D-Optimal Designs with Interaction Coverage

Covering arrays detect interactions among factors by ensuring that each combination of levels for at most t factors is tested in at least one run (the strength t is the maximum number of interacting fac20 tors examined). D-optimal designs ensure orthogonality rather than just coverage. Hence D-optimal designs lend themselves to measurement of interactions, while covering arrays may alias some of the interactions of strength at most t. This loss of estimation capability leads to a substantial difference in the number of runs, with D-optimal designs for strength t requiring far more runs that does a covering array of strength t. Despite their differences, both D-optimal designs and covering arrays attempt to disperse experimental test points throughout the design space as widely as possible, but use different criteria to do so. When a D-optimal design for a main effect, two-factor interaction, and three-factor interaction (ME+2FI+ 3FI) model has too many runs for practical experimentation, a D-optimal design for an ME+ 2FI model may be used despite the fact that significant 3-way interactions arise. However, interactions of strength three may then fail to be covered in any run. Instead a covering array could be employed. Because a covering array of strength three may have fewer runs than a D-optimal design for the ME+2FI model, hybrid arrays are developed that combine D-optimality and coverage to realize the benefits of both. With such a hybrid array, statistical estimation takes into account all three-way interactions and can produce better results. This paper examines methods to construct hybrid designs and makes comparisons to pure D-optimal designs. Such designs can be created without sacrificing too much D-efficiency.