Joint modeling of high-dimensional longitudinal data and survival using supervised low-rank tensor decomposition

Summary

High-dimensional longitudinal data are increasingly available in biomedical research, especially from omics platforms, but pose substantial challenges for joint modeling with survival outcomes. These challenges include modeling complex temporal dynamics, accommodating cross-feature dependencies, and maintaining computational feasibility. We propose a novel joint modeling framework that addresses these issues using supervised low-rank functional tensor decomposition to capture latent structure in multivariate longitudinal data and proportional hazards modeling for time-to-event outcomes. The longitudinal process is represented as a multivariate functional tensor, with a low-rank approximation that incorporates supervision from baseline covariates. Estimation is performed using a likelihood-based Monte Carlo Expectation-Maximization algorithm, enabling coherent inference and individualized prediction. Our method produces dynamic predictions of both longitudinal feature trajectories and survival probabilities. Simulation studies demonstrate substantial improvements in estimation accuracy and predictive performance over a standard two-stage approach, particularly under high censoring and limited sample sizes. In application to the Alzheimer’s Disease Neuroimaging Initiative lipidomics data, the proposed model explains over 99% of variation with four components, and identifies significant subject-level latent predictors of dementia onset. This framework provides a scalable and interpretable strategy for integrating high-dimensional longitudinal biomarkers into joint models for disease progression and risk stratification.

1. Introduction

Alzheimer’s disease (AD) is a progressive neurodegenerative disorder affecting over six million individuals in the United States, primarily those aged 65 years or older (Alzheimer’s Association 2024). It is characterized by the accumulation of amyloid-beta plaques and tau neurofibrillary tangles, leading to the deterioration of memory, cognition, and daily functioning (Jahn 2013; Luckett et al. 2023). Current diagnostic approaches rely on cerebrospinal fluid (CSF) assays, neuropsychological assessments, and neuroimaging, which are invasive, expensive, and difficult to scale (Weller and Budson 2018). These limitations have motivated the development of minimally invasive biomarkers and predictive models to support early detection and longitudinal monitoring (Galvin et al. 2020).

Recent advances in high-throughput technologies have enabled longitudinal profiling across multiple omics layers, including transcriptomics, proteomics, metabolomics, and lipidomics. These data reflect dynamic biological processes that often precede clinical symptom onset. Among these, peripheral lipidomic signatures have emerged as promising blood-based biomarkers for AD, offering a scalable alternative to central nervous system measures (Mapstone et al. 2014; Liu et al. 2021). However, the high dimensionality, irregular sampling intervals, and measurement noise inherent to omics data pose substantial analytical challenges. Longitudinal omics trajectories, particularly from lipidomics and proteomics, have shown strong associations with AD-related time-to-event outcomes, including cognitive decline and dementia diagnosis. Mounting evidence supports integrating these data to improve early detection and personalized risk prediction (Mapstone et al. 2014; Tynkkynen et al. 2018; Liu et al. 2021). These insights motivate the need for modeling frameworks that can accommodate temporal dynamics, cross-feature correlations, and censoring due to clinical events.

To leverage these data, we jointly model time to dementia onset and high-dimensional longitudinal biomarkers, while adjusting for baseline covariates. We introduce subject-specific latent variables to capture shared variation between the longitudinal and survival processes. Allowing these latent variables to depend on baseline covariates enhances interpretability and enables individualized prediction. A key complication arises when dementia diagnosis leads to informative dropout in the longitudinal data. For example, clinical diagnosis may affect future biomarker trajectories or reduce follow-up due to changes in care. These complexities call for a modeling approach that can extract predictive latent structure while accounting for event-related truncation.

To meet these challenges, we build on joint modeling (JM), a principled framework for analyzing longitudinal and survival data. Originally proposed by Wulfsohn and Tsiatis (1997), JM has become widely used to assess associations between repeated biomarkers and time-to-event outcomes. Classical implementations pair linear mixed-effects models with Cox/proportional hazards models. More recent extensions incorporate multivariate functional principal component analysis (MFPCA) and multivariate functional mixed models (MFMM) to accommodate irregular sampling and cross-feature dependence (Li et al. 2022; Zou et al. 2023, 2025; Gomon et al. 2024). However, these methods are not computationally scalable when the number of biomarkers far exceeds the number of subjects, as is typical in omics studies. Estimating large covariance structures or individual trajectories becomes impractical in such high-dimensional settings.

To overcome these limitations, we adopt a supervised extension of functional tensor singular value decomposition (FTSVD), termed supervised FTSVD (supFTSVD) (Alam et al. 2024), which incorporates baseline covariates into the decomposition to guide latent structure learning. supFTSVD provides a scalable solution for extracting dominant patterns in multivariate longitudinal data while accounting for individual-level predictors. We embed this decomposition within a joint modeling framework that links low-rank longitudinal representations to a proportional hazards model. This unified approach supports simultaneous feature selection, trajectory estimation, and survival modeling, enabling time-updated prediction of biomarker progression and dementia risk. The integration of baseline covariates into supFTSVD also facilitates indirect effects on survival by influencing latent structure. These flexibilities introduce challenges for parameter estimation, which we address using a Monte Carlo Expectation-Maximization (MCEM) algorithm designed to handle diverse parameter types and constraints.

The remainder of the manuscript is organized as follows. Section 2 introduces the joint modeling framework. Section 3 details the estimation algorithm. Section 4 presents methods for dynamic prediction. Section 5 reports simulation results evaluating estimation accuracy and predictive performance. Section 6 applies the proposed method to plasma lipidomics data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Section 7 concludes with a summary and future directions.

2. Joint modeling of longitudinal and time-to-event data via supFTSVD

Let $\left\{{\mathbf{x}}_i,{\left({\mathbf{Y}}_{ij},t_{ij}\right)}_{j=1}^{J_i},T_i,{\delta }_i\right\};i=\mathrm{1,2},\dots ,N$ denote the observed data from $N$ independent subjects. For subject $i,{\mathbf{x}}_i={\left[x_{i1},x_{i2},\dots ,x_{iQ}\right]}^{\prime }$ is a vector of baseline covariates, and ${\mathbf{Y}}_{ij}={\left\{Y_{ij}^p\right\}}_{p=1}^P$ represents the $P$-dimensional longitudinal feature vector observed at time $t_{ij}\in [0,\tau ]\subset \mathbb{R}$. The observed event time is $T_i=\text{min}\left({\tilde{T}}_i,C_i\right)$, where ${\tilde{T}}_i$ is the true failure time and $C_i$ is the censoring time, with event indicator ${\delta }_i=I\left({\tilde{T}}_i\le C_i\right)$. In the context of AD, $T_i$ denotes the time from baseline to either dementia diagnosis or end of study follow-up. We consider a high-dimensional setting where the number of longitudinal features $P$ is much larger than the number of subjects $N$. The number of repeated measurements $J_i$ and the observation times $\left\{t_{i1},\dots ,t_{i{J}_i}\right\}$ may vary across subjects, with the observation times typically irregularly spaced within $[0,\tau ]$. Moreover, $J_i$ is often small to moderate, so the multivariate longitudinal data ${\left({\mathbf{Y}}_{ij},t_{ij}\right)}_{j=1}^{J_i}$ can be viewed as sparsely observed functional data. Figure 1 illustrates this data structure using a hypothetical study with $N=7$ subjects, highlighting the key components of the observed design.

Figure 1.

Schematic representation of a hypothetical study design with $N=7$ subjects. Each subject is followed over time with irregular and subject-specific visit schedules, producing high-dimensional longitudinal omics data. The figure illustrates the timing of feature measurements and the occurrence (or censoring) of the event of interest.

Our proposed approach builds on supervised functional tensor singular value decomposition (supFTSVD) (Alam et al. 2024), which provides a scalable framework for modeling high-dimensional longitudinal data. supFTSVD generalizes the functional tensor singular value decomposition (FTSVD), itself an extension of canonical polyadic (CP) decomposition. In traditional CP decomposition, a tensor $𝒳$ of size $d_1\times \cdots \times d_M$ is approximated by a rank-$K$ sum of outer products: $𝒳\approx \sum _{k=1}^K{\lambda }_k\left({\mathbf{u}}_{k1}\otimes \cdots \otimes {\mathbf{u}}_{kM}\right)$, where each ${\mathbf{u}}_{km}$ is a loading vector for mode $m$. For multivariate longitudinal data represented as a three-mode tensor with modes corresponding to subjects, features, and time, CP decomposition necessarily treats the time mode as a collection of discrete indices, which fails to capture the underlying continuous functional structure of the trajectories. To address this, Han et al. (2024) introduced functional tensors—tensors with one infinite-dimensional mode—to preserve temporal structure. FTSVD modifies CP decomposition by replacing the time mode, which is typically the third mode of the tensor, with a singular function defined on $[0,\tau ]$. This produces a low-rank representation that respects the functional nature of longitudinal trajectories. Compared to multivariate functional principal component analysis (MFPCA) (Ramsay and Silverman 2002; Happ and Greven 2018), FTSVD offers greater computational scalability in high-dimensional settings.

supFTSVD extends FTSVD by incorporating subject-specific baseline covariates into the decomposition through supervised modeling of the subject-mode loadings ${\mathbf{u}}_{k1}$. Following Alam et al. (2024), we model these loadings using multiple linear regression, though other parametric or nonparametric models may be substituted. supFTSVD enables covariate-guided latent structure learning, quantifies covariate effects on feature trajectories, and separates subject-level from group-level variation (Li et al. 2016; Lock and Li 2018; Nie et al. 2018; Guan 2024). Within our joint modeling framework, we embed the supFTSVD-extracted subject loadings as covariates in a proportional hazards model, linking latent longitudinal structure to time-to-event outcomes. This integration allows us to leverage both the dynamic information from high-dimensional trajectories and baseline predictors for individualized risk prediction. Figure 2 provides a schematic overview of the proposed approach.

Figure 2.

Structural overview of the proposed joint modeling framework, which integrates supervised functional tensor decomposition of high-dimensional longitudinal data with proportional hazards regression for survival outcomes.

2.1. Modeling longitudinal trajectories with supFTSVD

Let ${\mathbf{V}}_i(\cdot )$ be a multivariate process defined over $[0,\tau ]$ such that ${\mathbf{V}}_i\left(t\right)={\left\{V_i^p\left(t\right)\right\}}_{p=1}^P$. We assume that the observed multivariate longitudinal data satisfy ${\mathbf{Y}}_{ij}={\mathbf{V}}_i\left(t_{ij}\right)+{ϵ}_{ij}$, where ${ϵ}_{ij}={\left\{{ϵ}_{ij}^p\right\}}_{p=1}^P$ denotes independent measurement errors. Each ${ϵ}_{ij}^p$ is assumed to be independent and identically distributed (iid) with mean zero and common variance ${\sigma }^2$. This homoscedastic assumption primarily reflects a parsimony-driven modeling choice. Allowing feature-specific variances would introduce $P$ additional parameters, which is undesirable in the high-dimensional setting where $P\gg N$ and the primary goal is to recover low-rank structure rather than to estimate individual noise levels. Although measurement variability may differ across lipid features, such differences typically have a negligible impact on the estimated latent components. Nonetheless, the model can be extended to incorporate feature-specific variances if needed.

We model ${\mathbf{V}}_i(\cdot )$ using a rank-$K$ supFTSVD as

$$\begin{gathered} V_i^p(t)=\sum _{k=1}^K{\lambda }_k{\zeta }_{ik}{\xi }_k^p{\psi }_k(t),t\in [0,\tau ], \\ {\zeta }_{ik}={\mathbf{x}}_i^{\prime }{\beta }_k+U_{ik},k=1,\dots ,K, \end{gathered}$$ (1)

where ${\lambda }_k$ is the $k\text{th}$ singular value, ${\zeta }_{ik}$ is the subject-specific loading for individual $i,{\xi }_k^p$ is the feature loading for feature $p$, and ${\psi }_k\left(t\right)$ is the $k\text{th}$ singular function. The fixed-effects vector ${\beta }_k\in {\mathbb{R}}^Q$ captures covariate-driven variation, and $U_{ik}$ represents a zero-mean random deviation independent of ${ϵ}_{ij}^p$ across $i,j$, and $p$. The second equation defines the subject-loading model, where ${\zeta }_{ik}$ connects the baseline covariates to the longitudinal process and enables assessment of indirect effects on the survival outcome. The latent quantities $U_{ik}$ provide subject-specific deviations that capture individual heterogeneity in longitudinal trajectories. Consequently, the subject-loading terms ${\zeta }_{ik}$ are random in supFTSVD, in contrast to the fixed subject-mode factors (eg ${\mathbf{u}}_{k1}$) assumed in standard CP decomposition. Combining both equations yields the unified supFTSVD-based model:

$$Y_{ij}^p=\sum _{k=1}^K{\lambda }_k\left({\mathbf{x}}_i^{\prime }{\beta }_k\right){\xi }_k^p{\psi }_k\left(t_{ij}\right)+\sum _{k=1}^K{\lambda }_k{U}_{ik}{\xi }_k^p{\psi }_k\left(t_{ij}\right)+{ϵ}_{ij}^p,$$ (2)

where $Y_{ij}^p$ denotes the $p\text{th}$ component of the observed longitudinal vector ${\mathbf{Y}}_{ij}$ at time $t_{ij}$.

Similar to the CP decomposition for tabular tensors, supFTSVD is identifiable up to permutation of $\{\mathrm{1,2},\dots ,K\}$ and sign changes of its components, subject to additional constraints. We impose unit norm constraints on both the feature loading vectors ($‖{\xi }_k‖=1$, where ${\xi }_k={\left({\xi }_k^1,{\xi }_k^2,\dots ,{\xi }_k^P\right)}^{\prime }$ and $\Vert \cdot \Vert$ denotes the Euclidean norm) and the singular functions (${‖{\psi }_k‖}_{\mathscr{H}}=1$, where $\Vert \cdot {\Vert }_{\mathscr{H}}$ is the Hilbert space norm). The Hilbert space $\mathscr{H}$ consists of square-integrable functions on $[0,\tau ]$, equipped with norm $\Vert f{\Vert }_{\mathscr{H}}=\sqrt{⟨f,f{⟩}_{\mathscr{H}}}$ for any $f\in \mathscr{H}$, where $⟨\cdot ,\cdot {⟩}_{\mathscr{H}}$ denotes the inner product. In contrast to the original reparameterization in Alam et al. (2024), which absorbs ${\lambda }_k$ into both ${\beta }_k$ and $U_{ik}$, we instead impose $‖{\beta }_k‖=1$ and $\text{Var}\left(U_{ik}\right)=1$, requiring estimation of ${\lambda }_k$ instead of ${\sigma }_k^2$.

To characterize the implied statistical structure of the observed data, we derive the mean, variance, and cross-covariance functions from Model (2). Model (2), introduced by Alam et al. (2024), extends FTSVD by incorporating supervision from baseline covariates into the low-dimensional representation. This supervised framework captures smoothly varying covariate effects across longitudinal features, revealing complex temporal relationships. The coefficient function $\sum _{k=1}^K{\lambda }_k{\beta }_k{\xi }_k^p{\psi }_k\left(t_{ij}\right)$ quantifies how unit changes in elements of ${\mathbf{x}}_i$ influence $E\left(Y_{ij}^p\right)$. Model (2) characterizes the response structure through: (1) the mean $E\left(Y_{ij}^p\right)=\sum _{k=1}^K{\lambda }_k\left({\mathbf{x}}_i^{\prime }{\beta }_k\right){\xi }_k^p{\psi }_k\left(t_{ij}\right)$; (2) the variance $\text{Var}\left(Y_{ij}^p\right)=\sum _{k=1}^K{\left\{{\lambda }_k{\xi }_k^p{\psi }_k\left(t_{ij}\right)\right\}}^2+\sum _{k\ne k^{\prime }=1}^K\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)\left\{{\lambda }_k{\xi }_k^p{\psi }_k\left(t_{ij}\right)\right\}\left\{{\lambda }_{k^{\prime }}{\xi }_{k^{\prime }}^p{\psi }_{k^{\prime }}\left(t_{ij}\right)\right\}+{\sigma }^2$; and (3) the cross-covariance $\text{Cov}\left(Y_{ij}^p,Y_{i{j}^{\prime }}^{p^{\prime }}\right)=\sum _{k=1}^K\sum _{k^{\prime }=1}^K\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)\left\{{\lambda }_k{\xi }_k^p{\psi }_k\left(t_{ij}\right)\right\}\left\{{\lambda }_{k^{\prime }}{\xi }_{k^{\prime }}^{p^{\prime }}{\psi }_{k^{\prime }}\left(t_{i{j}^{\prime }}\right)\right\}$. Although the general formulation permits component correlations (ie $\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)\ne 0$), practical implementations may assume independence ($\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)=0$ when $k\ne k^{\prime }$) to simplify likelihood construction and improve computational efficiency. Importantly, even under the independence assumption $\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)=0$ for $k\ne k^{\prime }$, the model still induces a cross-feature dependence, since $\text{Cov}\left(Y_{ij}^p,Y_{i{j}^{\prime }}^{p^{\prime }}\right)=\sum _{k=1}^K{\lambda }_k^2{\xi }_k^p{\xi }_k^{p^{\prime }}{\psi }_k\left(t_{ij}\right){\psi }_k\left(t_{i{j}^{\prime }}\right)$, which is generally nonzero for $p\ne p^{\prime }$. Thus, uncorrelated latent components do not imply uncorrelated observed features.

The first and second components in Model (2) correspond to the deterministic and random parts, respectively, reflecting the mean structure and subject-specific deviations induced by the supervised subject loadings. This modeling framework generalizes several well-known approaches in functional data analysis. Under specific constraints, it subsumes both the multivariate functional mixed model (MFMM) (Li et al. 2022) and the multivariate Karhunen–Loève (KL) expansion (Happ and Greven 2018). The MFMM can be recovered by imposing: (i) orthogonality of ${\psi }_k(\cdot )$ across $k$; (ii) subject-feature-specific scores $U_{ik}{\xi }_k^p=\int V_i^p\left(t\right){\psi }_k\left(t\right)dt$; (iii) no shared subject-specific effects across features; and (iv) unit scale parameters. The multivariate KL expansion is obtained under the constraints: (i) eigenfunctions of the form ${\psi }_k(\cdot )={\xi }_k{\psi }_k(\cdot )$ with orthogonality across $k$; and (ii) subject scores $U_{ik}=\sum _{p=1}^P\int {\xi }_k^p{V}_i^p\left(t\right){\psi }_k\left(t\right)dt$ for $k=\mathrm{1,2},\dots ,K$. Both models share the same mean function $\sum _{k=1}^K\left({\mathbf{x}}_i^{\prime }{\beta }_k\right){\xi }_k^p{\psi }_k(\cdot )$ for the $p\text{th}$ feature. Thus, Model (2) not only unifies and generalizes these classical decompositions but also embeds them within a supervised, multivariate framework that accommodates structured feature dependencies, temporal dynamics, and covariate guidance. This extension is well suited to high-dimensional omics data, where scalable inference, interpretability, and individualized prediction are critical modeling objectives.

2.2. Modeling survival using supervised latent trajectories

Let $h_i\left(t\right)$ denote the hazard function for subject $i$ at time $t$, formally defined as $h_i\left(t\right)={\text{lim}}_{\Delta t\to 0}P\left(T_i<\right.\left.t+\Delta t\mid T_i>t\right)/\Delta t$. In our joint modeling framework, we assume that $h_i\left(t\right)$ depends on the latent longitudinal process $V_i\left(t\right)$, which is unobserved and must be inferred from the data. We specify a proportional hazards model of the form

$$h_i\left(t\right)=h_0\left(t\right)\text{exp}\left\{{\mathbf{w}}_i^{\prime }\alpha +F\left({\mathbf{V}}_i,t\right)\right\},$$ (3)

where $h_0\left(t\right)$ is the unspecified baseline hazard, ${\mathbf{w}}_i\subseteq {\mathbf{x}}_i$ is a vector of baseline covariates for subject $i,\alpha$ is a vector of unknown coefficients associated with ${\mathbf{w}}_i$, and $F\left({\mathbf{V}}_i,t\right)$ is a functional capturing the influence of the latent trajectory ${\mathbf{V}}_i$ on the hazard at time $t$. The parameter $\alpha$ encodes the direct effects of the baseline covariates, while the function $F(\cdot )$ characterizes their indirect effects via the longitudinal process.

Model (3) allows for flexible specification of the dependency between the longitudinal and survival processes through the functional term $F\left({\mathbf{V}}_i,t\right)$. Different choices of $F(\cdot )$ induce varying levels of model complexity and interpretability. In this work, for both the simulation studies and real data analysis, we adopt $F\left({\mathbf{V}}_i,t\right)={\zeta }_i^{\prime }\nu$, where ${\zeta }_i={\left({\zeta }_{i1},{\zeta }_{i2},\dots ,{\zeta }_{iK}\right)}^{\prime }$ are the supervised low-rank subject-specific loadings obtained from the longitudinal model, and $v\in {\mathbb{R}}^K$ is a vector of unknown regression coefficients. This specification reduces the dimensionality of the survival component by projecting the high-dimensional, time-varying feature space into a low-dimensional subject-specific representation. Instead of modeling the effect of $P$ longitudinal features $(P\gg N)$, the survival model includes only $K$ derived components, enhancing both interpretability and computational feasibility. Here, interpretability refers to summarizing the high-dimensional longitudinal features through a small number of supervised latent components that capture the dominant temporal patterns across features, rather than requiring direct interpretation of hundreds of individual trajectories. However, this specification implies a proportional-hazards structure for the latent components, since their subject-specific loadings ${\zeta }_i$ enter the hazard model as time-invariant covariates.

To formalize this framework, we refer to the proposed joint modeling approach as supFTSVD-JM, which integrates the longitudinal model defined in Equation (2) with the survival model given in Equation (3). In this setup, the supervised low-rank decomposition captures latent subject-specific trajectories via ${\zeta }_i$, which are then used to model the hazard function through a low-dimensional representation. This construction enables efficient inference and dynamic prediction in settings with high-dimensional longitudinal data $(P\gg N)$. The supFTSVD-JM framework facilitates coherent modeling of both longitudinal and survival processes while maintaining scalability, interpretability, and statistical rigor.

Other specifications for $F\left({\mathbf{V}}_i,t\right)$ are also possible. For instance, one may define $F\left({\mathbf{V}}_i,t\right)=\sum _{k=1}^K{\zeta }_{ik}{\psi }_k\left(t\right)v_k\left(t\right)$, where $v_k\left(t\right)$ is a smooth coefficient function defined on $[0,\tau ]$. This formulation incorporates $K$ time-varying covariates into the survival model, yielding a concurrent modeling structure. In practice, model fitting under this specification would involve representing each $v_k\left(t\right)$ using basis functions such as B-splines. Alternatively, cumulative effects of the longitudinal trajectories can be captured by defining $F\left({\mathbf{V}}_i,t\right)=\sum _{k=1}^K{\int }_0^t{\zeta }_{ik}{\psi }_k\left(s\right)v_k\left(s\right)ds$, which allows the survival risk to depend on the historical values of the longitudinal process up to time $t$.

Various specifications are available for the baseline hazard $h_0\left(t\right)$ in Model (3). For instance, $h_0\left(t\right)$ may be assumed to follow a constant, linear, or more flexible nonlinear function of time. In the case of a nonlinear specification, it is common to represent log $h_0\left(t\right)$ using B-spline basis functions. The selection of $h_0\left(t\right)$ influences the overall model complexity, particularly in conjunction with the form of $F(\cdot ,\cdot )$. Across the simulation studies, we consider both constant and piecewise–constant baseline hazard specifications, while the ADNI analysis exclusively adopts a piecewise–constant baseline hazard.

3. Estimation of supFTSVD-JM parameters

The proposed supFTSVD-JM framework comprises two sub-models, each with distinct set of parameters. Estimating these parameters jointly from the observed data is essential for coherent inference. A straightforward strategy is to fit the two sub-models separately using their standard estimation procedures. Typically, the longitudinal sub-model defined in Equation (2) is fitted first to recover subject-specific random effects $\left(U_{ik}\right)$, which are then used as known covariates while fitting the survival sub-model defined in Equation (3) next. This approach, commonly referred to as two-stage (TS) estimation, offers computational simplicity but is known to introduce bias, especially when the estimated random effects are treated as fixed in the second stage. To address this limitation, we develop a penalized MCEM algorithm to jointly estimate all parameters in the supFTSVD-JM framework. In what follows, we describe our proposed joint estimation method.

Joint estimation is challenging due to the presence of unobserved shared random effects. Alam et al. (2024) proposed an EM algorithm for fitting supFTSVD to high-dimensional multivariate longitudinal data. We build on this methodology by extending it to a joint modeling context following the structure in Li et al. (2022). Specifically, we construct the complete-data log-likelihood assuming that the joint distribution of latent random effects is known, and then take its expectation conditional on the observed longitudinal and survival data. The survival component is modeled through the hazard function specified in Equation (3), which induces a survival likelihood via the cumulative hazard. The longitudinal component is modeled under the assumption that $U_{ik}~N\left(0,{\sigma }_k^2\right)$ and $\text{Cov}\left(U_{ik},U_{i{k}^{\prime }}\right)=0$ for $k\ne k^{\prime }$, while the measurement errors follow ${ϵ}_{ij}^p~N\left(0,{\sigma }^2\right)$. To ensure identifiability, we impose $\text{Var}\left(U_{ik}\right)=1$ and fix ${\sigma }_k^2=1$ for all $k$, thereby eliminating the need to estimate these variance parameters.

For likelihood construction, we define the following vectors and matrices: ${\mathbf{U}}_i={\left[U_{i1},U_{i2},\dots ,U_{iK}\right]}^{\prime },{\beta }_{{\mathbf{x}}_i}={\left[{\mathbf{x}}_i^{\prime }{\beta }_1,{\mathbf{x}}_i^{\prime }{\beta }_2,\dots ,{\mathbf{x}}_i^{\prime }{\beta }_K\right]}^{\prime },{\psi }_{ik}={\left[{\psi }_k\left(t_{i1}\right),{\psi }_k\left(t_{i2}\right),\dots ,{\psi }_k\left(t_{i{J}_i}\right)\right]}^{\prime },t_i={\left[t_{i1},t_{i2},\dots ,t_{i{J}_i}\right]}^{\prime },{\mathbf{Y}}_i=\left[{\mathbf{Y}}_{i1},{\mathbf{Y}}_{i2},\dots ,{\mathbf{Y}}_{i{J}_i}\right]$, and ${ℰ}_i=\left[{ϵ}_{i1},{ϵ}_{i2},\dots ,{ϵ}_{i{J}_i}\right]$. These notations facilitate a mixed-effects model-like representation of Equation (2). The matrices ${\mathbf{Y}}_i$ and ${ℰ}_i$ have dimensions $P\times J_i$, and we let $\text{vec}(\cdot )$ denote the column-wise vectorization operator. The longitudinal sub-model can be expressed compactly as ${\mathbf{y}}_i={\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i}+{\mathbf{H}}_i{\mathbf{U}}_i+{\mathbf{e}}_i$, where ${\mathbf{y}}_i=\text{vec}\left({\mathbf{Y}}_i\right),{\mathbf{e}}_i=\text{vec}\left({ℰ}_i\right)$, and ${\mathbf{H}}_i$ is an $M_i\times K$ matrix whose $k\text{th}$ column is given by ${\lambda }_k\left({\xi }_k\otimes {\psi }_{ik}\right)$, with $M_i=P{J}_i$. Let ${\theta }_K$ denote the full set of model parameters for the proposed rank-$K$ supFTSVD-JM, comprising the functional process parameters ${\left\{{\beta }_k,{\xi }_k,{\psi }_k(\cdot ),{\lambda }_k\right\}}_{k=1}^K$, noise variance ${\sigma }^2$, and survival parameters $\{\alpha ,\nu \}$.

The mixed-effects model-like representation facilitates the development of the proposed penalized MCEM algorithm. By treating random effects ${\mathbf{U}}_i$ as latent variables, we define the complete-data joint likelihood and denote its logarithm by $\ell \left({\theta }_K;{\left\{{\mathbf{y}}_i,t_i,{\mathbf{U}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N\right)$. Conditional on the shared random effects, the longitudinal and survival processes are independent. Under standard normality assumptions and assuming independence across subjects, the complete-data log-likelihood is given by

$$\begin{gathered} \ell \left({\theta }_K;{\left\{{\mathbf{y}}_i,t_i,{\mathbf{U}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N\right)=\sum _{i=1}^N\text{log}{\varphi }_{M_i}\left({\mathbf{y}}_i;{\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i}+{\mathbf{H}}_i{\mathbf{U}}_i,{\sigma }^2{\mathbf{I}}_{M_i}\right)+\sum _{i=1}^N\text{log}{\varphi }_K\left({\mathbf{U}}_i;\mathbf{0},{\mathbf{I}}_K\right) \\ +\sum _{i=1}^N{\delta }_i\left\{\text{log}{h}_0\left(T_i\right)+{\mathbf{w}}_i^{\prime }\alpha +F\left({\mathbf{V}}_i,T_i\right)\right\}+\sum _{i=1}^N\text{log}\left\{S_i\left(T_i\right)\right\}, \end{gathered}$$ (4)

where ${\varphi }_d(\mathbf{y};\mu ,\Sigma )$ denotes the $d$-variate normal density with mean $\mu$ and covariance matrix $\Sigma$, and ${\mathbf{I}}_K$ denotes the $K\times K$ identity matrix. The subject-specific survival function is given by $S_i\left(T_i\right)=\text{exp}\left[-{\int }_0^{T_i}{h}_0\left(t\right)\text{exp}\left\{{\mathbf{w}}_i^{\prime }\alpha +F\left({\mathbf{V}}_i,t\right)\right\}dt\right]$. The evaluation of this integral depends on the choice of $h_0\left(t\right)$, which affects computational complexity. In our simulation study, we assume a constant baseline hazard $h_0\left(t\right)=\kappa$. For the ADNI data analysis, we adopt a piecewise-constant hazard function defined by $h_0(t)=\sum _{b=1}^B{\kappa }_{b}I\left({\tau }_{b-1}<t\le {\tau }_b\right)$, where $0={\tau }_0<{\tau }_1<\dots <{\tau }_B=\tau$ and $I(\cdot )$ is the indicator function.

We aim to maximize the likelihood induced by the proposed joint model with respect to the components of ${\theta }_K$. This optimization is non-trivial due to several factors, including the latent nature of ${\mathbf{U}}_i$, the involvement of functional parameters (eg ${\psi }_k(\cdot )$), and the presence of parameter constraints such as unit-norms. To address these challenges, we develop a penalized MCEM algorithm. The procedure is iterative: in each iteration, the E-step approximates the conditional expectation of the complete-data log-likelihood given the observed data and the current parameter estimates; the M-step then updates ${\theta }_K$ by maximizing the expected complete-data log-likelihood with penalties or constraints as required. This approach enables stable and scalable estimation in the high-dimensional joint modeling framework.

3.1. E-step

Let ${\theta }_K^{\left(s\right)}$ denote the current estimate of the parameter set ${\theta }_K$ at iteration $s$ of the MCEM algorithm. The E-step involves computing the conditional expectation of the complete-data log-likelihood $\ell \left({\theta }_K;{\left\{{\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{U}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N\right)$ given the observed data ${\left\{{\mathbf{y}}_i,{\mathbf{t}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N$ and the current parameter estimate ${\theta }_K^{\left(s\right)}$. This expectation is analytically intractable because the conditional distribution $f\left({\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right)$ is unknown in closed form. However, following Li et al. (2022), for any smooth function $g(\cdot )$, the conditional expectation can be expressed as

$$E_{{\theta }_K^{\left(s\right)}}\left[g\left({\mathbf{U}}_i\right)\mid {\mathbf{y}}_i,{\mathbf{t}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right]=\frac{\int g\left({\mathbf{U}}_i\right)f_{{\theta }_K^{\left(s\right)}}\left(T_i,{\delta }_i\mid {\mathbf{U}}_i\right)f_{{\theta }_K^{\left(s\right)}}\left({\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i\right)d{\mathbf{U}}_i}{\int f_{{\theta }_K^{\left(s\right)}}\left(T_i,{\delta }_i\mid {\mathbf{U}}_i\right)f_{{\theta }_K^{\left(s\right)}}\left({\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i\right)d{\mathbf{U}}_i},$$ (5)

where $f_{{\theta }_K^{\left(s\right)}}(\cdot )$ denotes the probability density functions parameterized by the current estimates. This formulation enables numerical approximation of the expected log-likelihood via Monte Carlo integration. In practice, we apply this expression with $g\left({\mathbf{U}}_i\right)={\mathbf{U}}_i$ and $g\left({\mathbf{U}}_i\right)=\text{exp}\left({\mathbf{U}}_i^{\prime }v\right)$ to evaluate terms relevant to the E-step.

Evaluating the expectation in Equation (5) involves two main steps: (i) sampling from $f_{{\theta }_K^{\left(s\right)}}\left({\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i\right)$, and (ii) approximating the integral numerically. We begin by noting that step (i) reduces to generating samples from a multivariate normal (MVN) distribution. Under the assumed normality, we have ${\mathbf{U}}_i~N_K\left(\mathbf{0},{\mathbf{I}}_K\right)$ and ${\mathbf{e}}_i~N_{M_i}\left(\mathbf{0},{\sigma }^2{\mathbf{I}}_{M_i}\right)$, which implies that the marginal distribution of ${\mathbf{y}}_i$ is $N_{M_i}\left({\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i},{\mathbf{H}}_i{\mathbf{H}}_i^{\prime }+{\sigma }^2{\mathbf{I}}_{M_i}\right)$ and the conditional distribution ${\mathbf{y}}_i\mid {\mathbf{U}}_i$ is $N_{M_i}\left({\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i}+{\mathbf{H}}_i{\mathbf{U}}_i,{\sigma }^2{\mathbf{I}}_{M_i}\right)$. It follows that the conditional distribution ${\mathbf{U}}_i\mid {\mathbf{y}}_i$ is also MVN, with mean vector given by $E_{{\theta }_K}\left[{\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i\right]={\mathbf{H}}_i^{\prime }{\left({\sigma }^2{\mathbf{I}}_{M_i}+{\mathbf{H}}_i{\mathbf{H}}_i^{\prime }\right)}^{-1}\left({\mathbf{y}}_i-{\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i}\right)$ and covariance matrix ${\text{Cov}}_{{\theta }_K}\left[{\mathbf{U}}_i\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i\right]={\sigma }^2{\left({\sigma }^2{\mathbf{I}}_K+{\mathbf{H}}_i^{\prime }{\mathbf{H}}_i\right)}^{-1}$. Let ${\tilde{\mathbf{U}}}_{i1}^{\left(s\right)},{\tilde{\mathbf{U}}}_{i2}^{\left(s\right)},\dots ,{\tilde{\mathbf{U}}}_{i{N}_u}^{\left(s\right)}$ be a Monte Carlo sample of size $N_u$ drawn from this conditional MVN distribution. The expectation in Equation (5) can then be approximated by a weighted average $\sum _{h=1}^{N_u}g\left({\tilde{\mathbf{U}}}_{ih}^{\left(s\right)}\right)w_{ih}^{\left(s\right)}$, where the weights are given by $w_{ih}^{\left(s\right)}=f\left(T_i,{\delta }_i\right.|\left.{\tilde{\mathbf{U}}}_{ih}^{\left(s\right)},{\theta }_K^{\left(s\right)}\right)/\sum _{h^{\prime }=1}^{N_u}f\left(T_i,{\delta }_i\mid {\tilde{\mathbf{U}}}_{i{h}^{\prime }}^{\left(s\right)},{\theta }_K^{\left(s\right)}\right)$. These weights incorporate survival information by upweighting sampled values of ${\mathbf{U}}_i$ that better explain the observed event or censoring time under the current model estimate ${\theta }_K^{\left(s\right)}$.

To estimate the functional parameters ${\psi }_k(\cdot )$ for $k=\mathrm{1,2},\dots ,K$, we penalize their reproducing kernel Hilbert space (RKHS) norms. We define the penalized conditional expectation of the complete-data log-likelihood in Equation (4) as

$$Q\left({\theta }_K\right)=E_{{\theta }_K^{\left(s\right)}}\left\{\ell \left({\theta }_K;{\left\{{\mathbf{y}}_i,{\mathbf{U}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N\right)\mid {\left\{{\mathbf{y}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}}_{i=1}^N\right\}+\sum _{k=1}^K{\eta }_k{‖{\psi }_k‖}_{\mathscr{H}},$$ (6)

where ${\eta }_1,{\eta }_2,\dots ,{\eta }_K$ are tuning parameters that control the smoothness of the estimated functions ${\psi }_1(\cdot ),{\psi }_2(\cdot ),\dots ,{\psi }_K(\cdot )$, respectively. Let ${\tilde{\mathbf{U}}}_i^{\left(s\right)}$ and ${\Gamma }_i^{\left(s\right)}$ denote the numerical approximations of $E_{{\theta }_K^{\left(s\right)}}\left({\mathbf{U}}_i\mid \left\{{\mathbf{y}}_i,{\mathbf{t}}_i,T_i,{\delta }_i,{\mathbf{x}}_i\right\}\right)$ and ${\text{Cov}}_{{\theta }_K^{\left(s\right)}}\left({\mathbf{U}}_i\mid \left\{{\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i,T_i,{\delta }_i\right\}\right)$, respectively. With these notations and using $F\left({\mathbf{V}}_i,T_i\right)={\zeta }_i^{\prime }v$, we derive the simplified expression

$$\begin{gathered} Q\left({\theta }_K\right)=\frac{M}{2}\text{log}\left(1/{\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{i=1}^N\left[{‖{\mathbf{y}}_i-{\mathbf{H}}_i{\beta }_{{\mathbf{x}}_i}-{\mathbf{H}}_i{\tilde{\mathbf{U}}}_i^{\left(s\right)}‖}^2+\frac{1}{{\sigma }^2}\text{tr}\left\{\left({\mathbf{H}}_i^{\prime }{\mathbf{H}}_i+{\sigma }^2{\mathbf{I}}_K\right){\Gamma }_i^{\left(s\right)}\right\}+{‖{\tilde{\mathbf{U}}}_i^{\left(s\right)}‖}^2\right] \\ +\sum _{i=1}^N{\delta }_i\left[\text{log}{h}_0\left(T_i\right)+{\mathbf{w}}_i^{\prime }\alpha +{\beta }_{{\mathbf{x}}_i}^{\prime }v+{\left\{{\tilde{\mathbf{U}}}_i^{\left(s\right)}\right\}}^{\prime }v\right] \\ -\sum _{i=1}^N{H}_0\left(T_i\right)\text{exp}\left\{{\mathbf{w}}_i^{\prime }\alpha +{\beta }_{{\mathbf{x}}_i}^{\prime }v\right\}{E}_{{\theta }_K^{\left(s\right)}}\left[\text{exp}\left({\mathbf{U}}_i^{\prime }v\right)\mid {\mathbf{y}}_i,{\mathbf{t}}_i,{\mathbf{x}}_i,T_i,{\delta }_i\right]+\sum _{k=1}^K{\eta }_k{‖{\psi }_k‖}_{\mathscr{H}}, \end{gathered}$$ (7)

where $M=\sum _{i=1}^N{M}_i$, and $H_0\left(a\right)={\int }_0^a{h}_0\left(t\right)dt$ denotes the cumulative baseline hazard function.

3.2. M-step

At iteration $(s+1)$, we update the parameter estimates by maximizing the penalized expected complete-data log-likelihood $Q\left({\theta }_K\right)$ in Equation (7) with respect to ${\theta }_K$. This optimization is nontrivial, as closed-form solutions are unavailable for several components of ${\theta }_K$. For instance, the score equations for the fixed-effect vectors ${\beta }_k$ do not admit analytical solutions. In such cases, we employ a one-step Newton–Raphson update to obtain the new estimates. For the functional parameters ${\psi }_k(\cdot )(k=\mathrm{1,2},\dots ,K)$, we invoke the Representer Theorem to express each ${\psi }_k$ in a finite-dimensional basis and derive corresponding updates. All components of ${\theta }_K$ are updated sequentially within each M-step to obtain ${\theta }_K^{(s+1)}$.

To simplify the representation, let $P(\mathbf{A},\mathbf{a},\mathbf{b},c)=\Vert \text{vec}\left(\mathbf{A}\right)-\text{vec}(\mathbf{a}\otimes \mathbf{b})c{\Vert }^2$ for some $\mathbf{A}\in {\mathbb{R}}^{p\times m},\mathbf{a}\in {\mathbb{R}}^p,\mathbf{b}\in {\mathbb{R}}^m$, and $c\in \mathbb{R}$. Further, let ${\gamma }_{iK}^{\left(s\right)}\left({\beta }_k\right)={\widehat{H}}_0^{\left(s\right)}\left(T_i\right)\text{exp}\left\{{\mathbf{w}}_i^{\prime }{\widehat{\alpha }}^{\left(s\right)}+\sum _{l\ne k=1}^K\left({\mathbf{x}}_i^{\prime }{\widehat{\beta }}_l^{\left(s\right)}\right){\widehat{v}}_l^{\left(s\right)}+{\widehat{v}}_k^{\left(s\right)}\left({\mathbf{x}}_i^{\prime }{\beta }_k\right)\right\}{E}_{{\theta }_K^{\left(s\right)}}\left[\text{exp}\left({\mathbf{U}}_i^{\prime }v\right)\right]$, where ${\widehat{H}}_0^{\left(s\right)}(\cdot )$ is the estimate of $H_0(\cdot )$ from iteration $s$, and $v_l^{\left(s\right)}$ is an element of $v^{\left(s\right)}\in {\theta }_K^{\left(s\right)}$. For each $k$, we update ${\beta }_k^{(s+1)},{\xi }_k^{(s+1)}$, and ${\psi }_k^{(s+1)}(\cdot )$ by solving the following optimization problems:

$$\begin{gathered} \left(i\right)\underset{{\beta }_k}{\text{max}}\sum _{i=1}^N\left\{-\frac{1}{{\sigma }^2}P\left(R_{ik}^{\left(s\right)}-{\lambda }_k{U}_{ik}\left({\widehat{\xi }}_k^{\left(s\right)}\otimes {\widehat{\psi }}_{ik}^{\left(s\right)}\right),{\widehat{\xi }}_k^{\left(s\right)},{\widehat{\psi }}_{ik}^{\left(s\right)},{\lambda }_k{x}_i^{\prime }{\beta }_k\right)+{\delta }_i{v}_k{x}_i^{\prime }{\beta }_k-{\gamma }_{iK}^{\left(s\right)}\left({\beta }_k\right)\right\}, \\ \left(ii\right)\underset{{\xi }_k}{\text{min}}\sum _{i=1}^{N}P\left({\tilde{R}}_{ik}^{\left(s\right)},{\xi }_k,{\psi }_{ik}^{\left(s\right)},\sqrt{{\left({\tilde{U}}_{ik}^{\left(s\right)}+x_i^{\prime }{\beta }_k^{\left(s+1\right)}\right)}^2+{\Gamma }_{i,kk}^{\left(s\right)}}\right), \\ \left(iii\right)\underset{{\psi }_k\in {ℒ}^2\left(𝒯\right)}{\text{min}}\sum _{i=1}^{N}P\left({\tilde{R}}_{ik}^{\left(s\right)},{\xi }_k,{\psi }_{ik}^{\left(s\right)},\sqrt{{\left({\tilde{U}}_{ik}^{\left(s\right)}+x_i^{\prime }{\beta }_k^{\left(s+1\right)}\right)}^2+{\Gamma }_{i,kk}^{\left(s\right)}}\right)+\sum _{k=1}^K{\eta }_k‖{\psi }_k‖\mathscr{H}, \end{gathered}$$ (8)

where ${\mathbf{R}}_{ik}^{\left(s\right)}$ and ${\tilde{\mathbf{R}}}_{ik}^{\left(s\right)}$ are $P\times J_i$ matrices with elements

$$\begin{gathered} R_{ijk}^{p,\left(s\right)}=Y_{ij}^p-\sum _{l\ne k=1}^K\left({\tilde{U}}_{il}^{\left(s\right)}+{\mathbf{x}}_i^{\prime }{\widehat{\beta }}_l^{\left(s\right)}\right){\widehat{\xi }}_l^{p,\left(s\right)}{\widehat{\psi }}_l^{\left(s\right)}\left(t_{ij}\right), \\ {\tilde{R}}_{ijk}^{p,\left(s\right)}=\frac{R_{ijk}^{p,\left(s\right)}\left\{{\tilde{U}}_{ik}^{\left(s\right)}+{\mathbf{x}}_i^{\prime }{\widehat{\beta }}_k^{(s+1)}\right\}-\sum _{l\ne k=1}^K{\Gamma }_{i,kl}^{\left(s\right)}{\widehat{\xi }}_l^{p,\left(s\right)}{\widehat{\psi }}_l^{\left(s\right)}\left(t_{ij}\right)}{\sqrt{{\left\{{\tilde{U}}_{ik}^{\left(s\right)}+{\mathbf{x}}_i^{\prime }{\widehat{\beta }}_k^{(s+1)}\right\}}^2+{\Gamma }_{i,kk}^{\left(s\right)}}} \end{gathered}$$

at cell $(p,j)$, respectively.

An exact solution for optimization problem (i) in Equation (8) does not exist. We therefore apply a one-step Newton method to update ${\beta }_k$ using the current estimate, gradient, and information matrix. Specifically, we compute ${\tilde{\beta }}_k^{(s+1)}={\widehat{\beta }}_k^{\left(s\right)}+{\mathbf{I}}_{{\beta }_k}^{-1}{\mathbf{S}}_{{\beta }_k}$, where with some abuse of notation and suppressing dependencies on other parameters, ${\mathbf{S}}_{{\beta }_k}$ and ${\mathbf{I}}_{{\beta }_k}$ represent the gradient and information matrix for ${\beta }_k$ based on (i). Since this update does not automatically satisfy the unit norm condition for identifiability, we set ${\widehat{\beta }}_k^{(s+1)}={\tilde{\beta }}_k^{(s+1)}/‖{\tilde{\beta }}_k^{(s+1)}‖$.

Our joint modeling framework does not alter optimization problems (ii) and (iii), allowing us to adopt the procedure in Alam et al. (2024). The objective in (ii) is quadratic in ${\xi }_k$, so a closed-form solution for ${\widehat{\xi }}_k^{p,(s+1)}$ exists. For estimating ${\psi }_k(\cdot )$, we use the reproducing kernel Hilbert space (RKHS) regression approach, which yields an exact solution analogous to ridge regression. Specifically, we treat ${\psi }_k(\cdot )$ as an element of an RKHS $\mathscr{H}$ and optimize problem (iii) of Equation (8). By the Representer Theorem, the optimizer admits a finite-dimensional representation ${\psi }_k(t)=\sum _{\ell =1}^M{\alpha }_{k\ell }K\left(t,t_{\ell }\right)$, where $\left\{t_1,\dots ,t_M\right\}$ denotes the collection of all observed measurement times across all subjects, $K(\cdot ,\cdot )$ is the reproducing kernel, and ${\alpha }_{k\ell }$ are unknown coefficients. In practice, we use the rescaled Bernoulli polynomial kernel so that ${\psi }_k$ has square-integrable second derivatives, and estimation reduces to solving a linear system for ${\alpha }_k={\left({\alpha }_{k1},\dots ,{\alpha }_{kM}\right)}^{\prime }$ at each M-step. This approach assumes that the tuning parameters ${\eta }_k$ are known; in practice, we select them using the data-driven procedure in Section 3.3. For both optimization problems, we normalize the resulting estimates to satisfy the unit norm constraint.

Due to the constraint ${\sigma }_k^2=1$, we do not update these parameters. However, this imposes the requirement of updating ${\lambda }_k$ at each iteration. To update $\lambda =\left[{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_K\right]$ at iteration $(s+1)$, we minimize

$$\sum _{i=1}^N{‖{\mathbf{y}}_i-{\mathbf{Z}}_i^{(s+1)}\lambda ‖}^2+{\lambda }^{\prime }\left[\left\{{\left({\tilde{\mathbf{H}}}_i^{(s+1)}\right)}^{\prime }{\tilde{\mathbf{H}}}_i^{(s+1)}\right\}\odot {\Gamma }_i^{\left(s\right)}\right]\lambda ,$$

where ${\mathbf{Z}}_i^{(s+1)}$ and ${\tilde{\mathbf{H}}}_i$ are $M_i\times K$ matrices with the $k\text{th}$ column defined by $\left({\mathbf{x}}_i^{\prime }{\widehat{\beta }}_k^{(s+1)}+\right.\left.{\tilde{U}}_{ik}^{\left(s\right)}\right)\text{vec}\left({\widehat{\xi }}_k^{(s+1)}\otimes {\widehat{\psi }}_{ik}^{(s+1)}\right)$ and $\text{vec}\left({\widehat{\xi }}_k^{(s+1)}\otimes {\widehat{\psi }}_{ik}^{(s+1)}\right)$, respectively. The operator $\odot$ denotes the element-wise multiplication of two matrices with identical dimensions. The update formula then reduces to

$${\lambda }^{(s+1)}={\left[\sum _{i=1}^N{\left({\mathbf{Z}}_i^{(s+1)}\right)}^{\prime }{\mathbf{Z}}_i^{(s+1)}+\sum _{i=1}^N\left\{{\left({\tilde{\mathbf{H}}}_i^{(s+1)}\right)}^{\prime }{\tilde{\mathbf{H}}}_i^{(s+1)}\right\}\odot {\Gamma }_i^{\left(s\right)}\right]}^{-1}\left\{\sum _{i=1}^N{\left({\mathbf{Z}}_i^{(s+1)}\right)}^{\prime }{\mathbf{y}}_i\right\}.$$

We also have an exact updating formula for the error variance associated with the longitudinal sub-model:

$${\sigma }^{2,(s+1)}=\frac{1}{M}\sum _{i=1}^N\left[{‖{\mathbf{y}}_i-{\widehat{\mathbf{H}}}_i^{(s+1)}{\beta }_{{\mathbf{x}}_i}^{(s+1)}-{\widehat{\mathbf{H}}}_i^{(s+1)}{\tilde{\mathbf{U}}}_i^{\left(s\right)}‖}^2+\text{tr}\left\{{\left({\mathbf{H}}_i^{(s+1)}\right)}^{\prime }{\mathbf{H}}_i^{(s+1)}{\Gamma }_i^{\left(s\right)}\right\}\right],$$

where ${\widehat{\mathbf{H}}}_i^{(s+1)}$ and ${\widehat{\beta }}_{{\mathbf{x}}_i}^{(s+1)}$ are the updated estimates of ${\mathbf{H}}_i$ and ${\beta }_{{\mathbf{x}}_i}$, respectively. A feature-specific error variance, when assumed for the supFTSVD model (as in Section 2.1), can also be accommodated within this estimation framework. In that case, the same updating formula applies componentwise after reconstructing the feature-specific longitudinal vector and design matrix, ie by retaining only the entries of ${\mathbf{y}}_i$ and ${\widehat{\mathbf{H}}}_i^{(s+1)}$ corresponding to feature $p$. This yields a separate variance estimate ${\sigma }_p^2$, which is specific to feature $p$.

Alternative choices for $h_0\left(t\right)$ are discussed in Section 3. Among these, the constant and piecewise-constant baseline hazards are implemented in this manuscript. For the constant baseline hazard, the update formula for $\kappa$ reduces to

$${\widehat{\kappa }}^{(s+1)}=\frac{\sum _{i=1}^N{\delta }_i}{\sum _{i=1}^N{T}_i\text{exp}\left\{{\mathbf{w}}_i^{\prime }{\alpha }^{\left(s\right)}+{\left({\beta }_{{\mathbf{x}}_i}^{(s+1)}\right)}^{\prime }{v}^{\left(s\right)}\right\}{E}_{{\theta }_K^{\left(s\right)}}\left\{\text{exp}\left({\mathbf{U}}_i^{\prime }v\right)\right\}},$$ (9)

where $\kappa =h_0\left(t\right)$ for any $t>0$. In contrast, when the piecewise-constant baseline hazard $h_0\left(t\right)=\sum _{b=1}^B{\kappa }_{b}I\left({\tau }_{b-1}<t\le {\tau }_b\right)$ is considered, the update formula for ${\kappa }_b$ is given by

$${\widehat{\kappa }}_b^{(s+1)}=\sum _{i=1}^N\frac{{\delta }_{i}I\left({\tau }_{b-1}<t_i\le {\tau }_b\right)}{\sum _{i=1}^N\text{exp}\left\{{\mathbf{w}}_i^{\prime }{\alpha }^{\left(s\right)}+{\left({\beta }_{{\mathbf{x}}_i}^{(s+1)}\right)}^{\prime }{v}^{\left(s\right)}\right\}{E}_{{\theta }_K^{\left(s\right)}}\left\{\text{exp}\left({\mathbf{U}}_i^{\prime }v\right)\right\}\left(T_i\wedge {\tau }_b-{\tau }_{b-1}\right)I\left(T_i\ge {\tau }_{b-1}\right)},$$ (10)

where $c\wedge d=\text{min}(c,d)$. In this setting, the choice of change points ${\left\{{\tau }_b\right\}}_{b=1}^B$ determines the number of parameters to be estimated. Selecting ${\tau }_b$ as the distinct observed failure times may lead to an increased number of parameters. Therefore, in practice, it is often chosen at the deciles of the observed survival times, employing $B=9$. For the ADNI data analysis, we adopt this approach and specify the change points at the deciles of the observed survival times.

Finally, we update both $\alpha$ and $\nu$ using the one-step Newton method, following the approach used for updating ${\beta }_k$. Let ${\mathbf{S}}_{\alpha }$ and ${\mathbf{S}}_{\nu }$ denote the gradients, and ${\mathbf{I}}_{\alpha }$ and ${\mathbf{I}}_{\nu }$ the corresponding information matrices for $\alpha$ and $v$ at iteration $s$. The derivations for these quantities are provided in the Supplementary Material. With these notations, the update formulas are given by

$${\widehat{\alpha }}^{(s+1)}={\widehat{\alpha }}^{\left(s\right)}+{\mathbf{I}}_{\alpha }^{-1}{\mathbf{S}}_{\alpha }\text{and}{\widehat{v}}^{(s+1)}={\widehat{v}}^{\left(s\right)}+{\mathbf{I}}_v^{-1}{\mathbf{S}}_v,$$

respectively.

3.3. Cross-validation-based tuning of rank and smoothing parameters

The proposed joint modeling framework involves two sets of tuning parameters: the model rank $K$ and the smoothing parameters $\left\{{\eta }_1,{\eta }_2,\dots ,{\eta }_K\right\}$ associated with the $K$ singular functions. Because the number of smoothing parameters depends on the rank, we determine $K$ prior to model fitting. Both sets of parameters are selected using data-driven strategies.

To select $K$, we adopt a grid search approach based on $g$-fold cross-validation, following the procedure in Lock and Li (2018). We use the term “$g$-fold” to avoid conflict with existing notation. Subjects are randomly partitioned into $g$ folds. When $n$ is not divisible by $g$, we distribute the excess subjects randomly after assigning an equal number to each fold. Each fold serves once as the test set, while the remaining $g-1$ folds comprise the training data. For each candidate rank $K$, we fit the supFTSVD-JM using the training data and evaluate the conditional likelihood in Equation (7) on the test data. Averaging over all $g$ folds yields the mean conditional likelihood $\bar{Q}\left({\widehat{\theta }}_K\right)$. The rank is incremented until the relative gain $\left\{\left(\bar{Q}\left({\widehat{\theta }}_K\right)-\bar{Q}\left({\widehat{\theta }}_{K-1}\right)\right)/\bar{Q}\left({\widehat{\theta }}_{K-1}\right)\right\}$ falls below a pre-specified threshold ${\kappa }_0$. We then select $K_{\text{opt}}=K$.

With $K_{\text{opt}}$ fixed, we determine the smoothing parameters ${\eta }_k$ using a $d$-fold cross-validation scheme within the first $s_0$ iterations of the MCEM algorithm. At each iteration and for each $k$, we randomly split subjects into $d$ folds, repeating the earlier partitioning logic. For each training fold, we estimate ${\widehat{\psi }}_k^{(s+1)}(\cdot )$ over a grid of candidate ${\eta }_k$ values and compute the squared correlation between $\text{vec}\left({\tilde{\mathbf{R}}}_{ik}^{\left(s\right)}\right)$ from the corresponding test data and its predicted counterpart. The smoothing parameter ${\eta }_k^{\text{opt}}$ is set as the value that maximizes the average squared correlation across all $d$ test sets. To clarify, although we perform $d$-fold cross-validation at each of the first $s_0$ MCEM iterations, only the set of optimal tuning parameters $\left\{{\eta }_1^{\text{opt}},{\eta }_2^{\text{opt}},\dots ,{\eta }_K^{\text{opt}}\right\}$ obtained at iteration $s_0$ is retained and used in all subsequent MCEM iterations. Optimal tuning sets identified at earlier iterations $\left(1,\dots ,s_0-1\right)$ are discarded because the corresponding parameter estimates ${\widehat{\theta }}_K^{\left(s\right)}$ remain preliminary and may be too unstable to yield reliable smoothing choices.

3.4. Initialization strategy for the MCEM algorithm

The proposed penalized MCEM algorithm is an iterative procedure that requires initial values for the model parameters, denoted by ${\theta }_K^{\left(0\right)}$. These parameters include components associated with both the longitudinal and survival sub-models. We adopt a modular initialization strategy by treating these sub-models separately when determining their respective starting values.

To initialize the longitudinal sub-model, we follow the procedure of Alam et al. (2024). We first construct the matrix $\mathbf{Y}=\left[{\mathbf{Y}}_1,{\mathbf{Y}}_2,\dots ,{\mathbf{Y}}_N\right]$, where each ${\mathbf{Y}}_i\in {\mathbb{R}}^{P\times J_i}$ contains the longitudinal measurements for subject $i$. We then compute the first $K$ left singular vectors of $\mathbf{Y}$ and use them as initial estimates for ${\xi }_1,\dots ,{\xi }_K$. Next, for each $k$, we compute subject-specific aggregated signals $\sum _{p=1}^P{Y}_{ij}^p{\xi }_k^p$ at each time point $t_{ij}$ and average these values across time for subject $i$ to obtain initial estimates for the latent scores ${\zeta }_{ik}$. These are treated as noisy observations of ${\mathbf{x}}_i^{\prime }{\beta }_k+U_{ik}$. We then apply the RKHS regression approach from Section 3 to estimate the initial singular functions ${\psi }_k(\cdot )$ based on ${\zeta }_{ik}$ and ${\xi }_k$. Finally, the initialization is completed by using the update formulas in Section 3.2 to obtain initial values for ${\lambda }_k$ and ${\sigma }^2$.

For the survival sub-model, we fit a Cox model using ${\mathbf{w}}_i$ and the initialized ${\zeta }_{ik}$ values as covariates to obtain starting values for $\alpha$ and $\nu$. These estimates are derived by maximizing the corresponding partial likelihood. When a constant baseline hazard is assumed, its initial value is determined by fitting a linear regression of the Breslow estimator ${\widehat{H}}_0\left(t\right)$ against $t$, using observed event times $\left\{T_i:{\delta }_i=1\right\}$. The slope of this regression is used as the initial estimate of the baseline hazard rate $\kappa$. On the other hand, under the piecewise-constant baseline hazard assumption in supFTSVD-JM, we initialize the baseline hazard levels by computing a Breslow-type estimate within each interval $\left[{\tau }_{b-1},{\tau }_b\right]$. Specifically, for each interval, we sum the observed events occurring in that interval and divide by the aggregated risk-set contributions evaluated using the current parameter estimates. This yields stable and interpretable starting values for the hazard segments ${\kappa }_b,b\in \{\mathrm{1,2},\dots ,B\}$.

4. Dynamic prediction of longitudinal trajectories and survival outcomes

Given a fitted supFTSVD-JM with data-driven selection of rank $K$, we can generate predictions for both new subjects and existing participants as their longitudinal information accumulates over time. This latter scenario is referred to as dynamic prediction, wherein the subject-specific loadings ${\zeta }_{ik}$ are updated using the newly acquired longitudinal data. Because these shared latent variables ${\zeta }_{ik}$ appear in both the longitudinal and survival sub-models, the supFTSVD-JM naturally supports coherent prediction of future longitudinal feature values and time-to-event probabilities. These predictive tools are useful for multiple purposes. Group-level predictions are particularly valuable in clinical trial settings, such as estimating average treatment effects or identifying high-risk subgroups. In contrast, individual-level predictions are central to personalized medicine, including tailoring treatment regimens or clinical decision-making. In what follows, we outline the procedures for predicting longitudinal trajectories and survival risk for a new subject.

Let $\left\{{\mathbf{x}}_{\text{new}},{\left({\mathbf{Y}}_{\text{new},j},t_{\text{new},j}\right)}_{j=1}^{J_{\text{new}}}\right\}$ denote the observed data for a new subject who has not yet experienced the event by a specified landmark time $t$, implying ${\delta }_{\text{new}}=0$ and $t_{\text{new},J_{\text{new}}}\le t$. Given a fitted parameter set ${\widehat{\theta }}_K$, we first compute the matrix ${\widehat{\mathbf{H}}}_{\text{new}}$, which is then used to estimate ${\tilde{\mathbf{U}}}_{\text{new}}$, the conditional mean of the shared subject-specific effects. This requires sampling from the distribution $f_{{\widehat{\theta }}_K}\left(\cdot \mid \left\{{\mathbf{y}}_{\text{new}},T_{\text{new}}=t,{\delta }_{\text{new}}=0,{\mathbf{x}}_{\text{new}}\right\}\right)$, where ${\mathbf{y}}_{\text{new}}=\text{vec}\left({\mathbf{Y}}_{\text{new}}\right)$, and applying the formula given in Equation (5). Using these estimates, the predicted value of the $p\text{th}$ feature trajectory at time $t$ is given by

$${\widehat{Y}}_{\text{new}}^p\left(t\right)=\sum _{k=1}^K{\widehat{\lambda }}_k\left({\mathbf{x}}_{\text{new}}^{\prime }{\widehat{\beta }}_k+{\tilde{U}}_{\text{new},k}\right){\widehat{\xi }}_k^p{\widehat{\psi }}_k\left(t\right),$$

where ${\tilde{U}}_{\text{new},k}$ is the $k\text{th}$ element of ${\tilde{\mathbf{U}}}_{\text{new}}$.

Note that ${\delta }_{\text{new}}=0$ at time $t$ implies that $S_{\text{new}}\left(t\right)=1$. However, for a future time point $t^{\prime }>t$, the survival probability $S_{\text{new}}\left(t^{\prime }\right)$ is unknown and must be estimated conditional on survival up to time $t$. Let ${\pi }_{\text{new}}\left(t^{\prime }\mid t\right)$ denote the conditional survival probability that $T_{\text{new}}\ge t^{\prime }$ given $T_{\text{new}}>t$. We define the predictor ${\widehat{\pi }}_{\text{new}}\left(t^{\prime }\mid t\right)$ as

$$\begin{gathered} {\widehat{\pi }}_{\text{new}}\left(t^{\prime }\mid t,{\mathbf{y}}_{\text{new}},{\widehat{\theta }}_K\right)=P\left(T_{\text{new}}\ge t^{\prime }\mid T_{\text{new}}>t,{\mathbf{y}}_{\text{new}},{\widehat{\theta }}_K\right), \\ =\frac{S_{\text{new}}\left(t^{\prime }\mid {\mathbf{y}}_{\text{new}},{\widehat{\theta }}_K\right)}{S_{\text{new}}\left(t\mid {\mathbf{y}}_{\text{new}},{\widehat{\theta }}_K\right)}, \\ =\text{exp}\left[-\left\{H_0\left(t^{\prime }\right)-H_0\left(t\right)\right\}\text{exp}\left\{{\mathbf{w}}_{\text{new}}^{\prime }\widehat{\alpha }+\sum _{k=1}^K{\widehat{v}}_k\left({\mathbf{x}}_{\text{new}}^{\prime }{\widehat{\beta }}_k+{\tilde{U}}_{\text{new},k}\right)\right\}\right]. \end{gathered}$$

Because $H_0\left(t^{\prime }\right)-H_0\left(t\right)$ is non-decreasing in $t^{\prime }$, the conditional survival probability ${\widehat{\pi }}_{\text{new}}\left(t^{\prime }\mid t\right)$ is non-increasing for $t^{\prime }>t$.

5. Simulation study for estimation and dynamic prediction performance

We assess the estimation and dynamic prediction performance of the proposed supFTSVD-JM framework through a Monte Carlo simulation study. To evaluate the advantages of joint estimation, we compare our method against the two-stage (TS) estimation approach introduced at the beginning of Section 3. The supFTSVD-JM is implemented using a publicly available R package hosted at https://github.com/msalam14/supFTSVDJM. For the two-stage approach, we use the supFTSVD package available at https://github.com/msalam14/supFTSVD. All results are summarized over 100 independent replications.

5.1. Simulation design and data generation

We generate data $\left\{{\mathbf{x}}_i,{\left({\mathbf{Y}}_{ij},t_{ij}\right)}_{j=1}^{J_i},T_i,{\delta }_i\right\}$ for a given $N$ using the joint modeling framework defined by Equations (2) and (3). We set simulation parameter values to generate realistic longitudinal trajectories and survival outcomes with identifiable low-rank structure and representative signal-to-noise levels, rather than to mimic the ADNI application. In our numerical study, we set ${\mathbf{x}}_i={\left(x_{1i},x_{2i}\right)}^{\prime }$, where $x_{1i}~U\left(\mathrm{0,1}\right)$ and $x_{2i}~\text{Beta}\left(\mathrm{2.5,1.25}\right)$. We first generate the shared random effects ${\zeta }_i={\left({\zeta }_{i1},{\zeta }_{i2},{\zeta }_{i3}\right)}^{\prime }$ under a rank-3 supFTSVD-JM. For this, we set ${\beta }_1=(\mathrm{0.81,0.59}{)}^{\prime },{\beta }_2=(\mathrm{0.57,0.82})$, and ${\beta }_3=(-\mathrm{0.50,0.87})$, all satisfying the unit norm condition, and generate $U_{ik}~N\left(\mathrm{0,1}\right)$ for $k=\mathrm{1,2},3$, from which the shared random effects ${\zeta }_{ik}$ are constructed. Once the shared random effects ${\zeta }_i$ are generated for $i=\mathrm{1,2},\dots ,N$, we complete the data generation process. Survival data are generated first, followed by the longitudinal data. This sequence ensures $t_{i{m}_i}\le T_i$ for all $i$.

In our numerical study, we consider a constant baseline hazard with $h_0\left(t\right)=0.1$ for all $t$. Survival times are generated using the formula ${\tilde{T}}_i=S_i/\left\{0.1\text{exp}\left(3.50x_{1i}+2.60x_{2i}-{0.15}_{i1}-0.20{\zeta }_{i2}+\right.\right.\left.\left.0.15{\zeta }_{i3}\right)\right\}$, where $S_i~U\left(\mathrm{0,1}\right)$ represents the survival probability for subject $i$. Censoring times $C_i$ are generated independently from $U\left(\mathrm{0,5}\right)$, resulting in approximately 25% censoring. We set $T_i=\text{min}\left({\tilde{T}}_i,C_i\right)$ as the observed survival time. To mimic a realistic study duration, we consider [0, 1] as the observation window. Accordingly, any $T_i>1$ is truncated to 1, and the corresponding subject is marked as censored $({\delta }_i=0)$, along with all others for whom ${\tilde{T}}_i>C_i$. To evaluate performance under higher censoring, we also create an alternative scenario using $h_0\left(t\right)=0.05$, which yields approximately 40% censoring. We present the results from this scenario in Section S2 of the Supplementary Material.

For the longitudinal data, we generate $J_i-2$ time points from $U\left(0,T_i\right)$ for subject $i$ and sort them to form $t_{i2},t_{i3},\dots ,t_{i,J_i-1}$. We set $t_{i1}=0$ and $t_{i,J_i}=T_i$. From the unit sphere ${\mathbb{S}}^{P-1}$ with $P=500$, we generate feature loading vectors ${\xi }_k$ for $k=\mathrm{1,2},3$. We use ${\psi }_1\left(t\right)=\left(8+6t^8-3t^2-4t^3\right)/\sqrt{45.61},{\psi }_2\left(t\right)=10t^2\text{exp}\left(-t^5\right)/\sqrt{10-10\text{exp}(-2)}$, and ${\psi }_3\left(t\right)=\sqrt{2}\text{sin}\left(2.5\pi t\right)$ as singular functions, each normalized to unit $L_2$ norm over [0, 1]. Singular values are set as ${\lambda }_1=5.20,{\lambda }_2=4.80$, and ${\lambda }_3=3.35$. To obtain noisy observations $Y_{ij}^p$, we add iid errors ${ϵ}_{ij}^p~N\left(\mathrm{0,0.1}\right)$.

To assess model performance under different conditions, we consider the following experimental design. We evaluate performance at three different sample sizes: $n=\mathrm{100,200}$, and 300, and evaluate prediction at three landmark times: 0.2, 0.3, and 0.4, the last of which is used only for assessing dynamic prediction. In each Monte Carlo replicate, we generate one training set and three test sets to support model fitting and prediction at different landmarks. Each test set corresponds to one of the landmark times. The training set includes both longitudinal and survival data for $n$ subjects and is used exclusively for model estimation. Each test set contains 100 subjects and is kept entirely separate from the training data. For training subjects, we sample $J_i$ uniformly from $\{\mathrm{5,6},\dots ,10\}$; for test subjects, we fix $J_i=\mathrm{10,15}$, and 20 for the landmark times 0.2, 0.3, and 0.4, respectively. We fit the supFTSVD-JM model via the proposed MCEM algorithm and the TS approach described in Section 3 to the training data, and use the fitted models to predict subject-feature-specific longitudinal trajectories and conditional survival curves for subjects in the test datasets, separately for each landmark time.

We evaluate estimation performance for $\left\{{\xi }_k,{\psi }_k(\cdot )\right\}$ for all $k$ and $\{\alpha ,\nu \}$ for the survival sub-model. Specifically, we report Euclidean norms for ${\xi }_k,L_2$ norms for ${\psi }_k(\cdot )$, and mean squared errors for $\alpha$ and $\nu$. For longitudinal parameters, we account for the sign-indeterminacy in the estimation. To assess prediction accuracy, we compute the integrated mean squared prediction error (iMSPE) for both longitudinal trajectories and survival curves. For the longitudinal component, we define $\text{iMSPE}=\sum _{i=1}^{100}\sum _{p=1}^P\left[{\int }_0^1{\left\{V_i^p\left(t\right)-{\widehat{V}}_i^p\left(t\right)\right\}}^{2}dt\right]/100P$, where ${\widehat{V}}_i^p\left(t\right)$ is the predicted value of $V_i^p\left(t\right)$. A similar formula is used for the iMSPE of survival curves. Additionally, we evaluate time-dependent area under the curve (AUC) and Brier scores (BS), and summarize them using their integrated versions, denoted by iAUC and iBS, obtained by integrating over time, following Wang et al. (2017) and Zheng et al. (2012). Only the measures (iMSPE, iAUC, and iBS) that assess prediction performance were calculated using the test data.

Besides the primary simulation setting described above, we also investigate two further scenarios to assess robustness: (i) a non-informative observation-time design in which longitudinal visit times are generated independently of the event process and subsequently truncated by censoring, and (ii) settings in which the baseline hazard used for data generation differs from that assumed during estimation (Weibull or log-logistic generation with a fitted piecewise-constant hazard). Detailed descriptions and corresponding numerical results for these robustness analyses are provided in Sections S2.2 and S2.3 of the Supplementary Material.

5.2. Simulation results

Table 1 reports the estimation results for the feature loading vectors ${\xi }_k$ and the singular functions ${\psi }_k(\cdot )$ across the three rank-1 components. As expected, estimation errors decrease steadily with increasing sample size for all components and across both types of parameters. For the strongest component $(k=1)$, the JM and TS approaches perform similarly, reflecting the relatively high signal-to-noise ratio in this direction. For components $k=2$ and $k=3$, the relative performance of JM and TS differs across components and sample sizes: JM yields smaller errors for several settings, especially when the signal from the longitudinal component aligns strongly with the survival information, while TS attains slightly lower errors in a few cases where the latent component carries weaker longitudinal signal. These patterns are consistent with the roles of the two estimators—TS relies solely on the longitudinal likelihood when estimating the latent structure, whereas JM borrows strength across both sub-models, which can improve estimation in some directions while slightly increasing variability in others. Table 2 presents the mean squared errors for the survival sub-model parameters $\alpha$ and $\nu$ across sample sizes. Similar to the longitudinal results, estimation accuracy improves with increasing sample size. The largest gains from joint modeling are observed for $v_1,v_2$, and $v_3$, which are the coefficients associated with the latent factors, demonstrating the additional information contributed by the survival process when estimating shared structure.

Table 1.

Estimation accuracy for the feature loading vectors ${\xi }_k$ and singular functions ${\psi }_k(\cdot )$ across rank-1 components $k=\mathrm{1,2},3$, based on joint modeling (JM) and two-stage (TS) approaches.^a

		$\sqrt{{10}^3{‖{\widehat{\xi }}_k-{\xi }_k‖}_2}$		$\sqrt{\int {\left\{{\hat{\psi }}_k(t)-{\psi }_k(t)\right\}}^{2}dt}$
Component	n	JM	TS	JM	TS
$k=1$	100	0.021(0.010)	0.022(0.010)	0.012(0.006)	0.022(0.005)
	200	0.002(0.000)	0.002(0.000)	0.001(0.000)	0.001(0.000)
	300	0.001(0.000)	0.001(0.000)	0.001(0.000)	0.000(0.000)
$k=2$	100	0.669(0.042)	0.739(0.042)	0.931(0.080)	0.839(0.076)
	200	0.084(0.026)	0.058(0.020)	0.123(0.043)	0.076(0.035)
	300	0.029(0.014)	0.021(0.009)	0.020(0.015)	0.008(0.005)
$k=3$	100	0.714(0.043)	1.055(0.068)	1.519(0.064)	1.559(0.063)
	200	0.093(0.027)	0.090(0.028)	0.188(0.051)	0.177(0.047)
	300	0.052(0.028)	0.045(0.019)	0.095(0.034)	0.103(0.037)

^aReported values represent the Euclidean norm (for ${\xi }_k$) and $L_2$ norm (for ${\psi }_k$), averaged over 100 Monte Carlo replications, with SEs shown in parentheses.

Table 2.

Mean squared errors for estimated survival sub-model parameters $\alpha$ and $v$ under joint modeling (JM) and two-stage (TS) approaches.^a

	${\alpha }_1$		${\alpha }_2$		${\nu }_1$		$v_2$		${\nu }_3$
$n$	JM	TS	JM	TS	JM	TS	JM	TS	JM	TS
100	0.211	0.278	0.574	0.497	0.036	0.015	0.189	0.198	0.169	0.328
200	0.086	0.094	0.270	0.228	0.005	0.015	0.189	0.436	0.024	0.044
300	0.064	0.069	0.121	0.113	0.004	0.006	0.189	0.470	0.024	0.067

^aResults are averaged over 100 Monte Carlo replications. The parameters $v_1,v_2$, and $v_3$ correspond to the shared latent components derived from longitudinal data.

Although Table 2 shows that the two-stage estimator can occasionally achieve lower MSE for a fixed-effect parameter such as ${\alpha }_2$, this is a finite-sample consequence of conditioning on point estimates of the latent components and thus understating their uncertainty. In contrast, the joint model more accurately recovers the component-specific associations $v_1,v_2$, and $v_3$, underscoring the advantage of joint estimation when parameters are linked to the latent longitudinal structure. While the estimation results in Table 1 indicate that the relative estimation performance of JM and TS can differ across individual components, trajectory prediction depends primarily on how accurately the subject-specific latent loadings ${\zeta }_{ik}$ are recovered. Because the joint model updates these loadings using both longitudinal and survival information, it can achieve substantially better prediction accuracy even when TS exhibits slightly lower estimation error for a particular basis component.

Values in Table 3 represent the iMSPE observed while predicting individual feature trajectories and conditional survival curves at all combinations of three landmark times and three sample sizes. We observe improved prediction accuracy when the sample size increases, and for a fixed sample size, accuracy also improves as the landmark time grows, reflecting the utilization of more longitudinal observations. For trajectory prediction, JM substantially outperforms the TS approach across all settings. This gain is consistent with the JM formulation, which estimates the latent components using information from both the longitudinal and survival sub-models and therefore yields a more accurate representation of the underlying trajectories. In contrast, the TS approach derives the latent structure only from the longitudinal likelihood, resulting in less accurate predictions of the feature trajectories, especially at smaller sample sizes. For survival prediction, the two methods perform comparably, with iMSPE values nearly identical across all landmark times and sample sizes, because both approaches apply the same survival model once the latent components are estimated. Together, these results highlight the benefit of joint estimation for improving longitudinal trajectory prediction while maintaining competitive performance for survival curve prediction.

Table 3.

Integrated mean squared prediction error (iMSPE) for feature trajectories and conditional survival curves, based on joint modeling (JM) and two-stage (TS) approaches, across three landmark times ($t=0.20,0.30,0.40$) and three sample sizes ($n=\mathrm{100,200,300}$).^a

	Feature trajectory						Survival curve
	$t=0.2$		$t=0.3$		$t=0.4$		$t=0.2$		$t=0.3$		$t=0.4$
$n$	JM	TS	JM	TS	JM	TS	JM	TS	JM	TS	JM	TS
100	1.314	2.577	1.427	2.396	0.357	0.761	0.005	0.006	0.005	0.005	0.004	0.004
200	0.187	0.258	0.181	0.249	0.117	0.133	0.003	0.004	0.003	0.003	0.003	0.003
300	0.088	0.263	0.086	0.248	0.074	0.080	0.002	0.002	0.002	0.002	0.001	0.001

^aEach value is averaged over 100 Monte Carlo replicates.

We additionally investigate whether incorporating classical landmarking into the two-stage (TS) procedure improves survival prediction. In this variant, at a given landmark time $t$, model fitting is restricted to subjects with observed survival time $T_i>t$, excluding individuals who experience the event or are censored at or before $t$. For the retained subjects, longitudinal measurements up to time $t$ are used in the longitudinal model to estimate subject-specific latent components. Table ST12 compares the standard TS approach with its landmarked counterpart for conditional survival curve prediction. Implementation details of the landmarked two-stage procedure are provided in Supplementary Section S2.4. The results indicate that incorporating classical landmarking does not improve predictive accuracy in this setting.

To evaluate time-specific discrimination and calibration, we compare the joint modeling (JM) and two-stage (TS) approaches using time-dependent area under the curve (AUC) and Brier score (BS), which are summarized through their integrated versions, iAUC and iBS. These metrics assess classification performance for event occurrence at specified times. Both AUC and BS range from 0 to 1, with higher AUC indicating better discrimination and lower BS indicating better calibration and overall prediction quality. Table 4 presents the average iAUC and iBS values obtained from each method, along with corresponding benchmarks computed using the true subject-specific loadings ${\zeta }_{ik}$ employed in the data generation process. Overall, the estimated iAUC and iBS values closely match their true counterparts. Compared to TS, the JM approach consistently yields iAUC and iBS estimates that are more aligned with the ground truth, supporting the benefits of incorporating survival data in the estimation process. Both approaches demonstrate strong performance across all sample sizes and landmark times.

Table 4.

Comparison of integrated area under the curve (iAUC) and integrated Brier score (iBS) for joint modeling (JM) and two-stage (TS) approaches at varying sample sizes and landmark times.^a

		$t=0.2$			$t=0.3$			$t=0.4$
Criterion	$n$	JM	TS	True	JM	TS	True	JM	TS	True
iAUC	100	0.724	0.723	0.756	0.767	0.765	0.780	0.576	0.574	0.582
	200	0.709	0.708	0.703	0.712	0.713	0.713	0.598	0.597	0.595
	300	0.695	0.693	0.692	0.781	0.780	0.790	0.596	0.598	0.593
iBS	100	0.141	0.141	0.135	0.122	0.122	0.118	0.126	0.125	0.123
	200	0.153	0.154	0.153	0.167	0.167	0.166	0.167	0.168	0.165
	300	0.122	0.121	0.122	0.135	0.135	0.131	0.135	0.135	0.132

^aReported values are averages across 100 simulation replicates, with integration of the time-dependent measure performed within each replicate. The “True” column reflects values computed using the true subject-specific latent variables.

To understand the computational burden of the proposed joint modeling framework, we investigate the runtime of the supFTSVD–JM algorithm under the original simulation design described in Section 5.1. Figure SF1 summarizes the computation time for both the JM and the TS methods across sample sizes $n=\mathrm{100,200,300}$. As expected, JM requires additional computation because it jointly estimates the longitudinal and survival sub-models using a Monte Carlo EM procedure. Nonetheless, as shown in the figure, the additional computational cost of JM grows approximately linearly with sample size, and the overall runtime remains well within practical limits for all scenarios considered. This analysis demonstrates that, despite the complexities inherent to high-dimensional joint modeling, the proposed estimation procedure is computationally scalable.

Tables ST1–ST4 summarize the simulation results under a higher censoring scenario (approximately 40%) in Section S2.1. This is achieved by reducing the constant baseline hazard from 0.10 to 0.05. Despite the increased censoring, the proposed supFTSVD-JM continues to perform well in both estimation and prediction tasks. Table ST1 shows that the estimation errors for both feature loading vectors and singular functions remain small and decrease as the sample size increases. Table ST2 indicates that the joint modeling approach retains a clear advantage over the two-stage method for estimating survival sub-model parameters, especially those associated with latent components. Table ST3 demonstrates consistent improvements in prediction performance (lower iMSPE) with increasing sample size and landmark time. Finally, Table ST4 shows that the joint model maintains strong discriminatory ability (higher iAUC) and calibration (lower iBS), with estimates closely matching those based on the true subject-specific effects. These findings suggest that the proposed method is robust to moderate increases in censoring.

The Supplementary Material further reports results from two additional simulation scenarios not included in the main text. The first examines a non-informative observation-time design (Section S2.2), where longitudinal visit times are generated independently of the event time before censoring is applied. This addresses a potential source of informative observation in the original design and confirms that the proposed method maintains stable estimation and prediction performance under non-informative schedules. The second scenario (Section S2.3) evaluates robustness to baseline-hazard misspecification by generating survival times from Weibull and log-logistic models while fitting a misspecified piecewise-constant baseline hazard. Across both settings (see Tables ST5–ST11), the supFTSVD–JM method exhibits strong numerical stability, accurate recovery of the longitudinal structure, and competitive survival prediction performance. Together, these results reinforce the robustness of the proposed modeling framework under a broader range of data-generating mechanisms.

6. Application to longitudinal lipidomics data from ADNI

The Alzheimer’s Disease Neuroimaging Initiative (ADNI) is a large-scale, ongoing multicenter study that collects longitudinal biomarker data alongside clinical, genetic, imaging, and neuropathological assessments. AD is a progressive neurodegenerative disorder marked by synaptic and neuronal loss, and it benefits from early detection strategies that enable timely intervention. Given the invasiveness and cost of imaging or cerebrospinal fluid–based diagnostics, blood-based biomarkers—particularly lipidomic profiles—have garnered significant interest. Lipids are integral to brain structure and function, and changes in lipid composition have been associated with AD pathophysiology. Accordingly, lipidomics data have emerged as promising biomarkers for early-stage AD detection, offering a noninvasive and scalable tool for risk prediction and clinical monitoring.

In this analysis, we apply our proposed joint modeling framework to study whether plasma lipidomic trajectories can enhance prediction of conversion from mild cognitive impairment (MCI) to AD dementia. Specifically, we jointly model longitudinal lipidomics profiles and time-to-dementia onset using a dynamic prediction framework. Baseline covariates incorporated into both sub-models include age, sex (coded as 1 for female and 0 for male), years of education, and Apolipoprotein E (ApoE) $\epsilon 4$ allele count (0, 1, or 2). The dataset comprises subjects from ADNI 1, ADNI GO, and ADNI 2. For the survival sub-model, we specify a piecewise-constant baseline hazard with nine change points placed at the deciles of the observed event-time distribution. We prefer this specification over a constant baseline hazard because: (i) the risk of conversion from MCI to dementia is known to accelerate at later disease stages and is therefore unlikely to follow a constant-hazard pattern, and (ii) the ADNI follow-up schedule and sparse lipidomics measurements create uneven information across time, making a flexible baseline hazard better able to capture the underlying event-time structure. Longitudinal lipidomics data were generated by the Alzheimer’s Disease Metabolomics Consortium (ADMC) and processed at the Meikle Lab using mass spectrometry platforms, as described in Huynh et al. (2019).

ADNI continues follow-up after AD dementia diagnosis, resulting in post-diagnosis longitudinal data. For subjects who convert to dementia, we define the event time as the duration from baseline to the first diagnosis and exclude all post-diagnosis visits to maintain a pre-event modeling framework. For non-converters, all longitudinal visits are included, with censoring time defined as the duration from baseline to the last available visit. For this application, we include only those visits for which all lipid features were available for a given subject. Addressing scenarios where some features are missing at certain visits is an important direction for future work, but this is outside the scope of the present work. The final analysis includes 2,092 visits from 736 individuals, with 779 plasma lipid features measured per visit.

Prior to fitting the proposed supFTSVD-JM to the ADNI lipidomics data, we apply the five-fold cross-validation procedure described in Section 3.3 to select the model rank. In this procedure, the threshold parameter ${\kappa }_0$ is used as part of the stopping criterion, and for the ADNI analysis we chose ${\kappa }_0$ so that the selected rank explains more than 99% of the total variability in the longitudinal features under the fitted supFTSVD decomposition. The results indicate that a rank-4 model explains over 99% of the total variability in the lipidomic trajectories. We also examine nearby choices of ${\kappa }_0$ (eg targeting 95–99.5% explained variability) and find that the selected rank remains stable at $K=4$. Table 5 presents the parameter estimates (Coef.) from the survival sub-model of the fitted rank-4 joint model, including their exponentiated values, exp(Coef.), corresponding to hazard ratios (HR), standard errors (SE) based on the inverse observed information matrix, se(Coef.), and associated Wald test $p$-values. Among the baseline covariates, age and ApoE-$\epsilon$ 4 allele count emerge as significant predictors of dementia risk. Additionally, all four supervised lipidomics components incorporated into the survival model exhibit significant associations with the hazard of conversion from MCI to AD dementia. These findings are consistent with prior literature on AD risk factors.

Table 5.

Parameter estimates from the survival sub-model of the rank-4 supFTSVD-JM fitted to the ADNI data.^a

Parameters	Coef.	exp(Coef.)	se(Coef.)	$p$-value
Age	0.029	1.030	0.009	0.001
Gender	0.018	1.018	0.143	0.902
Education	0.032	1.033	0.022	0.135
ApoE-e4 alleles	0.578	1.782	0.091	0.000
${\zeta }_1$	0.214	1.238	0.032	0.000
${\zeta }_2$	0.216	1.242	0.027	0.000
${\zeta }_3$	−0.111	0.895	0.031	0.000
${\zeta }_4$	0.172	1.188	0.044	0.000

^aReported values include HR, SE, and p-values from Wald tests. Covariates include baseline demographic factors and the first four supervised lipidomic components.

Our results indicate that each one-year increase in baseline age is associated with a 3.0% increase in the hazard of dementia onset (HR = 1.030), controlling for other covariates. The presence of one or two ApoE-$\epsilon$ 4 alleles increases the hazard by 78.2% and 256.4%, respectively, relative to non-carriers. The estimated HRs for the four supervised lipidomic components show heterogeneity in direction and magnitude: the first, second, and fourth components are associated with elevated risk (increasing the hazard by approximately 24%, 24%, and 19%, respectively), while the third component appears to be protective, reducing the hazard by roughly 10%.

As the low-dimensional components ${\left\{{\zeta }_{ik}\right\}}_{k=1}^4$ extracted from the lipidomics data show significant associations with dementia risk, we further identify the top five contributing lipid features based on the aggregated ranks of ${\left\{\left|{\xi }_k^p\right|\right\}}_{k=1}^4$. Among these, three are cholesteryl esters (CE), specifically cholesteryl linoleate (CE 18:2), cholesteryl oleate (CE 18:1), and cholesteryl palmitate (CE 16:0), ranked first, second, and fourth, respectively. The remaining two are free cholesterol (COH), ranked third, and triglyceride (TG 52:2), ranked fifth. These findings align with previous reports linking CE species to AD pathology (Proitsi et al. 2015; Turri et al. 2023; Dakterzada et al. 2024), and TG 52:2 has also been identified as differentially expressed in AD patients relative to controls (de Leeuw et al. 2017). Furthermore, the well-established relationship between serum cholesterol and AD risk (Wood et al. 2014), together with the association of ApoE-$\epsilon$ 4 alleles with VLDL (very low-density lipoprotein) particle cholesterol content (Puglielli et al. 2003), supports the biological relevance of these lipid markers. Table ST3 confirms that all five features differ significantly at baseline between those who developed dementia and those who did not. A Hotelling’s $T^2$ test also reveals a significant multivariate group-level difference $(P=0.0007)$. Importantly, Table 6 demonstrates that these lipid-derived components enhance the accuracy of dynamic prediction: the joint model incorporating both lipidomics and baseline covariates achieves higher iAUC and lower iBS across all landmark time points compared to models that omit either covariate type.

Table 6.

Integrated area under the curve (iAUC) and integrated Brier score (iBS) for dynamic prediction at 12, 18, and 24 months.^a,b

	iAUC			iBS
$t$	JM(DE)	JM(w/o DE)	PHM	JM(DE)	JM(w/o DE)	PHM
12	0.696(0.014)	0.633(0.019)	0.671(0.014)	0.074(0.003)	0.077(0.003)	0.075(0.003)
18	0.691(0.012)	0.581(0.014)	0.681(0.013)	0.093(0.002)	0.100(0.002)	0.093(0.002)
24	0.644(0.012)	0.501(0.012)	0.692(0.011)	0.122(0.003)	0.129(0.003)	0.120(0.003)

^aEstimates are based on the fitted rank-4 supFTSVD-JM with and without baseline covariates in the survival sub-model, as well as on a proportional hazards model involving baseline covariates only. Values in parentheses indicate SEs.

^bJM(DE): supFTSVD-JM with the survival sub-model that includes both baseline covariates and derived low-dimensional components from lipidomics data, JM(w/o DE): supFTSVD-JM with the survival sub-model that includes only the derived low-dimensional components from lipidomics data, and PHM: proportional hazards model with baseline covariates only.

We assess the dynamic prediction performance of two variants of the rank-4 supFTSVD-JM: one that includes the direct effects of baseline covariates in the survival sub-model [denoted JM(DE)] and another that excludes them [denoted JM(w/o DE)]. For comparison, we also fit a proportional hazards model incorporating only baseline covariates (denoted PHM). Dynamic prediction is evaluated at three landmark times—12, 18, and 24 months. At each landmark, we use the longitudinal data available up to that time point to estimate the shared random effects, which are then used to compute survival probabilities at 25 equidistant time points over the subsequent 12 months. Predictive accuracy is quantified using the area under the curve (AUC) and Brier score (BS), both integrated over the 12-month prediction interval. Table 6 summarizes the mean integrated AUC (iAUC) and integrated Brier score (iBS) across 25 randomly sampled test datasets, each consisting of 20% of ADNI subjects. SEs are reported in parentheses.

Across all landmark times, the JM(DE) model consistently outperforms the JM(w/o DE) model, achieving higher iAUC and lower iBS values, indicating superior discrimination and calibration. Compared to the PHM model, which does not utilize longitudinal data, the proposed JM(DE) model generally performs better on both metrics. In particular, at the 12- and 18-month landmarks, JM(DE) yields higher iAUC and lower iBS values than PHM, highlighting the predictive advantage of incorporating time-varying biomarkers. However, at the 24-month landmark, JM(DE) exhibits a modest decline in performance, showing slightly lower iAUC and higher iBS relative to PHM. This reduction may stem from the skewed distribution of observed or truncated survival times in the ADNI dataset (see Fig. SF2). At this landmark, the test set consists of 20% of subjects who remained dementia-free at least 24 months, while the training set is dominated by subjects with shorter observed survival times. This imbalance may degrade the estimation of the singular functions in the longitudinal sub-model, ultimately affecting the predictive accuracy of the fitted supFTSVD-JM at later landmark times. We anticipate that with a larger sample of long-term survivors, the supFTSVD-JM would outperform PHM at all landmark times. For fair comparison, the PHM was fitted by maximizing the full likelihood using a piecewise-constant baseline hazard with change points identical to those used in the supFTSVD-JM so that differences in prediction arise solely from the use of longitudinal information.

Finally, we present subject-level dynamic predictions using the fitted supFTSVD-JM in Fig. 3. We consider three representative subjects, labeled A, B, and C, who have similar baseline ages but differ in ApoE-$\epsilon$ 4 allele count, allowing visual comparison across levels of genetic risk. All three subjects are approximately 60 years old. In the figure, columns correspond to landmark times (12, 18, and 24 months), while rows represent ApoE-$\epsilon$ 4 allele counts of 0, 1, and 2. Within each column, survival probabilities decrease from the top to the bottom row, reflecting the increasing genetic risk associated with a greater number of ApoE-$\epsilon$ 4 alleles. Conversely, as the landmark time increases from 12 to 24 months, survival probability generally improves, since these individuals have remained dementia-free for longer. We also provide analogous dynamic prediction plots for a second set of subjects aged 75 years and older in Fig. SF6. For this older cohort, we observe similar trends in predicted survival curves across ApoE-$\epsilon$ 4 allele counts, suggesting no evidence of an interaction between age group and ApoE-$\epsilon$ 4 status.

Figure 3.

Dynamic prediction of survival probabilities for three ADNI subjects using the proposed joint model (JM) at landmark times $t=\mathrm{12,18}$, and 24 months. Each row corresponds to a different subject with increasing ApoE-$\epsilon$ 4 allele counts (0, 1, and 2), while columns indicate increasing landmark times. The solid lines represent the predicted conditional survival probabilities $S_i\left(t^{\prime }\mid t\right)$ for future time $t^{\prime }>t$, and the dashed lines show the corresponding pointwise 95% CIs. The vertical dashed line in each panel marks the landmark time $t$. Subject A (top row) is 64.8 years old with no ApoE-$\epsilon$ 4 alleles, subject B (middle row) is 57.7 years old with one ApoE-$\epsilon$ 4 allele, and subject C (bottom row) is 57.7 years old with two ApoE-$\epsilon$ 4 alleles.

7. Conclusions

We have introduced a novel joint modeling framework, supFTSVD-JM, for analyzing high-dimensional multivariate longitudinal data jointly with time-to-event outcomes. This approach builds on canonical polyadic (CP) tensor decomposition to represent latent temporal and cross-feature structures in the longitudinal data, while incorporating covariate supervision for learning the longitudinal process and covariate adjustment for modeling the survival outcome within a unified low-rank representation. The resulting low-rank representation supports flexible modeling of baseline effects, individualized trajectory recovery, and dynamic survival prediction as new longitudinal data accumulate.

Through comprehensive simulation studies, we demonstrate that the proposed penalized MCEM algorithm yields superior estimation and prediction accuracy compared to the conventional two-stage approach. As sample size increases, the method exhibits improved efficiency and stability. Furthermore, supFTSVD-JM remains robust under moderate to high censoring. When applied to the ADNI lipidomics dataset, the data-driven rank-4 supFTSVD-JM explains over 99% of the variation in lipid profiles. The resulting latent factors show strong associations with dementia onset, highlighting the model’s capacity for clinically relevant biomarker discovery.

While supFTSVD-JM provides a scalable and interpretable framework for high-dimensional longitudinal modeling, the current formulation assumes a single-omics input. This limits its direct applicability to integrative multi-omics studies, where distinct data modalities may be asynchronously measured. A simple workaround involves concatenating multi-omics vectors at each visit, but this strategy presumes synchronous sampling and overlooks inter-omics heterogeneity. In addition, several modeling assumptions warrant further development. First, the survival sub-model currently relies on a proportional hazards structure induced by time-invariant latent loadings, which may be restrictive for applications requiring time-varying associations. Second, the longitudinal component assumes homoscedastic measurement error across features, and although this choice promotes parsimony, allowing feature-specific variances may offer additional flexibility in certain omics settings. Third, independence among latent components is imposed for computational tractability, yet relaxing this assumption could yield richer covariance structures and potentially improve estimation of cross-feature dependencies. Future work should extend this framework to accommodate modality-specific timing, shared and modality-specific latent structures, and missingness mechanisms inherent to multi-omics integration. Additional directions include incorporating time-varying effects into the survival sub-model, developing heteroscedastic error formulations, and exploring correlated latent components, all of which would broaden the applicability of supFTSVD-JM and enhance its interpretive capacity in high-dimensional biomedical settings.

Supplementary Material

Supplementary material is available at Biostatistics Journal online.