Conditional Cross-covariance Reduction Model (CCR; Biometrics, 2026)
Conditional Cross-covariance Reduction Model (CCR; Biometrics, 2026)
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S. H. Park, X. Zhang, E. Slate, S. Sun, & H. Yao (2026). Dimension Reduction for Characterizing Sexual Dimorphism in Biomechanics of the Temporomandibular Joint. Accepted for publication in Biometrics, 82(4), ujag181. DOI: https://doi.org/10.1093/biomtc/ujag181 view/download
Abstract:
Sexual dimorphism is a critical factor in many biological and medical research fields. In biomechanics and bioengineering, understanding sex differences is crucial for studying musculoskeletal conditions such as temporomandibular disorder (TMD). This paper focuses on the association between the skull and temporomandibular joint (TMJ) muscle attachments to discern sex differences. Data were collected from 10 male and 11 female cadaver heads to investigate sex-specific relationships between the skull and muscles. We propose a conditional cross-covariance reduction (CCR) model, designed to examine the dynamic association between two sets of random variables conditioned on a third binary variable (e.g., sex), highlighting the most distinctive sex-related relationships between skull and muscle attachments in the human cadaver data. Under the CCR model, we employ a sparse singular value decomposition algorithm and introduce a sequential permutation for selecting sparsity (SPSS) method to select important variables and to determine the optimal number of selected variables.
Contribution:
CCR model characterizes dynamic associations between two sets of variables conditioning on a third binary variable.
Our estimation method incorporates variable selection with hard thresholding.
SPSS method enhances robust feature selection particularly in small-sample settings.
Model formulation:
Explicit Description:
For continuous X, Y and binary Z, in (a), the CCR model differentiates associations by group membership defined by (Z).
Sparsity selection on X and Y is performed using the SPSS method with leave-two-out samples.
In (d), RGCCA (Tenenhaus and Tenenhaus, 2011) cannot capture group differences, which are common outcomes in CCA approaches.
In (b), GLAA(Li et al. 2023) also feasibly accounts for group differences; however, because it assumes Z is continuous and applies penalization, the method is subject to potential bias in estimation.
In (c), Bayesian CCA (BCCA; Klami et al. 2013) also specifies the group difference. However, it cannot estimate the correct differences.
Figure. Comparison of dynamic association by binary variable Z =1, 2 with competing methods. The methods under comparison are: conditional cross-covariance reduction (CCR) model, generalized liquid association analysis (GLAA), Bayesian canonical correlation analysis (BCCA), and regularized generalized canonical correlation analysis (RGCCA).
Figure. Linear combinations of the CCR model for the multi-group case, illustrating group-specific correlations with 200 samples in each group.
Extension:
The CCR model can be extended to a multicategorical variable Z (e.g., Z=1,2,3) with a simple modification.
Here, the caption values indicate the true correlations for each group.
Predictor Envelope Linear Mixed Model (PELMM; revision submitted to Biometrics)
Predictor Envelope Linear Mixed Model (PELMM; revision submitted to Biometrics)
Abstract:
The envelope approach in linear regression improves estimation efficiency by identifying a subspace of the parameter space that captures the variation in the response material to the covariates. We extend this idea to repeated-measures data by proposing a predictor envelope linear mixed model (PELMM), which imposes an envelope structure on the Fisher information matrix for the fixed effects. Under this formulation, the Fisher information asymptotically coincides with the covariance matrix of the transformed (whitened) fixed-effect covariates. By filtering out immaterial covariate variation and targeting the envelope subspace, PELMM isolates the response-relevant signal and improves the efficiency of fixed-effects estimation in longitudinal settings. Simulation studies demonstrate substantial efficiency gains over existing methods. We further illustrate the proposed method using data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) to investigate longitudinal associations with the clinical dementia rating–sum of boxes (CDRSB) score.
Main takeaway:
PELMM attains higher efficiency than REML by eliminating redundant variation in the estimation process.
In PELMM, time-dependent continuous fixed-effect covariates are modeled through envelope estimation, whereas time-independent fixed-effect covariates are incorporated without an envelope structure.
The efficiency gain in PELMM leads to the identification of more significant variables (with smaller p-values) in the linear mixed model.
Tensor Landmark Analysis Model (TLM; Stat, 2024)
Tensor Landmark Analysis Model (TLM; Stat, 2024)
S. H. Park, R. Zhou, X. Zhang, L. Li, & L. Liu (2024). Tensor Landmark Analysis with Application to ADNI data. Stat, 13(4), e70014.
Abstract:
Recent advancements in data collection have facilitated the use of multi-dimensional arrays, also known as tensors, in prediction of health outcomes. In this article, we introduce a tensor landmark model for predicting survival outcomes using multiple longitudinal biomarkers as tensor covariates through CANDECOMP/PARAFAC (CP) decomposition. An iteratively reweighted least squares estimation is adopted for components of the tensor coefficients in the landmark model. We also present empirical results of AIC and BIC for right-censored data to select the CP rank. Simulations and Alzheimer’s Disease Neuroimaging Initiative (ADNI) data analysis demonstrate that our proposed model accurately estimates survival coefficients and predicts survival probabilities. The implementation code can be found online.
Main takeaway:
We have proposed the TLM under a low-rank tensor structure.
The proposed IRLS solution performs well in both estimation and prediction.
Iterative likelihood updates using baseline and profile likelihood.
TLM can convey the trajectory information of risk factors.
AIC and BIC for rank selection consistently estimate the CP rank.
Introduction:
Instead of using covariates from each visit separately for risk prediction (as in landmark modeling or dynamic prediction), we represent multiple visits of covariates in matrix form and incorporate this matrix observation into the survival model for risk prediction.
Tensor Landmark Model:
Algorithm and CP Rank Selection:
We apply iteratively reweighted least squares estimation, which updates each component in the CP decomposition using least squares.
Moreover, we implement the AIC and BIC for censored data for CP rank selection as follows:
Summary of Simulation Result:
Table: Numerical assessments of parameter estimation with 100 replicates.
Competing method "LM-Vec" is concatenating longitudinal variables as vector and fit in the Landmark Analysis model.
Rank 2 is used for simulation.
Assessments: Ratio of Bias (RBias), Mean Squared Error (MSE), Frobenius norm loss (lossF), Spectral norm loss (losss).
TLM has better estimation result than LM-Vec.
Consistency by increase of sample size (n).
ADNI data analysis results:
Table: Assessments for the predictions on the year 1.5, 2 and 2.5 in 5-fold CV.
Compared TLM, LM (landmark analysis model), LM-Vec.
5-fold CV using stratified sampling for survival outcome.
CP rank is 1 (from AIC and BIC).
Assessments:
C-index (Harrell et al., 1982): 🙁 0 to 1 😃
Integrated IPCW Brier score (IBS): 🙁 1 to 0 😃
Blockwise Missing Data Imputation and Multiview Data Integration with Envelopes (EBI; manuscript available upon request)
Blockwise Missing Data Imputation and Multiview Data Integration with Envelopes (EBI; manuscript available upon request)
Abstract:
We propose a new envelope blockwise imputation (EBI) framework to address blockwise missing data in multiview data integration. The collected data often present blockwise missing structures; that is, data in one or more sources may be entirely unobserved for a subset of samples. Blockwise missing imputation is often needed as a preprocessing step for data integration. In this article, we develop a simultaneous envelope-based method that takes advantage of canonical correlations of different data sets to impute blockwise missing data. The method effectively estimates canonical subspaces by fitting a regression model with congregating envelope subspaces, reducing the number of parameters in both predictors and responses. As a by-product, the envelope blockwise imputation method also obtains a more accurate estimate of the canonical correlations than the standard canonical correlation analysis (CCA). We devise an expectation conditional maximization (ECM) algorithm to estimate the envelope parameters.
Sparse Longitudinal Canonical Correlation Analysis (SLoCCA)
Sparse Longitudinal Canonical Correlation Analysis (SLoCCA)
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