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.
