Abstract
The AVAS (Additivity And Variance Stabilization) algorithm of Tibshirani provides a non-parametric transformation of the response in a linear model to approximately constant variance. It is thus a generalization of the much used Box-Cox transformation. However, AVAS is not robust. Outliers can have a major effect on the estimated transformations both of the response and of the transformed explanatory variables in the Generalized Additive Model (GAM). We describe and illustrate robust methods for the non-parametric transformation of the response and for estimation of the terms in the model and report the results of a simulation study comparing our robust procedure with AVAS. We illustrate the efficacy of our procedure through a simulation study and the analysis of real data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atkinson, A. C., Riani, M., & Cerioli, A. (2010). The forward search: theory and data analysis (with discussion). Journal of the Korean Statistical Society, 39, 117–134. https://doi.org/10.1016/j.jkss.2010.02.007
Atkinson, A. C., Riani, M., & Corbellini, A. (2020). The analysis of transformations for profit-and-loss data. Applied Statistics, 69, 251–275. https://doi.org/10.1111/rssc.12389
Barlow, R. E., Bartholomew, D. J., Bremner, J. M., & Brunk, H. D. (1972). Statistical inference under order restrictions. Chichester: Wiley.
Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations (with discussion). Journal of the Royal Statistical Society, Series B, 26, 211–252.
Box, G. E. P., & Tidwell, P. W. (1962). Transformations of the independent variables. Technometrics, 4, 531–550.
Breiman, L. (1988). Comment on “Monotone regression splines in action” (Ramsey, 1988). Statistical Science, 3, 442–445.
Buja, A., Hastie, T., & Tibshirani, R. (1989). Linear smoothers and additive models. Annals of Statistics, 17, 453–510.
Friedman, J., & Stuetzle, W. (1982). Smoothing of scatterplots. Technical report, Department of Statistics, Stanford University, Technical Report ORION 003.
Hampel, F. R. (1975). Beyond location parameters: robust concepts and methods. Bulletin of the International Statistical Institute, 46, 375–382.
Hastie, T., & Tibshirani, R. (1986). Generalized additive models. Statistical Science, 1, 297–318.
Hastie, T. J., & Tibshirani, R. J. (1990). Generalized additive models. London: Chapman and Hall.
Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 52, 163–172.
Riani, M., Atkinson, A. C., & Cerioli, A. (2009). Finding an unknown number of multivariate outliers. Journal of the Royal Statistical Society, Series B, 71, 447–466.
Riani, M., Atkinson, A. C., & Corbellini, A. (2022). Automatic robust Box-Cox and extended Yeo-Johnson transformations in regression. Statistical Methods and Applications. https://doi.org/10.1007/s10260-022-00640-7.
Riani, M., Atkinson, A. C., & Corbellini, A. (2023). Robust transformations for multiple regression via additivity and variance stabilization. Journal of Computational and Graphical Statistics. https://doi.org/10.1080/10618600.2023.2205447.
Rousseeuw, P. J. (1984). Least median of squares regression. Journal of the American Statistical Association, 79, 871–880.
Tibshirani, R. (1988). Estimating transformations for regression via additivity and variance stabilization. Journal of the American Statistical Association, 83, 394–405.
Torti, F., Corbellini, A., & Atkinson, A. C. (2021). fsdaSAS: A package for robust regression for very large datasets including the Batch Forward Search. Stats, 4, 327–347.
Acknowledgements
We are very grateful to the editors and referees, whose comments greatly helped us to clarify the presentation of our work. Our research has benefited from the High Performance Computing (HPC) facility of the University of Parma. We acknowledge financial support from the University of Parma project “Robust statistical methods for the detection of frauds and anomalies in complex and heterogeneous data,” and the Project ECS00000033 “Ecosystem for Sustainable Transition in Emilia-Romagna”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Riani, M., Atkinson, A.C., Corbellini, A. (2023). Robust Response Transformations for Generalized Additive Models via Additivity and Variance Stabilization. In: Grilli, L., Lupparelli, M., Rampichini, C., Rocco, E., Vichi, M. (eds) Statistical Models and Methods for Data Science. CLADAG 2021. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham. https://doi.org/10.1007/978-3-031-30164-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-031-30164-3_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-30163-6
Online ISBN: 978-3-031-30164-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)