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Testing for lack of fit in blocked and split-plot response surface designs

Testing for lack of fit in blocked and split-plot response surface designs
Testing for lack of fit in blocked and split-plot response surface designs
Textbooks on response 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. It generalizes the one 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.
kenward-roger degrees of freedom, multi-stratum design, replication, residual, maximum likelihood (reml), split-split-plot design, treatment model
1-19
Gilmour, Steven G.
984dbefa-893b-444d-9aa2-5953cd1c8b03
Gilmour, Steven G.
984dbefa-893b-444d-9aa2-5953cd1c8b03

Gilmour, Steven G. (2012) Testing for lack of fit in blocked and split-plot response surface designs. Pre-print, 1-19. (Submitted)

Record type: Article

Abstract

Textbooks on response 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. It generalizes the one 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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More information

Submitted date: 21 June 2012
e-pub ahead of print date: 8 August 2012
Keywords: kenward-roger degrees of freedom, multi-stratum design, replication, residual, maximum likelihood (reml), split-split-plot design, treatment model
Organisations: Social Sciences

Identifiers

Local EPrints ID: 341926
URI: https://eprints.soton.ac.uk/id/eprint/341926
PURE UUID: ae10ab34-9ceb-4045-bb8f-e2b6cc0098e1

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Date deposited: 08 Aug 2012 11:05
Last modified: 18 Jul 2017 05:31

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