Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments

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 Toptimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.


Introduction
Response surface methodology (RSM) is a collection of mathematical and statistical tools or techniques that are useful for modeling and analysis of problems in which a response of interest is influence by several ingredients and the objective is to optimize this response, Montgomery (2001). Response Surface Methodology is an important subject in the statistical design and analysis of experiments. Mixture experiments are special type of RSM associated with the investigation of the m factors, assumed to influence the response only through proportions in which they are blended together. The mixture ingredients t 1 Under experimental condition m t T ∈ , the response t Y is taken to be a real-valued random variable. In a polynomial regression model the expected value ( ) t E Y is a polynomial function of t. The work done by Draper and Pukelsheim (1998) is being extended to polynomial regression model for third degree mixture model. Korir et al (2008) extended the work to third degree Kronecker model by use of equivalent theorem in calculation of weights, also Kerich et al (2014) studied the D-optimal designs for third degree Kronecker model mixture experiments with application in artificial sweetener experiment. In many applications of response surface methodology, good estimation of the derivatives of the response function is as important as estimation of the mean response. From the work of Hader and Park (1978), Huda and Al-Siha (1999) and Huda (2006), it is clear that most of the work has been done on central composite designs hence there was a need to extend the concept of slope to mixture experiments third degree Kronecker, this method was therefore used for proper identification of the ingredients ratio that leads to an optimal response. The S-polynomial is given as, and the homogeneous third-degree K-polynomial is in which the Kronecker powers for all , and k i j All observations taken in an experiment are assumed to be of equal unknown variance and uncorrelated. The moment The parameter subsystem K θ ′ of interest is a maximal parameter system in the full parameter model. The information matrix for the parameter subsystem is given by where L is the left inverse of coefficient matrix K and is defined by Thus the information matrices for K θ are linear transformation of moment matrices.
The coefficient matrix K for the four ingredients parameter subsystems of interest in (7) and (8)

Slope Designs
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 may be of greater interest, Herzberg (1967), Box and Draper (1980), Huda and Mukerjee (1984) and Huda (2006).

Optimal Values for Slope Information Matrices (SIM)
Optimal designs are experimental designs that are generated based on a particular optimality criterion and are generally optimal only for a specific statistical model. The optimality properties of designs are determined by their moment matrices, Pukelsheim (1993). The amount of information inherent to C k (M(η )) is provided by p ϕ -criteria with C k (M(η ))∈ PD(m), defined by: We obtained the optimal values for both Weighted Simplx Centroid (WSC) designs and Uniform Weighted Simplex Centroid Designs (UWSCD) for four ingredients mixture experiments. Uniformly Weighted Simplex Centroids Design (UWSCD) was observed to yield more optimal values than Weighted Simplex Centroid Designs (WSCD) at all points of the simplex centroids mixture experiments.

Conclusion
It was noted that Uniformly Weighted Simplex Centroid (UWSC) designs were more efficient than Weighted Simplex Centroid (WSC) designs. For more optimal results, the experimenter is advised to allocate weights in the mixture components uniformly. In UWSC design, the centroid point (¼, ¼, ¼, ¼) produced the most efficient results than any other point while WSC design yield the optimal results at point (0.333, 0.333, 0.333, 0) only.