The design of mixture experiments in the presence of covariates

Over the past three decades, the design and analysis of mixture experiments has been an active area of research, often driven by industrial applications. However, the construction of block designs for mixture experiments and trend-free orderings of the mixtures are problems that have been largely ignored until recently. These two problems form the principal subjects of this dissertation after presenting some key concepts in the design and analysis of mixture experiments;Block designs are constructed using combinatorial structures called symbolic and integral mixture mates of strength t. Certain pairs of Latin squares are a special case of symbolic mixture mates. One flexible method of constructing integral mixture mates of strength t uses the theory of trade-off for m-ary designs. In addition to mixture mates, block designs may be constructed via other methods. When the region of interest is a constrained subregion of the simplex, confounding in fractional factorial designs or asymmetrical orthogonal arrays may be used to produce orthogonal block designs. Methods for constructing non-orthogonal block designs utilizing factorial designs or orthogonal arrays in another manner are given. Finally, we formulate algorithms that allocate a given set of mixtures to blocks in such a way that an objective function is maximized;Trend-free mixture orderings allow uncorrelated estimators of mixture model parameters and deterministic trend parameters to be obtained. Deterministic trends may be induced by time effects or other lurking variables. Given a trend-free ordering of a factorial design in p - 1 factors, we illustrate how a trend-free order of mixtures can be found by transforming the p - 1 factors into p mixture variables, using one of the many transformations available, see Cornell (1991) for example. If the experimental region is a constrained subregion of the simplex, trend-free run orders are constructed using trend-free factorial designs in p - 1 factors as a template and incorporating a p-th factor by adjusting the levels of the other p - 1 components for each row. Nearly trend-free mixture orders are also found by ordering the mixtures according to an objective function