||Experiments with both quantitative and qualitative factors always complicate the selections of experimental settings and the statistical analysis for data. Response surface methodology (RSM) provides the systematic procedures such as the steepest ascent method to develop and improve the response models through the optimal settings of quantitative factors. However the sequential method lacks of exploring the direction of the maximum increase in the response among the qualitative levels. In this dissertation the optimal designs for experiments with both qualitative and quantitative factors are investigated. Focused on the second-order response surface model for quantitative factors, which is widely used in RSM as a good approximation for the true response surface, the approximate and exact D-optimal designs are proposed for the model containing the qualitative effects. On spherical design regions, the D-optimal designs have particular structures for considering the qualitative effects to be fixed or random.|
In this study, the exact D-optimal designs for a second-order response surface model on a circular design region with qualitative factors are proposed. For this model, the interactions between the quantitative and qualitative factors are assumed to be negligible. Based on this design region, an exact D-optimal design with regular polygon structure is made up according to the remainder terms of the numbers of experimental trials at each qualitative levels divided by 6. The complete proofs of exact D-optimality for models including two quantitative factors and one 2-level qualitative factor are presented as well as those for a model with only quantitative factors. When the qualitative factor has more than 2 levels, a method is proposed for constructing exact designs based on the polygonal structure with high efficiency. Furthermore, a procedure for minimizing the number of support points for the quantitative factors of exact D-optimal designs is also proposed for practical consideration. There are no more than 13 support points for the quantitative factors at an individual qualitative level.
When the effects between the quantitative and qualitative factors are taken into consideration, approximate D-optimal designs are investigated for models in which the qualitative effects interact with, respectively, the linear quantitative effects, or the linear effects and 2-factor interactions of the quantitative factors or quadratic effects of the quantitative factors. It is shown that, at each qualitative level, the corresponding D-optimal design consists of three portions as a central composite design but with different weights on the cube portion, star portion and center points. Central composite design (CCD) is widely applied in many fields to construct a second-order response surface model with quantitative factors to help to increase the precision of the estimated model. A chemical study is illustrated to show that the effects of the qualitative factor interacts with 2-factor interactions of the quantitative factors are important but absent in a second-order model including a qualitative factor treated as a coded variable.
The verification of the D-optimality for exact designs has become more and more intricate when the qualitative levels or the number of quantitative factors increase, even when the patterns of the exact optimal designs have been speculated. The efficient rounding method proposed by Pukelsheim and Rieder (1992) is a model-free approach and it generates an exact design by apportioning the number of trials on the same support points of a given design. For constructing the exact designs with high efficiencies, a modified efficient rounding method is proposed and is based on the polygonal structure of the approximate D-optimal design on a circular design region. This modification is still based on the same rounding approach by apportioning the number of trials to the concentric circles where the support points of the given design are standing on. Then a regular polygon design will be assigned on the circles by the apportionments. For illustration, the exact designs for a third-order response surface model with qualitative factors are presented as well as those for the second-order model. The results show that nearly D-optimal designs are obtained by the modified procedure and the improvement in D-efficiency is very significant.
When the factors with the levels selected randomly from a population, they are treated as with random effects. Especially for the qualitative effects caused by the experimental units that the experimenter is not interested in, one should consider the model with random block effects. In this model, the observations on the same unit are assumed to be correlated and they are uncorrelated between different units. Then the mean response surface is still considered as second-order for quantitative factors but the covariance matrix of the observations is different from the identity matrix. In the fourth part of this dissertation, the locally D-optimal designs on a circular design region are proposed for given the value of the correlations. These optimal designs with the structures based on the regular polygons are similar to the D-optimal designs for the uncorrelated model.