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arxiv: 2605.05371 · v1 · submitted 2026-05-06 · 📊 stat.ME

Multilevel Regression Modeling of Covariance Matrix Outcomes

Michelle Murphy Green , Xi Luo , Brian S. Caffo , Yi Zhao This is my paper

Pith reviewed 2026-05-08 16:14 UTC · model grok-4.3

classification 📊 stat.ME
keywords multilevel modelingcovariance regressionvon Mises-Fisher distributionfunctional connectivityprincipal regressionhierarchical likelihoodbrain networkslifespan studies
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The pith

Multilevel Covariate-Assisted Principal Regression identifies cluster-specific projections of covariance matrices and models them on the unit sphere to borrow information across groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops MCAP to perform regression of covariance matrix outcomes when observations are nested inside clusters, as occurs when subjects are grouped by age in lifespan studies. It determines a linear projection specific to each cluster so that the projected values can enter a generalized linear mixed effects model driven by subject-level covariates. The projections themselves are represented as points on the unit sphere and given a von Mises-Fisher distribution that pools strength across clusters while leaving covariate effects undistorted. Estimation proceeds by maximizing a hierarchical likelihood, asymptotic theory is supplied, and a two-stage bootstrap supplies inference. Simulations show gains over single-level covariance regression, and the method applied to nested connectivity data recovers a main spectral network modulated by age and sex together with coordinated lifespan patterns in cross-network regions.

Core claim

MCAP identifies, for each cluster, a linear projection of the covariance matrix such that the projected outcomes admit a generalized linear mixed effects model with the covariates. The cluster-specific projections are modeled as directions on the unit sphere via a von Mises-Fisher distribution, which enables principled borrowing of information across clusters. Model parameters are obtained by maximizing a hierarchical likelihood, asymptotic properties of the estimators are derived, and inference uses a two-stage bootstrap procedure. This multilevel construction yields substantially more accurate recovery of regression coefficients than single-level competitors.

What carries the argument

Cluster-specific linear projections of covariance matrices placed on the unit sphere and distributed according to the von Mises-Fisher law, which supplies the mechanism for information sharing while preserving covariate effects inside a generalized linear mixed model.

If this is right

  • Simulation studies show that MCAP substantially outperforms single-level competitors in estimating regression coefficients.
  • In applied lifespan neuroimaging data spanning ages five to ninety, MCAP identifies a dominant spectral brain network that captures age and sex effects on functional connectivity.
  • The fitted model reveals convergence of neural reorganization patterns in late adulthood.
  • The analysis shows coordinated lifespan modulation of cross-network regions linked to language and executive function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spherical projection step may prove useful for other nested high-dimensional outcomes where directional concentration captures shared cluster structure.
  • The two-stage bootstrap procedure could be examined for coverage properties under varying cluster sizes and numbers.
  • Relaxing the von Mises-Fisher assumption to a more flexible directional distribution might be tested as a robustness check in future applications.

Load-bearing premise

The cluster-specific projections of the covariance matrices behave as directions on the unit sphere that can be described by a von Mises-Fisher distribution without distorting the underlying covariate effects.

What would settle it

A simulation experiment in which the true cluster projections deviate markedly from von Mises-Fisher concentration on the sphere and MCAP loses its advantage over single-level methods would falsify the claimed superiority.

Figures

Figures reproduced from arXiv: 2605.05371 by Brian S. Caffo, Michelle Murphy Green, Xi Luo, Yi Zhao.

Figure 1
Figure 1. Figure 1: The profile of functional connectivity within the identified brain subnetwork over age, as view at source ↗
Figure 2
Figure 2. Figure 2: Result of the average projection of the component identified in the HCP study. (a) view at source ↗
read the original abstract

Covariance matrix outcomes arise naturally in neuroimaging experiments to study brain functional connectivity. It is also of interest to understand how brain network organization varies with subject-level covariates. Existing covariance regression methods operate in a single-level framework and do not accommodate the hierarchically nested data structure in which subjects are grouped into clusters, such as age cohorts in lifespan studies. A Multilevel Covariate-Assisted Principal Regression (MCAP) framework is introduced, which identifies, for each cluster, a linear projection such that a generalized linear mixed effects model can be formulated with the covariates. The cluster-specific projections are modeled on the unit sphere via a von Mises-Fisher distribution, enabling principled borrowing of information across clusters. Model parameters are estimated by maximizing a hierarchical likelihood. For inference, a two-stage bootstrap procedure is proposed. Asymptotic properties of the estimators are established. Simulation studies demonstrate that MCAP substantially outperforms single-level competitors in estimating regression coefficients. Applied to the Human Connectome Project Lifespan Study spanning ages from five to ninety, MCAP identifies a dominant spectral brain network capturing age and sex effects on functional connectivity, and reveals findings including the convergence of neural reorganization patterns in late adulthood and the coordinated lifespan modulation of cross-network regions linked to language and executive function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes the Multilevel Covariate-Assisted Principal Regression (MCAP) framework for covariance matrix outcomes in clustered data structures, such as subjects nested in age cohorts. For each cluster it identifies a linear projection of the covariance matrix, places the resulting unit-sphere vectors under a von Mises-Fisher distribution to enable information borrowing, formulates a generalized linear mixed-effects model for subject-level covariates, and estimates all parameters by maximizing a joint hierarchical likelihood. Asymptotic consistency and normality are claimed, a two-stage bootstrap is introduced for inference, simulations are used to show that MCAP outperforms single-level competitors in recovering regression coefficients, and the method is applied to the Human Connectome Project Lifespan Study (ages 5–90) to recover a dominant spectral network modulated by age and sex.

Significance. If the central modeling assumptions and performance claims hold, MCAP supplies a principled route to covariate-adjusted analysis of covariance outcomes under realistic multilevel sampling designs that are ubiquitous in neuroimaging and longitudinal studies. The explicit use of the von Mises-Fisher prior for borrowing strength across clusters, together with the reported simulation superiority and the concrete lifespan findings (convergence of reorganization patterns in late adulthood, coordinated modulation of language/executive cross-network regions), would constitute a substantive methodological advance over existing single-level covariance regression techniques.

major comments (3)
  1. [§2.3] §2.3 (von Mises-Fisher modeling of cluster projections): The hierarchical likelihood couples the GLM coefficients β directly to the vMF concentration and mean parameters; the manuscript provides no simulation experiments under misspecification (multimodal, heavy-tailed, or cluster-varying concentration distributions) that would demonstrate whether bias in β remains negligible when the vMF assumption is violated. Because this assumption is load-bearing for the claimed information-borrowing benefit, the outperformance result in §5 is not yet robustly supported.
  2. [§4] §4 (asymptotic theory): The abstract states that asymptotic properties are established, yet the regularity conditions on the vMF concentration κ, the projection dimension, and the cluster-size growth rate are not stated explicitly enough to verify that the two-stage bootstrap remains valid when the number of clusters is moderate (as in the HCP application). A concise statement of the theorem assumptions and a sketch of the proof would be required before the consistency claim can be accepted as load-bearing.
  3. [§5] §5 (simulation design): The claim that MCAP “substantially outperforms” single-level competitors rests on the reported coefficient recovery; however, the data-generation process, the precise definition of “exclusion rules,” and whether any simulation replicates included vMF misspecification are not described in sufficient detail to allow independent verification of the performance gap.
minor comments (2)
  1. [§2] Notation for the cluster-specific projection p_c and the GLM link function should be introduced once in §2 and used consistently thereafter; occasional redefinition of symbols interrupts readability.
  2. [Figure 3] Figure 3 (real-data network visualization): axis labels and color-bar scaling are too small for the dominant spectral network to be interpreted without magnification; enlarging or adding an inset would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the insightful comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§2.3] The hierarchical likelihood couples the GLM coefficients β directly to the vMF concentration and mean parameters; the manuscript provides no simulation experiments under misspecification (multimodal, heavy-tailed, or cluster-varying concentration distributions) that would demonstrate whether bias in β remains negligible when the vMF assumption is violated. Because this assumption is load-bearing for the claimed information-borrowing benefit, the outperformance result in §5 is not yet robustly supported.

    Authors: We agree that additional simulations under vMF misspecification would strengthen the robustness claims. Although the vMF is a standard and flexible model for spherical data, we will include new simulation scenarios in the revised manuscript where the true distribution of cluster projections deviates from vMF (e.g., using a mixture of vMFs to induce multimodality and varying concentration parameters). These will assess the bias in β estimates and whether MCAP still outperforms single-level methods under such violations. revision: yes

  2. Referee: [§4] The abstract states that asymptotic properties are established, yet the regularity conditions on the vMF concentration κ, the projection dimension, and the cluster-size growth rate are not stated explicitly enough to verify that the two-stage bootstrap remains valid when the number of clusters is moderate (as in the HCP application). A concise statement of the theorem assumptions and a sketch of the proof would be required before the consistency claim can be accepted as load-bearing.

    Authors: We appreciate this observation. The asymptotic consistency and normality results rely on standard conditions for M-estimators in hierarchical models, including κ bounded away from zero and infinity, fixed projection dimension p, and the number of clusters n growing such that cluster sizes m_i → ∞. We will add an explicit statement of these assumptions in Section 4, along with a brief proof sketch in the appendix. For the two-stage bootstrap with moderate n (as in the HCP Lifespan data with age cohorts), we will include a small-scale simulation study demonstrating empirical coverage rates to support its practical validity. revision: yes

  3. Referee: [§5] The claim that MCAP “substantially outperforms” single-level competitors rests on the reported coefficient recovery; however, the data-generation process, the precise definition of “exclusion rules,” and whether any simulation replicates included vMF misspecification are not described in sufficient detail to allow independent verification of the performance gap.

    Authors: We apologize for the insufficient detail in the simulation section. The data are generated by first sampling cluster-specific mean directions from a vMF, then constructing covariance matrices consistent with the projected structure, and adding subject-level noise. Exclusion rules discard projections with norm below a threshold to avoid degenerate cases. The main simulations assume the vMF model holds. In the revision, we will provide a detailed description including pseudocode, all parameter settings, and reference the new misspecification simulations from our response to the first comment to allow full verification. revision: yes

Circularity Check

0 steps flagged

No circularity: standard hierarchical model built from independent statistical components

full rationale

The MCAP framework combines principal regression on covariance matrices, a generalized linear mixed-effects model for covariate effects, and a von Mises-Fisher prior on unit-sphere cluster projections, with parameters obtained by direct maximization of the hierarchical likelihood. Asymptotic properties are derived under standard regularity conditions, and performance is assessed via external simulation benchmarks and real-data application rather than any internal redefinition of fitted quantities as predictions. No equation reduces the target regression coefficients β or the cluster projections to quantities defined solely by other fitted parameters within the paper; the vMF component is an explicit modeling assumption, not a self-definitional tautology. Any self-citations are to prior methodological work on principal regression or mixed models and do not carry the central claims.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of linear projection for covariance regression and the suitability of vMF for directional data on the sphere; no new free parameters beyond those in the vMF concentration and mixed-model variance components are explicitly introduced in the abstract.

free parameters (1)
  • vMF concentration parameter
    Controls the spread of cluster-specific projections around the mean direction and is estimated from data.
axioms (2)
  • domain assumption Covariance matrices admit a linear projection onto a lower-dimensional space suitable for subsequent generalized linear mixed modeling.
    Invoked to formulate the regression after projection.
  • domain assumption Cluster-specific projection directions are exchangeable and can be modeled as draws from a common von Mises-Fisher distribution.
    Enables the borrowing of information across clusters.

pith-pipeline@v0.9.0 · 5518 in / 1416 out tokens · 36017 ms · 2026-05-08T16:14:51.163359+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

58 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Acharyya, S., Zhang, Z., Bhattacharya, A., and Pati, D. (2023). Bayesian hierarchical modeling on covariance valued data. Stat , 12(1):e534

    2023

  2. [2]

    Anderson, T. (1973). Asymptotically efficient estimation of covariance matrices with linear structure. The Annals of Statistics , 1(1):135--141

    1973

  3. [3]

    A., Seidlitz, J., White, S

    Bethlehem, R. A., Seidlitz, J., White, S. R., Vogel, J. W., Anderson, K. M., Adamson, C., Adler, S., Alexopoulos, G. S., Anagnostou, E., Areces-Gonzalez, A., et al. (2022). Brain charts for the human lifespan. Nature , 604(7906):525--533

    2022

  4. [4]

    M., and Hyde, J

    Biswal, B., Zerrin Yetkin, F., Haughton, V. M., and Hyde, J. S. (1995). Functional connectivity in the motor cortex of resting human brain using echo-planar MRI . Magnetic Resonance in Medicine , 34(4):537--541

    1995

  5. [5]

    Boik, R. J. (2002). Spectral models for covariance matrices. Biometrika , 89(1):159--182

    2002

  6. [6]

    Y., Salat, D

    Bookheimer, S. Y., Salat, D. H., Terpstra, M., Ances, B. M., Barch, D. M., Buckner, R. L., Burgess, G. C., Curtiss, S. W., Diaz-Santos, M., Elam, J. S., et al. (2019). The lifespan human connectome project in aging: an overview. Neuroimage , 185:335--348

    2019

  7. [7]

    Y., Leonard, T., and Tsui, K.-W

    Chiu, T. Y., Leonard, T., and Tsui, K.-W. (1996). The matrix-logarithmic covariance model. Journal of the American Statistical Association , 91(433):198--210

    1996

  8. [8]

    A., Lavelle, A., Fitzsimons, C

    Dohm-Hansen, S., English, J. A., Lavelle, A., Fitzsimons, C. P., Lucassen, P. J., and Nolan, Y. M. (2024). The 'middle-aging' brain. Trends in Neurosciences , 47(4):259--272

    2024

  9. [9]

    S., Glasser, M

    Elam, J. S., Glasser, M. F., Harms, M. P., Sotiropoulos, S. N., Andersson, J. L., Burgess, G. C., Curtiss, S. W., Oostenveld, R., Larson-Prior, L. J., Schoffelen, J.-M., et al. (2021). The human connectome project: A retrospective. NeuroImage , page 118543

    2021

  10. [10]

    Flury, B. N. (1984). Common principal components in k groups. Journal of the American Statistical Association , 79(388):892--898

    1984

  11. [11]

    Fox, E. B. and Dunson, D. B. (2015). Bayesian nonparametric covariance regression. Journal of Machine Learning Research , 16:2501--2542

    2015

  12. [12]

    Franks, A. M. and Hoff, P. (2019). Shared subspace models for multi-group covariance estimation. Journal of Machine Learning Research , 20(171):1--37

    2019

  13. [13]

    P., Somerville, L

    Harms, M. P., Somerville, L. H., Ances, B. M., Andersson, J., Barch, D. M., Bastiani, M., Bookheimer, S. Y., Brown, T. B., Buckner, R. L., Burgess, G. C., et al. (2018). Extending the Human Connectome Project across ages: Imaging protocols for the Lifespan Development and Aging projects. Neuroimage , 183:972--984

    2018

  14. [14]

    Hoff, P. D. (2009). A hierarchical eigenmodel for pooled covariance estimation. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 71(5):971--992

    2009

  15. [15]

    Hoff, P. D. and Niu, X. (2012). A covariance regression model. Statistica Sinica , 22(2):729--753

    2012

  16. [16]

    A., Horn, R

    Horn, R. A., Horn, R. A., and Johnson, C. R. (1990). Matrix analysis . Cambridge university press

    1990

  17. [17]

    D., Elliott, M

    Ingalhalikar, M., Smith, A., Parker, D., Satterthwaite, T. D., Elliott, M. A., Ruparel, K., Hakonarson, H., Gur, R. E., Gur, R. C., and Verma, R. (2014). Sex differences in the structural connectome of the human brain. Proceedings of the National Academy of Sciences , 111(2):823--828

    2014

  18. [18]

    R., Simonsick, E

    Kaup, A. R., Simonsick, E. M., Harris, T. B., Satterfield, S., Metti, A. L., Ayonayon, H. N., Rubin, S. M., and Yaffe, K. (2014). Older adults with limited literacy are at increased risk for likely dementia. Journals of Gerontology Series A: Biomedical Sciences and Medical Sciences , 69(7):900--906

    2014

  19. [19]

    N., Roussel, M., Tasseel-Ponche, S., Godefroy, O., and Aarabi, A

    Khalilian, M., Toba, M. N., Roussel, M., Tasseel-Ponche, S., Godefroy, O., and Aarabi, A. (2024). Age-related differences in structural and resting-state functional brain network organization across the adult lifespan: A cross-sectional study. Aging Brain , 5:100105

    2024

  20. [20]

    A., Church, A., Meriwether, D., Mahurkar-Joshi, S., Li, V

    Kilpatrick, L. A., Church, A., Meriwether, D., Mahurkar-Joshi, S., Li, V. W., Sohn, J., Reist, J., Labus, J. S., Dong, T., Jacobs, J. P., et al. (2025). Differential brainstem connectivity according to sex and menopausal status in healthy male and female individuals. Biology of Sex Differences , 16(1):25

    2025

  21. [21]

    A covariate-dependent Cholesky decomposition for high-dimensional covariance regression

    Kim, R. and Zhang, E. J. (2026). A covariate-dependent Cholesky decomposition for high-dimensional covariance regression. arXiv preprint arXiv:2604.07547

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  22. [22]

    and Zhang, J

    Kim, R. and Zhang, J. (2025). High-dimensional covariance regression with application to co-expression QTL detection. Journal of the American Statistical Association , pages 1--14

    2025

  23. [23]

    Krzanowski, W. (1984). Principal component analysis in the presence of group structure. Journal of the Royal Statistical Society: Series C (Applied Statistics) , 33(2):164--168

    1984

  24. [24]

    and Nelder, J

    Lee, Y. and Nelder, J. A. (1996). Hierarchical generalized linear models. Journal of the Royal Statistical Society. Series B (Methodological) , pages 619--678

    1996

  25. [25]

    A., and Pawitan, Y

    Lee, Y., Nelder, J. A., and Pawitan, Y. (2006). Generalized linear models with random effects: unified analysis via H-likelihood . Chapman and Hall/CRC

    2006

  26. [26]

    M., Thung, K.-H., Taylor IV, H

    Li, G., Huynh, K. M., Thung, K.-H., Taylor IV, H. P., Lin, G., Lin, W., Ahmad, S., and Yap, P.-T. (2025). The effective connectome over a century of human life. Communications Biology , 8(1):1638

    2025

  27. [27]

    Livingston, G., Huntley, J., Sommerlad, A., Ames, D., Ballard, C., Banerjee, S., Brayne, C., Burns, A., Cohen-Mansfield, J., Cooper, C., et al. (2020). Dementia prevention, intervention, and care: 2020 report of the Lancet Commission . The Lancet , 396(10248):413--446

    2020

  28. [28]

    P., Babb, C., Nishino, T., and Barch, D

    Luby, J., Belden, A., Botteron, K., Marrus, N., Harms, M. P., Babb, C., Nishino, T., and Barch, D. (2013). The effects of poverty on childhood brain development: the mediating effect of caregiving and stressful life events. JAMA Pediatrics , 167(12)

    2013

  29. [29]

    A., Yeh, F.-C., and Astle, D

    Mousley, A., Bethlehem, R. A., Yeh, F.-C., and Astle, D. E. (2025). Topological turning points across the human lifespan. Nature Communications , 16(1):10055

    2025

  30. [30]

    Murphy, S. A. and Van der Vaart, A. W. (2000). On profile likelihood. Journal of the American Statistical Association , 95(450):449--465

    2000

  31. [31]

    Ng, C.-T., Huang, P.-H., Cho, Y.-C., Lee, P.-H., Liu, Y.-C., and Chang, T.-T. (2024). Frontoparietal and salience network synchronizations during nonsymbolic magnitude processing predict brain age and mathematical performance in youth. Human Brain Mapping , 45(11):e26777

    2024

  32. [32]

    Orindi, B., Quintero, A., Bruyneel, L., Li, B., and Lesaffre, E. (2023). A combined multilevel factor analysis and covariance regression model with mixed effects in the mean and variance structure. Statistics in Medicine

    2023

  33. [33]

    Orlichenko, A., Su, K.-J., Shen, H., Deng, H.-W., and Wang, Y.-P. (2024). Somatomotor-visual resting state functional connectivity increases after 2 years in the UK biobank longitudinal cohort. Journal of Medical Imaging , 11(2):024010--024010

    2024

  34. [34]

    C., McCoy, R

    Pan, Y., Bi, C., Kochunov, P., Shardell, M., Smith, J. C., McCoy, R. G., Ye, Z., Yu, J., Lu, T., Yang, Y., et al. (2024). Brain-wide functional connectome analysis of 40,000 individuals reveals brain networks that show aging effects in older adults. Imaging Neuroscience , 2:imag--2

    2024

  35. [35]

    J., and Park, T

    Pourahmadi, M., Daniels, M. J., and Park, T. (2007). Simultaneous modelling of the Cholesky decomposition of several covariance matrices. Journal of Multivariate Analysis , 98(3):568--587

    2007

  36. [36]

    Saberi, M., Khosrowabadi, R., Khatibi, A., Misic, B., and Jafari, G. (2021). Requirement to change of functional brain network across the lifespan. PloS One , 16(11):e0260091

    2021

  37. [37]

    and Holmes, S

    Seiler, C. and Holmes, S. (2017). Multivariate heteroscedasticity models for functional brain connectivity. Frontiers in Neuroscience , 11:696

    2017

  38. [38]

    Shen, C., Liu, C., Chen, N., and Qiu, A. (2025). Dedifferentiation of brain functional gradient captures cognition performance and stroke occurrence: A UK Biobank study. NeuroImage , 311:121183

    2025

  39. [39]

    M., Jenkinson, M., Woolrich, M

    Smith, S. M., Jenkinson, M., Woolrich, M. W., Beckmann, C. F., Behrens, T. E., Johansen-Berg, H., Bannister, P. R., De Luca, M., Drobnjak, I., Flitney, D. E., et al. (2004). Advances in functional and structural MR image analysis and implementation as FSL . NeuroImage , 23:S208--S219

    2004

  40. [40]

    H., Bookheimer, S

    Somerville, L. H., Bookheimer, S. Y., Buckner, R. L., Burgess, G. C., Curtiss, S. W., Dapretto, M., Elam, J. S., Gaffrey, M. S., Harms, M. P., Hodge, C., et al. (2018). The lifespan human connectome project in development: A large-scale study of brain connectivity development in 5--21 year olds. Neuroimage , 183:456--468

    2018

  41. [41]

    Sra, S. (2012). A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I_ s (x) . Computational Statistics , 27:177--190

    2012

  42. [42]

    J., and Dayan, E

    Stanford, W., Mucha, P. J., and Dayan, E. (2024). Age-related differences in network controllability are mitigated by redundancy in large-scale brain networks. Communications Biology , 7(1):701

    2024

  43. [43]

    Stern, Y. (2002). What is cognitive reserve? theory and research application of the reserve concept. Journal of the International Neuropsychological Society , 8(3):448--460

    2002

  44. [44]

    M., Bartr \'e s-Faz, D., Belleville, S., Cantilon, M., Chetelat, G., Ewers, M., Franzmeier, N., Kempermann, G., Kremen, W

    Stern, Y., Arenaza-Urquijo, E. M., Bartr \'e s-Faz, D., Belleville, S., Cantilon, M., Chetelat, G., Ewers, M., Franzmeier, N., Kempermann, G., Kremen, W. S., et al. (2020). Whitepaper: Defining and investigating cognitive reserve, brain reserve, and brain maintenance. Alzheimer's & Dementia , 16(9):1305--1311

    2020

  45. [45]

    Sun, L., Zhao, T., Liang, X., Xia, M., Li, Q., Liao, X., Gong, G., Wang, Q., Pang, C., Yu, Q., et al. (2025). Human lifespan changes in the brain's functional connectome. Nature Neuroscience , pages 1--11

    2025

  46. [46]

    Q., Prater, K., Amin, H., Greicius, M

    Supekar, K., Uddin, L. Q., Prater, K., Amin, H., Greicius, M. D., and Menon, V. (2010). Development of functional and structural connectivity within the default mode network in young children. Neuroimage , 52(1):290--301

    2010

  47. [47]

    Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., and Knight, K. (2005). Sparsity and smoothness via the fused lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 67(1):91--108

    2005

  48. [48]

    Van der Vaart, A. W. (1998). Asymptotic statistics , volume 3. Cambridge university press

    1998

  49. [49]

    C., Smith, S

    Van Essen, D. C., Smith, S. M., Barch, D. M., Behrens, T. E., Yacoub, E., Ugurbil, K., and Wu-Minn HCP Consortium (2013). The WU - Minn human connectome project: an overview. Neuroimage , 80:62--79

    2013

  50. [50]

    Wang, S., Zhang, X., Liu, Y., Xu, W., Tian, X., and Zhao, Y. (2026). Latent space-based network analysis for brain--behavior linking in neuroimaging. Nature Methods , 23(1):225--235

    2026

  51. [51]

    Wang, X., Huang, C.-C., Tsai, S.-J., Lin, C.-P., and Cai, Q. (2024). The aging trajectories of brain functional hierarchy and its impact on cognition across the adult lifespan. Frontiers in Aging Neuroscience , 16:1331574

    2024

  52. [52]

    H., and Northoff, G

    Yang, C., Li, G., Jing, X., Wang, Y., Yan, J. H., and Northoff, G. (2025). The lifelong nonlinear development of spatial variability of brain signals. Communications Biology , 8(1):500

    2025

  53. [53]

    D., Arbabshirani, M

    Zhang, C., Cahill, N. D., Arbabshirani, M. R., White, T., Baum, S. A., and Michael, A. M. (2016). Sex and age effects of functional connectivity in early adulthood. Brain Connectivity , 6(9):700--713

    2016

  54. [54]

    C., Baum, S

    Zhang, C., Dougherty, C. C., Baum, S. A., White, T., and Michael, A. M. (2018). Functional connectivity predicts gender: Evidence for gender differences in resting brain connectivity. Human Brain Mapping , 39(4):1765--1776

    2018

  55. [55]

    T., Cheng, W., and Feng, J

    Zhang, R., Rolls, E. T., Cheng, W., and Feng, J. (2024). Different cortical connectivities in human females and males relate to differences in strength and body composition, reward and emotional systems, and memory. Brain Structure and Function , 229(1):47--61

    2024

  56. [56]

    S., and Luo, X

    Zhao, Y., Caffo, B. S., and Luo, X. (2024). Longitudinal regression of covariance matrix outcomes. Biostatistics , 25(2):385--401

    2024

  57. [57]

    H., Caffo, B

    Zhao, Y., Wang, B., Mostofsky, S. H., Caffo, B. S., and Luo, X. (2021). Covariate assisted principal regression for covariance matrix outcomes. Biostatistics , 22(3):629--645

    2021

  58. [58]

    Zou, T., Lan, W., Wang, H., and Tsai, C.-L. (2017). Covariance regression analysis. Journal of the American Statistical Association , 112(517):266--281

    2017