from the following model
 |
10 |
where only 
and 
were included as covariates. The event times generation procedure was the same as the shared random effect structure.
Simulation scenarios
Our simulated dataset structure mimicked that from the real data in Sect. " Real Data Application" to allow for reasonable generalization. Three association structures (including the separate models) described in the previous section were considered to reflect various potential real-world relationships between the longitudinal and dropout processes. We generated the following six scenarios and summarized all parameter settings in Table 1. For each scenario, 200 datasets were generated and analyzed.
Scenario 1 (the shared random effect structure in Eq. (9)) indicated that a participant with a later disease stage was more likely to drop out.
Scenario 2 (the current value structure in Eq. (8)) implied that the hazard for dropout was associated with the unobserved current values of both longitudinal cognitive measures.
Scenario 3 (separate model structure in Eq. (10)) indicated that dropout times were only dependent on the observed covariates
and 
, unrelated to the values of longitudinal measures.Scenarios 4, 5, and 6 were based on the shared random effect structure, resembling Scenario 1. In Scenario 4, we evaluated the impact of a smaller sample size on model performance by setting n = 500. In Scenario 5, the latent time shift was assumed to have a shorter span. In Scenario 6, the event rate was increased from 22 to 50% by varying baseline hazard parameters.
Table 1.
Parameter settings for all six scenarios in the simulation study
| Parameters | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 5 | Scenario 6 |
|---|---|---|---|---|---|---|
| Longitudinal | ||||||
| N sample size | 1000 | 1000 | 1000 | 500 | 1000 | 1000 |
 |
−0.06 | −0.06 | −0.06 | −0.06 | −0.06 | −0.06 |
 |
2.15 | 2.15 | 2.15 | 2.15 | 2.15 | 2.15 |
 |
0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |
 |
0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
 |
−0.02 | −0.02 | −0.02 | −0.02 | −0.02 | −0.02 |
 |
2.1 | 2.1 | 2.1 | 2.1 | 2.1 | 2.1 |
 |
0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 |
 |
0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
| β | −2 | −2 | −2 | −2 | −2 | −2 |
 |
8.5 | 8.5 | 8.5 | 8.5 | 4 | 8.5 |
 |
0.45 | 0.45 | 0.45 | 0.45 | 0.45 | 0.45 |
 |
0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |
 |
0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 |
 |
0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |
 |
0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
| Survival | ||||||
| Event rate | 22% | 22% | 22% | 22% | 22% | 50% |
| b | 4.85 | 2.8 | 3.6 | 4.85 | 4.9 | 4.95 |
 |
−16 | −10 | −12 | −16 | −16 | −15 |
 |
−0.5 | −0.2 | −0.5 | −0.5 | −0.5 | −0.5 |
 |
−0.05 | 0.03 | −0.1 | −0.05 | −0.05 | −0.05 |
 |
0.15 | −1.1 | NA | 0.15 | 0.15 | 0.15 |
 |
0.02 | 0.15 | NA | 0.02 | 0.02 | 0.02 |
 |
−0.12 | NA | NA | −0.12 | −0.12 | −0.12 |
Scenario 1, shared random effect association structure; Scenario 2, current value association structure; Scenario 3, separate (independent) model structure; Scenario 4, shared random effect association structure with N = 500; Scenario 5, shared random effect association structure with a smaller span of latent time shift of 4; Scenario 6, shared random effect association structure with a higher event rate of 50%
Analysis models
The following three analysis models were applied to analyze each scenario, where the MCDP model described in Eq. (7) was the longitudinal sub-model and the three survival sub-models in Eqs. (8)- (10) were considered:
MCDP + RE: MCDP + the shared random effect association structure in Eq. (9).
MCDP + CV: MCDP + the current value association structure in Eq. (8).
Separate models: MCDP + the independent association structure in Eq. (10).
We used weakly informative priors described in the Methods section and obtained 6000 MCMC sample iterations from each of the 3 parallel chains with the first 4000 iterations as a burn-in phase. Convergence was evaluated through the potential scale reduction statistic 
. For a simulated dataset of 1000 participants, fitting each proposed joint model required approximately 3–5 h on average.
Simulation results
The objectives of our simulation study were twofold. First, to demonstrate that our proposed joint models effectively accounted for informative dropout by recovering the true mean disease progression trajectory. Second, to illustrate that the estimated latent time shifts from our joint models were more accurate than those derived from the MCDP model, which ignored informative dropout by using separate models.
Scenario 1: shared random effect data structure
In one of the simulated datasets (Fig. 3), no clear trajectory patterns for the longitudinal measures were observed using the original timescale, since the true mean trajectories were obscured. After implementing the MCDP model, the participants were realigned according to a global disease timeline, allowing the true disease progression trajectories to be reflected in the realigned data.
Fig. 3.
A simulated example of trajectories plotted on original and disease time scales. The simulated dataset is generated based on MCDP + RE in Scenario 1. The eight panels illustrate the simulated trajectories for two longitudinal cognitive measures. The y-axis in each panel denotes the values of the cognitive measures, and the x-axis represents the simulated time. The two panels in the first column denotes the trajectories aligned on the original timescale. The other panels represent the trajectories that are realigned on the disease time by fitting different models with latent time shifts. MCDP + RE, the joint model with the shared random effect association structure; MCDP + CV, the joint model with the current value association structure
Simulation results confirmed that the joint model with the correct association structure, i.e., MCDP + RE, gave rise to the best performance with the lowest bias (< 5% for most parameters), coverage probabilities around the nominal rate of 95%, and the smallest WAIC, consistent with our simulating model (Table 2). Both joint models that addressed informative dropout (MCDP + RE and MCDP + CV) outperformed the separate models that treated dropout as noninformative in the survival process in terms of bias, coverage and WAIC. The association structures in MCDP + RE and MCDP + CV had different model parametrizations and interpretations, so they were not comparable. However, MCDP + CV indicated that a higher dropout risk was linked to lower current values of the first longitudinal measure (
, while the effect for the second longitudinal measures was small and not significant. This suggested that participants with these characteristics were likely at a later disease stage, which was consistent with the assumptions for MCDP + RE. Therefore, MCDP + CV allowed for retrieving the directionality of the association between two processes despite the misspecified association structure.
Table 2.
Simulation results for Scenario 1 (shared random effect association structure)
True parameter true value, Mean posterior mean, %Bias percent bias, CP coverage probability, MCDP + RE the joint model with the shared random effect association structure, MCDP + CV the joint model with the current value association structure, Separate the separate models of the MCDP model and the proportional hazards model, Reported WAIC values represent the average WAIC computed across all simulated datasets. The percent bias and coverage probability columns for and in MCDP + CV are reported as ‘NA’ because the true generating model is MCDP + RE, and therefore no directly corresponding true values exist for and in MCDP + CV
We compared the mean longitudinal trajectories for two measures estimated by three models (Fig. 4). The joint model with the correct association structure, which was consistent with model that simulated the data, i.e., MCDP + RE, produced trajectories that aligned with the true mean trajectories, indicating superior model fit. The estimated trajectories in the separate models displayed the flattest curve, resulting in the greatest deviation from the true trajectories. The underestimated downward trend from the separate models provided a misleadingly optimistic view of disease progression, as it failed to account for informative dropout associated with worsening cognitive conditions. MCDP + CV, which addressed informative dropout through a misspecified association structure, produced intermediate estimates (Fig. 4).
Fig. 4.
Mean disease progression trajectories estimated by joint models and separate models in Scenarios 1, 2, and 3. The panels in the left, middle and right columns represent the mean disease trajectories in Scenario 1, 2, and 3, respectively. These mean trajectories were generated using the estimated parameters averaged from the model results of all simulated datasets. The y-axis in each panel represents the mean values of cognitive measure, and the x-axis denote the disease time estimated from the models. Four curves in each panel show trajectories estimated by different models: MCDP + RE (blue), MCDP + CV (orange), separate models (green), and true trajectories (red). Note that if a curve overlaps with the true trajectory, only the red curve representing the true trajectory is displayed. Scenario 1, shared random effect association structure; Scenario 2, current value association structure; Scenario 3, separate (independent) model association structure; MCDP + RE, the joint model with the shared random effect association structure; MCDP + CV, the joint model with the current value association structure; Separate, the separate models of the MCDP model and the proportional hazards model
To examine whether our proposed joint models produced more accurate disease stages through latent time shifts, we compared three models using three evaluation metrics for the estimated latent time shifts: root-mean-square errors (RMSE), median absolute deviations (MAD), and Spearman correlations between estimated and true latent time shifts across all simulated datasets (Fig. 5). MCDP + RE yielded the most accurate estimation of latent time shifts with the smallest RMSE and MAD, as well as the highest correlations with the true values, while MCDP + CV produced intermediate results (average RMSE: 4.88 for MCDP + RE versus 5.20 for MCDP + CV and 5.37 for separate models; average MAD: 2.92 for MCDP + RE versus 3.09 for MCDP + CV and 3.27 for separate models; average correlation: 0.82 for MCDP + RE versus 0.79 for MCDP + CV and 0.78 for separate models).
Fig. 5.
Evaluation metrics of latent time shifts for three models in Scenarios 1, 2, and 3. Each of the 9 panels in this figure displays a density plot for one of the evaluation metrics across all simulations. The columns in the figure correspond to different scenarios: the left column shows density plots for Scenario 1, the middle column for Scenario 2, and the right column for Scenario 3. In each scenario, the top, middle, and bottom panels display the distributions for RMSE, MAD, and correlation, respectively. Scenario 1, shared random effect association structure; Scenario 2, current value association structure; Scenario 3, separate (independent) association structure; MCDP + RE, the joint model with the shared random effect association structure; MCDP + CV, the joint model with the current value association structure; Separate, the separate models of the MCDP model and the proportional hazards model; RMSE, root mean square error; MAD, median absolute deviations
Scenario 2: current value data structure
When dropout was related to the true unobserved current values of the longitudinal measures, the correct model was MCDP + CV, which achieved the most optimal performance (Additional File 1: Table S1). Both MCDP + CV and MCDP + RE outperformed the separate models. Despite the misspecified association structure, the direction of the association between two processes was captured by MCDP + RE. The estimated mean cognitive trajectories from MCDP + CV overlapped with the true trajectories (Fig. 4). In contrast, MCDP + RE produced slightly steeper curves, indicating an overcorrection for informative dropout. The separate models generated trajectories with a slower acceleration in their rate of decline compared to the joint models, providing overoptimistic estimates (Fig. 4). In the evaluation of latent time shifts, most parts of the distributions for the three models overlapped, suggesting minor differences among the models (Fig. 5). Nevertheless, the latent time shifts from MCDP + CV were the most accurate.
Scenario 3: separate model structure
When dropouts were not informative, all three models produced unbiased estimates (bias < 5%) and comparable WAIC (Additional File 1: Table S2). Notably, the estimated association parameters in MCDP + RE and MCDP + CV were close to 0. The mean trajectories estimated by all three models closely matched the true trajectories, as shown in Fig. 4. Furthermore, the evaluation metrics for the latent time shift were nearly identical across the three models (Fig. 5). These findings suggested that all models performed well, irrespective of assumptions about informative missingness.
Other scenarios
Scenarios 4, 5, and 6 adopted the shared random effect structure as used in Scenario 1 but differed by having a smaller sample size (Scenario 4: n = 500 versus 1000), a smaller standard deviation of the latent time shift (
versus 8.5), and a higher event rate (Scenario 6: 50% versus 22%). We observed that reducing sample size, limiting the span of latent time shift, or increasing event rate caused more biased estimates in the longitudinal sub-model for all three models compared with Scenario 1 (Additional File 1: Tables S3-S5), especially for the effect of the random effect of the first longitudinal measure due to its very small true effect size (close to the null) as well as potential multicollinearity. Nevertheless, the conclusions regarding model comparison from Scenario 1 also applied to Scenarios 4, 5, and 6 (Additional File 1: Figures S1- S2). Notably, lowering the span of latent time shift minimized the difference between the joint model with misspecified structure (MCDP + CV) and the separate models. In addition, a larger proportion of informative dropout more clearly highlighted the superior performance of the correctly specified model (MCDP + RE) compared to the other two models.
Real data application
In this section, we analyzed longitudinal data from NP tests, along with dementia and death data, from the FHS Offspring cohort to demonstrate our models.
Data description
The FHS is a multigenerational cohort study initiated in 1948 by enrolling 5209 residents from Framingham, Massachusetts, in the Original cohort [39]. In 1971, 5214 participants who were the offspring of the Original cohort and the spouses of these offspring were included in the Offspring cohort [40]. Beginning in 1999, the surviving participants in the Offspring cohort were invited to join an ancillary study, where they were administered a battery of NP tests approximately every four years [41]. Participants identified as having potential cognitive impairment were invited to undergo additional, annual neurologic and NP tests. The NP tests are used to measure participants’ cognitive changes over time in four domains including memory and language [42]. A complete list of tests included in the memory and language domains were summarized in Additional File 1: Table S6. For each participant at a visit, we computed a memory z-score and a language z-score by averaging the z-scores from all individual NP tests within the memory domain and the language domain, respectively. If any NP test within a given domain was missing at a visit, the corresponding domain z-score for that visit was set to missing, and the visit was not considered to have valid cognitive data for that domain. A dementia review panel with at least one neurologist and one neuropsychologist determined whether the participants had dementia, as well as the dementia type and the date of onset [41, 43]. We excluded the participants who had dementia onset before the age of 60 at baseline, as well as those who had less than three visits with complete NP data in both memory and language domains. According to Raket et al. (2020), disease progression models predict individual disease stage from temporal patterns of repeated cognitive measurements. Disease stage cannot be reliably inferred from a single assessment, and very sparse follow-up provides limited information for progression modeling [10]. In addition, a review of longitudinal dementia studies reported that the number of completed measurement waves typically exceeds two [6]. Therefore, we required at least three visits to ensure sufficient within-subject temporal information for stable trajectory estimation.
Our final sample included 1117 participants with 4162 observations. Of the 1117 participants, 86 were diagnosed with dementia and 231 died during follow-up. At baseline, the participants included in our final sample were around 60 years old and slightly more women (56.7%) with an average of 14.5 years of education. The participants were followed for a median of 14 years, with a median of three study visits. The mean Z-scores for the memory and language domains were 0.206 and 0.161, respectively (Additional File 1: Tables S7).
Model specification
In the longitudinal part of the joint model, we used the z-scores in the memory and language domains as two longitudinal measures. The MCDP model was shown as follows:
 |
11 |
where 
denoted the age centered at 60 years for participant i at visit j. 
represented memory z-scores for k=1, and language z-scores for k=2. Covariates included sex (
) and education years (
). β denoted the fixed effect of sex, which had an impact on the estimation of the latent time shift, while 
was the random part in the latent time shift with 
. Education had an influence on the mean cognitive levels with coefficients 
. The random intercepts for the memory and language domains, 
and 
, followed a bivariate normal distribution: 
.
The following survival sub-models with three association structures were considered:
MCDP + RE
 |
12 |
MCDP + CV
 |
13 |
Separate models
 |
14 |
Time in the survival sub-model was the age at the occurrence of dropout subtracted by 60 years to be consistent with the timescale in the longitudinal part. We considered three scenarios of informative dropout: (i) dropout due to dementia only (event rate: 7.7%); (ii) dropout due to death only (event rate: 14.8%); (iii) combined dropout due to either dementia or death (event rate: 22.5%). Both sex and education were included as covariates. All other settings were the same as in our simulation study.
In the sensitivity analysis, we compared WAIC values from the proposed joint model using the nonlinear exponential function with the shared random effects association structure (MCDP + RE) with those from two alternative joint models: (1) a quadratic form of cognitive decline with the shared random effects association structure (MCDP + RE Quadratic) (2) a linear form of cognitive decline with the shared random effects association structure (MCDP + RE Linear). Specifically, the longitudinal sub-model for MCDP + RE Quadratic is
 |
and the longitudinal sub-model for MCDP + RE Linear is
 |
where 
for two longitudinal cognitive measures. All random effects and error terms were defined in the same manner as in the previous sections. The survival sub-models for these two joint models were the same as the survival sub-model in MCDP + RE.
Results
In all three dropout scenarios, the WAIC values favored MCDP + RE, followed by MCDP + CV, indicating the real data likely exhibited a shared random effect data structure between the longitudinal and survival processes (Table 3, Additional File 1: Tables S8-S9). For brevity, the following parameter interpretations were mainly based on MCDP + RE when both dementia and death were combined as a composite time-to-event outcome in scenario (iii) (Table 3). Under each scenario, fitting each proposed joint model required approximately 3–5 h on average.
Table 3.
Real data results using a composite survival outcome of dementia and death
Mean Posterior mean, Lower Lower bound of 95% credible interval, Upper Upper bound of 95% credible interval, MCDP + RE the joint model with the shared random effect association structure, MCDP + CV the joint model with the current value association structure, Separate the separate models of the MCDP model and the proportional hazards model
Trajectories for cognitive measures of memory and language domains were re-aligned along a new global disease timeline after applying our proposed models with latent time shifts (Additional File 1: Figure S3). For the memory domain, the realigned mean disease progression trajectories showed a more pronounced downward trend compared with those on the original timescale. The change for the language domain was less distinct, but the slight upward trend observed in the original data was mitigated after the application of latent time shifts.
Longer education duration was associated with higher cognitive measures in both the memory and language domains (
, 95% CI: 0.061 to 0.084; 
, 95% CI: 0.121 to 0.144). On average, women had a smaller disease age compared to men given the same actual chronological age (β =-2.07, 95% CI: −3.76 to −0.40). The standard deviation for the random latent time shift was estimated to be 8.47 years, implying participants with similar actual chronological ages may exhibited considerable variation in disease stages. The random intercepts showed a moderate positive relationship between the baseline values of the two cognitive domains (
=0.57). In the survival sub-model, women displayed a smaller hazard of developing dementia or death than men after adjusting for other factors (HR = 0.59, 95% CI: 0.42 to 0.82). In addition, the significant positive association between the random latent time shift and the hazard of dementia or death confirmed that dropouts due to dementia or death represented informative missingness. Specifically, for two participants with the same actual chronological age, sex, and education level, the one with a greater disease age faced a higher hazard of informative dropout (HR = 1.17, 95% CI: 1.13 to 1.21). When the data was modeled by MCDP + CV, the significant negative association between the current value of cognitive measures for the memory domain and the hazard of dropout also indicated informative missingness in the longitudinal measures (HR = 0.33, 95% CI: 0.24 to 0.45). Most of the findings from the composite event of dementia and death also applied to the other two scenarios (Additional File 1: Tables S8-S9).
In the sensitivity analysis, WAIC values indicated that our proposed joint model with a nonlinear exponential form (MCDP + RE; WAIC = 5441) had superior model fit compared with the joint models assuming linear (MCDP + RE Linear; WAIC = 6748) and quadratic forms (MCDP + RE Quadratic; WAIC = 5948) (Additional File 1: Table S10).
Discussion
In this paper, we developed a joint model that integrated a multivariate nonlinear disease progression model with latent time shifts. To evaluate the model’s capacity to handle informative dropout, we compared joint models with two association structures to separate models that treated the longitudinal and dropout processes independently, in which the estimates from the longitudinal sub-model were equivalent to those from a standard MCDP model. Our simulation showed that all three models incorporating latent time shifts can realign individual trajectories along a new global disease timeline. In the presence of informative dropout, the joint models with correctly specified association structures outperformed the those with misspecifications, providing the most precise estimates of mean disease progression trajectories and latent time shifts. Even the joint model with a misspecified association structure captured the relationship between two processes and yielded intermediate results, offering advantages over the separate models. Compared to the joint models, the separate models tended to provide over-optimistic estimates with respect to mean disease progression. When dropout was unrelated to the longitudinal data, all joint and separate models performed similarly, showing unbiased results. The application in the FHS Offspring cohort suggested that the relationship between longitudinal cognitive measures and time to dementia or death may resemble a shared random effect association structure. The latent time shift and the unobserved true value of longitudinal measures were significantly associated with the hazard for dementia or death, confirming that dropouts due to dementia or death were informative. Furthermore, we showed sex influenced the latent time shift and the risk for dementia or death, while education was associated with average cognitive levels.
Our findings were consistent with prior studies. First, we showed that joint models outperformed separate models in the presence of informative dropout, aligned with most of the previous studies on joint models, both linear and nonlinear [18–22, 24, 26]. Second, in our study, the separate models that ignored informative dropout showed over-optimistic estimates for the disease progression trend, consistent with conclusions drawn in previous studies [20–22]. It was shown that subjects who dropped out due to excessively high or low longitudinal measures were downweighed during the estimation procedure, so they contributed weakly to the likelihood function in the mixed models. For example, mixed models might estimate a slower rate of cognitive decline, because participants with lower cognitive scores, who are more likely to drop out, had a reduced influence on the model's estimation process due to informative missingness. Therefore, failing to simultaneously consider the dropout process can lead to bias. Third, our application of joint models in the dementia study using the FHS data revealed that the hazard of dementia or death was associated with the true unobserved values of cognitive measures through the current value structure [23, 24].
Our study had some limitations. First, the MCDP model assumed an exponential curve for longitudinal cognitive measures, which represented a monotonic decreasing trend over time. However, this assumption may not apply to short-term observations, as participants can show cognitive improvements within months [44]. The assumption of a monotonic exponential curve fit our data, which were collected every four to five years, but may not generalize well to short-term scenarios. Second, we considered time to dementia or death as a composite survival outcome instead of modeling them as competing or semi-competing risks. Previous studies showed the competing risks joint model that distinguished between two causes of dropout may outperform the standard joint model that treated all the dropouts due to different causes equally in general [20, 24, 26]. However, the primary focus of the current work is on accounting for informative dropout in the longitudinal process rather than on investigating the mechanisms underlying different survival outcomes. In addition, we examined semi-competing risks in greater detail in another related work [45], and we studied how different longitudinal cognitive measures are associated with the hazards for dementia, death, and death after dementia.
Our study had some strengths. First, our proposed joint model was an extension of a previously proposed multivariate nonlinear mixed model with latent time shifts for disease progression [2]. To the best of our knowledge, this was the first integration of such a disease progression model into the joint model framework. Second, we compared joint models with different association structures, as well as with separate models of the longitudinal and dropout processes, across various data structure scenarios. Few prior studies compared the performance of multivariate nonlinear joint models using both shared random effect and current value association structures. Third, previous disease progression models with latent time shifts were primarily applied to specific disease cohorts, like the Alzheimer's Disease Neuroimaging Initiative [2, 8, 46]. In contrast, we evaluated our model using a community cohort, offering new insights into dementia progression, including Alzheimer's disease, and highlighting the generalizability of our results to the general population.
Our study can be extended in the following ways. First, our proposed joint model can be further adapted to account for different shapes of dementia progression curves, such as sigmoid or quadratic functions [9]. Other models have postulated a changepoint during disease progression as an alternative to describe the accelerated rate of cognitive decline at a later disease stage [47–49]. The exponential function used in our joint model is mainly guided by the findings from previous studies on cognitive measures [2, 10]. In practice, when trajectory shapes are unknown in priori, model comparison tools, such as WAIC, can be used to guide the choice of trajectory shape in more exploratory settings. For example, WAIC values can be compared across models assuming linear, quadratic, and exponential trajectories. Second, our model can be extended to distinguish dropouts due to dementia, death, and other causes by treating them as competing risks or semi-competing risks [20, 24, 26]. Third, in aging research, dementia onset was usually treated as interval-censored [15, 50]. Our joint model can be extended to account for interval censoring, which may help reduce bias in estimating both longitudinal and dropout processes.
In conclusion, we incorporated a multivariate nonlinear mixed-effects model for disease progression with latent time shifts into a joint model framework. To handle informative missingness in the longitudinal data, we evaluated and compared different association structures within the joint models. In addition, we compared the joint models with the separate models that ignored informative dropout. We demonstrated the improvement of our proposed joint model over the disease progression model with latent time shifts across various scenarios. Future extensions could explore alternative parameterizations of the longitudinal sub-model and incorporate competing risks with interval censoring into the survival sub-model.
Supplementary Information
Acknowledgements
We acknowledge the FHS participants for their contribution to the data.
Abbreviations
- MNAR
Missing not at random
- MAR
Missing at random
- MCDP
Multivariate continuous-time disease progression
- CV
Current value
- RE
Shared random effect
- MCMC
Markov Chain Monte Carlo
- HMC
Hamiltonian Monte Carlo
- WAIC
Widely Applicable Information Criterion
- RMSE
Root-mean-square error
- MAD
Median absolute deviations
- NP
Neuropsychological
- FHS
Framingham Heart Study
- HR
Hazard ratio
Authors’ contributions
XW and CL conceptualized the study. XW performed the simulation study and statistical analysis under the supervision of CL. CL, YT, and ML provided useful suggestions and feedback throughout the process of the simulation study and statistical analysis. XW drafted the paper with input from CL, YT, and ML. CL substantively revised it. All authors read and approved the final manuscript.
Funding
Data collection in the Framingham Heart Study was supported by N01-HC-25195 and HHSN268201500001 from the National Heart, Lung, and Blood Institute (NHLBI).
Data availability
The real data that support the findings of this study can be requested at https://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/document.cgi?study_id=phs000007.v33.p14&phv=523294&phd=4398&pha=4313&pht=12882&phvf=&phdf=&phaf=&phtf=&dssp=1&consent=&temp=1. Simulated data and code during the current study are available from the following GitHub page https://github.com/xzwang19/JM-LTS.
Declarations
Ethics approval and consent to participate
This study was conducted in accordance with the principles outlined in the Declaration of Helsinki. The protocol of the FHS was approved by the Institutional Review Board at the Boston University Medical Center. All study participants provided informed consent for participation in the study.
Consent for publication
All study participants provided informed consent for the publication.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.Cipriani G, Danti S, Picchi L, Nuti A, Fiorino MD. Daily functioning and dementia. Dement Neuropsychol. 2020;14(2):93–102. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 2.Kuhnel L, Berger AK, Markussen B, Raket LL. Simultaneous modeling of Alzheimer’s disease progression via multiple cognitive scales. Stat Med. 2021;40(14):3251–66. [DOI] [PubMed] [Google Scholar]
- 3.Korten AE, Henderson AS, Christensen H, Jorm AF, Rodgers B, Jacomb P, et al. A prospective study of cognitive function in the elderly. Psychol Med. 1997;27(4):919–30. [DOI] [PubMed] [Google Scholar]
- 4.Backman L, Jones S, Small BJ, Aguero-Torres H, Fratiglioni L. Rate of cognitive decline in preclinical Alzheimer’s disease: the role of comorbidity. J Gerontol B Psychol Sci Soc Sci. 2003;58(4):P228-236. [DOI] [PubMed] [Google Scholar]
- 5.Dik MG, Pluijm SM, Jonker C, Deeg DJ, Lomecky MZ, Lips P. Insulin-like growth factor I (IGF-I) and cognitive decline in older persons. Neurobiol Aging. 2003;24(4):573–81. [DOI] [PubMed] [Google Scholar]
- 6.Karr JE, Graham RB, Hofer SM, Muniz-Terrera G. When does cognitive decline begin? A systematic review of change point studies on accelerated decline in cognitive and neurological outcomes preceding mild cognitive impairment, dementia, and death. Psychol Aging. 2018;33(2):195–218. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 7.Lespinasse J, Dufouil C, Proust-Lima C. Disease progression model anchored around clinical diagnosis in longitudinal cohorts: example of Alzheimer’s disease and related dementia. BMC Med Res Methodol. 2023;23(1):199. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 8.Li D, Iddi S, Thompson WK, Donohue MC. Alzheimer’s Disease Neuroimaging I: Bayesian latent time joint mixed effect models for multicohort longitudinal data. Stat Methods Med Res. 2019;28(3):835–45. [DOI] [PubMed] [Google Scholar]
- 9.Donohue MC, Jacqmin-Gadda H, Le Goff M, Thomas RG, Raman R, Gamst AC, et al. Estimating long-term multivariate progression from short-term data. Alzheimers Dement. 2014;10(5 Suppl):S400-410. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 10.Raket LL. Statistical disease progression modeling in Alzheimer disease. Front Big Data. 2020;3:24. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 11.Saulnier T, Fabbri M, Pavy-Le Traon A, Le Goff M, Helmer C, Peran P, et al. Disease progression in multiple system atrophy: the value of clinical cohorts with long follow-up. Mov Disord. 2023;38(8):1567–9. [DOI] [PubMed] [Google Scholar]
- 12.Lorenzi M, Filippone M, Frisoni GB, Alexander DC, Ourselin S. Alzheimer’s Disease Neuroimaging I: Probabilistic disease progression modeling to characterize diagnostic uncertainty: Application to staging and prediction in Alzheimer’s disease. Neuroimage. 2019;190:56–68. [DOI] [PubMed] [Google Scholar]
- 13.Yoshioka H, Jin R, Hisaka A, Suzuki H. Disease progression modeling with temporal realignment: an emerging approach to deepen knowledge on chronic diseases. Pharmacol Ther. 2024;259:108655. [DOI] [PubMed] [Google Scholar]
- 14.Young AL, Bragman FJS, Rangelov B, Han MK, Galban CJ, Lynch DA, et al. Disease Progression Modeling in Chronic Obstructive Pulmonary Disease. Am J Respir Crit Care Med. 2020;201(3):294–302. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 15.Rouanet A, Joly P, Dartigues JF, Proust-Lima C, Jacqmin-Gadda H. Joint latent class model for longitudinal data and interval-censored semi-competing events: application to dementia. Biometrics. 2016;72(4):1123–35. [DOI] [PubMed] [Google Scholar]
- 16.Ibrahim JG, Molenberghs G. Missing data methods in longitudinal studies: a review. Test (Madr). 2009;18(1):1–43. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 17.Rizopoulos D. Joint models for longitudinal and time-to-event data with applications in R. 1st edn. CRC Press; 2012. [Google Scholar]
- 18.Ghosh P, Vaida F. Random changepoint modelling of HIV immunologic responses. Stat Med. 2007;26(9):2074–87. [DOI] [PubMed] [Google Scholar]
- 19.Elashoff RM, Li G, Li N. A joint model for longitudinal measurements and survival data in the presence of multiple failure types. Biometrics. 2008;64(3):762–71. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 20.Gueorguieva R, Rosenheck R, Lin H. Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial. J R Stat Soc A Stat Soc. 2012;175(2):417–33. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 21.Pantazis N, Touloumi G, Walker AS, Babiker AG. Bivariate modelling of longitudinal measurements of two human immunodeficiency type 1 disease progression markers in the presence of informative drop-outs. J Roy Stat Soc: Ser C (Appl Stat). 2005;54:405–23. [Google Scholar]
- 22.Touraine C, Cuer B, Conroy T, Juzyna B, Gourgou S, Mollevi C. When a joint model should be preferred over a linear mixed model for analysis of longitudinal health-related quality of life data in cancer clinical trials. BMC Med Res Methodol. 2023;23(1):36. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 23.Davis-Plourde KL, Mayeda ER, Lodi S, Filshtein T, Beiser A, Gross AL, et al. Joint models for estimating determinants of cognitive decline in the presence of survival bias. Epidemiology. 2022;33(3):362–71. [DOI] [PubMed] [Google Scholar]
- 24.van Oudenhoven FM, Swinkels SHN, Soininen H, Kivipelto M, Hartmann T, Rizopoulos D, et al. A competing risk joint model for dealing with different types of missing data in an intervention trial in prodromal Alzheimer’s disease. Alzheimers Res Ther. 2021;13(1):63. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 25.Gomeni R, Lavergne A, Merlo-Pich E. Modelling placebo response in depression trials using a longitudinal model with informative dropout. Eur J Pharm Sci. 2009;36(1):4–10. [DOI] [PubMed] [Google Scholar]
- 26.Cuer B, Conroy T, Juzyna B, Gourgou S, Mollevi C, Touraine C. Joint modelling with competing risks of dropout for longitudinal analysis of health-related quality of life in cancer clinical trials. Qual Life Res. 2022;31(5):1359–70. [DOI] [PubMed] [Google Scholar]
- 27.Cuer B, Mollevi C, Anota A, Charton E, Juzyna B, Conroy T, et al. Handling informative dropout in longitudinal analysis of health-related quality of life: application of three approaches to data from the esophageal cancer clinical trial PRODIGE 5/ACCORD 17. BMC Med Res Methodol. 2020;20(1):223. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 28.Vonesh EF, Greene T, Schluchter MD. Shared parameter models for the joint analysis of longitudinal data and event times. Stat Med. 2006;25(1):143–63. [DOI] [PubMed] [Google Scholar]
- 29.Furgal AKC, Sen A, Taylor JMG. Review and comparison of computational approaches for joint longitudinal and time-to-event models. Int Stat Rev. 2019;87(2):393–418. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 30.Hickey GL, Philipson P, Jorgensen A, Kolamunnage-Dona R. Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues. BMC Med Res Methodol. 2016;16(1):117. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 31.Lewandowski D, Kurowicka D, Joe H. Generating random correlation matrices based on vines and extended onion method. J Multivar Anal. 2009;100(9):1989–2001. [Google Scholar]
- 32.Stan Development Team. RStan: the R Interface to Stan. R package version 2.32.7 https://mc-stan.org/
- 33.Hoffman MD, Gelman A. The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J Mach Learn Res. 2014;15:1593–623. [Google Scholar]
- 34.Kerioui M, Mercier F, Bertrand J, Tardivon C, Bruno R, Guedj J, et al. Bayesian inference using Hamiltonian Monte-Carlo algorithm for nonlinear joint modeling in the context of cancer immunotherapy. Stat Med. 2020;39(30):4853–68. [DOI] [PubMed] [Google Scholar]
- 35.Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian data analysis. 3 edn. Chapman and Hall/CRC; 2013. [Google Scholar]
- 36.Watanabe S. Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research. 2010;11. https://www.jmlr.org/papers/v11/watanabe10a.html.
- 37.Vehtari A, Gabry J. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Stat Comput. 2017;27:1413–32. [Google Scholar]
- 38.Bender R, Augustin T, Blettner M. Generating survival times to simulate Cox proportional hazards models. Stat Med. 2005;24(11):1713–23. [DOI] [PubMed] [Google Scholar]
- 39.Dawber TR, Meadors GF, Moore FE Jr. Epidemiological approaches to heart disease: the Framingham study. Am J Public Health Nations Health. 1951;41(3):279–81. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 40.Feinleib M, Kannel WB, Garrison RJ, McNamara PM, Castelli WP. The Framingham Offspring Study. Design and preliminary data. Prev Med. 1975;4(4):518–25. [DOI] [PubMed] [Google Scholar]
- 41.Satizabal CL, Beiser AS, Chouraki V, Chene G, Dufouil C, Seshadri S. Incidence of dementia over three decades in the Framingham Heart Study. N Engl J Med. 2016;374(6):523–32. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 42.Yang J, Ang TFA, Lu S, Liu X, Devine S, Au R, et al. Establishing cognitive baseline in three generations: Framingham Heart Study. Alzheimers Dement Diagn Assess Dis Monit. 2023;15(1):e12416. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 43.Binder N, Balmford J, Schumacher M. A multi-state model based reanalysis of the Framingham Heart Study: is dementia incidence really declining? Eur J Epidemiol. 2019;34(11):1075–83. [DOI] [PubMed] [Google Scholar]
- 44.Qin Y, Han H, Li Y, Cui J, Jia H, Ge X, et al. Estimating bidirectional transitions and identifying predictors of mild cognitive impairment. Neurology. 2023;100(3):e297–307. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 45.Wang X, Tripodis Y, LaValley M, Liu C: Joint Model with Random Changepoints for Longitudinal Measures and Semi-competing Risks. medRxiv. 2024. X, Larson MG, Tripodis Y, LaValley MP, Liu C: Joint Model with Random Changepoint for Longitudinal Measures and Semi-competing Risks. In. medRxiv; 2024. 10.1101/2024.06.26.24309540Wang. [DOI]
- 46.Li D, Iddi S, Thompson WK, Rafii MS, Aisen PS, Donohue MC. Bayesian latent time joint mixed-effects model of progression in the Alzheimer’s Disease Neuroimaging Initiative. Alzheimers Dement Diagn Assess Dis Monit. 2018;10:657–68. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 47.Yu B, Ghosh P. Joint modeling for cognitive trajectory and risk of dementia in the presence of death. Biometrics. 2010;66(1):294–300. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 48.Terrera GM, Piccinin AM, Johansson B, Matthews F, Hofer SM. Joint modeling of longitudinal change and survival: an investigation of the association between change in memory scores and death. J Gerontopsychol Geriatr Psychiatry. 2011;24(4):177–85. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 49.Jacqmin-Gadda H, Commenges D, Dartigues JF. Random change point model for joint modeling of cognitive decline and dementia. Biometrics. 2006;62(1):254–60. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 50.Leffondre K, Touraine C, Helmer C, Joly P. Interval-censored time-to-event and competing risk with death: is the illness-death model more accurate than the Cox model? Int J Epidemiol. 2013;42(4):1177–86. [DOI] [PubMed] [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The real data that support the findings of this study can be requested at https://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/document.cgi?study_id=phs000007.v33.p14&phv=523294&phd=4398&pha=4313&pht=12882&phvf=&phdf=&phaf=&phtf=&dssp=1&consent=&temp=1. Simulated data and code during the current study are available from the following GitHub page https://github.com/xzwang19/JM-LTS.