The analysis of designed experiments and longitudinal data by using smoothing splines
Abstract
SUMMARY: In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatment effects. Modelling treatment effects can be complex in the presence of other sources of variation. Three examples are presented to illustrate an approach to analysis in such cases. The first example is a longitudinal experiment on the growth of cows under a factorial treatment structure where serial correlation and variance heterogeneity complicate the analysis. The second example involves the calibration of optical density and the concentration of a protein DNase in the presence of sampling variation and variance heterogeneity. The final example is a multienvironment agricultural field experiment in which a yield–seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity between environments and variation between varieties all need to be incorporated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with fixed and random effects, random coefficients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and flexible empirical model building in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitioned into smooth and non-smooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.
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The natural cubic smoothing spline can be formulated within a LMM framework.
“This allows the natural cubic smoothing spline to be formulated within a LMM framework (Verbyla et al., 1999).”
From Incorporating environmental covariates to explore genotype × environment × management (G × E × M) interactions: A one-stage predictive model (2023) · cites this paper as Verbyla et al., 1999
Incorporation of smoothing splines enables EC model terms to be partitioned into linear and non-linear components, allowing significance testing of these components independently.
“The incorporation of smoothing splines (Verbyla et al., 1999) also enables the EC model terms to be partitioned into linear and non-linear components (De Faveri et al., 2022), allowing for significance testing of these components to be performed independently (Tables 5 and 4 respectively).”
From Incorporating environmental covariates to explore genotype × environment × management (G × E × M) interactions: A one-stage predictive model (2023) · cites this paper as Verbyla et al., 1999
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- Authors
- Verbyla, A. P., Cullis, B. R., Kenward, M. G., & Welham, S. J.
- Affiliations
- University of Adelaide and South Australian Research and Development Institute , Glen Osmond , Australia
- New South Wales Agriculture , Wagga Wagga , Australia
- University of Kent (GB)
- Rothamsted Research (GB)
- Venue
- Journal of the Royal Statistical Society Series C: Applied Statistics
- Publisher
- Oxford University Press (OUP)
- Published
- 1999-09-01
- Volume
- 48
- Issue
- 3
- Pages
- 269-311
- ISSN
- 0035-9254 (ISSN-L)
- 1467-9876
- OpenAlex
- W2153287720
- Cited by
- 570
- References
- 73
- Citations by year
- 2026: 3 · 2025: 2 · 2024: 6 · 2023: 11 · 2022: 13 · 2021: 12 · 2020: 23 · 2019: 22 · 2018: 11 · 2017: 20 · 2016: 24 · 2015: 29 · 2014: 37 · 2013: 29 · 2012: 25
- FWCI
- 26.38
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model (vor)
- Locations
- Journal of the Royal Statistical Society Series C (Applied Statistics) (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Kent Academic Repository (University of Kent) (submittedVersion)
- Journal of the Royal Statistical Society Series A (Statistics in Society)
- Funders
- Grains Research and Development Corporation
- University of Adelaide
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
- Life Sciences › Agricultural and Biological Sciences › Ecology, Evolution, Behavior and Systematics › Agriculture, Soil, Plant Science
- Keywords
- smoothing splines, longitudinal data, designed experiments, linear mixed models, random coefficients, heteroscedasticity, serial correlation, cubic spline smoothing, seeding rate, spatial variability, fixed and random effects
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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