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 lackof-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.