Covering arrays and optimal designs

Covering arrays are combinatorial objects that have been used in software testing for determination of interactions. The use of these arrays provides a qualitative look at how factors interact with each other, and is intended to provide detection of interactions but not a numerical quantification of such. Optimal designs have statistical analysis capabilities and have been in use for more than 30 years in statistics and industrial engineering. They provide a way of analyzing experiments where common factorial and fractional factorials fall short. Such designs do not provide reduced experimentation run sets when the factors are categorical in nature and have more than two levels for each factor. These designs in turn can be analyzed after the experiment is run to determine quantitative relationships between factors and levels for the model being estimated. In this dissertation we explore the capabilities of covering arrays, not just to conduct interaction discovery, but also within the context of design and analysis of statistical experiments. Covering arrays have suffered limited use in statistical analysis because they can lack full rank and evenness of coverage. Metrics are developed that allow the conversion of these arrays into designs that have full rank and even coverage. The resulting arrays are competitive with D-optimal designs when approximating full factorial data, the ultimate goal. Three metrics for the comparison of covering arrays to D-optimal designs are introduced; in exhaustive comparisons as well as comparisons using commercial generators, covering arrays are competitive with D-optimal. This provides a statistical basis for the use of covering arrays in the measurement of interactions, not just their detection. Covering arrays and D-optimal designs exhibit many important similarities, but covering arrays offer some useful properties that D-optimal designs do not. Hence hybrid designs are defined that are capable of providing interaction coverage at a level one deeper than the model being estimated without any additional runs over that required for a D-optimal design. Lastly, designs with fewer runs than required for D-optimal designs are explored for the case when an interaction is considered negligible using process knowledge. Methods are shown to target the removal of runs while maintaining reasonable approximation capabilities using properties of covering arrays.