Multi-Factor Designs of First Order

The problem discussed arises when it is possible to choose in advance the N combinations of levels at which a set of quantitative factors are to be held in a set of N experiments to determine the slopes of a regression surface (assumed planar). It is shown that the minimum variance property of an ‘optimum’ design arises from the shape of the design pattern and is independent of its orientation. This fact may be utilized as follows: (1) When prior knowledge of the response surface exists the design may be rotated to reduce possible bias. (2) The design may be rotated so that systematic effects, such as polynomial time trends and block effects, are eliminated without loss of efficiency. (3) Subject to the conditions imposed by (2), the orientation of the design may be chosen at random. This has the effect of making the usual normal theory tests exact and completely independent of the distribution of the observations.