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Modular regression - a Lego system for building structured additive distributional regression models with tensor product interactions

Author

Listed:
  • Thomas Kneib

    (Georg-August-Universität Göttingen)

  • Nadja Klein

    (Humboldt Universität zu Berlin)

  • Stefan Lang

    (Universität Innsbruck)

  • Nikolaus Umlauf

    (Universität Innsbruck)

Abstract

Semiparametric regression models offer considerable flexibility concerning the specification of additive regression predictors including effects as diverse as nonlinear effects of continuous covariates, spatial effects, random effects, or varying coefficients. Recently, such flexible model predictors have been combined with the possibility to go beyond pure mean-based analyses by specifying regression predictors on potentially all parameters of the response distribution in a distributional regression framework. In this paper, we discuss a generic concept for defining interaction effects in such semiparametric distributional regression models based on tensor products of main effects. These interactions can be assigned anisotropic penalties, i.e. different amounts of smoothness will be associated with the interacting covariates. We investigate identifiability and the decomposition of interactions into main effects and pure interaction effects (similar as in a smoothing spline analysis of variance) to facilitate a modular model building process. The decomposition is based on orthogonality in function spaces which allows for considerable flexibility in setting up the effect decomposition. Inference is based on Markov chain Monte Carlo simulations with iteratively weighted least squares proposals under constraints to ensure identifiability and effect decomposition. One important aspect is therefore to maintain sparse matrix structures of the tensor product also in identifiable, decomposed model formulations. The performance of modular regression is verified in a simulation on decomposed interaction surfaces of two continuous covariates and two applications on the construction of spatio-temporal interactions for the analysis of precipitation on the one hand and functional random effects for analysing house prices on the other hand.

Suggested Citation

  • Thomas Kneib & Nadja Klein & Stefan Lang & Nikolaus Umlauf, 2019. "Modular regression - a Lego system for building structured additive distributional regression models with tensor product interactions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 1-39, March.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:1:d:10.1007_s11749-019-00631-z
    DOI: 10.1007/s11749-019-00631-z
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    References listed on IDEAS

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    1. Simon N. Wood, 2006. "Low-Rank Scale-Invariant Tensor Product Smooths for Generalized Additive Mixed Models," Biometrics, The International Biometric Society, vol. 62(4), pages 1025-1036, December.
    2. John Hughes & Murali Haran, 2013. "Dimension reduction and alleviation of confounding for spatial generalized linear mixed models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 139-159, January.
    3. Brian J. Reich & James S. Hodges & Vesna Zadnik, 2006. "Effects of Residual Smoothing on the Posterior of the Fixed Effects in Disease-Mapping Models," Biometrics, The International Biometric Society, vol. 62(4), pages 1197-1206, December.
    4. Reto Stauffer & Jakob W. Messner & Georg J. Mayr & Nikolaus Umlauf & Achim Zeileis, 2016. "Spatio-Temporal Precipitation Climatology over Complex Terrain Using a Censored Additive Regression Model," Working Papers 2016-07, Faculty of Economics and Statistics, Universität Innsbruck.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, September.
    6. Paciorek, Christopher J., 2007. "Bayesian Smoothing with Gaussian Processes Using Fourier Basis Functions in the spectralGP Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 19(i02).
    7. Simon N. Wood, 2008. "Fast stable direct fitting and smoothness selection for generalized additive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 495-518, July.
    8. Michael L. Lavine & James S. Hodges, 2012. "On Rigorous Specification of ICAR Models," The American Statistician, Taylor & Francis Journals, vol. 66(1), pages 42-49, February.
    9. Brezger, Andreas & Lang, Stefan, 2006. "Generalized structured additive regression based on Bayesian P-splines," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 967-991, February.
    10. R. A. Rigby & D. M. Stasinopoulos, 2005. "Generalized additive models for location, scale and shape," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(3), pages 507-554, June.
    11. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, September.
    12. Fahrmeir, Ludwig & Kneib, Thomas, 2011. "Bayesian Smoothing and Regression for Longitudinal, Spatial and Event History Data," OUP Catalogue, Oxford University Press, number 9780199533022.
    13. Marra, Giampiero & Radice, Rosalba, 2017. "Bivariate copula additive models for location, scale and shape," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 99-113.
    14. Giampiero Marra & Simon N. Wood, 2012. "Coverage Properties of Confidence Intervals for Generalized Additive Model Components," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(1), pages 53-74, March.
    15. Inyoung Kim & Noah D. Cohen & Raymond J. Carroll, 2003. "Semiparametric Regression Splines in Matched Case-Control Studies," Biometrics, The International Biometric Society, vol. 59(4), pages 1158-1169, December.
    16. J. Besag & D. Higdon, 1999. "Bayesian analysis of agricultural field experiments," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(4), pages 691-746.
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    Cited by:

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    2. Kneib, Thomas & Silbersdorff, Alexander & Säfken, Benjamin, 2023. "Rage Against the Mean – A Review of Distributional Regression Approaches," Econometrics and Statistics, Elsevier, vol. 26(C), pages 99-123.

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