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Order-of-Addition (OofA) experiments are widely used in a number of areas including chemistry, film, and food science. This type of experiments is to study the order effect when $m$ components are sequentially added into a system. Full designs require $m!$ runs, thus an efficient fraction is required. Van Nostrand (1995) is proposed the Pairwise-order (PWO) framework to study these designs. There was no further research in the area until Voelkel (2017) proposed criteria for evaluating efficient fractions, and algorithmically found "fully-efficient" or nearly fully-efficient PWO designs.
In this work we investigate the theoretically properties of OofA designs. We show that the orthogonality of PWO designs is restricted due to the transitive property of ordering relationship. We proved that no PWO design can be perfectly orthogonal. We show that any fractional PWO design cannot be "more orthogonal" than the full design in terms of $E(r^2)$. We introduce a close-form construction of fully-efficient fractions, of which the correlation structure is the same as the full design. We develop a general algorithm for finding nearly-fully efficient fractions.
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