Two Combined Alphabetic Optimality Criteria for Second Order Rotatable Designs Constructed Using Balanced Incomplete Block Design in Four Dimensions

The theory of optimal experimental designs is concerned with the construction of designs that are optimum with respect to some statistical criteria. Some of these criteria include the alphabetic optimality criteria such as; D-, A-, E-, T-, Gand Ccriterion. Compound optimality criteria are those that combine two or more alphabetic optimality criteria. Design that require optimality criteria have specific desired properties that do very well in one design and at the same time perform poorly in another design. Thus, a compound optimality criterion gives a balance to the desirability of any two or more alphabetic optimality criteria. The present paper aims to introduce CDand DTcriteria which are compound optimality criteria for second order rotatable designs constructed using Balanced Incomplete Block Designs (BIBDs) in four dimensions.


Introduction
Design experts have come to a realization that a design can perform very well in terms of a particular statistical characteristic and still perform poorly in terms of a rival characteristic. In the field of life sciences optimal designs are required in order to cut on cost of experimentation. Kussmaul [15] introduced method that allows for an efficient consideration of nonlinear constraints.
An experimenter is therefore advised to make the choice of a design to be used prior to carrying out any experiment. In statistics, Response Surface Methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and K. B. Wilson [1]. The main idea of RSM is to use a sequence of designed experiments to obtain an optimal response. Box and Wilson [1] suggest using a second-degree polynomial model to do this. They acknowledge that this model is only an approximation, but they use it because such a model is easy to estimate and apply, even when little is known about the process. Statistical approaches such as RSM can be employed to maximize the production of a special substance by optimization of operational factors. In contrast to conventional methods, the interaction among process variables can be determined by statistical techniques [2].
According to Box and Draper [3], RSM is either used to explore response surfaces or to estimate the parameters of a model. Bose and Draper [4] point out that the technique of fitting a response surface is one widely used to aid in the statistical analysis of experimental work in which the response of a product depends in some unknown factors on one or more controllable variables. A particular selection of settings or factor levels at which observations are to be taken is called a design. Designs are usually selected to satisfy some desirable criteria chosen by the experimenter.
The proper meaning of optimal depends on the situation and can include cost effective, minimum variance and minimum bias. Youdim [13] Correctly chosen D-optimum designs provide efficient experimental schemes when the aim of the investigation is to obtain precise estimates of parameters. The commonly used classical optimality criteria Rotatable Designs Constructed Using Balanced Incomplete Block Design in Four Dimensions which were introduced and widely discussed by Pukelsheim [5] includes, Determinant criterion (D-), the average variance criterion (A-), the smallest Eigen value (E-) and the trace criterion (T-). Many results on optimal designs of experiments are derived under the assumption that the statistical model is known at the design stage. Nguyen [14] proposed to use a compound optimality criterion based on the expected population Fisher information matrix in nonlinear mixed effect models. However, rarely it is known in advance which model is the most appropriate. Box and Hunter [6] introduced rotatable designs in order to explore the response surfaces. They developed second order rotatable designs through Schlaflian vectors and matrices. Mylona [12] allowed for a more powerful statistical inference than traditional optimal designs. According to Draper [7] a second order rotatable design aids the fitting of a second order surface and provides spherical information contours and a third order rotatable design aids the fitting of a third order surface. Thus, the goal of an experiment should be dual: to choose an appropriate design and the most adequate model.
A second degree response model with k factors is represented as follows where is the intercept is the linear coefficient for the i th factor is the quadratic coefficient for the i th factors is the cross product coefficient for the i th and j th factors is the level of the i th factor is the level of the i th and j th factor

Evaluation of C-Criterion in Four Dimensions
The C-criterion for the second order rotatable design in four dimensions is obtained through minimizing the variance of the linear unbiased estimator of the integral function / ( / ) . This was defined by Elfying [8] as; and $ = is the number of factors in the design for this case the factors are four.

Design Matrix
The generalized design matrix for the second order rotatable design is given by The vector in (3) is partitioned in the following order; the pure quadratic, the linear and the interaction effects. Consequently, the moment matrix is also partitioned as shown below.
The coefficients of w / in (3) are the diagonal elements of a k matrix in the parameter system of interest.

Information Matrix
Mwan and Rambaei [9] used the moment matrix for second order model to determine the information matrix for Using the elements of the inverse of the moment matrix in (7), (8) and (9) respectively (3) is obtained.
The computation for the C-criterion was portioned into three parts; the linear effects the pure quadratic and the interaction effects which were denoted as β GH . For the 64 points the parts are β , β and β with the help of matlab software.

D -Criterion for 2 nd Degree Design with Sixty Four Points
For k = 4 factors, the information matrix is given as; ( ) Thus the determinant criterion is ( ) ( ) Now from (13), we have, for the designs with k = 4, we substitute the following to (14)

T -Criterion for 2 nd Degree Design with Sixty Four Points
The trace criterion is given as; ( ) ( )

C-Criterion for 2 nd Degree Design with Sixty Four Points
Substituting 6 and 6 7 given in (15) to (7) Again by Substituting λ given in (15) to (8) Taking only the linear terms in (21) Taking only the interactions terms of vector in (21) The C-criterion for a design with 64 points is; β +β + β =36.62092 (30)

DT-Optimality
This paper combines two alphabetic optimality criteria Dand T-by using the concept that was introduced by Atkinson [10], where DT optimality criterion is a combination of Doptimality criterion for parameter estimation with the Toptimality criterion for discriminating between models. The DT-criterion provides a specified balance between model discrimination and parameter estimation.
The Generalized Determinant and Trace Criteria are given as; The DT-criterion is given by the formula;

CD-Criterion for 64 Points in Four Dimension
The CD-optimality that combines C-optimality for a model selection and D-optimality for parameter estimation which was introduced by Atkinson [11], provides a specified balance between model discrimination and parameter estimation too. The criterion to be maximized was; The designs maximizing (35) are called CD-optimality. The quantities in (2) and (31) were substituted in (35) to obtain the CD-optimality criterion.
The Determinant criterion was given in (16) and the C criterion in (30) for k= 4 using the compound formula stated in (35) gave the CD-compound optimality criterion as;

Conclusion
The study concludes by combining D-and T-optimality to get DT-(compound optimality). The design under consideration is said to be better than the alphabetic optimality design in four factors constructed by Mwan, kosgei and Rambaei [9]. The D-, T-and DT-optimality criteria are compared and there is a clear balance brought by the DTcombination. Again the analysis of the two alphabetic criteria and the compound criterion above show that design experts will prefer characteristics from the D-optimality criteria. However, when the experiment requires the utilization of the two properties the compound optimality serves the deal. This is from the result obtained above for the D-criterion the value was 0.3541807443, the T-criterion become 0.5440474286 but the combination of the two gave a value of 0.5355039691 which is in between the two criteria. Hence, a balance between parameter estimation and model discrimination is achieved. Again, the result obtained above for the D-criterion the value was 0.3541807443, the C-criterion become 36.62092 but the combination of the two gave a more homogenous value tending to zero 4.4336 as compared to single optimality criterion. This clearly brought a balance between parameter estimation and model discrimination.