Bayesian two-stage modeling of longitudinal and time-to-event data with an integrated fractional Brownian motion covariance structure

ABSTRACT

It is difficult to characterize complex variations of biological processes, often longitudinally measured using biomarkers that yield noisy data. While joint modeling with a longitudinal submodel for the biomarker measurements and a survival submodel for assessing the hazard of events can alleviate measurement error issues, the continuous longitudinal submodel often uses random intercepts and slopes to estimate both between- and within-patient heterogeneity in biomarker trajectories. To overcome longitudinal submodel challenges, we replace random slopes with scaled integrated fractional Brownian motion (IFBM). As a more generalized version of integrated Brownian motion, IFBM reasonably depicts noisily measured biological processes. From this longitudinal IFBM model, we derive novel target functions to monitor the risk of rapid disease progression as real-time predictive probabilities. Predicted biomarker values from the IFBM submodel are used as inputs in a Cox submodel to estimate event hazard. This two-stage approach to fit the submodels is performed via Bayesian posterior computation and inference. We use the proposed approach to predict dynamic lung disease progression and mortality in women with a rare disease called lymphangioleiomyomatosis who were followed in a national patient registry. We compare our approach to those using integrated Ornstein-Uhlenbeck or conventional random intercepts-and-slopes terms for the longitudinal submodel. In the comparative analysis, the IFBM model consistently demonstrated superior predictive performance.

1. INTRODUCTION

Models to characterize temporal biomarker data and time-to-event data alongside demographic/clinical covariates have a rich history in joint longitudinal-survival modeling (Tsiatis and Davidian (2004)) and, foundationally, in longitudinal data analysis (Fitzmaurice et al. (2008)). In two-stage and simultaneous modeling approaches, the random intercepts-and-slopes model by Laird and Ware (1982) is often used for the biomarker process represented in the longitudinal submodel. While this use of random effects has been a breakthrough for mixed effects modeling, it has become increasingly apparent through advanced longitudinal data analysis developments that random slopes may not accurately depict the longitudinal process. By assuming a linear disease trajectory with a constant slope, it is more likely that clinically relevant fluctuations will go unnoticed or undetected. As a remedy, researchers have proposed a stochastic process term for modeling the continuous longitudinal outcome that may boost its ability to explain highly variable biological processes (Diggle et al. (2014); Taylor et al. (1994); Stirrup et al. (2016)). In Stirrup et al. (2016), scaled fractional Brownian motion (FBM) introduced by Mandelbrot and Van Ness (1968) was used as a more flexible and plausible process to represent the natural evolution of CD4 cell counts in HIV-positive individuals before treatment. This work demonstrated that FBM accommodates more unpredictable fluctuations over time than the Brownian motion (BM) process or random intercepts-and-slopes models.

A complementary advantage of the use of a random walk as opposed to a slope in these BM-type models is the ability to create target-based probabilities by studying individualized underlying rates of change in the longitudinal process. For example, Diggle et al. (2014) employed integrated BM (IBM) to calculate predictive probabilities for monitoring progression toward renal failure as measured by estimated glomerular filtration rate. They calculated the predictive probabilities of losing kidney function at a clinically relevant rate. Szczesniak et al. (2020) applied the same nonstationary stochastic process to predict real-time rapid lung disease progression in cystic fibrosis. They used target functions defined as clinically meaningful drops in underlying rates of lung function decline. Asar et al. (2020) further developed a computationally efficient method and an R package to adjust for non-Gaussian random effects, stochastic process, and measurement error terms.

Given these advantages and the clinical interest in predicting rapid lung disease progression and characterizing its association with mortality, we propose embedding the integrated FBM (hereafter, IFBM) as the stochastic process in a longitudinal submodel of a two-stage longitudinal-survival model. We develop novel target functions to estimate clinically impactful declines in lung function for women with a rare lung disease called lymphangioleiomyomatosis (LAM). The information from the longitudinal process is connected to the event process through a shared parameter. We examine the hazard of mortality in LAM in our event submodel using the proportional hazards model by Cox (1972). The rest of the article is organized as follows: Section 2 provides model setup and methods. In Section 3, we illustrate the target functions using data obtained from women with LAM who were followed over time in the National Heart, Lung, and Blood Institute (NHLBI) LAM Patient Registry. In Section 4, we present the Monte Carlo simulation study results. We conclude in Section 5 with a discussion of the findings, limitations, improvements, and future areas of extension for the approach.

2. MODEL AND IMPLEMENTATION

2.1. IFBM model for longitudinal data and notation

Within the context of our motivating example, we model the longitudinal process of lung function as

(1)

where refers to the measured lung function for the patient at time for ; is the row of the design matrix that corresponds to patient i at time consisting of observed demographic/clinical covariates, some of which may be time-varying; is the vector of fixed effects parameters corresponding to covariates in . The patient-specific random intercept is denoted by , where (assumed to be independent between patients). is a realization of the IFBM process evaluated at time , where we assume is independent between patients, and the collection of terms follows a multivariate normal distribution with mean vector and variance-covariance matrix , that is,  . For given time points s and t where , the can be expressed as

(2)

where refers to the Hurst index, which defines the characteristics of the process, and is a positive scale parameter. Measurement error is defined and assumed to be independent and identically distributed as . Finally, we assume that , , and terms are mutually independent.

To provide some interpretation for the behavior of the IFBM process and covariance in Equation (2), it is worth noting that the stochastic process is continuous-time, Gaussian, non-Markov , non-stationary and has non-stationary increments. When , IFBM defaults to the widely used IBM process Diggle et al. (2014). Abundo and Pirozzi (2019) have shown that lower values of H, such as , yield lower variance than higher values like . They also showed that for 2 fixed time points, s and t  , the covariance function takes a parabolic shape with a minimum (see Figure 3 in Abundo and Pirozzi (2019)) when it is examined with respect to H. Because the next measurements in a patient’s lung function trajectory may be dependent on the current measurement, the non-Markov property of the IFBM process and its increments improve biological process modeling. Furthermore, the readily available derivative process of IFBM (ie, the FBM process) simplifies the calculation of predictive probabilities for medically meaningful drops in the rates of change in lung function, which can also be seen as a biological process of a complex nature.

In FBM, when , successive increments of the process are negatively correlated. Consequently, the trajectory’s path becomes characterized by significant fluctuations and the process realizations tend to gravitate toward a mean of zero. Conversely, when , successive increments of the FBM process exhibit a positive correlation, leading to a smoother trajectory path. Additionally, the realizations of the process display deviations from the mean zero. Simulated realizations of IFBM and FBM processes produced using different H values are depicted in Web Figure 1. Therefore, this adjustable nature of the characteristic parameter, H in this family of stochastic processes can be considered well-suited to model biological processes. The FBM process has previously found applications in mathematical modeling across various fields such as hydrology, computer network traffic, finance, and linear mixed-effects models for HIV (Stirrup et al. (2016)). However, IFBM has not been employed to model biological data before. Notably, its special instance at H=1/2 (IBM) has been used in modeling biological processes due to its demonstrated improvements in dynamic prediction accuracy (Diggle et al. (2014); Szczesniak et al. (2020)). Consequently, our endeavor involves utilizing the IFBM to model longitudinal data, while employing FBM to generate predictions for the underlying rate of change process corresponding to this longitudinal data.

2.1.1. Predictions with IFBM

It is of clinical interest to make individualized predictions about the underlying rate of change in the lung function process, particularly when this process depicts what is known as “rapid decline.” We let be the threshold that constitutes a rapid decline in lung function. By taking the derivative with respect to time t in Equation (1), we construct the requisite target function corresponding to the threshold:

(3)

Above, refers to the patient-specific rate of change of the IFBM process (ie, is an FBM process by construction).

Let, contain the first k longitudinal measurements for the patient and be the nowcasted trajectory value at time using available data up to time . To acquire the predictive probabilities corresponding to a specific threshold , we consider the conditional distributions, , , , and , which are generated together by using properties of the multivariate normal distribution, as well as the distributional properties shown below, in which the collection of longitudinal parameters are represented as :

(4)

(5)

(6)

where,

and

The term in (4) is the total variance accounted for by the patient and in (5) and (6) is the total variance of the first k longitudinal measurements for patient i. Similarly, we use properties of the multivariate normal distribution for dynamically updating the longitudinal trajectories and predictive probabilities. The stochastic process terms at time , that is, with lead-time u, can be calculated by replacing with in (5) and (6). Further, the 95% credible intervals can be found using and quantiles of .

The predictive probabilities of rapid lung function decline can be calculated as:

(7)

where is the threshold for identifying rapid decline.

2.1.2. Predictive performance of the longitudinal submodel

To evaluate the predictive performance of the model in Equation (1), we use root mean square error (RMSE), mean absolute percentage error (MAPE), and mean absolute error (MAE) for point predictions of the outcome variable, and the area under the receiver operating characteristic curve (AUC) for the predictive probabilities in Equation (7). To calculate the AUCs for predictive probabilities, first, a flag is defined when the observed rate of lung function change exceeds a specific threshold, (1 indicates exceeding, 0 otherwise). This flag serves as the observed response, while the calculated predictive probability using Equation (7) serves as the predicted value. Subsequently, the Youden index is employed to maximize sensitivity and specificity, ultimately deriving the AUC.

2.2. Survival model and notation

Upon fitting the model in Equation (1), we fit event data using a Cox proportional hazards model as the event submodel. For patient i, let be the true failure time, and be the censoring time. Then, be the observed failure time and be the event indicator. We express the model as:

(8)

Here, represents the baseline hazard function at time t, for which we assume a Weibull baseline hazard. that is, ; and are shape and scale parameters, respectively. By assuming the risk of an event may depend on the history of the longitudinal process through the current value of the error-free lung function for the patient i, and its slope, , they are shared with the survival submodel as endogenous time-varying covariates. Here, and are obtained from Equation (1). Then, and are the shared parameters that represent the association between the true response trajectory and the risk of the event and the slope of the true response trajectory and the risk of the event, respectively. The are a set of baseline covariates, which may or may not have been included in the longitudinal submodel, and are their corresponding parameters. Given the shared information, we assume that the longitudinal process is conditionally independent of the survival process.

2.2.1. Predictive performance of the event submodel

The survival submodel’s discriminative performance is evaluated using time-dependent AUC (Rizopoulos (2016); Andrinopoulou et al. (2018); Long and Mills (2018)). To assess the accuracy of the model, we use the prediction error (PE) estimate proposed by Henderson et al. (2002) and further used by Andrinopoulou et al. (2018). By using PE, we can compare the predicted probability of survival of a patient to that patient’s observed event.

2.3. Prior distributions

Prior distributions for the parameters are chosen so that the distributions have large variances. As a result, these priors would express partial information about the corresponding variable. In particular, for , , and in the longitudinal submodel, we use Inverse-Gamma priors. Normal priors are used for , logit-transformed H, and . For , and Gamma priors are employed. Detailed information on the chosen prior distributions may be found in Web Appendix B.

2.4. Implementation

We fit the model in 2 stages under a Bayesian framework where we utilize the Gibbs sampler with the Metropolis-Hastings (MH) algorithm by Hastings (1970). In the first stage, we fit the longitudinal submodel in (1). Then, fitted values ( and ) are used in the event submodel (8) for estimation in the second stage. Under the event submodel, we utilized the same approaches taken by Wang and Taylor (2001); Wang (1998); Struthers and Mcleish (2011) to approximate each time-varying covariate, including the IFBM terms, using a piecewise constant step function. We choose ; grid points; to contain all the time points where marker measurements are taken for patient i, so that the time-varying covariates can be considered constant over the time interval . That is, within that time interval. These grid points are not required to be the same between patients when longitudinal measurements are observed at different time points, which is common in registry studies. Therefore, we choose these grid points at each patient‘s recorded longitudinal measurement. That is, in our study, = time at which each longitudinal measurement was taken.

The full conditional distributions of and have inverse-Gamma and normal distributions. Since all other parameters have non-normalized full conditional distributions, we use the MH algorithm for them. We assume normal proposal densities for logit-transformed H, , and log-normal proposal densities for , . The current value and logarithm of the current value in each iteration are used as the means of the normal and log-normal proposal densities, respectively. The SDs of the proposal densities are chosen so that the acceptance rate is 20%-50%. Web Appendix C shows the derivation of the likelihood, the joint posterior distribution, and the full conditional distributions.

The initial values for , and are obtained by running separate models for longitudinal and survival data using existing ‘lmenssp’ and ‘epa’ R packages, respectively. For all other parameters, we draw starting values from their corresponding prior distributions. We assess convergence by using Gelman-Rubin (GR) diagnostics on samples following the burn-in period. Bayesian inferences are made using MCMC samples collected after convergence. R software (R Core Team (2021)) is used for all implementations (see Supplementary  Materials for R code).

3. NHLBI LAM REGISTRY COHORT APPLICATION

Lymphangioleiomyomatosis, commonly known as LAM, is a rare neoplasm that predominantly affects women and can be seen either in conjunction with the inheritable condition called tuberous sclerosis complex (TSC-LAM) or occur sporadically in women without TSC (S-LAM). LAM is caused by one of the 2 known TSC genes. This causes a deficit or dysfunction of the protein products of the genes, resulting in abnormal cell development and proliferation, notably of smooth muscle-like cells known as LAM cells, which infiltrate the lungs and lymphatics. The LAM cells, together with other secondary immune cells, secrete matrix-degrading enzymes and cause progressive destruction of the lung tissue over time. The natural history of disease progression in LAM is characterized by a progressive decline in lung function, which is measured as forced expiratory volume in 1 second (FEV) in liters, and early mortality (Gupta et al. (2019)). As a result, monitoring and preserving lung function over time is essential for survival in those affected by the disease.

We apply the proposed two-stage approach to data from the NHLBI LAM Registry, which enrolled women with LAM (1998-2003) (Ryu et al. (2006)). Select demographic and clinical characteristics were collected approximately every 12 months. Among the 246 participants, 16 (6.5) were excluded from our application due to a partially substantiated or unverified LAM diagnosis, and another 13 (5.3) who received a lung transplant prior to follow-up were omitted from the analysis. Using the predictive mean matching (van Buuren and Groothuis-Oudshoorn (2011)) method on the available cohort, missing covariate values are imputed a priori to any statistical inference. We excluded one patient with a recording error of the LAM diagnosis date, as well as an additional 25 patients who had only one longitudinal observation after completing imputation. A spaghetti plot of the analysis data set is included in Web  Figure 2.

The time variable considered for this study is the time since baseline, where the baseline is defined as the time at the first recorded pulmonary function test (PFT) measurement. The covariates used for our application were measured at baseline: namely, age at LAM diagnosis, menopausal status (pre or postmenopausal), percent predicted FEV, and percent predicted diffusing capacity for carbon monoxide (DLCO). The longitudinal submodel outcome is FEV. The event of interest in the Cox submodel is death or lung transplant.

3.1. Primary Results

The final analysis cohort consisted of 191 patients who collectively contributed a total of 730 observations and 602.2 years of follow-up. The mean (SD) of LAM diagnosis age was 40.9 (9.8) years, and 128 (67%) of them were postmenopausal. Mean (SD) of baseline FEV (in L and %predicted) were 2.1 (0.8) L and 71.8 (23.7) %, respectively, while DLCO %predicted was 64.3 (23.9) mL/mmHg/min. The patients in the data set had an average (SD) of 3.2 (1.2) years of follow-up, and each patient had a median (range) of 4 (2-6) PFT measurements. There were 55 (29%) of patients who experienced death or lung transplantation.

The posterior median is reported as model parameter estimates with 2.5% and 97.5% quantiles as 95% credible intervals in Table 1. The results for the longitudinal submodel show that the lung function declined at an average of 84 mL/yr. Adjusted for all other covariates, being older at the time of diagnosis was linked to a lower overall FEV, and higher FEV % predicted and DLCO %predicted at baseline corresponded to a greater overall FEV. The hazard ratios reported in Table 1 for the parameters in the survival submodel show that having a higher DLCO percent predicted and higher lung function was associated with reduced risk of lung transplant/death. Residual diagnostics showed no clear patterns, except that the data exhibited a heavier upper tail than a normal distribution (see Web Appendix D). As a sensitivity analysis, we conducted survival parameter estimation using a finer fixed grid across individuals. The details of this approach, along with the corresponding results, can be found in Web Appendix E for reference.

TABLE 1.

Parameter estimates (95% credible intervals) for LAM registry cohort.

	IFBM	IOU	Random effects
Longitudinal submodel
(Intercept)	0.61 (0.41,0.79)	0.64 (0.46,0.82)	0.65 (0.47,0.84)
(Time)	-0.08 (-0.10,-0.07)	-0.08 (-0.10,-0.07)	-0.08 (-0.1,-0.07)
(LAM diagnosis age)	-0.018 (-0.022,-0.014)	-0.018 (-0.022,-0.014)	-0.018 (-0.022,-0.014)
(Post-menopausal)	0.06 (-0.03,0.14)	0.06 (-0.03,0.14)	0.04 (-0.05,0.12)
(%predicted FEV)	0.028 (0.026,0.030)	0.029 (0.027,0.031)	0.028 (0.026,0.030)
(%predicted DLCO)	0.003 (7x,0.005)	0.002 (4x,0.004)	0.002 (5x,0.004)
(Measurement error)	0.016 (0.014,0.019)	0.011 (0.008,0.015)	0.131 (0.122,0.141)
(Between patient: random-intercept)	0.05 (0.04,0.06)	0.04 (0.03,0.07)	0.27 (0.25,0.31)
(Between patient: random-slope)			0.11 (0.09,0.13)
(Correlation: random-slope and -intercept)			0.71 (0.59,0.82)
(IFBM: scale parameter)	0.014 (0.010,0.019)
(IOU: scale parameter)		1.25 (0.03,9.74)
(Hurst index)	0.07 (0.01,0.18)
(Characteristic parameter: IOU)		3.84 (1.05,20.44)
Survival submodel
(LAM diagnosis age)	0.70 (0.47,1.003)	0.76 (0.53,1.07)	0.64 (0.44,0.94)
(Post-menopausal)	0.58 (0.3,1.13)	0.61 (0.32,1.15)	0.57 (0.3,1.08)
(%predicted FEV)	1.98 (0.63,6.23)	1.31 (0.51,3.26)	2.55 (0.83,7.49)
(%predicted DLCO)	0.45 (0.26,0.74)	0.43 (0.25,0.7)	0.44 (0.26,0.74)
(Shape: Weibull BH)	1.42 (0.81,2.09)	1.58 (0.96,2.22)	1.34 (0.74,2.00)
(Scale: Weibull BH)	0.72 (0.01,7.61)	2.23 (0.15,10.77)	0.29 (0.001,4.16)
(Association: value)	0.14 (0.03,0.68)	0.26 (0.07,0.93)	0.1 (0.02,0.47)
(Association: slope)	15.76 (0.02,28874.99)	0.64 (0.01,56.27)	1047.90 (1.02,1652154.21)

Abbreviations: BH = baseline hazard; IFBM = integrated fractional Brownian motion; IOU = integrated Ornstein-Uhlenbeck; LAM = lymphangioleiomyomatosis; FEV = forced expiratory volume in 1 second; DLCO = diffusing capacity for carbon monoxide.

3.2. Comparative results

The predictive capability of the longitudinal submodel with the IFBM process was compared to the model with the integrated Ornstein-Uhlenbeck (IOU) process and a model with no flexible covariance structure, that is, a random intercepts-and-slopes model using 5-fold cross-validation. The IOU process has a zero mean vector, and its variance-covariance function can be defined for 2 time points s and t () as, . Here, generates realizations that we consider as IOU process terms. is a scale parameter, and is the parameter that determines the strength of the pull toward the mean of the process.

The estimates of the fixed effects and baseline covariates of the survival submodel differed somewhat between the three models. However, the parameter of the IOU model had a very large 95% credible interval, which may lead to wider posterior predictive distributions for the outcome variable, indicating the inadequacy of the IOU model compared to the IFBM model. Wang (1998); Wang and Taylor (2001) mentioned similar issues when handling this parameter. The results of the 5-fold cross-validation are detailed in Table 2. These findings clearly indicate the superior performance of models incorporating stochastic process terms compared to the random effects model. Moreover, in terms of longitudinal predictions, the IFBM model exhibits a slight advantage over the IOU model. We further explored the scatter plots of observed and predicted FEV to investigate the adequacy of the IOU model and a random-slope random-intercept model in comparison to an IFBM model (Figure 1). We saw that the models with a stochastic process (Figures 1 a and b) provided more accurate predicted values than the random intercepts-and-slopes model (Figure 1c), and the best predictions were made by the model with the IFBM process.

TABLE 2.

The mean (SD) of model performance metrics in the comparative study are reported. AUCs for predictive probabilities for three thresholds ( = 100, 75, and 60 mL/yr) of rapid lung function decline are calculated. A total of 2 time-windows ((2, 4], and (3, 4] yrs) are considered for time-dependent AUC (tAUC) and PE calculations.

	Train			Test
	IFBM	IOU	Random effects	IFBM	IOU	Random effects
Longitudinal submodel
RMSE (L)	0.08 (0.01)	0.09 (0.02)	0.28 (0.01)	0.08 (0.02)	0.09 (0.02)	0.27 (0.04)
MAPE (%)	2.99 (0.19)	3.32 (0.81)	10.56 (0.62)	3.07 (0.37)	3.29 (0.85)	10.48 (2.14)
MAE (L)	0.05 (0.005)	0.06 (0.01)	0.18 (0.01)	0.05 (0.005)	0.06 (0.01)	0.17 (0.02)
Survival submodel
tAUC	0.86 (0.02)	0.85 (0.06)	0.84 (0.04)	0.89 (0.06)	0.89 (0.05)	0.86 (0.08)
tAUC	0.85 (0.02)	0.83 (0.07)	0.80 (0.24)	0.90 (0.07)	0.89 (0.07)	0.87 (0.11)
PE	0.12 (0.01)	0.13 (0.01)	0.12 (0.01)	0.13 (0.04)	0.14 (0.04)	0.12 (0.03)
PE	0.09 (0.01)	0.10 (0.01)	0.08 (0.01)	0.07 (0.05)	0.08 (0.06)	0.07 (0.05)

Abbreviations: AUC = Area under the receiver operating curve; IFBM = integrated fractional Brownian motion; IOU = integrated Ornstein-Uhlenbeck; L = liters; MAPE = mean absolute percentage error; MAE = mean absolute error; PE = prediction error; RMSE = root mean square error; yrs = years.

FIGURE 1.

Plot of observed vs. predicted FEV (in liters) values using a) IFBM, b) IOU, and c) random effects models for the full NHLBI LAM registry data. The unity line is indicated by a blue dashed line, while the Loess smoothing curve is indicated by a red dotted dash line. RMSE, MAPE, and MAE are as follows: a) 0.08 L, 3%, 0.05 L; b) 0.11 L, 4%, 0.06 L; and c) 0.28 L, 10%, 0.17 L. Abbreviations: FEV = forced expiratory volume in 1 second; IFBM = integrated fractional Brownian motion; IOU = integrated Ornstein-Uhlenbeck; LAM = lymphangioleiomyomatosis; L = liters; MAPE = mean absolute percentage error; MAE = mean absolute error; NHLBI = National Heart, Lung, and Blood Institute; RMSE = root mean square error.

We also compared the fitted and observed values using variograms. The variograms for IFBM, IOU, and random effects models are now included in Web Figures 5, 6, and 7, respectively. The variogram of the IFBM model demonstrates that the correlation diminishes with increasing time lag. Conversely, for the IOU model, the variogram indicates a decrease in correlation that eventually levels off as the time lag increases. While the former demonstrates the anticipated trend, the latter deviates from the characteristic behavior expected in a lung function process within the context of LAM. In contrast to models with stochastic processes, the random effects model reveals an unexpected increase in correlation as the time lag extends. This finding contradicts the natural behavior of lung function in LAM, suggesting a potential misalignment of the model with the underlying biological process. When it comes to predicting survival outcomes, the IFBM model slightly outperformed both the IOU and random effects models.

We calculated predictive probabilities, for three distinct, clinically meaningful target functions with = 100, 75, and 60 mL/yr using the IFBM model. All three target functions had AUCs greater than 0.80 for predictive probabilities (see Web Table 2). The individualized lung function trajectory predictions and predictive probabilities for the three target functions for 2 patients from the test cohort are shown in Figure 2.

FIGURE 2.

Predictions of lung function (FEV in liters) loss for 2 women (Patients A and B). On the left, measured FEV values in liters (red dots) are plotted with the fitted trend (solid black line), and the 95% credible interval (CI) for the fitted FEV trajectory using the IFBM model (gray area) are shown. The right plots illustrate the real-time predicted probabilities () that the underlying rate of change of lung function falls below the clinically relevant thresholds, =100, 75, and 60 mL/yr denoted by red, blue dashed, and black dot-dash lines, respectively. Abbreviations: FEV = forced expiratory volume in 1 second; IFBM = integrated fractional Brownian motion; mL = milliliters; yr = year.

4. SIMULATION STUDY

4.1. Study design

We set up our simulation study to follow the ADEMP framework by Morris et al. (2019). The aim of the simulation study was to evaluate the performance of the proposed two-stage model. We investigated how well the population parameters in our model can be estimated. Further, we evaluated how the death or lung transplant rate in our data set would affect the accuracy of the model parameter estimates.

We designed this simulation study to represent the structure of the LAM application. The step-by-step simulation data generation method is given in Web Appendix H. This method followed the approach in Wang (1998). We carried out 100 replications of each simulation setting. Each data set consisted of 200 subjects who were followed up for a maximum of 5 years with yearly longitudinal measurements, which will mirror the NHLBI LAM registry data set. Further, to investigate the impact of event rates on the accuracy of the parameter estimates, we considered 2 event rates: ”low” and ”high”, where ”low” simulation setting would have a mean (SD) of 27 (3)% and the ”high” setting would have a mean (SD) of 38 (3)% event rate. We considered longitudinal measurements until the patient experienced the event or until the end of follow-up.

For simplicity, we only considered an intercept and a time variable as fixed effects along with random intercept and stochastic process terms in the longitudinal submodel (1), and in the survival submodel, only one baseline covariate along with the underlying longitudinal trajectory in (8) were considered. Therefore, the estimands, we were interested in this study were the fixed effects (, ), between-subject variation (), scale parameter of the IFBM process (), Hurst index (H), measurement error variance (), shape () and scale () parameters of Weibull baseline hazard, the parameter corresponding to the baseline covariate in the survival submodel (), and the association parameter between the true longitudinal trajectory and the survival process ().

The posterior median is recorded as the parameter estimate, naïve standard error (ie, posterior sample variance over the number of samples as defined in Plummer et al. (2006)), 2.5% quantile and 97.5% quantile are recorded for each parameter. The accuracy of a parameter estimate is assessed in terms of bias and coverage rate of 100% credible intervals.

4.1.1. Sensitivity analyses

To evaluate the robustness of our model and explore the impact of potential deviations from its main assumptions, we conducted an additional set of simulations. In joint modeling, a Gaussian distribution is conventionally assumed for the marker distribution. Deviating from this assumption is of primary concern, given that the random effects play a pivotal role in depicting the association between the biomarker and the event processes. Furthermore, to accurately estimate the aforementioned association, the individualized trajectory must be effectively captured by the model. Therefore, in this simulation study, we specifically focused on examining the effects of misspecification of the marker distribution and marker trajectory on the association parameter estimate.

We maintained consistency with our previous simulation setup (described in Section 4.1), including the maximum follow-up time and follow-up frequency. Additionally, in this particular simulation setup, we chose the ”high” event rate setting described in Section 4.1, and the event times were generated using the same data generation method as in our previous simulations (see Web Appendix H). To assess marker distribution misspecification, we employed the proposed IFBM model used in the previous simulation setup, incorporating random intercept terms generated from a bimodal normal distribution. To explore marker trajectory misspecification, we utilized a model featuring random intercept, measurement error term, the time variable, along with its quadratic terms as fixed effects. We conducted 100 replications for both of these assessments. As metrics for evaluating the robustness of our model, we used bias and coverage of the association parameter.

4.2. Results

Coverage probabilities using 100(1–)% credible intervals and the biases with their Monte Carlo SEs (Morris et al. (2019)) are shown in Figures 3 and 4, respectively. The coverage probabilities for all parameters in both submodels, except for , exceed 90%. Further, the coverage probability for is lower in the ”high” setting compared to the ”low” setting. The bias of most longitudinal parameter estimates is slightly higher in the ”high” event rate setting than in the ”low” event rate setting. In contrast, the bias of all parameter estimates in the survival submodel in the ”high” event rate setting has decreased compared to the low event rate setting, except for the association parameter. We observed improved MCMC convergence in when the event rate in the simulated dataset was higher. The low coverage rate of in the ”high” event rate setting may be explained by the data-generating mechanism used in the simulation study (Web Appendix H). Specifically, during data generation, longitudinal trajectories are initially generated using the selected set of parameter values for the longitudinal submodel. Subsequently, when survival data is generated, these trajectories are truncated at the event time. Therefore, when the event rate is high, the coverage rate of longitudinal parameter estimates may decrease, while the bias may increase. True parameter values, parameter estimates, and other performance measures such as mean square error and relative bias are presented in Web Appendix I.

FIGURE 3.

Summary of the probability of coverage for all parameters across simulation settings. The black dots represent the mean, while the solid black horizontal lines reflect the 95% Monte Carlo confidence intervals of coverage probability derived from 100 replications. Abbreviations: yr = year.

FIGURE 4.

Summary of the bias for all parameters across simulation settings. The horizontal solid black lines reflect the 95% Monte Carlo confidence intervals, while the black dots represent the mean bias of parameter estimates from 100 replicates. Abbreviations: yr = year.

In Web Table 3, it is evident that the evaluation metrics employed to assess the predictive capabilities of our model consistently yielded comparable results in both simulation scenarios. Specifically, the performance assessment of the longitudinal submodel within our simulation study indicates the model’s ability to forecast lung function with an RMSE (SD) of 36 (2) mL, a value falling below the threshold indicative of rapid lung function decline in LAM. Furthermore, when compared with the performance metrics derived from the application of the NHLBI LAM registry data, our findings demonstrate superior performance metric values for both submodels compared to those observed within the NHLBI LAM registry data context. Given that the simulated datasets closely mimic the NHLBI LAM registry dataset, these results show that our approach accurately predicts lung function trajectories and time-to-event outcomes.

In our sensitivity analysis, we examined 2 different sample sizes: one consisting of 100 subjects and the other of 200 subjects. The simulation results, detailed in Web Table 6, reveal a significant improvement in the relative bias of the association parameter when misspecifying the marker trajectory as the sample size increased, decreasing from 12% to 6%. Moreover, the coverage of the association parameter consistently remained above 98%. Consequently, it is evident that the estimate of the association between a longitudinal marker and the event process remains largely unaffected by the misspecification of the random effect distribution, provided the sample size is sufficiently large. When we assessed the impact of misspecifying the marker trajectory, we observed a similar trend. The relative bias of the association parameter decreased from 36% to 30% with the increase in sample size, while the coverage probability consistently exceeded 96%. These findings emphasize the critical importance of accurately specifying the longitudinal trajectory of a marker when employing a joint model for estimating the association between a longitudinal biomarker and time-to-event.

5. DISCUSSION

In this research, we developed a more flexible two-stage model using an IFBM process that captures substantial biological heterogeneity, and we generated clinically meaningful, target-based probabilities by exploring individualized underlying rates of change in the longitudinal process. The IFBM stochastic process was chosen as a more generalized form of the IBM process or conventional inclusion of random intercepts and slopes, which are commonly employed in longitudinal studies. Another important consideration is that the IFBM process and its increments are non-Markov, which is advantageous for modeling biological processes, given that successive measurements in a biological process may show a notable degree of dependence on the preceding measurement. Finally, IFBM has a readily available derivative (FBM), which, as demonstrated, simplifies the calculation of predictive probabilities for medically relevant target functions. Comparative analyses in our application showed that longitudinal predictions from the proposed IFBM submodel were more accurate than those of the IOU or random intercepts-and-slopes model. Although the IOU process used by Wang and Taylor (2001) is a nonstationary Gaussian stochastic process, unlike IFBM, it has the Markov property. So, intuitively, IFBM may be considered to explain biological processes more effectively. Additionally, the IFBM’s characteristic parameter has better convergence and is easier to manage than that of the IOU. Similar issues with the IOU process were noted in Wang and Taylor (2001); Wang (1998).

The exclusion of patients with only 1 longitudinal observation is an inherent drawback of survival analysis that could be addressed in future extensions pertinent to the event submodel in (8). Exploring alternative association structures, which link the risk of the event with other attributes of longitudinal trajectories (Rizopoulos (2012)), proves to be a promising avenue of future research. For instance, a lagged effects association structure can be employed if the hazard of an event at the time t is presumed to be associated with the longitudinal measure at a preceding time point . Furthermore, if the association between the biomarker and the survival event changes temporally, the association parameter can be made time-varying, which may lead to improved dynamic predictions of survival events (Andrinopoulou et al. (2018)). Given more recent implementations with joint modeling available in JMbayes2 (Rizopoulos et al. (2022)), albeit with more conventional stochastic processes, one may adjust for immortality bias in the joint model by changing the distribution assumption for the longitudinal lung function (eg, assuming a Gamma in place of the normal distribution).

The complexity of the joint modeling often involves identifiability problems due to a large number of parameters and the requirement of numerical integration in calculating marginal likelihood and survival functions without closed form, thus making the inferential process more time-consuming. Therefore, two-stage modeling is utilized as a classic alternative to alleviate the complexity of joint modeling, which circumvents identifiability issues and improves computational time. However, the two-stage approach comes with the drawback of giving biased survival parameter estimates. This issue may be alleviated by using informative priors in the Bayesian framework. Moreover, our sensitivity analyses indicate that the two-stage modeling approach remains adequate when the model is accurately specified. This had been previously studied by Huang et al. (2009); Rizopoulos et al. (2008). It is worth noting that previous simulation studies in literature (Arisido et al. (2019); Crowther et al. (2016)) have reported biased estimates although with good coverage probabilities for association parameters while investigating the impacts of trajectory misspecification. Specifically, Crowther et al. (2016) emphasizes that the current value association structure exhibits greater robustness to trajectory misspecification compared to the rate of change association structure. We can speculate that the substantial bias in the trajectory misspecification simulation scenario may be a result of the complexity of the true trajectory we considered (Arisido et al. (2019); Crowther et al. (2016)) and the intricate nature of the covariance structure in our model, which interacts with the trajectory assumptions. Additionally, a joint modeling study in the context of pulmonary diseases has found that bias observed in the association parameter can also be attributable to immortal cohort bias (Szczesniak et al. (2023)). Consequently, we believe that addressing bias stemming from trajectory misspecification could serve as a potential avenue for future research.

The diagnostic plots from the application to LAM data provide a reliable model with room to improve non-normal tails, which could be a future topic of interest. The predictions may be improved for the biomarker-specific models with stochastic processes by using the simultaneous approach to estimate the joint model parameters. However, when we used the simultaneous approach, we encountered computational resource constraints and convergence concerns. Inferring from past joint modeling research, these issues were most likely caused by a large number of random effects Rizopoulos (2012), not to mention the inclusion of stochastic process terms. Developing methodologies to fit our proposed class of joint models simultaneously is an area requiring further investigation.

While incorporating IFBM or any other stochastic process enhances the goodness-of-fit and predictive capabilities in contrast to the random effects model, it’s important to note that this approach indeed demands heightened computing resources and time due to the intricate nature of the stochastic process. Nonetheless, when compared to the random effects model, the number of parameters requiring estimation is approximately equivalent. Moreover, considering the aforementioned benefits of including a stochastic process term rather than relying solely on a random slope in the model, the advantages of incorporating a stochastic process outweigh the benefits of excluding it when modeling biological processes.

Under the current approach, we generated dynamic predictions for the longitudinal submodel that were utilized to account for longitudinal lung function changes in the event submodel. While dynamic survival predictions warrant additional research, our application identified correlates for the hazard of LAM mortality. The framework developed here using IFBM and target functions provides a path toward enhanced prediction and monitoring of biomarkers and mortality in biomedical research.

Supplementary Material

ujae011_Supplemental_Files

Web Appendices, Tables, and Figures referenced in Sections 2-4 and a simulated dataset with sample R codes used in Section 4 are available with this paper at the Biometrics website on Oxford Academic.

FUNDING

This work was supported by the National Institutes of Health/ National Heart, Lung, and Blood Institute [R01 HL141286 to R.S.]; and The LAM Foundation [LAM0153P01-22 to R.S.].

CONFLICT OF INTEREST

None declared.

DATA AVAILABILITY

The authors do not have permission to provide the National Heart, Lung, and Blood Institute Lymphangioleiomyomatosis Registry dataset. Further, this dataset is not publicly available due to privacy or ethical restrictions.