Optimal factorial designs with robust properties

Fractional factorial designs are used in a wide variety of disciplines as a means of studying how changes in the settings of a set of factors influence a response variable. Two important considerations in choosing a fractional factorial design are identifying which effects can be jointly estimated and how the effects not estimated influence the estimation. Orthogonal arrays with clear two-factor interactions provide a class of designs robust to nonnegligible effects. In the first part of this thesis, we introduce the concept of partially clear interactions which leads to a richer class of robust designs when specific interactions are known to be negligible a priori. We develop several methods to construct designs that allow for additional factors to be studied in comparison to designs with clear two-factor interactions. When used in conjunction with non-regular designs, the results become even more powerful as they provide additional flexibility and retain the robust properties. In some situations, the experimenter would like to study factors at more than two levels, such as when curvature has the potential to occur within the experimental region. The second part of this thesis focuses on the estimation of main effects and specified interactions for designs with more than two levels. As designs with more than two levels have additional complications, results are provided that aid in the search for efficient designs that also have robust properties. For two-level designs, the criteria of $G$ and $G_2$-aberration are based on $J$-characteristics and they provide measures of the projection properties of a design. For multi-level designs, extension to $G_2$ was previously done without the use of $J$-characteristics. The $J$-characteristics for multi-level designs are introduced in the last part of this thesis as an intuitively appealing means to measure lower-dimensional properties, which leads to more natural definitions of $G$ and $G_2$-aberration. We show how the properties of a design can be gleaned by using an analysis of variance as taught in introductory statistics courses.