RESPONSE-SURFACE MODELS WITH RANDOM BLOCK EFFECTS

In many experimental situations, a response surface design is divided into several blocks to control an extraneous source of variation. The traditional approach in most response surface applications is to treat the block effect as fixed in the assumed model. There are, however, situations in which it is more appropriate to consider the block effect as random. This article is concerned with inference about a response surface model in the presence of a random block effect. Since this model also contains fixed polynomial effects, it is considered to be a mixed-effects model. The main emphasis of the proposed analysis is on estimation and testing of the fixed effects. A two-stage mixed-model procedure is developed for this purpose. The variance components due to the random block effect and the experimental error are first estimated and then used to obtain the generalized least squares estimator of the fixed effects. This procedure produces the so-called Yates combined intra- and inter-block estimator. By contrast, the Yates intra-block estimator is the one obtained when the block effect is treated as fixed. In particular, if the response surface design blocks orthogonally, then the two estimators are shown to be identical. An experiment on bonding galvanized steel bars is used to motivate the problem and illustrate the results.