OPTIMAL EXPERIMENTAL DESIGN FOR TWO VARIABLE QUADRATIC LOGISTIC MODELS

Abstract

This research work developed optimal design of experiment for two variable quadratic logistic regression models which is an extension of existing works in the literature of one variable quadratic or multiple variables linear Logistic models. Determining an optimal design for this model is complicated by the fact that the optimal design is dependent on the unknown true parameters. Methods to obtain locally D, A, V and G optimal designs for two variable quadratic logistic models are illustrated. A special case of four types of response curves namely high wide with initial parameter guess ∝=(2 0 0−0.1−0.1)T , high narrow with initial parameter guess ∝=(2 0 0−4−4)T , low wide with initial parameter guess ∝=(−2 0 0−0.1−0.1)T and low narrow with initial parameter guess ∝=(−2 0 0 −4 −4)T was considered. The cases were chosen to represent any position of parameter set in the design space. The optimal designs were established via the general equivalent theorem and surface plot of the design. Using the prediction error variance (PEV), it is shown how the number of design points varies depending on the parameter being estimated. The small sample performance of the locally optimal designs is compared to the performances of some non-optimal designs in a simulation study of ten thousand (10000) experimental runs. A grid search approach was adopted to determine the predictive capability of the model for prediction error variance. The result of the experiment had demonstrated that the PEV for the optimal designs is 0.556 and that of non-optimal designs range from 1.0 to 3.5. The research was able to determine optimal design for two variable quadratic logistic models and evaluated model parameter performance of D, A, V and G optimality criteria. The model was applied to real life data and the optimality of the design was also confirmed through the general equivalence theorem and prediction error variance at nine equally weighted support points. The values of D-optimal=1.43, V-optimal=0.42, G-optimal=0.556 and A-optimal= 1.89 conformed with those obtained in the simulation. At optimal region, the number of peaks of the surface was five which is equal to the number of the parameters in the model and the optimal region is located in the innermost contour. The relative efficiency of the D-optimal design to non-optimal designs was considered. D-optimal design was 100 percent efficient relative to other designs.