BAYESIAN HIERARCHICAL MODELING WITH ADAPTIVE REGULARIZATION FOR COMPLEX MULTIVARIATE DATA ANALYSIS
DOI:
https://doi.org/10.69980/8zazm946Abstract
Complex multivariate longitudinal data often contain correlated outcomes, repeated measurements, and nested dependence structures that require flexible statistical methods capable of handling heterogeneity and parameter uncertainty. Cross-sectional analyses of a secondary longitudinal dataset (4,519 observations from 1,270 people observed at four time points) with seven stress-domain outcomes were completed. The proposed framework employed multivariate hierarchical regression and random effects at the participant and group level and global-local horseshoe shrinkage to dampen the coefficients of the outcomes while retaining more power on the positive side. Posterior inference was carried out by Markov Chain Monte Carlo with the No-U-Turn Sampler (NUTS); model adequacy was assessed by convergence diagnostics (CD), credible intervals, posterior predictive checks (PPC), and predictive performance measures (PPM). They revealed significant positive links between the stress domains, unique temporal trends, and considerable inter-individual variability. Adaptive shrinkage focused posterior signal in a small number of predictors and caused weak effects to shrink to zero. The time, male and time-by-GPA interactions were significant for various outcomes. The intraclass correlation at the participant level was higher than the intraclass correlation at higher levels of the group, reinforcing the need to consider hierarchical dependence. Overall, the framework presented a statistically meaningful and interpretable way to accommodate multivariate covariance, longitudinal variation, uncertainty quantification, and adaptive regularization in complex repeated-measures analyses. The method was also consistent under moderate predictor dimensionality conditions.
References
1. Bhadra, A., Datta, J., Li, Y., & Polson, N. (2020). Horseshoe regularisation for machine learning in complex and deep models 1. International Statistical Review, 88(2), 302-320.
2. Biziaev, T., Kopciuk, K., & Chekouo, T. (2025). Using prior-data conflict to tune Bayesian regularized regression models. Statistics and Computing, 35(2), 53.
3. Bürkner, P. C., Gabry, J., & Vehtari, A. (2021). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-t models: P.-C. Bürkner et al. Computational Statistics, 36(2), 1243-1261.
4. Grinsztajn, L., Semenova, E., Margossian, C. C., & Riou, J. (2021). Bayesian workflow for disease transmission modeling in Stan. Statistics in medicine, 40(27), 6209-6234.
5. Högberg, B. (2021). Educational stressors and secular trends in school stress and mental health problems in adolescents. Social science & medicine, 270, 113616.
6. Kohns, D., & Szendrei, T. (2024). Horseshoe prior Bayesian quantile regression. Journal of the Royal Statistical Society Series C: Applied Statistics, 73(1), 193-220.
7. Kulakow, S., Raufelder, D., & Hoferichter, F. (2021). School‐related pressure and parental support as predictors of change in student stress levels from early to middle adolescence. Journal of adolescence, 87(1), 38-51.
8. Kundu, D., Mitra, R., & Gaskins, J. T. (2021). Bayesian variable selection for multioutcome models through shared shrinkage. Scandinavian Journal of Statistics, 48(1), 295-320.
9. Mameli, V., Slanzi, D., Griffin, J. E., & Brown, P. J. (2025). Multivariate Bayesian Global–Local Shrinkage Methods for Regularisation in the High-Dimensional Linear Model. Mathematics, 13(11), 1812.
10. Nezlek, J.B., & Mroziński, B. (2020). Applications of the multilevel model in psychological sciences: intensive repeated measures plans. L'Année psychologique , 120 (1), 39-72.
11. Okano, R., Hamura, Y., Irie, K., & Sugasawa, S. (2024). Locally adaptive Bayesian isotonic regression using half shrinkage priors. Scandinavian Journal of Statistics, 51(1), 109-141.
12. Palma, M., El Makkaoui, O., Keogh, R. H., Carr, S. B., Szczesniak, R., Taylor‐Robinson, D., ... & Barrett, J. K. (2026). A Bayesian Location‐Scale Joint Model for Time‐To‐Event and Multivariate Longitudinal Data With Association Based on Within‐Individual Variability. Statistics in medicine, 45(10-12), e70596.
13. Shen, L. L., Del Priore, L. V., & Warren, J. L. (2022). A hierarchical Bayesian entry time realignment method to study the long-term natural history of diseases. Scientific Reports, 12(1), 4869.
14. Silva, L. A., & Zanella, G. (2024). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Journal of the American Statistical Association, 119(547), 2369-2381.
15. Sivula, T., Magnusson, M., & Vehtari, A. (2023). Unbiased estimator for the variance of the leave-one-out cross-validation estimator for a Bayesian normal model with fixed variance. Communications in Statistics-Theory and Methods, 52(16), 5877-5899.
16. Smith, A. N., & Griffin, J. E. (2023). Shrinkage priors for high-dimensional demand estimation. Quantitative Marketing and Economics, 21(1), 95-146.
17. Song, Q., & Liang, F. (2023). Nearly optimal Bayesian shrinkage for high-dimensional regression. Science China Mathematics, 66(2), 409-442.
18. Trevethan, M., Jain, A. T., Shatiyaseelan, A., Luebbe, A. M., & Raval, V. V. (2022). A longitudinal examination of the relation between academic stress and anxiety symptoms among adolescents in India: The role of physiological hyperarousal and social acceptance. International Journal of Psychology, 57(3), 401-410.
19. van Loon, A. W., Creemers, H. E., Vogelaar, S., Saab, N., Miers, A. C., Westenberg, P. M., & Asscher, J. J. (2022). Trajectories of adolescent perceived stress and symptoms of depression and anxiety during the COVID-19 pandemic. Scientific Reports, 12(1), 15957.
20. Veenman, M., Stefan, A. M., & Haaf, J. M. (2024). Bayesian hierarchical modeling: An introduction and reassessment. Behavior Research Methods, 56(5), 4600-4631.
21. Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., & Bürkner, P. C. (2021). Rank-normalization, folding, and localization: An improved R ̂ for assessing convergence of MCMC (with discussion). Bayesian analysis, 16(2), 667-718.
22. Wang, B., & Zou, H. (2021). Honest leave‐one‐out cross‐validation for estimating post‐tuning generalization error. Stat, 10(1), e413.
23. Wang, Z., Murray, T. A., Xiao, M., Lin, L., Alemayehu, D., & Chu, H. (2023). Bayesian hierarchical models incorporating study‐level covariates for multivariate meta‐analysis of diagnostic tests without a gold standard with application to COVID‐19. Statistics in medicine, 42(28), 5085-5099.
24. Williams, D. R., Martin, S. R., Liu, S., & Rast, P. (2020). Bayesian multivariate mixed-effects location scale modeling of longitudinal relations among affective traits, states, and physical activity. European Journal of Psychological Assessment, 36(6), 981.
25. Wong, M. (2025). RISE Up Study–stress (Version 3) [Data set]. Mendeley Data. https://doi.org/10.17632/btzdd3fy9t.3
26. Wuthrich, V. M., Belcher, J., Kilby, C., Jagiello, T., & Lowe, C. (2021). Tracking stress, depression, and anxiety across the final year of secondary school: A longitudinal study. Journal of school psychology, 88, 18-30.
27. Zhang, R., & Ghosh, M. (2022). Ultra high-dimensional multivariate posterior contraction rate under shrinkage priors. Journal of Multivariate Analysis, 187, 104835.
28. Zhang, Y. D., Naughton, B. P., Bondell, H. D., & Reich, B. J. (2022). Bayesian regression using a prior on the model fit: The R2-D2 shrinkage prior. Journal of the American Statistical Association, 117(538), 862-874.
29. Zou, H., Zeng, D., Xiao, L., & Luo, S. (2023). Bayesian inference and dynamic prediction for multivariate longitudinal and survival data. The annals of applied statistics, 17(3), 2574.
