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CN 34-1054/N

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An approach to estimating nonlinear sufficient dimension reduction subspace for censored survival data

  • Department of Statistics and Finance, School of Management, University of Science and of Technology of China,Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.09.001
  • Received Date: 03 March 2015
  • Accepted Date: 21 May 2015
  • Rev Recd Date: 21 May 2015
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    An approach was proposed to estimating the nonlinear sufficient dimension reduction (SDR) subspace for survival data with censorship. Based on the theory of reproducing kernel Hilbert spaces (RKHS) and the double slicing procedure，the joint nonlinear sufficient dimension reduction central subspace was estimated by means of the generalized eigen-decomposition equation. And the weight function was estimated by the definition and property of SDR central subspace. The efficiency was improved by the iteration method while the algorithm was being implemented. Finally， the performance of the proposed method was illustrated on simulated data.

    Abstract

    An approach was proposed to estimating the nonlinear sufficient dimension reduction (SDR) subspace for survival data with censorship. Based on the theory of reproducing kernel Hilbert spaces (RKHS) and the double slicing procedure，the joint nonlinear sufficient dimension reduction central subspace was estimated by means of the generalized eigen-decomposition equation. And the weight function was estimated by the definition and property of SDR central subspace. The efficiency was improved by the iteration method while the algorithm was being implemented. Finally， the performance of the proposed method was illustrated on simulated data.
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    • References(17)
  • [1]
    Cook R D. Regression Graphics: Ideas for Studying Regressions Through Graphics[M]. New York: Wiley, 1998: 215.
    [2]
    Ma Y, Zhu L. A review on dimension reduction[J]. International Statistical Review, 2013, 81(1):134-150.
    [3]
    Li K C, Wang J L, Chen C H. Dimension reduction for censored regression data[J]. The Annals of Statistics, 1999, 27(1): 1-23.
    [4]
    Li K C. Sliced inverse regression for dimension reduction[J]. Journal of the American Statistical Association, 1991, 86(414):316-327.
    [5]
    Li L, Li H. Dimension reduction methods for microarrays with application to censored survival data[J]. Bioinformatics, 2004, 20(18): 3 406-3 412.
    [6]
    Shevlyakova M, Morgenthaler S. Sliced inverse regression for survival data[J]. Statistical Papers, 2014, 55(1):209-220.
    [7]
    Wen X M. On sufficient dimension reduction for proportional censorship model with covariates[J]. Computational Statistics & Data Analysis, 2010, 54(8): 1 975-1 982.
    [8]
    Xia Y, Zhang D,Xu J. Dimension reduction and semiparametric estimation of survival models[J]. Journal of the American Statistical Association, 2010, 105(489): 278-290.
    [9]
    Lu W, Li L. Sufficient dimension reduction for censored regressions[J]. Biometrics, 2011, 67(2):513-523.
    [10]
    Bender R,Augustin T, Blettner M. Generating survival times to simulate Cox proportional hazards models[J]. Statistics in Medicine, 2005, 24(11):1 713-1 723.
    [11]
    Wu H M. Kernel sliced inverse regression with applications to classification[J]. Journal of Computational and Graphical Statistics, 2008, 17(3): 590-610.
    [12]
    Aronszajn N. Theory of reproducing kernels[J]. Transactions of the American Mathematical Society, 1950, 68(3): 337-404.
    [13]
    Schlkopf B, Smola A J. Learning With Kernels: Support Vector Machines, Regularization, Optimization,and Beyond[M]. Cambridge, MA: MIT Press, 2001.
    [14]
    Zhong W, Zeng P, Ma P, et al. RSIR: Regularized sliced inverse regression for motif discovery[J]. Bioinformatics, 2005, 21(22):4 169-4 175.
    [15]
    Ferr W L, Villa N. Multilayer perceptron with functional inputs: An inverse regression approach[J]. Scandinavian Journal of Statistics, 2006, 33(4): 807-823.
    [16]
    Li L, Yin X. Sliced inverse regression with regularizations[J]. Biometrics, 2008, 64(1):124-131.
    [17]
    Cox D R. Regression models and life-tables[C]// Breakthroughs in Statistics. New York: Springer, 1992: 527-541.
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    [1]
    Cook R D. Regression Graphics: Ideas for Studying Regressions Through Graphics[M]. New York: Wiley, 1998: 215.
    [2]
    Ma Y, Zhu L. A review on dimension reduction[J]. International Statistical Review, 2013, 81(1):134-150.
    [3]
    Li K C, Wang J L, Chen C H. Dimension reduction for censored regression data[J]. The Annals of Statistics, 1999, 27(1): 1-23.
    [4]
    Li K C. Sliced inverse regression for dimension reduction[J]. Journal of the American Statistical Association, 1991, 86(414):316-327.
    [5]
    Li L, Li H. Dimension reduction methods for microarrays with application to censored survival data[J]. Bioinformatics, 2004, 20(18): 3 406-3 412.
    [6]
    Shevlyakova M, Morgenthaler S. Sliced inverse regression for survival data[J]. Statistical Papers, 2014, 55(1):209-220.
    [7]
    Wen X M. On sufficient dimension reduction for proportional censorship model with covariates[J]. Computational Statistics & Data Analysis, 2010, 54(8): 1 975-1 982.
    [8]
    Xia Y, Zhang D,Xu J. Dimension reduction and semiparametric estimation of survival models[J]. Journal of the American Statistical Association, 2010, 105(489): 278-290.
    [9]
    Lu W, Li L. Sufficient dimension reduction for censored regressions[J]. Biometrics, 2011, 67(2):513-523.
    [10]
    Bender R,Augustin T, Blettner M. Generating survival times to simulate Cox proportional hazards models[J]. Statistics in Medicine, 2005, 24(11):1 713-1 723.
    [11]
    Wu H M. Kernel sliced inverse regression with applications to classification[J]. Journal of Computational and Graphical Statistics, 2008, 17(3): 590-610.
    [12]
    Aronszajn N. Theory of reproducing kernels[J]. Transactions of the American Mathematical Society, 1950, 68(3): 337-404.
    [13]
    Schlkopf B, Smola A J. Learning With Kernels: Support Vector Machines, Regularization, Optimization,and Beyond[M]. Cambridge, MA: MIT Press, 2001.
    [14]
    Zhong W, Zeng P, Ma P, et al. RSIR: Regularized sliced inverse regression for motif discovery[J]. Bioinformatics, 2005, 21(22):4 169-4 175.
    [15]
    Ferr W L, Villa N. Multilayer perceptron with functional inputs: An inverse regression approach[J]. Scandinavian Journal of Statistics, 2006, 33(4): 807-823.
    [16]
    Li L, Yin X. Sliced inverse regression with regularizations[J]. Biometrics, 2008, 64(1):124-131.
    [17]
    Cox D R. Regression models and life-tables[C]// Breakthroughs in Statistics. New York: Springer, 1992: 527-541.
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