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Model-Based Inverse Regression and Its Applications

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Festschrift in Honor of R. Dennis Cook

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Abstract

One fundamental concept in regression is dimension reduction, the basic idea being to reduce the dimension of the predictor space without loss of information on the regression. To avoid the curse of dimensionality, many methods in this field restrict attention to inverse reduction in the framework of inverse regression. This review focuses on model-based inverse regression. First, we consider sufficient reduction for multivariate count data in different contexts, on the basis of the multinomial distribution and its generalizations. Second, we take a different perspective on model-based inverse reduction. Sufficient reduction is achieved in the dual sample-based space, rather than in the primal predictor-based space. The results extend the known duality between principal component analysis and principal coordinate analysis. Finally, we consider an application of inverse modeling to testing the independence between the microbiome composition and a continuous outcome. An adaptive test is presented based on a dynamic slicing technique.

Tao Wang was supported in part by National Natural Science Foundation of China (11971017), National Key R&D Program of China (2018YFC0910500), Shanghai Municipal Science and Technology Major Project (2017SHZDZX01), SJTU Trans-med Awards Research Young Faculty Grant (YG2019QNA26, YG2019QNA37), and Neil Shen’s SJTU Medical Research Fund. Lixing Zhu was supported by a grant from the University Grants Council of Hong Kong.

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References

  • K.P. Adragni, R.D. Cook, Sufficient dimension reduction and prediction in regression. Philos. Trans. R. Soc. A 367(1906), 4385–4405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • J. Aitchison, The Statistical Analysis of Compositional Data (Chapman and Hall, London, 1986)

    Book  MATH  Google Scholar 

  • A. Antoniadis, S. Lambertlacroix, F. Leblanc, Effective dimension reduction methods for tumor classification using gene expression data. Bioinformatics 19(5), 563–570 (2003)

    Article  Google Scholar 

  • D. Billheimer, P. Guttorp, W.F. Fagan, Statistical interpretation of species composition. J. Am. Stat. Assoc. 96(456), 1205–1214 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bura, R.D. Cook, Estimating the structural dimension of regressions via parametric inverse regression. J. R. Stat. Soc. Ser. B 63(2), 393–410 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bura, L. Forzani, Sufficient reductions in regressions with elliptically contoured inverse predictors. J. Am. Stat. Assoc. 110(509), 420–434 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bura, R.M. Pfeiffer, Graphical methods for class prediction using dimension reduction techniques on DNA microarray data. Bioinformatics 19(10), 1252–1258 (2003)

    Article  Google Scholar 

  • E. Bura, S. Duarte, L. Forzani, Sufficient reductions in regressions with exponential family inverse predictors. J. Am. Stat. Assoc. 111(515), 1313–1329 (2016)

    Article  MathSciNet  Google Scholar 

  • F. Chiaromonte, J. Martinelli, Dimension reduction strategies for analyzing global gene expression data with a response. Bellman Prize Math. Biosci. 176(1), 123–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • P. Clifford, Markov random fields in statistics, in Disorder in Physical Systems: A Volume in Honour of John M. Hammersley (Clarendon Press, Oxford, 1990)

    Google Scholar 

  • R.D. Cook, Using dimension-reduction subspaces to identify important inputs in models of physical systems, in Proceedings of the section on Physical and Engineering Sciences (American Statistical Association, Alexandria, VA, 1994), pp. 18–25

    Google Scholar 

  • R.D. Cook, Regression Graphics: Ideas for Studying Regressions Through Graphics (Wiley, New York, 1998)

    Book  MATH  Google Scholar 

  • R.D. Cook, Fisher lecture: dimension reduction in regression. Stat. Sci. 22(1), 1–26 (2007)

    MathSciNet  MATH  Google Scholar 

  • R.D. Cook, Principal components, sufficient dimension reduction, and envelopes. Annu. Rev. Stat. Appl. 5, 533–559 (2018)

    Article  MathSciNet  Google Scholar 

  • R.D. Cook, L. Forzani, Likelihood-based sufficient dimension reduction. J. Am. Stat. Assoc. 104(485), 197–208 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • R.D. Cook, L. Li, Dimension reduction in regressions with exponential family predictors. J. Comput. Graph. Stat. 18(3), 774–791 (2009)

    Article  MathSciNet  Google Scholar 

  • R.D. Cook, L. Ni, Sufficient dimension reduction via inverse regression: a minimum discrepancy approach. J. Am. Stat. Assoc. 100(470), 410–428 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • R.D. Cook, L. Orzani, Principal fitted components for dimension reduction in regression. Stat. Sci. 23(4), 485–501 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • R.D. Cook, S. Weisberg, Comment. J. Am. Stat. Assoc. 86(414), 328–332 (1991)

    Google Scholar 

  • R.D. Cook, L. Forzani, A.J. Rothman, Estimating sufficient reductions of the predictors in abundant high-dimensional regressions. Ann. Stat. 40(1), 353–384 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • L. Forzani, R.G. Arancibia, P. Llop, D. Tomassi, Supervised dimension reduction for ordinal predictors. Comput. Stat. Data Anal. 125, 136–155 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • J.C. Gower, Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325–338 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  • J.C. Gower, Adding a point to vector diagrams in multivariate analysis. Biometrika 55(3), 582–585 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  • R. Heller, Y. Heller, S. Kaufman, B. Brill, M. Gorfine, Consistent distribution-free K-sample and independence tests for univariate random variables. J. Mach. Learn. Res. 17(29), 1–54 (2016)

    MathSciNet  MATH  Google Scholar 

  • B. Jiang, C. Ye, J.S. Liu, Nonparametric K-sample tests via dynamic slicing. J. Am. Stat. Assoc. 110(510), 642–653 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • P.S. La Rosa, J.P. Brooks, E. Deych, E.L. Boone, D.J. Edwards, Q. Wang, et al., Hypothesis testing and power calculations for taxonomic-based human microbiome data. PLoS One 7(12), e52078 (2012)

    Google Scholar 

  • S.L. Lauritzen, Graphical Models (Clarendon Press, Oxford, 1996)

    MATH  Google Scholar 

  • K.-Y. Lee, B. Li, F. Chiaromonte, A general theory for nonlinear sufficient dimension reduction: formulation and estimation. Ann. Stat. 41(1), 221–249 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • K.-C. Li, Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86(414), 316–327 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • K.-C. Li, On principal Hessian directions for data visualization and dimension reduction: another application of Stein’s lemma. J. Am. Stat. Assoc. 87(420), 1025–1039 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • L. Li, Survival prediction of diffuse large-b-cell lymphoma based on both clinical and gene expression information. Bioinformatics 22(4), 466–471 (2006)

    Article  Google Scholar 

  • B. Li, Sufficient Dimension Reduction: Methods and Applications with R (CRC Press, Boca Raton, 2018a)

    Book  MATH  Google Scholar 

  • L. Li, Sufficient Dimension Reduction. Wiley StatsRef: Statistics Reference Online (2018b)

    Google Scholar 

  • L. Li, X. Yin, Sliced inverse regression with regularizations. Biometrics 64(1), 124–131 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • B. Li, S. Wang, On directional regression for dimension reduction. J. Am. Stat. Assoc. 102(479), 997–1008 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • B. Li, H. Zha, F. Chiaromonte, Contour regression: a general approach to dimension reduction. Ann. Stat. 33(4), 1580–1616 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Ma, L. Zhu, A semiparametric approach to dimension reduction. J. Am. Stat. Assoc. 107(497), 168–179 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Ma, L. Zhu, A review on dimension reduction. Int. Stat. Rev. 81(1), 134–150 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • B.H. Mcardle, M.J. Anderson, Fitting multivariate models to community data: a comment on distance based redundancy analysis. Ecology 82(1), 290–297 (2001)

    Article  Google Scholar 

  • N. Meinshausen, P. Bühlmann, High-dimensional graphs and variable selection with the lasso. Ann. Stat. 34(3), 1436–1462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • P.A. Naik, M.R. Hagerty, C. Tsai, A new dimension reduction approach for data-rich marketing environments: sliced inverse regression. J. Market. Res. 37(1), 88–101 (2000)

    Article  Google Scholar 

  • S.S. Roley, R.M. Newman, Predicting Eurasian watermilfoil invasions in Minnesota. Lake Reserv. Manage. 24(4), 361–369 (2008)

    Article  Google Scholar 

  • Y. Song, H. Zhao, T. Wang, An adaptive independence test for microbiome community data. Biometrics 76(2), 414–426 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Q. Sun, R. Zhu, T. Wang, D. Zeng, Counting process-based dimension reduction methods for censored outcomes. Biometrika 106(1), 181–196 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • M. Taddy, Multinomial inverse regression for text analysis. J. Am. Stat. Assoc. 108(503), 755–770 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • M. Taddy, Distributed multinomial regression. Ann. Appl. Stat. 9(3), 1394–1414 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • D. Tomassi, L. Forzani, S. Duarte, R. Pfeiffer, Sufficient dimension reduction for compositional data. Biostatistics (2019). https://doi.org/10.1093/biostatistics/kxz060.

  • T. Wang, Dimension reduction via adaptive slicing. Stat. Sin. (2019) https://doi.org/10.5705/ss.202019.0102.

  • T. Wang, Graph-assisted inverse regression for count data and its application to sequencing data. J. Comput. Graph. Stat. 29(3), 444–454 (2020)

    Article  MathSciNet  Google Scholar 

  • T. Wang, P. Xu, On supervised reduction and its dual. Stat. Sin. (2019) https://doi.org/10.5705/ss.202017.0532.

  • T. Wang, L. Zhu, Sparse sufficient dimension reduction using optimal scoring. Comput. Stat. Data Anal. 57(1), 223–232 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • T. Wang, L. Zhu, Flexible dimension reduction in regression. Stat. Sin. 28(2), 1009–1029 (2018)

    MathSciNet  MATH  Google Scholar 

  • T. Wang, X. Guo, L. Zhu, P. Xu, Transformed sufficient dimension reduction. Biometrika 101(4), 815–829 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • T. Wang, M. Chen, H. Zhao, L. Zhu, Estimating a sparse reduction for general regression in high dimensions. Stat. Comput. 28(1), 33–46 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • T. Wang, C. Yang, H. Zhao, Prediction analysis for microbiome sequencing data. Biometrics 75, 875–884 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Xia, H. Tong, W.K. Li, L. Zhu, An adaptive estimation of dimension reduction space. J. R. Stat. Soc. Ser. B 64(3), 363–410 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • F. Xia, J. Chen, W.K. Fung, H. Li, A logistic normal multinomial regression model for microbiome compositional data analysis. Biometrics 69(4), 1053–1063 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Zhang, D. Yeung, J.T. Kwok, E.Y. Chang, Sliced coordinate analysis for effective dimension reduction and nonlinear extensions. J. Comput. Graph. Stat. 17(1), 225–242 (2012)

    Article  MathSciNet  Google Scholar 

  • W. Zhong, P. Zeng, P. Ma, J.S. Liu, Y. Zhu, RSIR: regularized sliced inverse regression for motif discovery. Bioinformatics 21(22), 4169–4175 (2005)

    Article  Google Scholar 

  • L. Zhu, T. Wang, L. Zhu, L. Ferré, Sufficient dimension reduction through discretization-expectation estimation. Biometrika 97(2), 295–304 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, Shanghai Jiao Tong University, Shanghai, China

    Tao Wang

  2. SJTU-Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University, Shanghai, China

    Tao Wang

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Lixing Zhu

  4. Center for Statictics and Data Science, Beijing Normal University at Zhuhai, Guangdong, China

    Lixing Zhu

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  1. Tao Wang
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Correspondence to Lixing Zhu .

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Editors and Affiliations

  1. Applied Statistics, Vienna University of Technology, Vienna, Wien, Austria

    Efstathia Bura

  2. Department of Statistics, The Pennsylvania State University, University Park, PA, USA

    Bing Li

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Wang, T., Zhu, L. (2021). Model-Based Inverse Regression and Its Applications. In: Bura, E., Li, B. (eds) Festschrift in Honor of R. Dennis Cook. Springer, Cham. https://doi.org/10.1007/978-3-030-69009-0_6

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