Tensor Cubic Smoothing Splines in Designed Experiments Requiring Residual Modelling

  • Arūnas P. Verbyla
  • Joanne De Faveri
  • John D. Wilkie
  • Tom Lewis


Modelling response surfaces using tensor cubic smoothing splines is presented for three designed experiments. The aim is to show how the analyses can be carried out using the asreml software in the R environment, and details of the analyses including the code to do so are presented in a tutorial style. The experiments were all run over time and involve an explanatory quantitative treatment variable; one experiment is a field trial which has a spatial component and involves an additional treatment. Thus, in addition to the response surface for the time by explanatory variable, modelling of temporal and, for the third experiment, of temporal and spatial effects at the residual level is required. A linear mixed model is used for analysis, and a mixed model representation of the tensor cubic smoothing spline is described and seamlessly incorporated in the full linear mixed model. The analyses show the flexibility and capacity of asreml for complex modelling.

Supplementary materials accompanying this paper appear online.


asreml Cubic smoothing spline Mixed models Spatial variation Temporal variation Tensor spline 



The full woodlot data were kindly provided by Rob Stevens. The suggestions and corrections by the Editor, Associate Editor and Reviewers improved the paper considerably and we thank them all.

Supplementary material (203 kb)
Supplementary material 1 (zip 202 KB)


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Copyright information

© International Biometric Society 2018

Authors and Affiliations

  1. 1.Data61CSIROAthertonAustralia
  2. 2.Queensland Department of Agriculture and FisheriesMareebaAustralia
  3. 3.Queensland Department of Agriculture and FisheriesWollongbarAustralia
  4. 4.Queensland Department of Agriculture and FisheriesUniversity of the Sunshine CoastSippy DownsAustralia

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