Effect of blocking on estimation of response surface

Abstract

The conditions under which experimental trials are performed may not, in general, be homogeneous. However, it may be possible to achieve a greater degree of homogeneity by carying out limited groups—or block—of trials. In this case, when fitting a response surface model, the least squares estimates of the model's parameters will generally depend on how the response surface design is blocked. The purpose of this article is to demonstrate the effects of the blocks on estimating the mean response, on the prediction variance and on the optimum of the response surface. These are all shown to be affected by the sizes of the blocks and the allocation of experimental runs to the blocks. In particular, the prediction variance increases as a result of blocking. In the special case of an orthogonally blocked design, the least squares estimates of the fitted model's parameters remain unchanged by the block effect, except possibly for the intercept. The increase in the prediction variance in this case depends only on the sizes of the blocks. A numerical example is also given.