Longitudinal Sparse Single-Omics Factor Analysis for High-Dimensional Blood Biomarkers in Alzheimer's Disease.

Longitudinal Sparse Single-Omics Factor Analysis for High-Dimensional Blood Biomarkers in Alzheimer’s Disease - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Search Log in Primary site navigation Search Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice Stat Med . Author manuscript; available in PMC: 2026 May 31. Published in final edited form as: Stat Med. 2026 May;45(10-12):e70601. doi: 10.1002/sim.70601 Longitudinal Sparse Single-Omics Factor Analysis for High-Dimensional Blood Biomarkers in Alzheimer’s Disease Haotian Zou 1 , Rima Kaddurah-Daouk 2, 3, 4 , Sheng Luo 1 , Alzheimer’s Disease Neuroimaging Initiative Author information Copyright and License information 1 Department of Biostatistics and Bioinformatics, Duke University, Durham, North Carolina, USA 2 Department of Psychiatry and Behavioral Sciences, Duke University, Durham, North Carolina, USA 3 Duke Institute of Brain Sciences, Duke University, Durham, North Carolina, USA 4 Department of Medicine, Duke University, Durham, North Carolina, USA Author Contributions Haotian Zou: methodology, software, formal analysis, data curation, writing – original draft, writing – review and editing, visualization. Rima Kaddurah-Daouk: project administration, funding acquisition, data curation, writing – review and editing. Sheng Luo: conceptualization, methodology, resources, writing – review and editing, supervision, project administration, funding acquisition. ✉ Correspondence: Sheng Luo ( sheng.luo@duke.edu ) PMC Copyright notice PMCID: PMC13222108  NIHMSID: NIHMS2178999  PMID: 42136096 The publisher's version of this article is available at Stat Med Abstract Alzheimer’s disease (AD) is a progressive neurodegenerative disorder whose molecular mechanisms involve multiple biological pathways. Longitudinal blood-based omics data, such as lipidomics and metabolomics profiles, offer promising noninvasive biomarkers for early detection and prognosis, yet they are high-dimensional, sparse, and exhibit complex temporal and cross-feature correlations. The primary goal of this study is to identify which omics data types are most strongly associated with time to dementia onset in patients with mild cognitive impairment (MCI) at baseline. To address this, we propose a longitudinal sparse single-omics factor analysis (LS-SOFA) framework that models each omics view through view-specific latent factors and feature-weight matrices, with temporal dynamics captured by functional principal component analysis (FPCA). The resulting functional principal component (FPC) scores are incorporated into a survival model to test whether each omics view is associated with time to dementia onset. An efficient covariance-based estimation algorithm substantially reduces computational and memory cost, enabling large-scale application in the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study. In simulations, LS-SOFA achieves higher longitudinal estimation accuracy and more stable hypothesis testing than competing methods. Applied to five blood-based omics views from ADNI, LS-SOFA identified plasma lipidomics and serum metabolomics from FIA and UPLC as significantly associated with dementia risk after FDR adjustment, with nominal evidence of association for gut microbial metabolomics from serum. The top features within each omics view reveal biologically interpretable metabolic pathways that may serve as blood-based biomarkers for AD progression. Keywords: ADNI, biomarker discovery, covariance modeling, functional principal component analysis, high-dimensional data, survival association 1 |. Introduction Alzheimer’s disease (AD) is a progressive neurodegenerative disorder characterized by the gradual decline of cognitive and functional abilities [ 1 , 2 ]. Among individuals diagnosed with mild cognitive impairment (MCI), a heterogeneous clinical syndrome that can precede AD and other dementias, accurately predicting progression to dementia is essential for early intervention and for evaluating emerging disease-modifying therapies [ 3 ]. Large longitudinal studies such as the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and the National Alzheimer’s Coordinating Center (NACC) have collected extensive multimodal data, including neuroimaging, neuropsychological testing, genetics, and blood-based omics, to facilitate biomarker discovery and improve prognostic modeling in AD research [ 4 – 6 ]. Blood-based omics platforms, including plasma lipidomics, serum metabolomics, proteomics, and DNA methylation profiling [ 7 – 9 ], generate circulating biomarkers such as plasma p-tau181, p-tau217, and amyloid- β 42/40 ratios, which have been shown to associate with AD pathology and cognitive decline [ 10 – 14 ]. Longitudinal evidence further indicates that dynamic changes in these biomarkers track disease progression and clinical conversion [ 15 , 16 ]. Despite these advances, modeling longitudinal blood-based omics data remains statistically and computationally challenging due to (i) complex correlation structures both within individuals over time and across thousands of molecular features, (ii) sparse and irregular measurement schedules that lead to nonlinear trajectories, and (iii) high computational demands in large cohorts (sample size N ≈ 600 ) with high-dimensional omics profiles ( P ≈ 750 features per view). Here N denotes the number of participants and P the number of molecular features per view. Addressing these challenges requires methods that are both statistically flexible and computationally scalable. Several frameworks have been developed for integrating or modeling multi-omics and longitudinal omics data. The multi-omics factor analysis with smooth temporal structure (MEFISTO) model [ 17 ] extends multi-view factor analysis by introducing smooth latent processes through Gaussian-process priors, enabling cross-modal and temporal integration. The timeOmics pipeline [ 18 ] aligns time-course data across omics layers via spline-based regression, while the pathway activity level modeling–omics (PALMO) framework [ 19 ] uses latent-factor representations for omics integration and pathway enrichment. Although these approaches have demonstrated utility in moderate-sized studies, they typically assume parametric covariance structures or explicit time alignment and have cubic complexity in sample size, which limits scalability and robustness when applied to high-dimensional, irregular longitudinal data in large AD cohorts. These limitations motivate the use of functional data analysis (FDA), which models longitudinal trajectories as smooth stochastic functions over time. Functional principal component analysis (FPCA) decomposes subject-specific trajectories into orthogonal eigenfunctions and corresponding FPC scores, providing an efficient data-driven representation of within-subject temporal variation [ 20 ]. Subsequent extensions have incorporated hierarchical and multivariate structures to accommodate complex longitudinal outcomes [ 21 – 24 ]. These methodological advances highlight the versatility of FPCA for high-dimensional biomedical data, providing a principled foundation for the framework developed in this study. Building upon these developments, we propose a longitudinal sparse single-omics factor analysis (LS-SOFA) framework that models each omics view separately while capturing both temporal and cross-feature correlations. In LS-SOFA, the observed longitudinal omics data are decomposed into view-specific latent loading functions that evolve smoothly over time and feature-specific weight matrices that encode cross-feature covariance. The latent loading functions are modeled nonparametrically using FPCA, allowing flexible representation of nonlinear temporal trajectories without requiring explicit time alignment. To address the computational burden of large-scale longitudinal omics data, we develop a covariance-based estimation algorithm with computational complexity on the order of 𝒪 P ⋅ R 3 , where R denotes the number of tensor-product B-spline bases. The LS-SOFA framework offers three main methodological novelties: (1) a data-adaptive, nonparametric FPCA decomposition for latent factors that flexibly captures nonlinear longitudinal trajectories; (2) a covariance-based estimation algorithm that is computationally efficient, memory-lean, and scalable to high-dimensional omics data; and (3) a two-stage inference pipeline that integrates estimated FPC scores into a Cox model to formally test whether each omics view is associated with time to dementia onset. We evaluate LS-SOFA through extensive simulation studies and apply it to five longitudinal blood-based omics datasets from the ADNI to identify omics views associated with time to dementia onset. The remainder of this article is organized as follows. Section 2 describes the motivating ADNI study. Section 3 presents the LS-SOFA model and estimation algorithm. Section 4 reports simulation results comparing LS-SOFA with existing approaches. Section 5 applies LS-SOFA to the ADNI dataset. Section 6 concludes with discussion and future directions. 2 |. Motivating Study The motivating dataset is derived from the ADNI, a multicenter longitudinal cohort designed to identify and validate biomarkers for AD progression. ADNI has completed multiple phases (ADNI-1, ADNI-GO, ADNI-2, ADNI-3) and is currently in its ADNI-4 phase. Across phases, ADNI has collected comprehensive multimodal data, including demographic and clinical information, genetic and genomic profiles, neuroimaging measures, longitudinal neuropsychological assessments, and blood-based omics data. These rich longitudinal data provide an opportunity to investigate dynamic molecular signatures associated with the risk of dementia onset among individuals with MCI at baseline. In this study, we included N = 591 participants who were diagnosed with MCI at baseline and had available longitudinal blood-based omics data. Five omics views were analyzed: (1) plasma lipidomics; (2) gut microbial metabolomics from serum; (3) serum lipoproteins and metabolites; (4) serum metabolomics from flow-injection analysis (FIA); and (5) serum metabolomics from ultra-performance liquid chromatography (UPLC). Each view comprises a high-dimensional set of molecular features (up to P ≈ 750 per view) measured at irregularly spaced follow-up visits. Detailed descriptions of the preprocessing, quality control, and feature selection procedures for each omics view are provided in Supporting Information Section S3.1 . Table 1 summarizes the platform, number of features (before and after preprocessing), and data characteristics for each view. TABLE 1 |. Description of the five longitudinal omics views analyzed in this study. Omics view Omics type Features (after/before processing) Analytical platform View 1 Plasma lipidomics 749/781 LC–MS, Baker Heart and Diabetes Institute View 2 Gut microbial metabolomics (serum) 104/104 UPLC–MS/MS, University of Hawaii Cancer Center View 3 Serum lipoproteins and metabolites 173/173 NMR, Nightingale Health View 4 Serum metabolomics (FIA) 518/518 FIA–MS, Duke University View 5 Serum metabolomics (UPLC) 73/106 UPLC–MS/MS, Duke University Open in a new tab Abbreviations: FIA–MS, flow-injection analysis–mass spectrometry; LC–MS, liquid chromatography–mass spectrometry; NMR, nuclear magnetic resonance; UPLC–MS/MS, ultra-performance liquid chromatography–tandem mass spectrometry. The clinical endpoint of interest is time to the first diagnosis of dementia. For each participant, the event time is defined as the time from baseline to the first dementia diagnosis, and right-censoring occurs when a participant has not converted to dementia at the last follow-up visit. Among the 591 participants, 256 (43.3%) experienced dementia onset during the follow-up period. The cohort included 249 females (42.1%) with a mean baseline age of 73.2 years (standard deviation = 7.5 years). The distribution of apolipoprotein E (ApoE) ε 4 alleles was 307 (0 copies), 224 (1 copy), and 60 (2 copies). These data provide a good setting to evaluate the proposed LS-SOFA framework for modeling high-dimensional, irregular longitudinal omics trajectories and identifying omics views associated with progression to dementia. 3 |. Methods In AD studies, longitudinal multi-omics data can be represented as Y i ( v ) t i k , where v = 1 , … , V indexes omics views, i = 1 , … , N indexes subjects, and t i k k = 1 , … , K v i denotes the observation time points for subject i in view v . Here, V is the number of omics data modalities, N is the number of subjects, and K v i is the number of visits for subject i and view v . Each observation Y i ( v ) t i k is a row vector of length P v , where the p th element y i p ( v ) t i k corresponds to a feature (e.g., metabolite, lipid, or protein). Thus, P v denotes the number of molecular features in omics view v . Throughout the Methods, k and h index repeated observation times within subject i and omics view v , whereas p and p ′ index molecular features within the same view. We suppress the superscript on t i k for notational simplicity unless view-specific time grids must be emphasized. 3.1 |. Longitudinal Sparse Single-Omics Factor Analysis (LS-SOFA) We propose an LS-SOFA model to characterize the dynamic latent structure within a single omics view v : Y i ( v ) ( t ) = μ ( v ) ( t ) + Z i ( v ) ( t ) W v ⊤ + ϵ i ( v ) ( t ) , (1) The view-specific mean function μ ( v ) ( t ) = μ p ( v ) ( t ) p = 1 P v is a row vector of length P v that captures the average temporal pattern across features. The view-specific random profile Z i ( v ) ( t ) = z i f ( v ) ( t ) f = 1 F v represents the latent variation arising from F v factors, where F v denotes the number of latent factors (optimized as described in Section 3.2.4 ). Each latent factor z if ( v ) ( t ) provides a low-dimensional representation of the longitudinal dynamics within the omics view. The weight matrix W v = w v p f ∈ R P v × F v assigns feature-specific weights, linking observed molecular features to latent factors. The random noise vector ϵ i ( v ) ( t ) = ϵ i p ( v ) ( t ) p = 1 P v , also of length P v , is assumed to satisfy ϵ i p ( v ) ( t ) ~ N 0 , σ v p 2 and is independent of z i f ( v ) ( t ) . The LS-SOFA model in Equation (1) reduces to a baseline exploratory factor analysis when t = 0 . Each latent factor z if ( v ) ( t ) is assumed to follow a Gaussian process with covariance function C f ( v ) ( s , t ) = ∑ l = 1 ∞ d f l ( v ) ϕ f l ( v ) ( s ) ϕ f l ( v ) ( t ) , where s and t are two time points, d f l ( v ) are nonincreasing eigenvalues, and ϕ f l ( v ) ( t ) are orthonormal eigenfunctions satisfying ∫ ϕ f l ( v ) ( t ) ϕ f l ′ ( v ) ( t ) d t = I l = l ′ [ 20 ]. By Mercer’s theorem, the process admits the expansion z if ( v ) ( t ) = ∑ l = 1 ∞ ξ i f l ( v ) ϕ f l ( v ) ( t ) , where ξ i f l ( v ) ~ N 0 , d f l ( v ) are FPC scores, independent across f and l . In practice, we approximate the process by truncating to the first L v f FPCs, z i f ( v ) ( t ) ≈ ∑ l = 1 L v f ξ i f l ( v ) ϕ f l ( v ) ( t ) , where L v f is selected using the proportion of variance explained (PVE) criterion [ 21 ]. 3.2 |. Estimation of Functional Components in the LS-SOFA Framework We develop a covariance-based estimation algorithm to estimate the functional components of the LS-SOFA model in Equation (1) . The procedure consists of four sequential steps, each targeting specific model components: (1) estimation of the mean functions μ ( v ) ( t ) ; (2) estimation of the auto- and cross-covariance matrices within each omics view; (3) estimation of the weight matrix W v and the factor-specific covariance functions C f ( v ) ( s , t ) ; and (4) estimation of the subject-specific time-varying factor loadings Z i ( v ) ( t ) and the corresponding FPC scores. The optimization of the number of latent factors F v for each omics view v = 1 , … , V is described separately in Section 3.2.4 . In Step (1), the mean function μ ^ p ( v ) ( t ) for each feature p = 1 , … , P v is estimated using a penalized cubic B-spline smoother [ 25 ]. Step (2) estimates the auto-covariance matrix C p p ( v , v ) ( s , t ) , which characterizes temporal covariance of feature p within omics view v , and the cross-covariance matrix C 1 p ( v , v ) ( s , t ) , which captures covariance between a base feature and another feature p in the same view. These covariance estimates form the foundation for subsequent parameter estimation. Step (3) estimates the weight matrix W v , the factor covariance function C f ( v ) ( s , t ) for each latent factor f = 1 , … , F v , and the associated eigenfunctions. Step (4) applies the conditional expectation method to estimate the time-dependent factor loadings z if ( v ) ( t ) and performs sparse FPCA to derive the FPC scores ξ i f l ( v ) . Each of these steps is detailed in Sections 3.2.1 – 3.2.3 , which together define the complete LS-SOFA estimation procedure. 3.2.1 |. Estimation of Covariance Matrices We estimate the covariance structures of the omics data to characterize both temporal and inter-feature dependencies within each view v . Specifically, C p p v , v s , t = cov y i p v s , y i p v t = ∑ f = 1 F v w v p f 2 C f v s , t + σ v p 2 I s = t , (2) C p p ′ ( v , v ) ( s , t ) = cov y i p ( v ) ( s ) , y i p ′ ( v ) ( t ) = ∑ f = 1 F v w v p f w v p ′ f C f ( v ) ( s , t ) , p ≠ p ′ , (3) where C p p ( v , v ) ( s , t ) denotes the feature-level auto-covariance and C p p ′ ( v , v ) ( s , t ) denotes the within-view cross-covariance between features p and p ′ . These covariance functions are estimated using residual-based unbiased estimators. For each feature p , the residual is defined as r i p ( v ) t i k = y i p ( v ) t i k - μ p ( v ) t i k , where μ p ( v ) ( t ) is the mean function estimated in Step (1). Since E r i p ( v ) ( s ) r i p ( v ) ( t ) = C p p ( v , v ) ( s , t ) , the empirical product r i p ( v ) ( s ) r i p ( v ) ( t ) provides an unbiased estimate of the auto-covariance. In practice, we replace μ p ( v ) ( t ) with its spline estimate μ ^ p ( v ) ( t ) and compute residuals r ^ i p ( v ) t i k = y i p ( v ) t i k - μ ^ p ( v ) t i k . The covariance surface C p p ( v , v ) ( s , t ) is approximated by a smooth function F ( v ) ( s , t ) , modeled via a tensor-product spline representation, F ( v ) ( s , t ) = ∑ r 1 = 1 R ∑ r 2 = 1 R θ r 1 r 2 ( v ) B r 1 ( s ) B r 2 ( t ) for s ≠ t , where B r 1 ( ⋅ ) and B r 2 ( ⋅ ) are B-spline basis functions, θ r 1 r 2 ( v ) are coefficients, and R is the number of basis functions in each direction. Let Θ ( v ) = θ r 1 r 2 ( v ) r 1 , r 2 = 1 R denote the R × R coefficient matrix. We estimate Θ ( v ) by minimizing a penalized least squares criterion, min Θ ( v ) ∑ i = 1 N ∑ h = 1 K v i ∑ k = 1 K v i F ( v ) t i h , t i k - r ^ i p ( v ) t i h r ^ i p ( v ) t i k 2 + τ g Θ ( v ) , h ≠ k , where g ( ⋅ ) is a smoothness penalty and τ controls the level of regularization. We adopt a thin-plate regression spline with shrinkage [ 26 ]. This choice was motivated by three considerations. First, thin-plate regression splines provide a low-rank smoother that avoids manual knot placement, which is advantageous for sparse and irregular longitudinal observation times. Second, the shrinkage penalty regularizes weak covariance-surface components and improves numerical stability when estimating many feature-specific covariance surfaces. Third, in our sensitivity checks, cubic B-splines with shrinkage yielded nearly identical numerical results; therefore, thin-plate regression splines were selected as the default because they require fewer user-specified tuning choices and performed stably across omics views. The estimated covariance function is then given by C ^ p p ( v , v ) ( s , t ) = ∑ r 1 = 1 R ∑ r 2 = 1 R θ ^ r 1 r 2 ( v ) B r 1 ( s ) B r 2 ( t ) for s ≠ t . Similarly, to estimate the within-view cross-covariance C 1 p ′ ( v , v ) ( s , t ) for p ′ = 2 , … , P v , we use the residual-based product r ^ i 1 ( v ) ( s ) r ^ i p ′ ( v ) ( t ) and apply the same penalized spline method, ensuring consistent estimation across all covariance components. 3.2.2 |. Estimation of Weight Matrix, Error Variance, and Factor Covariance for Each View This subsection describes the iterative estimation of the weight matrix W v , error variance matrix Ψ ( v ) , and factor covariance matrix C f ( v ) ( s , t ) for each omics view v . Estimation is based on the baseline omics data y i ( v ) ( 0 ) = y i p ( v ) ( 0 ) p = 1 , … , P v and its sample covariance matrix Σ ^ ( v ) . The error variance matrix Ψ ( v ) is defined as a P v × P v diagonal matrix with elements σ v p 2 , assuming that the residual components ε i p ( v ) ( t ) are mutually independent across features and time within each view and have constant variance σ v p 2 across subjects. Hence, Ψ ( v ) captures feature-specific measurement noise that is time-invariant but view-specific. These assumptions are standard in factor-analytic covariance models and are required for the identifiability of W v and Z i ( v ) ( t ) . In latent factor models, Z i ( v ) ( t ) , W v is identifiable only up to orthogonal rotations and column-wise sign changes. To ensure identifiability and interpretability, we impose conventions consistent with exploratory factor analysis; a varimax rotation is applied to W ^ v to enhance loading sparsity without affecting the likelihood or fitted covariance and its sign is set such that the largest loading is positive. All subsequent steps, including estimation of C f ( v ) ( s , t ) and FPCA, are based on this rotated and oriented W ^ v , denoted simply by W v for notational simplicity. To anchor the factor scale at baseline and ensure comparability of trajectories across views, we impose a variance constraint cov z i f ( v ) ( 0 ) = ∑ l = 1 ∞ d f l ( v ) ϕ f l ( v ) ( 0 ) 2 = 1 for f = 1 , … , F v . Under this constraint, the covariance matrix of the baseline omics data is expressed as Σ ( v ) = cov y i ( v ) ( 0 ) = W v W v ⊤ + Ψ ( v ) . We adopt an iterative procedure to estimate Ψ ( v ) and W v . For clarity, the superscript ( v ) and subscript v are omitted unless necessary. The procedure is summarized below and in Algorithm 1 . Initialization of Covariance Estimates. Compute the estimated auto-covariance C ^ p p ( v , v ) ( s , t ) for s ≠ t , p = 1 , … , P v , and cross-covariance C ^ 1 p ( v , v ) ( s , t ) for p = 2 , … , P v . Initialization of Error Variance. Set Ψ ( 1 ) as a diagonal matrix with small values (e.g., σ v p 2 = 0.01 ) and initialize the convergence flag ϵ = 1 . Iterative Refinement of Weight Matrix and Error Variance. Compute the residual covariance Σ ^ * = Σ ^ - Ψ ( 1 ) and perform eigendecomposition Σ ^ * = C D C ⊤ , where C contains eigenvectors and D is a diagonal matrix of ordered eigenvalues. Truncate D to its first F v eigenvalues and update W ( 1 ) = C D 1 / 2 . Update the error variance matrix as the diagonal of Σ ^ - W ( 1 ) W ( 1 ) ⊤ . Compute ϵ = max Ψ ( 2 ) - Ψ ( 1 ) , replace Ψ ( 1 ) with Ψ ( 2 ) , and iterate until ϵ < 10 - 6 . Identifiability Adjustment. Apply the varimax rotation to W ^ v for interpretability and set its sign such that the largest loading is positive. This enforces identifiability and ensures consistent scaling across views. Estimation of Factor Covariance Matrix. Using least squares, estimate C ^ f ( v ) ( s , t ) for time-dependent factors z if ( v ) ( t ) . For each ( s , t ) pair, define Y = Y 1 ⊤ , Y 2 ⊤ ⊤ , where Y 1 contains C ˆ p p ( v , v ) ( s , t ) - σ ˆ v p 2 I ( s = t ) for p = 1 , … , P v , and Y 2 contains C ˆ 1 p ( v , v ) ( s , t ) for p = 2 , … , P v . Construct X = X 1 ⊤ , X 2 ⊤ ⊤ , where X 1 ∈ R P v × F v has rows w ^ v p f 2 , and X 2 ∈ R P v - 1 × F v has rows w ^ v 1 f w ^ v p f . Eigendecomposition of covariance functions. Perform eigendecomposition on C ^ f ( v ) ( s , t ) to obtain the first L v f eigenvalues d ^ f l ( v ) and eigenfunctions ϕ ^ f l ( v ) ( t ) for l = 1 , … , L v f , where L v f is determined by the PVE criterion using the 95% threshold. ALGORITHM 1 |. Iterative estimation algorithm for the weight matrix W v , error variance matrix Ψ ( v ) , and factor covariance matrix C f ( v ) ( s , t ) for each latent factor z if ( v ) ( t ) . Input: Estimated auto-covariance C ^ p p ( v , v ) ( s , t ) p = 1 , … , P v and cross-covariance C ^ 1 p ( v , v ) ( s , t ) p = 2 , … , P v . Output: Estimated weight matrix w ^ v , error variance Ψ ^ ( v ) , and factor covariance C ^ f ( v ) ( s , t ) . 1: Compute the sample covariance Σ ^ ( v ) for baseline omics data y i ( v ) ( 0 ) 2: for v = 1 to V do 3: Initialize Ψ ( 1 ) = diag ( 0.01 ) and set ϵ = 1 4: while ϵ > 10 − 6 do 5: Compute Σ ^ * = Σ ^ − Ψ ( 1 ) . 6: Perform eigendecomposition Σ ^ * = C D C ⊤ . 7: Truncate D to the first F v eigenvalues and update W ( 1 ) = CD 1 / 2 . 8: Update Ψ ( 2 ) = diag Σ ^ − W ( 1 ) W ( 1 ) ⊤ . 9: Set ϵ = max Ψ ( 2 ) − Ψ ( 1 ) and replace Ψ ( 1 ) with Ψ ( 2 ) . 10: end while 11: Apply varimax rotation to w ^ v and orient signs for interpretability. 12: Estimate C f ( v ) ( s , t ) using least squares. 13: end for Open in a new tab This iterative procedure yields consistent estimates of the rotated and normalized loading matrices under standard factor-model assumptions and stabilizes cross-view comparability in multi-omics applications. 3.2.3 |. Estimation of Time-Dependent Factor Loadings and FPC Scores In the final estimation step, the time-dependent factor loadings z if ( v ) ( t ) are estimated via the conditional expectation approach, followed by sparse FPCA to obtain the corresponding FPC scores. For each omics view v , the observed omics data y i p ( v ) ( t ) are first demeaned to remove the estimated mean function, yielding y i p ( v ) * ( t ) = y i p ( v ) ( t ) - μ ^ p ( v ) ( t ) . Note that y i p ( v ) * ( t ) corresponds to the previously defined residuals r ˆ i p ( v ) ( t ) used for covariance estimation; we use this notation here to emphasize that the demeaned measurements serve as input for estimating the subject- and time-specific latent factors. For subject i , the demeaned omics measurements are vectorized as Y i ( v ) * = Y i 1 ( v ) * , … , Y i P v ( v ) * ⊤ , where Y i p ( v ) * = y i p ( v ) * t i 1 , … , y i p ( v ) * t i K v i ⊤ represents the time series of feature p . These demeaned data serve as the input for estimating the subject- and time-specific latent factors. Under the LS-SOFA framework, the joint distribution of the demeaned omics vector Y i ( v ) * and the latent factor z if ( v ) t i k at time t i k follows a multivariate normal distribution: Y i ( v ) * z i f ( v ) t i k ~ N 0 0 , Σ 11 Σ 12 Σ 12 ⊤ Σ 22 , (4) where Σ 11 denotes the covariance matrix of Y i ( v ) * , Σ 12 represents the covariance between Y i ( v ) * and z if ( v ) t i k , and Σ 22 is the variance of z if ( v ) t i k . Based on the conditional expectation formula for multivariate normal distributions, the conditional mean of the latent factor is given by E z i f ( v ) t i k ∣ Y i ( v ) * = Σ 12 ⊤ Σ 11 - 1 Y i ( v ) * . In practice, this conditional expectation is approximated using estimated covariance components obtained from Section 3.2.2 : z ^ i f ( v ) t i k = Σ ^ 12 ⊤ Σ ^ 11 - 1 Y i ( v ) * , where Σ ^ 11 and Σ ^ 12 are the empirical estimates of the population covariance matrices. The resulting z ^ i f ( v ) t i k represents the estimated time-dependent latent factor loading for subject i , factor f , and view v . Finally, sparse FPCA [ 20 ] is applied to the estimated loading trajectories z ^ if ( v ) ( t ) across all subjects to extract the subject-specific FPC scores ξ ˆ i f l ( v ) , l = 1 , … , L v f These FPC scores form the basis for subsequent longitudinal association analyses with time-to-event outcomes. Detailed derivations of Σ 11 , Σ 12 , and Σ 22 are provided in Supplemental Section S1.1 . In the primary analyses, L v f was selected using the 95% PVE criterion. Because L v f determines the dimension of the FPC-score vector entering the Cox model, we report the selected L v f values for simulation Scenarios 1–7 in Table S3 and for the ADNI application in Section 5.2 . We also evaluate sensitivity to alternative PVE thresholds of 90% and 99% in Supporting Information Section S3.3 and Tables S5 and S6 . 3.2.4 |. Selection of the Number of Latent Factors The number of latent factors F v for each omics view v = 1 , … , V is selected using a mean squared error (MSE) criterion. For a fixed F v , the estimation algorithm described in Sections 3.2.1 – 3.2.3 is applied to obtain the estimated mean function μ ˆ ( v ) ( t ) , error variance matrix Ψ ^ ( v ) , weight matrix W ^ v , and functional components d ˆ f l ( v ) , ϕ ˆ f l ( v ) ( t ) . The time-dependent latent factor loadings z ^ i f ( v ) ( t ) are estimated using the conditional expectation approach introduced in Section 3.2.3 . Given these estimated components, the reconstructed longitudinal omics data are defined as Y ˆ i ( v ) ( t ) = μ ^ ( v ) ( t ) + Z ^ i ( v ) ( t ) W ^ v ⊤ . The model fit is evaluated by computing the mean squared error between the reconstructed and observed omics data, averaged across subjects and time: MSE F v = 1 N ∑ i = 1 N 1 K v i ∑ k = 1 K v i Y i ( v ) t i k - Y ^ i ( v ) t i k 2 . (5) We vary F v over a prespecified range and select the optimal value F ^ v that minimizes MSE F v . This criterion balances model complexity and reconstruction accuracy, providing a data-driven and computationally efficient choice for the number of latent factors in each omics view. We used the reconstruction MSE criterion because the primary role of F v is to determine the dimension of the latent factor representation needed to recover the longitudinal covariance structure within each omics view. Information criteria are difficult to define in this setting because the effective degrees of freedom depend jointly on covariance smoothing, factor rotation, FPCA truncation, and sparse irregular observation times. Full cross-validation is possible but computationally expensive for high-dimensional omics data because the covariance estimation and eigendecomposition steps must be repeated for each candidate F v and each fold. The reconstruction MSE criterion provides a direct, interpretable, and computationally efficient measure of model fit. In the ADNI application, we further used a plateau rule to select the smallest factor number with near-minimal MSE, balancing reconstruction accuracy and parsimony. 3.3 |. Hypothesis Testing for the Association With Survival Outcome After obtaining the estimated FPC scores ξ ˆ i f l ( v ) , we assess whether each omics view v is significantly associated with the time-to-event outcome. For subject i , let T i , δ i denote the observed survival data, where T i = min T i * , C i is the observed event or censoring time, δ i = I T i * ≤ C i is the event indicator, T i * is the true failure time, and C i is the censoring time. The baseline covariate vector w i includes demographic or clinical covariates (e.g., age, sex, or APOE ε 4 status). We fit a Cox model to relate the survival outcome to the FPC scores from omics view v : h i ( t ) = h 0 ( t ) exp β ⊤ w i + γ v ⊤ ξ ˆ i ( v ) , where h i ( t ) is the hazard function for subject i , h 0 ( t ) is the baseline hazard, and ξ ˆ i ( v ) = ξ ˆ i f ( v ) f = 1 , … , F v is the subject-specific vector of estimated FPC scores for omics view v . To formally test whether omics view v contributes to the risk of dementia, we test the null hypothesis H 0 : γ v = 0 against the alternative H 1 : γ v ≠ 0 . The Wald statistic is computed as T v = γ ˆ v ⊤ cov γ ˆ v - 1 γ ˆ v → d χ L v 2 , where cov γ ˆ v is the estimated covariance matrix of γ ˆ v , and L v = ∑ f = 1 F v L v f is the total number of FPC scores included for view v . The corresponding p -value, p v , is compared to the nominal significance level α = 0.05 . A significant result ( p v ≤ α ) indicates that omics view v is associated with the survival outcome and is thus considered informative in explaining time to dementia onset. Because the ADNI application evaluates five omics views, we also considered multiplicity across view-level tests. The primary analysis reports nominal Wald-test p -values to preserve the exploratory, view-screening objective of the analysis. As a multiplicity-aware sensitivity analysis, we additionally report Benjamini-Hochberg false discovery rate (FDR) adjusted q -values across the five omics-view tests. This adjustment is applied at the omics-view level; the subsequent feature ranking is descriptive and intended for biological interpretation rather than formal feature-wise inference. This Wald test is based on a two-stage plug-in procedure. Specifically, the FPC scores are first estimated from the LS-SOFA model and then treated as covariates in the Cox model. Consequently, the model-based covariance matrix cov γ ^ v from the Cox model does not analytically propagate uncertainty from the first-stage estimation of ξ ˆ i ( v ) . To evaluate the operating characteristics of this plug-in procedure, the simulation study repeats the full two-stage pipeline, including LS-SOFA estimation, FPC-score extraction, Cox model fitting, and Wald testing. Thus, the empirical Type I error and power reported in Section 4 reflect the proposed inference procedure rather than an oracle analysis using true latent scores. 3.4 |. Identification of Informative Omics Features To identify individual molecular features that most strongly contribute to the latent structure captured by LS-SOFA, we quantify the overall importance of each feature based on its factor loadings. For omics view v , let w ^ v p = w ^ v p f f = 1 , … , F v denote the estimated loading vector for feature p from Equation (1) . We define a feature importance measure as the ℓ 1 -norm of its loading vector, n ( v , p ) = ∑ f = 1 F v w ^ v p f , p = 1 , … , P v , which summarizes the overall magnitude of association between feature p and the latent factors within view v . Features with larger n ( v , p ) values contribute more strongly to the shared variance structure. We rank features in descending order of n ( v , p ) and designate the top K features as informative, where K is a user-specified threshold (e.g., K = 10 , 50 , or 100 ). These selected features are subsequently examined for biological interpretability, including their pathway annotations and known relevance to AD pathophysiology. 4 |. Simulation Study 4.1 |. Simulation Design We conducted a comprehensive simulation study to evaluate the longitudinal estimation performance of the proposed LS-SOFA framework under multiple data-generating mechanisms. Three primary scenarios were designed, each repeated 100 times. In all scenarios, three omics views (A, B, and C) were simulated from the LS-SOFA model (1), with N = 100 subjects in the training set and N = 200 subjects in the testing set. Scenario 1: Independent views. A, B, and C are mutually independent. View A has three latent factors F 1 , F 2 , F 3 ; view B has two factors F 4 , F 5 ; and view C has two factors F 6 , F 7 . Scenario 2: Partially dependent views. A and B share one latent factor F 1 , but both are independent of C. View A has F 1 , F 2 , F 3 , view B has ( F 1 , F 4 , F 5 ), and view C has ( F 6 , F 7 ). Scenario 3: Fully correlated views. All three views share the latent factor F 1 . View A has F 1 , F 2 , F 3 , view B has F 1 , F 4 , F 5 , and view C has ( F 1 , F 6 , F 7 ). Estimation performance was evaluated using the MSE between the estimated and true omics trajectories (excluding random error), averaged over time and subjects for each view. We computed both within-sample (training) MSE and out-of-sample (testing) MSE. The computation of out-of-sample MSE is described in Supporting Information Section S1.2 . The LS-SOFA results were compared with four existing approaches ( Supporting Information Section S2.1 ): (1) MEFISTO [ 17 ]; (2) FTSVD [ 27 ]; (3) FPCA [ 20 ]; and (4) Canonical Polyadic (CP) tensor decomposition [ 28 ]. The corresponding out-of-sample MSEs are summarized in Table S1 . To evaluate the effect of sample size, we further repeated Scenarios 1–3 with N = 200 and N = 400 subjects in the training set. The results, presented in Table S2 , demonstrate the performance of LS-SOFA across varying sample sizes. We also designed two robustness scenarios in which the true data-generating mechanism followed (i) the MEFISTO model (Scenario 4) and (ii) the FTSVD model (Scenario 5), to assess the robustness of LS-SOFA under model misspecification ( Supporting Information Sections S2.3 – S2.4 ). Finally, we extended the simulation framework to include survival outcomes for evaluating hypothesis testing performance (Scenarios 6 and 7) and factor selection performance (Scenarios 8 and 9). Scenarios 6 and 7 assess the Type I error and power of the survival association test, respectively, whereas Scenarios 8 and 9 evaluate the mean square error–based selection of the optimal number of factors F v for each omics view. Detailed settings for these scenarios are provided in Supporting Information Sections S2.5 – S2.7 . 4.2 |. Simulation Specification We generated longitudinal omics data from the LS-SOFA model (1) to illustrate Scenarios 1–3, with additional settings for Scenarios 4–9 provided in Supporting Information Sections S2.3 – S2.7 . Each simulated dataset contained three omics views ( V = 3 ), each with P = 200 features. For subject i and view v , observation times were defined as t i ( v ) = t i 0 , t i 1 ( v ) , t i 1 ( v ) + 0.1 , … , t i 1 ( v ) + 0.9 , where t i 0 = 0 is the baseline time and t i 1 ( v ) is drawn from a discrete uniform distribution on { 0.01,0.02 , … , 0.10 } . Time grids were generated independently across views, and the mean function μ p ( v ) ( t ) was set to zero for all features and views. The shared latent factor F 1 was simulated from a Gaussian process representation z i 1 ( t ) = ∑ l = 1 2 ξ i 1 l ϕ 1 l ( t ) , where ξ i 1 l ~ N 0 , d 1 l with d 1 l = ( 2,0.5 ) , ϕ 11 ( t ) = 2 sin ( 3 π t ) , and ϕ 12 ( t ) = 2 cos ( 3 π t ) . Feature weights for the shared factor in view A were generated as w 1 p 1 ~ Bin ( 1,0.5 ) × N ( 0,0.25 ) for p = 1 , … , 200 . Scenario 1 assumed no cross-view sharing of F 1 w 2 p 1 = w 3 p 1 = 0 ), Scenario 2 allowed sharing between A and B (same generation for w 2 p 1 as w 1 p 1 ), and Scenario 3 extended sharing to all three views (same generation for w 2 p 1 and w 3 p 1 ). For view-specific factors F 2 and F 3 , we generated z i f ( t ) = ∑ l = 1 2 ξ i f l ϕ f l ( t ) for f = 2 , 3 , where ξ i f l ~ N 0 , d f l , d 2 l = ( 8,0.5 ) , and d 3 l = ( 4,0.5 ) . The eigenfunctions were set as ϕ 21 ( t ) = 2 sin ( π t ) , ϕ 22 ( t ) = 2 cos ( π t ) , ϕ 31 ( t ) = 2 sin ( 2 π t ) , and ϕ 32 ( t ) = 2 cos ( 2 π t ) . The feature weights w 1 p f for f = 2 , 3 were drawn from Bin ( 1,0.5 ) × N ( 0,0.25 ) , and the same scheme was applied for view-specific factors in views B and C. Random errors were independently sampled as ϵ i p ( v ) ( t ) ~ N 0 , σ v p 2 with σ v p ~ Uniform ( 0.6,1.0 ) , ensuring heterogeneous measurement variability across features and views. 4.3 |. Simulation Results We evaluated the longitudinal estimation performance of the proposed LS-SOFA framework against four benchmark methods (MEFISTO, FTSVD, FPCA, and CP) across Scenarios 1–5. For each method, the longitudinal omics trajectories were estimated, and the within-sample mean squared error (MSE) was computed as the average squared deviation between the estimated and true trajectories (excluding random error), averaged across subjects, time points, and features. Supporting Information Section S2.1 presents the model specifications for each candidate method, including hyperparameter settings and selection criteria. Table 2 summarizes the results from 100 simulation replications. When the true data-generating mechanism followed the LS-SOFA model (Scenarios 1–3), LS-SOFA achieved substantially smaller MSEs than all competing approaches across all omics views, confirming its superior estimation accuracy. Under model misspecification (Scenarios 4–5), where data were generated from the MEFISTO or FTSVD model, LS-SOFA retained competitive accuracy, closely matching MEFISTO and FTSVD but outperforming FPCA and CP. The lower panel of Table 2 reports the average computation time for Scenario 1, averaged over 100 simulation replications. LS-SOFA required 21.5 s on average, compared with 273.0 s for MEFISTO, 25.7 s for FTSVD, and 130.8 s for FPCA. Although the CP model was the fastest (5.3 s), its estimation accuracy was markedly inferior, as reflected in substantially higher MSE values. Overall, LS-SOFA achieved an optimal balance between estimation accuracy and computational efficiency, demonstrating scalability for large-scale longitudinal omics data. TABLE 2 |. Within-sample mean squared error (MSE) and average computation time based on 100 simulation replications. Sample size View MSE of LS-SOFA MSE of MEFISTO MSE of FTSVD MSE of FPCA MSE of CP Scenario 1 N = 100 A 0.204 2.720 1.982 0.249 1.116 B 0.150 2.699 2.018 0.175 0.851 C 0.143 2.640 1.957 0.171 0.842 Scenario 2 N = 100 A 0.213 2.692 1.946 0.247 1.087 B 0.197 2.698 1.960 0.251 1.065 C 0.149 2.636 1.979 0.172 0.852 Scenario 3 N = 100 A 0.198 2.591 1.911 0.244 1.054 B 0.178 2.633 1.942 0.245 1.066 C 0.195 2.652 1.975 0.250 1.092 View MSE of LS-SOFA MSE of MEFISTO MSE of FTSVD MSE of FPCA MSE of CP Scenario 4 N = 100 A 0.020 0.027 0.075 0.059 0.102 B 0.018 0.026 0.075 0.058 0.103 Scenario 5 N = 100 A 0.016 0.017 0.007 0.020 0.135 B 0.016 0.018 0.007 0.020 0.135 Average computing time for Scenario 1 (in seconds) 21.5 273.0 25.7 130.8 5.3 Open in a new tab Note: Scenarios 1–3 were generated from the LS-SOFA model, and Scenarios 4–5 were generated from the MEFISTO and FTSVD models, respectively. The computation times (in seconds) correspond to the average per replication for Scenario 1, evaluated on a 13th-Gen Intel i7-13 700 2.10 GHz CPU. Out-of-sample predictive performance was assessed using 200 testing subjects. Table S1 displays the out-of-sample MSE from LS-SOFA and the four competing methods, based on 100 simulation replications. LS-SOFA consistently achieved the lowest or near-lowest prediction errors across Scenarios 1–3 and maintained strong predictive accuracy even when the true model followed MEFISTO (Scenario 4) or FTSVD (Scenario 5). To evaluate the impact of sample size, additional simulations with N = 200 and N = 400 training subjects ( Table S2 ) demonstrated that the within-sample MSE of LS-SOFA approached zero under all scenarios, confirming convergence and scalability of the estimation procedure. Table S3 reports the number of retained FPC scores L v f selected by the 95% PVE criterion for simulation Scenarios 1–7. Scenarios 8 and 9 evaluated the MSE-based procedure for selecting the optimal F v over candidate values from 1 to 10; hence, the corresponding L v f values are not separately reported for these scenarios. We further examined the hypothesis testing performance under survival outcomes using Scenarios 6 and 7. Type I error control was evaluated under the null hypothesis ( γ v = 0 ), and statistical power was assessed under the alternative hypothesis ( γ v ≠ 0 ) across varying effect sizes. In Scenarios 6 and 7, the complete two-stage analysis pipeline was repeated in each simulation replication: LS-SOFA was first fitted to the simulated longitudinal omics data, estimated FPC scores were obtained, and these estimated scores were then used in the Cox model for Wald testing. Therefore, the empirical Type I error and power reflect the impact of estimating the FPC scores, rather than relying on the true latent scores used in data generation. Table 3 summarizes the empirical Type I error and power based on 100 simulation replications. The LS-SOFA test maintained appropriate Type I error rates near the nominal 0.05 level for all sample sizes ( N = 100,200,400 ) and demonstrated increasing power with larger effect sizes, approaching 1 when γ v = 0.2 . These findings confirm the reliability of the LS-SOFA-based hypothesis testing in survival analyses. TABLE 3 |. Empirical Type I error (Scenario 6) and power (Scenario 7) of the LS-SOFA hypothesis testing procedure across varying effect sizes γ v and sample sizes ( N = 100,200,400 ), based on 100 simulation replications. Scenario 6 (Type-I error) Scenario 7 (Power) N = 100 N = 200 N = 400 N = 100 N = 200 N = 400 γ v = 0 0.07 0.06 0.06 γ v = 0.05 0.10 0.27 0.46 γ v = 0.06 0.14 0.31 0.62 γ v = 0.07 0.19 0.40 0.80 γ v = 0.08 0.24 0.56 0.92 γ v = 0.09 0.31 0.65 0.97 γ v = 0.1 0.41 0.72 0.98 γ v = 0.11 0.49 0.84 1 γ v = 0.12 0.55 0.94 1 γ v = 0.13 0.64 0.97 1 γ v = 0.14 0.72 0.99 1 γ v = 0.15 0.77 0.99 1 γ v = 0.16 0.83 0.99 1 γ v = 0.17 0.88 1 1 γ v = 0.18 0.90 1 1 γ v = 0.19 0.94 1 1 γ v = 0.2 0.98 1 1 Open in a new tab Finally, Scenarios 8 and 9 evaluated the MSE-based procedure for selecting the optimal number of factors F v for each omics view. Three omics views (A, B, and C) were simulated, with A and B associated with the survival outcome and C unassociated. Table 4 summarizes the frequency of selecting each F v and the corresponding hypothesis testing rejection rates. When the true number of factors was F v = 3 (Scenario 8) or F v = 5 (Scenario 9), the MSE criterion selected at least the correct number of factors with probability near 1, and the rejection rates remained close to 1 for associated views (A and B) and near 0.05 for the null view (C). These results indicate that the MSE-based selection approach yields stable and accurate identification of informative factor dimensions. TABLE 4 |. Frequency of selecting the number of latent factors using the MSE criterion and the corresponding rejection rates in Scenarios 8–9, each based on 100 simulation replications. Scenario 8 (True F v = 3 ) Scenario 9 (True F v = 5 ) MSE Testing MSE Testing View A B C A B C A B C A B C F v = 1 0 0 0 1 1 0.04 0 0 0 1 1 0.01 F v = 2 0 0 0 1 1 0.01 0 0 0 1 1 0 F v = 3 0.05 0.02 0.04 1 1 0.01 0 0 0 1 1 0.01 F v = 4 0.05 0.08 0.07 1 1 0.02 0 0 0 1 1 0 F v = 5 0.04 0.03 0.03 1 1 0.05 0.03 0.05 0.01 1 1 0 F v = 6 0.03 0.02 0.03 1 1 0.04 0.06 0.05 0.07 1 1 0.01 F v = 7 0.07 0.03 0.04 1 1 0.04 0.13 0.03 0.14 1 1 0.01 F v = 8 0.04 0.09 0.05 1 1 0.05 0.08 0.14 0.06 1 1 0.03 F v = 9 0.14 0.16 0.18 1 1 0.03 0.21 0.13 0.17 1 1 0 F v = 10 0.58 0.57 0.56 0.99 1 0.04 0.49 0.6 0.55 1 1 0.01 Open in a new tab 5 |. Application to ADNI Longitudinal Multi-Omics Data We applied the LS-SOFA framework to the ADNI cohort, which includes five longitudinal blood-based omics views: (1) plasma lipidomics, (2) serum gut microbial metabolomics, (3) serum lipoproteins and metabolites, (4) serum metabolomics from FIA, and (5) serum metabolomics from UPLC. The analysis included 591 participants with MCI at baseline, contributing 1634, 1617, 1637, 1639, and 1639 total visits across the five omics views, respectively. During follow-up, 256 participants progressed to dementia, corresponding to an event rate of 43.3%. 5.1 |. Selection of the Optimal Number of Factors For each omics view v , the number of latent factors F v was selected using an MSE–based criterion applied to the reconstructed longitudinal omics data. For each candidate F v , we computed MSE as in Equation (5) . To balance model fit and parsimony, we selected the smallest F v whose MSE F v was within 2% of the minimum observed value (the “elbow” or plateau rule). This approach provides stable factor selection across omics views and guards against overfitting. We evaluated F v values from 1 to 20 and computed the MSE ratio relative to a reference configuration, MSE F v / MSE F v ref , where F v ref was initially set to 5, 7, 5, 7, and 12 for Views 1–5, respectively. As shown in Figure 1 , the MSE ratios for Views 1–4 flatten after approximately 3–7 factors, and for View 5 after 10–12 factors, indicating minimal incremental improvement beyond these points. Accordingly, the optimal numbers of factors were selected as 3, 5, 3, 5, and 10 for Views 1–5, respectively. A sensitivity analysis using alternative factor counts is provided in Supporting Information Section S3.2 and Table S4 . FIGURE 1 |. Open in a new tab Ratios of mean squared error (MSE) to the reference value across candidate factor numbers for each omics view. Reference factor numbers were set to 5, 7, 5, 7, and 12 for Views 1–5, respectively. The curves illustrate diminishing improvements in reconstruction accuracy beyond the selected factor counts. 5.2 |. Results of the ADNI Analysis Using the optimal numbers of latent factors (3, 5, 3, 5, and 10 for omics Views 1–5, respectively), we estimated the FPC scores and tested their associations with time to dementia onset. Table 5 summarizes the hypothesis testing results. At the nominal 0.05 level, four omics views were associated with dementia risk: plasma lipidomics (View 1), gut microbial metabolomics from serum (View 2), serum metabolomics from FIA (View 4), and serum metabolomics from UPLC (View 5), with p -values of 8.70 × 10 −3 , 4.38 × 10 −2 , 8.40 × 10 −5 , and 2.94 × 10 −5 ), respectively. After Benjamini-Hochberg FDR adjustment across the five omics-view tests, Views 1, 4, and 5 remained significant at FDR 0.05, whereas View 2 showed suggestive evidence of association but did not remain below the FDR 0.05 threshold. View 3, serum lipoproteins and metabolites, did not reach statistical significance under either criterion. Under the 95% PVE criterion, the selected numbers of FPC scores for Views 1–5 were L 1 = ( 2,2 , 2 ) , L 2 = ( 2,2 , 2,2 , 2 ) , L 3 = ( 2,1 , 2 ) , L 4 = ( 2,2 , 2,2 , 2 ) , and L 5 = ( 2,1 , 1,2 , 2,2 , 3,1 , 2,2 ) , respectively. We also conducted sensitivity analyses using 90% and 99% PVE thresholds for selecting L v j ; the corresponding hypothesis testing results are provided in Supporting Information Section S3.3 and Tables S5 and S6 . TABLE 5 |. Summary of the hypothesis testing results in the real data analysis using the Wald test. Omics view Omics type Test statistic Degree of freedom p BH-FDR q -value PVE View 1 Plasma lipidomics 17.2 6 8.70E-03 1.45E-02 53.0% View 2 Gut microbial metabolomics from serum 18.7 10 4.38E-02 5.48E-02 78.9% View 3 Serum lipoproteins and metabolites 7.7 5 1.74E-01 1.74E-01 87.5% View 4 Serum metabolomics from FIA 36.0 10 8.40E-05 2.10E-04 77.8% View 5 Serum metabolomics from UPLC 52.7 18 2.94E-05 1.47E-04 89.6% Open in a new tab Note: PVE stands for the proportion of variance explained. Nominal p -values and Benjamini-Hochberg FDR-adjusted q -values are reported across the five omics-view tests. Significant p -values and FDR-adjusted q -values below 0.05 are boldfaced. To illustrate the longitudinal factor trajectories, Figure 2 shows the estimated factors z ^ i 1 ( 1 ) ( t ) , z ^ i 2 ( 1 ) ( t ) , and z ^ i 3 ( 1 ) ( t ) for View 1 (plasma lipidomics), smoothed using locally weighted scatterplot smoothing (LOWESS). Among participants who converted to dementia (AD-converters), Factor 1 showed an early decline after year 1, whereas non-converters exhibited a gradual increase after year 3. Factor 2 declined in converters after year 3 but increased in non-converters after year 4. Factor 3 increased steadily in converters between years 3–6, while remaining largely stable in non-converters. These patterns suggest heterogeneous molecular progression trajectories between converters and non-converters. Estimated eigenfunctions for View 1 are provided in Figure S2 , and analogous plots for Views 2, 4, and 5 appear in Figures S4 – S12 . FIGURE 2 |. Open in a new tab Estimated longitudinal factor trajectories z ^ i 1 ( 1 ) ( t ) , z ^ i 2 ( 1 ) ( t ) , and z ^ i 3 ( 1 ) ( t ) for plasma lipidomics (View 1), smoothed using locally weighted scatterplot smoothing (LOWESS) and truncated at 7.5 years. Blue lines represent non-converters, and red lines represent AD-converters. We next identified the most informative omics features within each view based on their feature-specific weight magnitudes. For each feature p in View 1, we computed the L1-norm n ( 1 , p ) = ∑ f = 1 F 1 w ˆ 1 p f and ranked all features in descending order. The top six features, CE(20:5) (cholesteryl ester), TG(54:7) [NL-22:6] (triglycerides, neutral loss), TG ( 54 : 6 ) [NL-22:6], TG(56:9) [NL-22:6], DG(16:0_22:6) (diacylglycerol), and TG(52:5) [NL-20:5], displayed distinct longitudinal profiles between converters and non-converters ( Figure S3 ). For example, CE(20:5) decreased in converters from year 1 to 3 but stabilized in non-converters during the same period. Similar heterogeneous temporal trends were observed for the remaining top-ranked lipids. Corresponding feature-level trajectories for Views 2, 4, and 5 are presented in Figures S6 , S9 , and S12 . 6 |. Discussion In this study, we propose a LS-SOFA framework for modeling high-dimensional longitudinal blood-based omics data in AD research. The model integrates FPCA to capture nonlinear temporal trajectories through latent factors and introduces a scalable estimation algorithm that efficiently handles large sample sizes and complex data structures. Through extensive simulations, LS-SOFA consistently outperformed MEFISTO, FTSVD, FPCA, and CP in reconstruction accuracy under the true model and remained robust under model misspecification. The Type I error rate was well controlled, statistical power increased appropriately with effect size, and the mean squared error (MSE)-based factor selection procedure was stable across scenarios. Application of LS-SOFA to the ADNI dataset revealed that plasma lipidomics and serum metabolomics profiled by FIA and UPLC remained associated with time to dementia onset after FDR adjustment, whereas gut microbial metabolomics from serum showed nominal evidence of association and the serum lipoprotein and metabolite view was not significant. LS-SOFA also identified top longitudinally varying metabolites with heterogeneous temporal patterns between converters and non-converters. These findings align with a growing body of ADNI and ADMC studies demonstrating that peripheral lipid and bile-acid dysregulation plays a central role in AD pathogenesis and progression. Recent multi-omics analyses have shown that serum bile acids improve prediction of AD progression in a sex-dependent manner [ 29 ] and that bile-acid profiles correlate with CSF and PET measures of amyloid and tau pathology [ 30 ]. Earlier ADMC work also reported consistent alterations in plasma cholesteryl esters, diacylglycerols, and triacylglycerols in relation to AD diagnosis and cognitive decline [ 31 , 32 ]. Moreover, network-level metabolomics studies revealed that ammonia and energy-related metabolic disturbances contribute to AD heterogeneity across cohorts [ 33 ]. Collectively, these findings support that lipid, bile-acid, and related metabolic pathways form a reproducible biochemical axis associated with AD risk, consistent with the FDR-significant associations observed for Views 1, 4, and 5 and the nominal association observed for View 2 in our analysis. In contrast, serum lipoprotein subclasses have shown weaker or inconsistent associations with clinical progression [ 34 ], supporting our observation that the lipoprotein and metabolite panel was not significantly related to dementia conversion. This study carries important implications for AD research. LS-SOFA provides a unified and computationally scalable framework for analyzing high-dimensional longitudinal omics data, capable of handling irregular sampling and large multi-view settings. The hypothesis testing was conducted by evaluating the association between each omics view and dementia risk, with nominal Wald-test p -values reported for exploratory omics-view screening. We also report Benjamini–Hochberg FDR-adjusted q -values across the five view-level tests as a multiplicity-aware sensitivity analysis. The subsequent feature-ranking analysis is descriptive and intended for biological interpretation rather than formal feature-wise inference. In future confirmatory analyses, hierarchical FDR procedures could be used when both view-level and feature-level hypotheses are formally tested. The identified metabolites, including cholesteryl esters, triacylglycerols, diacylglycerols, and bile acids, have repeatedly been implicated in AD progression across independent studies and may represent viable therapeutic or diagnostic targets. The estimated FPCA scores also provide interpretable summaries of longitudinal variation that can be used directly in association analyses of clinical endpoints such as time to dementia onset. Several limitations suggest directions for future research. LS-SOFA currently adopts a two-stage plug-in design in which estimated FPC scores are treated as observed covariates in the Cox model. Although our simulation study evaluates the full two-stage pipeline and shows appropriate empirical Type I error and power, the Cox model-based variance does not analytically propagate uncertainty from the first-stage LS-SOFA estimation. A joint modeling framework that simultaneously accounts for longitudinal factor estimation and survival association testing could further improve uncertainty quantification and inference efficiency. The Gaussian assumption can be relaxed to accommodate transcriptomic or microbiome count data under Poisson or multinomial distributions. In addition, the present analysis treats omics views independently, though biological pathways are often shared across modalities. Future extensions may incorporate cross-view dependencies through approaches such as joint and individual variance explained (JIVE) or multi-view tensor factorization. Finally, longitudinal cognitive outcomes have also been shown to predict dementia onset in AD [ 22 , 35 ]. Integrating LS-SOFA with multivariate functional mixed models (MFMM) could enable simultaneous modeling of cognitive and omics trajectories [ 22 , 23 ], further elucidating how molecular and clinical changes jointly drive disease progression. Overall, LS-SOFA provides a statistically rigorous, computationally efficient, and biologically interpretable framework for longitudinal multi-omics analysis that produces findings consistent with recent high-impact ADNI and ADMC studies implicating lipid and bile-acid metabolism in AD pathophysiology [ 36 ]. Supplementary Material supplement NIHMS2178999-supplement-supplement.pdf (6.7MB, pdf) Additional supporting information can be found online in the Supporting Information section. Table S1 : Out-of-sample mean squared error (MSE) from 100 simulation replications across five scenarios. Scenarios 1–3 were generated under the LS-SOFA model, and Scenarios 4–5 were generated under the MEFISTO and FTSVD models, respectively. For each scenario, N = 100 subjects were used for training and N test = 200 for evaluation. Each column reports the average MSE for the LS-SOFA, MEFISTO, FTSVD, FPCA, and CP methods. Table S2 : Within-sample mean squared error (MSE) of the LS-SOFA model under Scenarios 1–3 with training sample sizes N = 200 and N = 400. Each scenario includes three omics views (A–C), and MSE values are averaged across 100 simulation replications. Table S3 : Number of retained FPC scores Lvf selected by the 95% PVE criterion for the LS-SOFA method in simulation Scenarios 1–7. Entries give the selected Lvf values for each latent factor within each omics view. Figure S1 : Empirical power curves under Scenario 7 across sample sizes N = 100, N = 200, and N = 400, based on 100 simulation replications and effect sizes γ v ranging from 0.05 to 0.20. Table S4 : Sensitivity analysis of p -values for the association between each omics view and dementia risk using different numbers of latent factors (Fv = 3–10) in the real data analysis. Table S5 : Summary of the hypothesis testing results in the real data analysis using a 90% PVE threshold for selecting the number of FPC scores Lvf. Table S6 : Summary of the hypothesis testing results in the real data analysis using a 99% PVE threshold for selecting the number of FPC scores Lvf. Figure S2 : Estimated eigenfunctions for View 1 (plasma lipidomics data), truncated at 7.5 years. Figure S3 : Smoothed trajectories of the top six plasma lipid features y i p ( 1 ) ( t ) in View 1, obtained using locally weighted scatterplot smoothing (LOWESS), truncated at 7.5 years. The top features include CE(20:5) (cholesteryl ester family), TG(54:7) [NL-22:6] (triglycerides, neutral loss), TG(54:6) [NL-22:6], TG(56:9) [NL-22:6], DG(16:0_22:6) (diacylglycerol family), and TG(52:5) [NL-20:5]. Figure S4 : Estimated longitudinal factors z ^ if ( 2 ) ( t ) , f = 1 , ⋯ , 5 , for View 2 (gut microbial metabolomics data from serum). Each curve is smoothed using LOWESS and truncated at 7.5 years (few observations beyond 7.5 years lead to wider confidence intervals). Blue lines indicate non-converters; red lines indicate AD converters. Figure S5 : Estimated eigenfunctions for View 2 (gut microbial metabolomics data from serum), truncated at 7.5 years. Figure S6 : Smoothed trajectories of the top six serum metabolites y i p ( 2 ) ( t ) in View 2 using LOWESS, truncated at 7.5 years. The top features include CA, CDCA, BUDCA, GUDCA, TDCA, and ALLOLCA, all belonging to the bile acid family. Figure S7 : Estimated longitudinal factors z ^ if ( 4 ) ( t ) , f = 1 , ⋯ , 5 , for View 4 (serum metabolomics from FIA), smoothed using LOWESS and truncated at 7.5 years. Blue lines represent non-converters and red lines represent AD converters. Figure S8 : Estimated eigenfunctions for View 4 (serum metabolomics data from FIA), truncated at 7.5 years. Figure S9 : Smoothed trajectories of the top six serum metabolites y i p ( 4 ) ( t ) in View 4 obtained via LOWESS, truncated at 7.5 years. The top features include TG.17.2_36.4, TG.16.1_34.3, CE.20.5, TG.17.2_38.7, TG.17.1_38.7, and TG.20.5_34.0, primarily belonging to the triacylglycerol and cholesteryl ester classes. Figure S10 : Estimated longitudinal factors z ^ if ( 5 ) ( t ) , f = 1 , ⋯ , 10 , for View 5 (serum metabolomics from UPLC), smoothed using LOWESS and truncated at 7.5 years. Blue lines indicate non-converters; red lines indicate AD converters. Figure S11 : Estimated eigenfunctions for View 5 (serum metabolomics data from UPLC), truncated at 7.5 years. Figure S12 : Smoothed trajectories of the top six serum metabolites y i p ( 5 ) ( t ) in View 5 using LOWESS, truncated at 7.5 years. The top features include HipAcid (carboxylic acids), CA (bile acids), SDMA (amino acid-related), Indole and Ind. SO 4 (indole derivatives), and GUDCA (bile acids). Acknowledgments Data used in preparation of this article were obtained from the ADNI database ( adni.loni.usc.edu ). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in the analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf . Data used in preparation of this article were generated by the Alzheimer’s Disease Metabolomics Consortium (ADMC). As such, the investigators within the ADMC provided data but did not participate in the analysis or writing of this report. A complete listing of ADMC investigators can be found at: https://sites.duke.edu/adnimetab/team/ . Funding The research of Sheng Luo was supported by National Institute on Aging (grant numbers: R01AG064803 and P30AG072958). The Alzheimer’s Gut Microbiome Project (AGMP), the Alzheimer’s Disease Metabolomics Consortium (ADMC) and the Mood Disorders Precision Medicine Consortia (MDPMC) are funded wholly or in part by the following grants and supplements awarded to Dr. Kaddurah-Daouk at Duke University in partnership with a large number of academic institutions: R01AG046171, RF1AG051550, RF1AG057452, R01AG059093, U19AG063744, 3U19AG063744-04S1, RF1AG058942, 1U01AG088562, U01AG061359, R01MH108348, R01AG081322, FNIH: #DAOU16AMPA and FNIH: AMP AD 2.0 (AMP AD 2.0 is a public-private partnership managed by the Foundation for the National Institutes of Health and funded by the National Institute on Aging (NIA) in partnership with the private sector. For up-to-date information on the partners, visit https://fnih.org/our-programs/accelerating-medicines-partnership-amp/amp-alzheimers-disease-2-0/ ). As such, the investigators within the AGMP, the ADMC and MDPMC, not listed specifically in this publication’s author’s list, provided analysis-ready data, but did not participate in designing the study, conducting the analyses or writing of this manuscript. A listing of AGMP Investigators can be found at https://alzheimergut.org/meet-the-team/ . A listing of ADMC investigators can be found at: https://sites.duke.edu/adnimetab/team/ . A listing of MDPMC investigators can be found at https://sites.duke.edu/mdpmc/files/2022/06/MDPMC-Members2022.pdf . Footnotes Conflicts of Interest Dr. Kaddurah-Daouk is an inventor on a series of patents on use of metabolomics for the diagnosis and treatment of CNS diseases and holds equity in Metabolon Inc., Chymia, and Metabosensor. Other authors declare no potential conflicts of interest. 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Supplementary Materials supplement NIHMS2178999-supplement-supplement.pdf (6.7MB, pdf) Data Availability Statement The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions. ACTIONS View on publisher site PDF (631.6 KB) Cite Collections Permalink RESOURCES Similar articles Cited by other articles Links to NCBI Databases Follow NCBI NCBI on X (formerly known as Twitter) NCBI on Facebook NCBI on LinkedIn NCBI on GitHub NCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter) NLM on Facebook NLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top

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