A Time-Varying and Covariate-Dependent Correlation Model for Multivariate Longitudinal Studies

Abstract

In multivariate longitudinal studies, associations between outcomes often exhibit time-varying and individual level heterogeneity, motivating the modeling of correlations as an explicit function of time and covariates. However, most existing methods for correlation analysis fail to simultaneously capture the time-varying and covariate-dependent effects. We propose a Time-Varying and Covariate-Dependent (TiVAC) correlation model that jointly allows covariate effects on correlation to change flexibly and smoothly across time. TiVAC employs a bivariate Gaussian model where the covariate-dependent correlations are modeled semiparametrically using penalized splines. We develop a penalized maximum likelihood-based Newton-Raphson algorithm, and inference on time-varying effects is provided through simultaneous confidence bands. Simulation studies show that TiVAC consistently outperforms existing methods in accurately estimating correlations across a wide range of settings, including binary and continuous covariates, sparse to dense observation schedules, and across diverse correlation trajectory patterns. We apply TiVAC to a psychiatric case study of 291 bipolar I patients, modeling the time-varying correlation between depression and anxiety scores as a function of their clinical variables. Our analyses reveal significant heterogeneity associated with gender and nervous-system medication use, which varies with age, revealing the complex dynamic relationship between depression and anxiety in bipolar disorders.

Figures (5)

• Figure 1: PHQ-9 and GAD-7 trajectories and correlation heterogeneity.(A) Irregular longitudinal trajectories of PHQ-9 (depression; blue) and GAD-7 (anxiety; red) for two representative BP-I patients. The concurrent, within-patient Pearson correlation is high for the top patient ($\rho=0.88$) and low for the bottom patient ($\rho=-0.15$). (B) Three-dimensional correlation surface showing how the PHQ–GAD correlation (z-axis) varies with age (x-axis) and antipsychotic-use frequency (y-axis). Color encodes the correlation value through rainbow scale.
• Figure 2: Representative curve estimations in Scenario 1. Comparison between the estimated correlation curves from four methods (TiVAC, Empirical, CovReg and CoCoA-REML) and the true correlation curve when covariate $X = 1$. The figure presents nine independent cases, representing three types of coefficient functions and three settings for the number of time points per individual. Each case shows a representative example from the first run of the simulation out of 50 replications (using a fixed random seed for all cases). The TiVAC, Empirical, CovReg and CoCoA-REML methods are depicted as blue, orange, purple and green curves, respectively, while the true trajectory is represented by a dashed black curve.
• Figure 3: Representative heatmap estimations under $\mathscr{T}_{Moderate}$ setting in Scenario 2. Comparison between the estimated correlation heatmaps from three methods (TiVAC, CovReg and CoCoA-REML) and the true correlation heatmap under $\mathscr{T}_{Moderate}$ setting. In each heatmap, correlation values are color-coded, with red indicating higher correlations and blue indicating lower correlations. Each heatmap displays correlation patterns over time for different values of the continuous covariate. The figure presents three independent cases, each representing one of the three types of coefficient functions. Each case shows a representative example from the first run of the simulation (using a fixed random seed for all cases).
• Figure 4: Results of TiVAC coefficient functions on PLS-BD data. TiVAC coefficient functions over age for seven covariates, measuring their effects on the correlation between PHQ-9 and GAD-7 in PLS-BD. For each covariate, the coefficient function $\beta(t)$ is shown in blue, with a gray band representing the 95% simultaneous confidence band. The dashed black line at zero indicates no effect.
• Figure 5: Results of TiVAC correlations on PLS-BD data. Visualization of TiVAC correlations between PHQ-9 and GAD-7 for four covariates with significant or near-significant age periods. For gender (top-left subplot), the TiVAC correlation curve over age is shown in red for females and blue for males. For the other three heatmaps, the corresponding covariates are continuous, with the covariate values on the y-axis, age on the x-axis, and heatmap colors representing the TiVAC correlation values.