Electronic Journal of Statistics

Generalized functional additive mixed models

Fabian Scheipl, Jan Gertheiss, and Sonja Greven

Full-text: Open access

Abstract

We propose a comprehensive framework for additive regression models for non-Gaussian functional responses, allowing for multiple (partially) nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data as well as linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. Our implementation handles functional responses from any exponential family distribution as well as many others like Beta- or scaled and shifted $t$-distributions. Development is motivated by and evaluated on an application to large-scale longitudinal feeding records of pigs. Results in extensive simulation studies as well as replications of two previously published simulation studies for generalized functional mixed models demonstrate the good performance of our proposal. The approach is implemented in well-documented open source software in the pffr function in R-package refund.

Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 1455-1492.

Dates
Received: October 2015
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1464710238

Digital Object Identifier
doi:10.1214/16-EJS1145

Mathematical Reviews number (MathSciNet)
MR3507370

Zentralblatt MATH identifier
1341.62242

Citation

Scheipl, Fabian; Gertheiss, Jan; Greven, Sonja. Generalized functional additive mixed models. Electron. J. Statist. 10 (2016), no. 1, 1455--1492. doi:10.1214/16-EJS1145. https://projecteuclid.org/euclid.ejs/1464710238


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References

  • [1] Brockhaus, S. (2015)., FDboost: Boosting functional regression models. R package version 0.0-8.
  • [2] Brockhaus, S., F. Scheipl, T. Hothorn, and S. Greven (2015). The functional linear array model., Statistical Modelling 15(3), 279–300.
  • [3] Gertheiss, J., V. Maier, E. F. Hessel, and A.-M. Staicu (2015). Marginal functional regression models for analyzing the feeding behavior of pigs., Journal of Agricultural, Biological, and Environmental Statistics 20(3), 353–370.
  • [4] Goldsmith, J., V. Zipunnikov, and J. Schrack (2015). Generalized multilevel function-on-scalar regression and principal component analysis., Biometrics 71(2), 344– 353.
  • [5] Guillemet, R., J. Y. Dourmad, and M. C. Meunier-Salaün (2006). Feeding behavior in primiparous lactating sows: Impact of a high-fiber diet during pregnancy., Journal of Animal Science 84, 2474–2481.
  • [6] Hall, P., H.-G. Müller, and F. Yao (2008). Modelling sparse generalized longitudinal observations with latent Gaussian processes., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70(4), 703–723.
  • [7] Huang, L., F. Scheipl, J. Goldsmith, J. Gellar, J. Harezlak, M. W. McLean, B. Swihart, L. Xiao, C. Crainiceanu, and P. Reiss (2015)., refund: Regression with Functional Data. R package version 0.1-12.
  • [8] Hyun, Y., M. Ellis, F. K. McKeith, and E. R. Wilson (1997). Feed intake pattern of group-housed growing-finishing pigs monitored using a computerized feed intake recording system., Journal of Animal Science 75, 1443–1451.
  • [9] Li, H., J. Staudenmayer, and R. J. Carroll (2014). Hierarchical functional data with mixed continuous and binary measurements., Biometrics 70(4), 802–811.
  • [10] Marra, G. and S. N. Wood (2012). Coverage properties of confidence intervals for generalized additive model components., Scandinavian Journal of Statistics 39(1), 53–74.
  • [11] Maselyne, J., W. Saeys, B. D. Ketelaere, K. Mertens, J. Vangeyte, E. F. Hessel, S. Millet, and A. Van Nuffel (2014). Validation of a high frequency radio frequency identification (HF RFID) system for registering feeding patterns of growing-finishing pigs., Computers and Electronics in Agriculture 102, 10–18.
  • [12] Montgomery, G. W., D. S. Flux, and J. R. Carr (1978). Feeding patterns in pigs: The effects of amino acid deficiency., Physiology & Bahavior 20, 693–698.
  • [13] Morris, J. S. (2015). Functional regression., Annual Review of Statistics and its Applications 2, 321–359.
  • [14] Pya, N. and S. N. Wood (2016). A note on basis dimension selection in generalized additive modelling. Technical report., http://arxiv.org/abs/1602.06696.
  • [15] R Development Core Team (2015)., R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • [16] Reiss, P. T. and R. T. Ogden (2009). Smoothing parameter selection for a class of semiparametric linear models., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71(2), 505–523.
  • [17] Rigby, R. A. and D. M. Stasinopoulos (2005). Generalized additive models for location, scale and shape., Journal of the Royal Statistical Society: Series C (Applied Statistics) 54(3), 507–554.
  • [18] Ruppert, D. (2002). Selecting the number of knots for penalized splines., Journal of computational and graphical statistics 11(4), 735–757.
  • [19] Scheipl, F., A.-M. Staicu, and S. Greven (2015). Functional additive mixed models., Journal of Computational and Graphical Statistics 24(3), 477–501.
  • [20] Serban, N., A.-M. Staicu, and R. J. Carroll (2013). Multilevel cross-dependent binary longitudinal data., Biometrics 69(4), 903–913.
  • [21] Stan Development Team (2014)., rstan: R interface to Stan, Version 2.6.0.
  • [22] Tashman, L. J. (2000). Out-of-sample tests of forecasting accuracy: an analysis and review., International Journal of Forecasting 16(4), 437–450.
  • [23] The MathWorks, Inc. (2012)., MATLAB 7.12.0.635. The MathWorks, Inc., Natick, Massachusetts.
  • [24] Tidemann-Miller, B. A. (2014)., Statistical Modeling of Multivariate Functional Data that Exhibit Complex Correlation Structures. Ph. D. thesis, North Carolina State University.
  • [25] van der Linde, A. (2009). A Bayesian latent variable approach to functional principal components analysis with binary and count data., AStA Advances in Statistical Analysis 93(3), 307–333.
  • [26] Wang, B. and J. Q. Shi (2014). Generalized Gaussian process regression model for non-gaussian functional data., Journal of the American Statistical Association 109(507), 1123 –1133.
  • [27] Wood, S. N. (2006)., Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC.
  • [28] Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models., Journal of the Royal Statistical Society: Series B 73(1), 3–36.
  • [29] Wood, S. N. (2013). On p-values for smooth components of an extended generalized additive model., Biometrika 100(1), 221–228.
  • [30] Wood, S. N. (2014). General smooth additive modeling. In T. Kneib, F. Sobotka, J. Fahrenholz, and H. Irmer (Eds.), Proceedings of the 29th International Workshop on Statistical Modelling (IWSM), Volume 1, pp. 55–60. Georg-August-Universität Göttingen.
  • [31] Wood, S. N., N. Pya, and B. Säfken (2015). Smoothing parameter and model selection for general smooth models. Technical report., http://arxiv.org/abs/1511.03864.
  • [32] Zhu, H., P. J. Brown, and J. S. Morris (2011). Robust, adaptive functional regression in functional mixed model framework., Journal of the American Statistical Association 106(495), 1167–1179.

Supplemental materials

  • Code to reproduce results for Section 4. Contains code for the simulations and replication studies in Section 4 as well an R Markdown document with graphs of typical results for the Binomial data in Section 4.1.