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Open AccessOpen Access JUSTC Mathematicts 01 April 2024

Alternative modified Cholesky decomposition of the precision matrix of longitudinal data

  • College of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

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https://doi.org/10.52396/JUSTC-2023-0127
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  • Author Bio:

    Fei Lu is currently a lecturer at the College of Science, Zhejiang Sci-Tech University. He received his Ph.D. degree from Beijing University of Technology in 2020. His research mainly focuses on longitudinal data analysis

  • Corresponding author: E-mail: lufeiby@163.com
  • Received Date: 28 August 2023
  • Accepted Date: 28 November 2023
    • Available Online: 01 April 2024
    Abstract

  • Abstract

    The correlation matrix might be of scientific interest for longitudinal data. However, few studies have focused on both robust estimation of the correlation matrix against model misspecification and robustness to outliers in the data, when the precision matrix possesses a typical structure. In this paper, we propose an alternative modified Cholesky decomposition (AMCD) for the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances. A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proven to be consistent and asymptotically normally distributed. Furthermore, a double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed. The simulation studies and real data analysis demonstrate the effectiveness of the proposed AMCD method.

    Graphical abstract

    The framework of the double robust Laplace joint modeling model for longitudinal data.

    Abstract

    The correlation matrix might be of scientific interest for longitudinal data. However, few studies have focused on both robust estimation of the correlation matrix against model misspecification and robustness to outliers in the data, when the precision matrix possesses a typical structure. In this paper, we propose an alternative modified Cholesky decomposition (AMCD) for the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances. A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proven to be consistent and asymptotically normally distributed. Furthermore, a double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed. The simulation studies and real data analysis demonstrate the effectiveness of the proposed AMCD method.

    • Public Summary

    • We propose an alternative modified Cholesky decomposition (AMCD) of the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances.
    • A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proved to be consistent and asymptotically normally distributed.
    • A double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed.

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    • References(28)
  • [1]
    Diggle P J, Heagerty P J, Liang K Y, et al. Analysis of Longitudinal Data. Oxford: Oxford University Press, 2002 .
    [2]
    Diggle P J, Verbyla A P. Nonparametric estimation of covariance structure in longitudinal data. Biometrics, 1998, 54 (2): 401–415.
    [3]
    Pourahmadi M. Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika, 1999, 86 (3): 677–690. doi: 10.1093/biomet/86.3.677
    [4]
    Zhang W, Leng C. A moving average Cholesky factor model in covariance modelling for longitudinal data. Biometrika, 2012, 99 (1): 141–150. doi: 10.1093/biomet/asr068
    [5]
    Chen Z, Dunson D B. Random effects selection in linear mixed models. Biometrics, 2003, 59 (4): 762–769. doi: 10.1111/j.0006-341X.2003.00089.x
    [6]
    Pourahmadi M. Cholesky decompositions and estimation of a covariance matrix: orthogonality of variance-correlation parameters. Biometrika, 2007, 94 (4): 1006–1013. doi: 10.1093/biomet/asm073
    [7]
    Maadooliat M, Pourahmadi M, Huang J Z. Robust estimation of the correlation matrix of longitudinal data. Statistics and Computing, 2013, 23: 17–28. doi: 10.1007/s11222-011-9284-6
    [8]
    Zhang W, Leng C, Tang C Y. A joint modelling approach for longitudinal studies. Journal of the Royal Statistical Society Series B:Statistical Methodology, 2015, 77 (1): 219–238. doi: 10.1111/rssb.12065
    [9]
    Lin T I, Wang Y J. A robust approach to joint modeling of mean and scale covariance for longitudinal data. Journal of Statistical Planning and Inference, 2009, 139 (9): 3013–3026. doi: 10.1016/j.jspi.2009.02.008
    [10]
    Guney Y, Arslan O, Gokalp-Yavuz F. Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm. Journal of Multivariate Analysis, 2022, 191: 105026. doi: 10.1016/j.jmva.2022.105026
    [11]
    Pourahmadi M. Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika, 2000, 87 (2): 425–435. doi: 10.1093/biomet/87.2.425
    [12]
    Anderson D N. A multivariate Linnik distribution. Statistics & Probability Letters, 1992, 14 (4): 333–336. doi: 10.1016/0167-7152(92)90067-F
    [13]
    Ernst M D. A multivariate generalized Laplace distribution. Computational Statistics, 1998, 13 (2): 227–232.
    [14]
    Fernández C, Osiewalski J, Steel M F. Modeling and inference with υ -spherical distributions. Journal of the American Statistical Association, 1995, 90 (432): 1331–1340. doi: 10.1080/01621459.1995.10476637
    [15]
    Portilla J, Strela V, Wainwright M J, et al. Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing, 2003, 12 (11): 1338–1351. doi: 10.1109/TIP.2003.818640
    [16]
    Kotz S, Kozubowski T J, Podgórski K. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Boston, MA: Birkhäuser, 2001.
    [17]
    Press W H, Teukolsky S A, Vetterling W T, et al. Numerical recipes: The art of scientific computing. 3rd ed. Cambridge: Cambridge University Press, 2007 .
    [18]
    Pan J, Pan Y. jmcm: An R package for joint mean-covariance modeling of longitudinal data. Journal of Statistical Software, 2017, 82: 1–29. doi: 10.18637/jss.v082.i09
    [19]
    Kenward M G. A method for comparing profiles of repeated measurements. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1987, 36 (3): 296–308. doi: 10.2307/2347788
    [20]
    Pan J, Mackenzie G. On modelling mean-covariance structures in longitudinal studies. Biometrika, 2003, 90 (1): 239–244. doi: 10.1093/biomet/90.1.239
    [21]
    Belenky G, Wesensten N J, Thorne D R, et al. Patterns of performance degradation and restoration during sleep restriction and subsequent recovery: a sleep dose-response study. Journal of Sleep Research, 2003, 12 (1): 1–12. doi: 10.1046/j.1365-2869.2003.00337.x
    [22]
    Lin T I, Wang W L. Bayesian inference in joint modelling of location and scale parameters of the t distribution for longitudinal data. Journal of Statistical Planning and Inference, 2011, 141 (4): 1543–1553. doi: 10.1016/j.jspi.2010.11.001
    [23]
    Lee K, Baek C, Daniels M J. ARMA Cholesky factor models for the covariance matrix of linear models. Computational Statistics & Data Analysis, 2017, 115: 267–280. doi: 10.1016/j.csda.2017.05.001
    [24]
    Zhang W, Xie F, Tan J. A robust joint modeling approach for longitudinal data with informative dropouts. Computational Statistics, 2020, 35: 1759–1783. doi: 10.1007/s00180-020-00972-6
    [25]
    Yu J, Nummi T, Pan J. Mixture regression for longitudinal data based on joint mean-covariance model. Journal of Multivariate Analysis, 2022, 190: 104956. doi: 10.1016/j.jmva.2022.104956
    [26]
    Chiu T Y M, Leonard T, Tsui K W. The matrix-logarithmic covariance model. Journal of the American Statistical Association, 1996, 91 (433): 198–210. doi: 10.1080/01621459.1996.10476677
    [27]
    Rubin H. Uniform convergence of random functions with applications to statistics. The Annals of Mathematical Statistics, 1956, 27 (1): 200–203. doi: 10.1214/aoms/1177728359
    [28]
    Royden H L, Fitzpatrick P. Real Analysis. New York:Macmillan, 1968.
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    Figure  1.  Cattle data: sample regressograms and fitted curves for (a) log-innovation variances and (b) autoregressive coefficients (solid lines, curves fitted by the proposed AMCD method; dashed lines, 95% pointwise confidence intervals using the bootstrapping method).

    Figure  2.  Sleep dose-response data: (a) trajectories of average reaction time; (b) sample regressograms for log-innovation variances; (c) sample regressograms for autoregressive coefficients.

    [1]
    Diggle P J, Heagerty P J, Liang K Y, et al. Analysis of Longitudinal Data. Oxford: Oxford University Press, 2002 .
    [2]
    Diggle P J, Verbyla A P. Nonparametric estimation of covariance structure in longitudinal data. Biometrics, 1998, 54 (2): 401–415.
    [3]
    Pourahmadi M. Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika, 1999, 86 (3): 677–690. doi: 10.1093/biomet/86.3.677
    [4]
    Zhang W, Leng C. A moving average Cholesky factor model in covariance modelling for longitudinal data. Biometrika, 2012, 99 (1): 141–150. doi: 10.1093/biomet/asr068
    [5]
    Chen Z, Dunson D B. Random effects selection in linear mixed models. Biometrics, 2003, 59 (4): 762–769. doi: 10.1111/j.0006-341X.2003.00089.x
    [6]
    Pourahmadi M. Cholesky decompositions and estimation of a covariance matrix: orthogonality of variance-correlation parameters. Biometrika, 2007, 94 (4): 1006–1013. doi: 10.1093/biomet/asm073
    [7]
    Maadooliat M, Pourahmadi M, Huang J Z. Robust estimation of the correlation matrix of longitudinal data. Statistics and Computing, 2013, 23: 17–28. doi: 10.1007/s11222-011-9284-6
    [8]
    Zhang W, Leng C, Tang C Y. A joint modelling approach for longitudinal studies. Journal of the Royal Statistical Society Series B:Statistical Methodology, 2015, 77 (1): 219–238. doi: 10.1111/rssb.12065
    [9]
    Lin T I, Wang Y J. A robust approach to joint modeling of mean and scale covariance for longitudinal data. Journal of Statistical Planning and Inference, 2009, 139 (9): 3013–3026. doi: 10.1016/j.jspi.2009.02.008
    [10]
    Guney Y, Arslan O, Gokalp-Yavuz F. Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm. Journal of Multivariate Analysis, 2022, 191: 105026. doi: 10.1016/j.jmva.2022.105026
    [11]
    Pourahmadi M. Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika, 2000, 87 (2): 425–435. doi: 10.1093/biomet/87.2.425
    [12]
    Anderson D N. A multivariate Linnik distribution. Statistics & Probability Letters, 1992, 14 (4): 333–336. doi: 10.1016/0167-7152(92)90067-F
    [13]
    Ernst M D. A multivariate generalized Laplace distribution. Computational Statistics, 1998, 13 (2): 227–232.
    [14]
    Fernández C, Osiewalski J, Steel M F. Modeling and inference with υ -spherical distributions. Journal of the American Statistical Association, 1995, 90 (432): 1331–1340. doi: 10.1080/01621459.1995.10476637
    [15]
    Portilla J, Strela V, Wainwright M J, et al. Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing, 2003, 12 (11): 1338–1351. doi: 10.1109/TIP.2003.818640
    [16]
    Kotz S, Kozubowski T J, Podgórski K. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Boston, MA: Birkhäuser, 2001.
    [17]
    Press W H, Teukolsky S A, Vetterling W T, et al. Numerical recipes: The art of scientific computing. 3rd ed. Cambridge: Cambridge University Press, 2007 .
    [18]
    Pan J, Pan Y. jmcm: An R package for joint mean-covariance modeling of longitudinal data. Journal of Statistical Software, 2017, 82: 1–29. doi: 10.18637/jss.v082.i09
    [19]
    Kenward M G. A method for comparing profiles of repeated measurements. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1987, 36 (3): 296–308. doi: 10.2307/2347788
    [20]
    Pan J, Mackenzie G. On modelling mean-covariance structures in longitudinal studies. Biometrika, 2003, 90 (1): 239–244. doi: 10.1093/biomet/90.1.239
    [21]
    Belenky G, Wesensten N J, Thorne D R, et al. Patterns of performance degradation and restoration during sleep restriction and subsequent recovery: a sleep dose-response study. Journal of Sleep Research, 2003, 12 (1): 1–12. doi: 10.1046/j.1365-2869.2003.00337.x
    [22]
    Lin T I, Wang W L. Bayesian inference in joint modelling of location and scale parameters of the t distribution for longitudinal data. Journal of Statistical Planning and Inference, 2011, 141 (4): 1543–1553. doi: 10.1016/j.jspi.2010.11.001
    [23]
    Lee K, Baek C, Daniels M J. ARMA Cholesky factor models for the covariance matrix of linear models. Computational Statistics & Data Analysis, 2017, 115: 267–280. doi: 10.1016/j.csda.2017.05.001
    [24]
    Zhang W, Xie F, Tan J. A robust joint modeling approach for longitudinal data with informative dropouts. Computational Statistics, 2020, 35: 1759–1783. doi: 10.1007/s00180-020-00972-6
    [25]
    Yu J, Nummi T, Pan J. Mixture regression for longitudinal data based on joint mean-covariance model. Journal of Multivariate Analysis, 2022, 190: 104956. doi: 10.1016/j.jmva.2022.104956
    [26]
    Chiu T Y M, Leonard T, Tsui K W. The matrix-logarithmic covariance model. Journal of the American Statistical Association, 1996, 91 (433): 198–210. doi: 10.1080/01621459.1996.10476677
    [27]
    Rubin H. Uniform convergence of random functions with applications to statistics. The Annals of Mathematical Statistics, 1956, 27 (1): 200–203. doi: 10.1214/aoms/1177728359
    [28]
    Royden H L, Fitzpatrick P. Real Analysis. New York:Macmillan, 1968.
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