Testing for Lack of Fit in Blocked, Split-Plot, and Other Multi-Stratum Designs
Open accessAbstract
Textbooks on reponse surface methodology emphasize the importance of lack-of-fit tests when fitting response surface models and stress that, to be able to test for lack of fit, designed experiments should have replication and allow for pure-error estimation. In this paper, we show how to obtain pure-error estimates and how to carry out a lack-of-fit test when the experiment is not completely randomized, but a blocked experiment, a split-plot experiment, or any other multi-stratum experiment. Our approach to calculating pure-error estimates is based on residual maximum likelihood (REML) estimation of the variance components in a full treatment model (sometimes also referred to as a cell means model). It generalizes the approach suggested by Vining et al. (2005) in the sense that it works for a broader set of designs and for replicates other than center-point replicates. Our lack-of-fit test also generalizes the test proposed by Khuri (1992) for data from blocked experiments because it exploits replicates other than center-point replicates and works for split-plot and other multi-stratum designs as well. We provide analytical expressions for the test statistic and the corresponding degrees of freedom and demonstrate how to perform the lack-of-fit test in the SAS procedure MIXED. We re-analyze several published data sets and discover a few instances in which the usual response surface model exhibits significant lack of fit.
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The resulting designs enable unbiased and efficient inferences and provide efficient pure-error estimates of the variance components required by the lack-of-fit test proposed by Goos and Gilmour (2017).
“The resulting designs enable unbiased and ecient inferences and provide ecient pure-error estimates of the variance components required by the lackof-t test proposed by Goos and Gilmour (2017) for data from blocked and split-plot experiments.”
From Optimal Blocked and Split-Plot Designs Ensuring Precise Pure-Error Estimation of the Variance Components (2020) · cites this paper as Goos and Gilmour (2017)
The full treatment model does not exhibit any lack of fit, and the variance component estimates resulting from it are pure-error estimates.
“Therefore, the model does not exhibit any lack of t, and the variance component estimates resulting from it are pure-error estimates.”
From Optimal Blocked and Split-Plot Designs Ensuring Precise Pure-Error Estimation of the Variance Components (2020) · cites this paper as Goos and Gilmour (2017)
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D-optimal design of split-split-plot experiments
Jones, B., & Goos, P. (2009) · Biometrika · doi:10.1093/biomet/asn070 · Cited by 47 · cited 4× · see citing works
“For lack of published data sets from industrial split-split-plot response surface experiments, we simulated data for a design presented by Trinca and Gilmour (2017) as an alternative to a D-optimal split-split-plot design computed by Jones and Goos (2009).”
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Research works printed but not linked: 8
- Almimi, A. A., M. Kulahci, and D. C. Montgomery (2009). Checking the adequacy of fit of models from split-plot designs. Journal of Quality Technology 41, 272-284.
- Box, G. E. P. and N. R. Draper (2007). Response Surfaces, Mixtures and Ridge Analyses (2nd ed.). New York: Wiley. Commission internationale de l'éclairage (1986). Colorimetry, CIE publication 15.2. (2nd ed.). Vienna: Bureau Central CIE.
- Gilmour, S. G. and L. A. Trinca (2006). Response surface experiments on processes with high variation. In A. I. Khuri (Ed.), Response Surface Methodology and Related Topics, pp. 19-46. Singapore: World Scientific.
- McCulloch, C. E., S. R. Searle, and J. M. Neuhaus (2008). Generalized, Linear, and Mixed Models (2nd ed.). New York: Wiley.
- Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. i. block structure and the null analysis of variance. Proceedings of the Royal Society of London, Series A 283, 147-162.
- Parker, P., S. M. Kowalski, and G. G. Vining (2007). Unbalanced and minimal point equivalent estimation second-order split-plot designs. Journal of Quality Technology 39, 376-388.
- Vining, G. G. and S. M. Kowalski (2008). Exact inference for response surface designs within a split-plot structure. Journal of Quality Technology 40, 394-406.
- Vining, G. G., S. M. Kowalski, and D. C. Montgomery (2005). Response surface designs within a split-plot structure. Journal of Quality Technology 37, 115-128.
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Mylona, K., Gilmour, S. G., & Goos, P. (2020) · Technometrics · doi:10.1080/00401706.2019.1595153 · Cited by 13 · see citing works
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- Authors
- Goos, P., & Gilmour, S. G.
- Affiliations
- University of Antwerp (BE)
- King's College London (GB)
- Venue
- Journal of Quality Technology
- Publisher
- Informa UK Limited
- Published
- 2017-10
- Volume
- 49
- Issue
- 4
- Pages
- 320-336
- ISSN
- 0022-4065 (ISSN-L)
- 2575-6230
- OpenAlex
- W2608566179
- Cited by
- 9
- References
- 1
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 3 · 2022: 1 · 2019: 2 · 2018: 1
- FWCI
- 0.73
- Type
- journal article
- Language
- en
- Open access
- green
- kclpure.kcl.ac.uk
- Licence
- other-oa
- Locations
- Journal of Quality Technology (publishedVersion)
- Research Portal (King's College London) (submittedVersion, other-oa)
- Lirias (submittedVersion, other-oa)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- lack-of-fit test, response surface methodology, restricted maximum likelihood, variance components, split-plot design, degrees of freedom
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