Optimal Designs For Third Degree Kronecker Model Mixture Experiments With Application In Blending Of Chemicals For Control Of Mites In Strawberries

Abstract

Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD were obtained. Although mixtures experiments are usually intended to predict the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done on I-optimal third degree Kronecker designs. I-optimal designs were also studied in order to predict the optimal response(s) for all possible formulations or ingredients in the simplex centroid. The general equivalence theorem for I-optimality was used to test optimality of different mixture formulations. UWSCD was found to perform better than WSCD in terms of slope and average prediction variance with most formulations satisfying general equivalent theorem for I-optimality. The pure blend (1,0,0,0) for Vendex (V) and four mixture pesticides ( 1⁄4, 1⁄4, 1⁄4, 1⁄4 ) for Vendex (V), Omite (O), Kelthane (K) and Dibrom (D) for UWSC designs yielded more optimal results therefore recommended for use in mites eradication in straw berries plants. R- program was used in the analysis of data. This work could also be extended to cover mixture-process experiment Kronecker model and graphical methods for evaluating mixture designs with respect to slope such as slope along Cox direction.