When experimental units exhibit a smooth trend over time or in space, random allocation of treatments may no longer be appropriate. Instead, systematic run orders may have to be used to reduce or eliminate the effects of such a trend. The resulting designs are referred to as trend-free designs. We consider here, in particular, linear trend-free designs for factorial treatment structures such that estimates of main effects and two-factor interactions are trend-free. In addition to trend-freeness we incorporate a general cost structure and propose methods of constructing optimal or near-optimal full or fractional factorial designs. Building upon the generalized foldover scheme (GFS) introduced by Coster and Cheng (1988) we develop a procedure of selection of foldover vectors (SFV) which is a construction method for an appropriate generator matrix. The final optimal or near-optimal design can then be developed from this generator matrix. To achieve a reduction in the amount of work, i.e., a reduction of the large number of possible generator matrices, and to make this whole process easier to use by a practitioner, we introduce the systematic selection of foldover vectors (SSFV). This method does not always produce optimal designs but in all cases practical compromise designs. The cost structure for factorial designs can be modeled according to the number of level changes for the various factors. In general, if cost needs to be kept to a minimum, factor level changes will have to be kept at a minimum. This introduces a covariance structure for the observations from such an experiment. We consider the consequences of this covariance structure with respect to the analysis of trend-free factorial designs. We formulate an appropriate underlying mixed linear model and propose an AIC-based method using simulation studies, which leads to a useful practical linear model as compared to the theoretical model, because the theoretical model is not always feasible. Overall, we show that estimation of main effects and two-factor interactions, trend-freeness, and minimum cost cannot always be achieved simultaneously. As a consequence, compromise designs have to be considered, which satisfy requirements as much as possible and are practical at the same time. The proposed methods achieve this aim. / Ph. D.

Sequential experimentation is often employed in process optimization wherein a series of small experiments are run successively in order to determine which experimental factor levels are likely to yield a desirable response. Although there currently exists a framework for identifying optimal follow-up designs after an initial experiment has been run, the accepted methods frequently point to multiple designs leaving the practitioner to choose one arbitrarily. In this thesis, we apply preposterior analysis and Bayesian model-averaging to develop a methodology for further discriminating between optimal follow-up designs while controlling for both parameter and model uncertainty.

Two-level factorial experiments are widely used in experimental design because they are simple to construct and interpret while also being efficient. However, full factorial designs for many factors can quickly become inefficient, time consuming, or expensive and therefore fractional factorial designs are sometimes preferable since they provide information on effects of interest and can be performed in fewer experimental runs. The disadvantage of using these designs is that when using fewer experimental runs, information about effects of interest is sometimes lost. Although there are methods for selecting fractional designs so that the number of runs is minimized while the amount of information provided is maximized, sometimes the design must be augmented with a follow-up experiment to resolve ambiguities. Using a fractional factorial design augmented with an optimal follow-up design allows for many factors to be studied using only a small number of additional experimental runs, compared to the full factorial design, without a loss in the amount of information that can be gained about the effects of interest. This thesis looks at discovering regularities in the number of follow-up runs that are needed to estimate all aliased effects in the model of interest for 4-, 5-, 6-, and 7-factor resolution III and IV fractional factorial experiments. From this research it was determined that for all of the resolution IV designs, four or fewer (typically three) augmented runs would estimate all of the aliased effects in the model of interest. In comparison, all of the resolution III designs required seven or eight follow-up runs to estimate all of the aliased effects of interest. It was determined that D-optimal follow-up experiments were significantly better with respect to run size economy versus fold-over and semi-foldover designs for (i) resolution IV designs and (ii) designs with larger run sizes.