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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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“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
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“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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Citations
Cites
Random-Effects Models for Longitudinal Data
Laird, N. M., & Ware, J. H. (1982) · Biometrics · doi:10.2307/2529876 · Cited by 8,923
Brown, R. A., Hastie, T. J., & Tibshirani, R. J. (1991) · Biometrics · doi:10.2307/2532174 · Cited by 8,263
Everitt, B. S. (2001) · Psychology Press eBooks · doi:10.4324/9781410605542-11 · Cited by 6,877
Approximate Inference in Generalized Linear Mixed Models
Breslow, N. E., & Clayton, D. G. (1993) · Journal of the American Statistical Association · doi:10.1080/01621459.1993.10594284 · Cited by 4,662
Small sample inference for fixed effects from restricted maximum likelihood
Kenward, M. G., & Roger, J. H. (1997) · Biometrics · doi:10.2307/2533558 · Cited by 4,552
Self-citationFlexible smoothing with B-splines and penalties
Eilers, P., & Marx, B. (1996) · Statistical Science · doi:10.1214/ss/1038425655 · Cited by 4,159
Recovery of Inter-Block Information when Block Sizes are Unequal
Patterson, H. D., & Thompson, R. (1971) · Biometrika · doi:10.1093/biomet/58.3.545 · Cited by 4,073
Self, S., & Liang, K.-Y. (1987) · Journal of the American Statistical Association · doi:10.1080/01621459.1987.10478472 · Cited by 2,861
Nonparametric Regression and Generalized Linear Models
Green, P. J., & Silverman, B. W. (1994) · Springer eBooks · doi:10.1007/978-1-4899-4473-3 · Cited by 1,935
Approximate Inference in Generalized Linear Mixed Models
(1993) · doi:10.2307/2290687 · Cited by 1,800
Nonlinear Mixed Effects Models for Repeated Measures Data
Lindstrom, M. J., & Bates, D. M. (1990) · Biometrics · doi:10.2307/2532087 · Cited by 1,793
That BLUP is a Good Thing: The Estimation of Random Effects
Robinson, G. K. (1991) · Statistical Science · doi:10.1214/ss/1177011926 · Cited by 1,694
Genstat 5 Release 3 Reference Manual.
Roger, J., & Payne, R. W. (1994) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.2307/2983543 · Cited by 1,560
Markov Chain Monte Carlo in Practice.
Kass, R. E., Gilks, W. R., Richardson, S. T., & Spiegelhalter, D. J. (1997) · Journal of the American Statistical Association · doi:10.2307/2965438 · Cited by 1,380
Estimation in Generalized Linear Models with Random Effects
Schall, R. (1991) · Biometrika · doi:10.1093/biomet/78.4.719 · Cited by 1,230
Nonlinear Models for Repeated Measurement Data
Young, D. A., Davidian, M., & Giltinan, D. M. (1997) · Journal of the American Statistical Association · doi:10.2307/2965730 · Cited by 1,226
Analysis of Longitudinal Data.
Have, T. R. T., Diggle, P. J., Liang, K., & Zeger, S. L. (1995) · Journal of the American Statistical Association · doi:10.2307/2291352 · Cited by 1,171
Linear Smoothers and Additive Models
Buja, A., Hastie, T. J., & Tibshirani, R. J. (1989) · The Annals of Statistics · doi:10.1214/aos/1176347115 · Cited by 998
Variance components testing in the longitudinal mixed effects model
Stram, D. O., & Lee, J. W. (1994) · Biometrics · doi:10.2307/2533455 · Cited by 992
Estimating Regression Models with Multiplicative Heteroscedasticity
Harvey, A. C. (1976) · Econometrica · doi:10.2307/1913974 · Cited by 966
Hierarchical Generalized Linear Models
Lee, Y., & Nelder, J. A. (1996) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1996.tb02105.x · Cited by 958
A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines
Kimeldorf, G., & Wahba, G. (1970) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177697089 · Cited by 953
Gilmour, A. R., Thompson, R., & Cullis, B. R. (1995) · Biometrics · doi:10.2307/2533274 · Cited by 915
Self-citationSymbolic Description of Factorial Models for Analysis of Variance
Wilkinson, G. N., & Rogers, C. E. (1973) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2346786 · Cited by 906
Markov chain Monte Carlo method and its application
Brooks, S. (1998) · Journal of the Royal Statistical Society: Series D (The Statistician) · doi:10.1111/1467-9884.00117 · Cited by 843
Nonlinear Models for Repeated Measurement Data
Timm, N. H., Davidian, M., & Giltinan, D. M. (1996) · Technometrics · doi:10.2307/1270420 · Cited by 819
On the Distribution of the Likelihood Ratio
Chernoff, H. (1954) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177728725 · Cited by 808
Accounting for natural and extraneous variation in the analysis of field experiments
Gilmour, A. R., Cullis, B. R., Verbyla, A. P., & Verbyla, A. P. (1997) · Journal of Agricultural, Biological, and Environmental Statistics · doi:10.2307/1400446 · Cited by 752
Self-citationSpline models for observational data
de Boor, C. (1991) · Journal of Approximation Theory · doi:10.1016/0021-9045(91)90041-8 · Cited by 729
Bayesian “Confidence Intervals” for the Cross-Validated Smoothing Spline
Wahba, G. (1983) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1983.tb01239.x · Cited by 638
Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression
Wahba, G. (1978) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1978.tb01050.x · Cited by 560
Wahba, G. (1985) · The Annals of Statistics · doi:10.1214/aos/1176349743 · Cited by 524
Nonparametric Regression and Generalized Linear Models
Green, P. J., & Silverman, B. W. (1993) · Chapman & Hall/CRC eBooks · doi:10.1201/b15710 · Cited by 522
Recovery of Inter-Block Information when Block Sizes are Unequal
(1971) · doi:10.2307/2334389 · Cited by 520
Analysis of Growth and Dose Response Curves
Grizzle, J. E., & Allen, D. M. (1969) · Biometrics · doi:10.2307/2528795 · Cited by 505
An Approach to the Analysis of Repeated Measurements
Diggle, P. J. (1988) · Biometrics · doi:10.2307/2531727 · Cited by 498
Zeger, S. L., & Diggle, P. J. (1994) · Biometrics · doi:10.2307/2532783 · Cited by 482
Self, S. G., & Liang, K. (1987) · Journal of the American Statistical Association · doi:10.2307/2289471 · Cited by 473
Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves
Brumback, B., & Rice, J. A. (1998) · Journal of the American Statistical Association · doi:10.1080/01621459.1998.10473755 · Cited by 438
How Far Are Automatically Chosen Regression Smoothing Parameters From Their Optimum?
Härdle, W. K., Hall, P. M., & Marron, J. S. (1988) · Journal of the American Statistical Association · doi:10.1080/01621459.1988.10478568 · Cited by 400
Analysing data with repeated observations on each experimental unit
Rowell, J. G., & Walters, D. E. (1976) · The Journal of Agricultural Science · doi:10.1017/s0021859600027763 · Cited by 326
The Quantitative Relationships Between Plant Population And Crop Yield
Willey, R. W., & Heath, S. B. (1969) · Advances in Agronomy · doi:10.1016/s0065-2113(08)60100-5 · Cited by 310
A Method for Comparing Profiles of Repeated Measurements
Kenward, M. G. (1987) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2347788 · Cited by 303
Self-citationSemiparametric Stochastic Mixed Models for Longitudinal Data
Zhang, D., Lin, X., Raz, J. A., & Sowers, M. R. (1998) · Journal of the American Statistical Association · doi:10.1080/01621459.1998.10473723 · Cited by 301
The Signal Extraction Approach to Nonlinear Regression and Spline Smoothing
Wecker, W. E., & Ansley, C. F. (1983) · Journal of the American Statistical Association · doi:10.1080/01621459.1983.10477935 · Cited by 295
Modelling Variance Heterogeneity in Normal Regression Using GLIM
Aitkin, M. A. (1987) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2347792 · Cited by 287
Penalized Likelihood for General Semi-Parametric Regression Models
Green, P. J. (1987) · International Statistical Review · doi:10.2307/1403404 · Cited by 287
Estimation in Generalized Mixed Models
McGilchrist, C. A. (1994) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1994.tb01959.x · Cited by 273
Applications of Multivariate Analysis of Variance to Repeated Measurements Experiments
Cole, J. W. L., & Grizzle, J. E. (1966) · Biometrics · doi:10.2307/2528076 · Cited by 273
Statistical Analysis of Field Experiments Using Neighbouring Plots
Besag, J., & Kempton, R. (1986) · Biometrics · doi:10.2307/2531047 · Cited by 246
Gilks, W. R., Richardson, S. T., & Spiegelhalter, D. (1995) · Chapman & Hall/CRC eBooks · doi:10.1201/b14835-25 · Cited by 215
GROWTH-RATE DETERMINATIONS IN NUTRITION STUDIES WITH THE BACON PIG, AND THEIR ANALYSIS
WISHART, J. (1938) · Biometrika · doi:10.1093/biomet/30.1-2.16 · Cited by 210
Mixed Effects Smoothing Spline Analysis of Variance
Wang, Y. (1998) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/1467-9868.00115 · Cited by 193
Maximum-likelihood estimation in non-standard conditions
Moran, P. A. P. (1971) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100050088 · Cited by 175
Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood
Welham, S. J., & Thompson, R. (1997) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/1467-9868.00092 · Cited by 147
Self-citationModelling Variance Heterogeneity: Residual Maximum Likelihood and Diagnostics
Verbyla, A. P. (1993) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1993.tb01918.x · Cited by 128
Self-citationSemi-parametric Generalized Linear Models
Green, P. J., & Yandell, B. S. (1985) · Lecture notes in statistics · doi:10.1007/978-1-4615-7070-7_6 · Cited by 125
Nonparametric Estimation of Covariance Structure in Longitudinal Data
Diggle, P. J., & Verbyla, A. P. (1998) · Biometrics · doi:10.2307/3109751 · Cited by 125
Self-citationAsymptotic Properties of Maximum Likelihood Estimators Based on Conditional Specification
Sen, P. K. (1979) · The Annals of Statistics · doi:10.1214/aos/1176344785 · Cited by 119
The Performance of Cross-Validation and Maximum Likelihood Estimators of Spline Smoothing Parameters
Kohn, R., Ansley, C. F., & Tharm, D. (1991) · Journal of the American Statistical Association · doi:10.1080/01621459.1991.10475150 · Cited by 80
Some Simple Methods for Estimating Intraindividual Variability in Nonlinear Mixed Effects Models
Davidian, M., & Giltinan, D. M. (1993) · Biometrics · doi:10.2307/2532602 · Cited by 78
The Signal Extraction Approach to Nonlinear Regression and Spline Smoothing
Wecker, W. E., & Ansley, C. F. (1983) · Journal of the American Statistical Association · doi:10.2307/2287113 · Cited by 69
Prediction of wheat yield loss due to competition by wild oats ( Avena spp.)
Martin, R. J., Cullis, B. R., & McNamara, D. (1987) · Australian Journal of Agricultural Research · doi:10.1071/ar9870487 · Cited by 67
Self-citationEstimating Regression Models with Multiplicative Heteroscedasticity
Harvey, A. C. (1977) · Econometrica · doi:10.2307/1914080 · Cited by 66
Linear Models for Field Trials, Smoothing and Cross-Validation
Green, P. J. (1985) · Biometrika · doi:10.1093/biomet/72.3.527 · Cited by 59
Mead, R. (1970) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2346843 · Cited by 48
Genotype by Environment Variance Heterogeneity in a Two-Stage Analysis
Frensham, A. B., Cullis, B. R., & Verbyla, A. P. (1997) · Biometrics · doi:10.2307/2533504 · Cited by 45
Self-citationA Model for the Analysis of Growth Data from Designed Experiments
Cullis, B. R., & McGilchrist, C. A. (1990) · Biometrics · doi:10.2307/2531636 · Cited by 40
Self-citationLead discussion on 'Varying coefficient models' by Hastie & Tibshirani
Green, P. J. (1993) · Cited by 40
Cross-over designs in bioassay
Finney, D. J. (1956) · Proceedings of the Royal Society B Biological Sciences · doi:10.1098/rspb.1956.0017 · Cited by 35
Generalized linear mixed models
Gilks, W. R., Richardson, S. T., & Spiegelhalter, D. (1995) · Chapman & Hall/CRC eBooks · doi:10.1201/b14835-21 · Cited by 30
Growth-Rate Determinations in Nutrition Studies with the Bacon Pig, and Their Analysis
Wishart, J. (1938) · Biometrika · doi:10.2307/2332221 · Cited by 26
The Nonparametric Estimation of Branching Curves
Silverman, B. W., & Wood, J. T. (1987) · Journal of the American Statistical Association · doi:10.1080/01621459.1987.10478465 · Cited by 22
The Performance of Cross-Validation and Maximum Likelihood Estimators of Spline Smoothing Parameters
Kohn, R., Ansley, C. F., & Tharm, D. (1991) · Journal of the American Statistical Association · doi:10.2307/2290523 · Cited by 21
The Analysis of Multistratum and Spatially Correlated Repeated Measures Data
Verbyla, A. P., & Cullis, B. R. (1992) · Biometrics · doi:10.2307/2532695 · Cited by 15
Self-citationSmoothing biological data with splines
White, I., Cullis, B. R., Gilmour, A. R., & Thompson, R. D. (1998) · Rothamsted Repository (Rothamsted Repository) · Cited by 8
Self-citationThe Nonparametric Estimation of Branching Curves
Silverman, B. W., & Wood, J. T. (1987) · Journal of the American Statistical Association · doi:10.2307/2289464 · Cited by 6
Choosing an Appropriate Covariance Function in Bayesian Smoothing
Upsdell, M. P. (1996) · doi:10.1093/oso/9780198523567.003.0055 · Cited by 1
Flagged first, then by citations
Research works printed but not linked: 6
- [19]19. Gilmour, A. R., Thompson, R., Cullis, B. R. and Welham, S. J. (1996) ASREML. Biometric Bulletin. New South Wales Agriculture, Wagga Wagga.
- [28]28. Kenward, M. G. and Welham, S. J. (1995) Use of splines in random coecient regression models for the analysis of repeated measurements. In Proc. Symp. Statistical Software 95, pp. 95±112. Utrecht: University of Utrecht.
- [42]42. Searle, S. R., Casella, G. and McCulloch, C. E. (1992) Variance Components. New York: Wiley.
- [60]60. Box, G. E. P. and Draper, N. R. (1987) Empirical Modelling and Response Surfaces. New York: Wiley.
- [65]65. Diggle, P. J., Liang, K.-Y. and Zeger, S. L. (1994) Analysis of Longitudinal Data. Oxford: Clarendon.
- [66]66. Durban, M. (1999) Modelling spatial trends and local competition eects using semiparametric additive models. PhD Thesis. Heriot-Watt University, Edinburgh.
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Hierarchical generalized additive models in ecology: an introduction with mgcv
Pedersen, E. J., Miller, D. L., Simpson, G. L., & Ross, N. (2019) · PeerJ · doi:10.7717/peerj.6876 · Cited by 1,225
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Lin, X., & Zhang, D. (1999) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/1467-9868.00183 · Cited by 653
Low‐Rank Scale‐Invariant Tensor Product Smooths for Generalized Additive Mixed Models
Wood, S. N. (2006) · Biometrics · doi:10.1111/j.1541-0420.2006.00574.x · Cited by 613
Gilmour, A. R., Gogel, B. J., Cullis, B. R., Welham, S. J., & Thompson, R. (2002) · Rothamsted Repository (Rothamsted Repository) · Cited by 564
Self-citation
Metadata
- 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 › Agronomy and Crop Science › 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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