Robust And Efficient Factorial Designs Under Baseline Parametrization

Factorial designs are widely used in various experiments to study whether the responses are significantly affected by factors.A full factorial design allows us to estimate all the main effects and interactions of factors.However,as the number of factors increases,it may be too expensive to run an experiment with a full factorial design.Then a fractional factorial design which has a smaller run size is considered.A key issue is how to choose the “optimal” fraction from the full factorial design.Most of the existing work on fractional factorial designs focuses on orthogonal parametrization,and some optimal criteria have been proposed by researchers to select designs.However,a non-orthogonal baseline parametrization can arise quite naturally in many situations.Due to the nonorthogonality of baseline parametrization,the criteria under orthogonal parametrization are no longer applicable to baseline parametrization.Under baseline parametrization,there is still a lack of research on finding optimal fractional factorial designs to estimate parameters as accurately as possible.Based on Mukerjee and Huda(2016),the thesis tries to obtain an efficient exact design for a given model under the D-optimal criterion,and get a robust design when the model is misspecified.Under baseline parametrization,to estimate the parameters as accurately as possible for a given linear model including all main effects and some specified interactions,the thesis constructs efficient factorial designs based on the D-optimal criterion,which minimized the generalized variance of the parameter estimates.The work of Mukerjee and Huda(2016)was based on the A-optimal criterion,which minimizes the average variance of the parameter estimates.The D-optimal criterion has great difference from the A-optimal criterion.It is intricate to find the exact design directly,so we use a more circuitous way: find the approximate design,and then use discretization to get the exact design.First of all,the thesis obtains optimal approximate designs through the approximate theory for a given model.Then,transforms them into the exact design via rounding off and discretization procedures.However,it is well known that the D-optimal criterion is a model-based criterion.Since the model is pre-specified based on prior information,prediction bias will occur if the model is misspecified.At this time,it is not rigorous enough to only consider the variance of parameter estimates,and the mean square error needs to be considered.Accordingly,the thesis applies the D-optimal minimax criterion to overcome this difficulty.Through examples of two-level,three-level,and mixed-level factor designs and simulations,the thesis shows that the obtained designs have high efficiency.