By Anthony Atkinson, Barbara Bogacka, Anatoly A. Zhigljavsky

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**Sample text**

2). 3 gives all the ×Ø degree 1, 2 and 3 is the same as is expected from Lemma 2, namely the constant term plus two second degree terms and three third degree terms. Each initial ordering potentially leads to different sets of « . Now, using the method, write the new homogenised orthogonal polynomial model as ´¼ ½ ¾ ½ ¾µ ´ µ «¾Ä « « ´Üµ ¼ ·È « ½ « « ·È « ¾ « « · The claim is that interpretation of the analysis for mixture experiments is considerably clarified. Here is a summary of features: (i) An orthogonal analysis of « together with appropriate plots; (ii) An analysis of variance based on degree; (iii) A clear definition of an “effect”.

In this note, we present optimal and efficient designs for paired comparisons in a response surface setting where the alternatives are modeled by a linear, quadratic or cubic polynomial. These components can be readily used as bricks for constructing efficient designs for more than one active effect. 1. GENERAL SETTING As in the monograph by van Berkum (1987) we confine ourselves to the solution of the design problem for the approximating linear model. To be more specific we consider the general linear model ´Øµ ¬¼ · ´Øµ ¬ with an explicit constant term ¬¼ for modeling the value of an alternative Ø ¾ Ì .

In the Ø ordering È È È È « Ø ¬ if and only if ½« ½ ¬ or ½« ½ ¬ and the ordering depends right-most non-zero entry of « ¬ is negative. Thus the Ø on the ordering in which the indeterminates are arranged. We call this ordering initial ordering. For example, if the initial ordering of the three factors Ð × is Ð × then the ordered list of monomial terms will be: ½ Ð × Ð¾ Ð× ×¾ Ð × ¾ Ð¿ If, instead, the variables are initially ordered so that ½ Ð × ¾ Ð Ð¾ × Ð× ×¾ ¿ Ð¾ × Ð Ð¾ × then we obtain: ¾ Ð ¾ × Notice that the Ø term-ordering first takes into account the total degree of monomials and then orders according to the initial ordering, giving priority to the monomials with highest power of the leftmost variables.