Some Third Order Rotatable Designs In Six Dimensions

Abstract

In agriculture, science and technology experiments must be performed at predetermined levels of the controllable factors meaning that an experimental design must be selected prior to experimentation. A cyclical group of a certain order in a particular degree of a polynomial forms a rotatable design if it satisfies both moment and non-singularity conditions of the rotatability method or criterion. In agriculture and science we observe what happens, based on these observations form a theory as to what may be true; test the theory by further observations and by experiments; and watch to see if the predictions based on the theory are fulfilled while technology is the application of agricultural and scientific knowledge to practical tasks in all of which statistical theory is crucial in the formulation of theories or hypotheses and evolution of predictions. Suppose that an experimenter is interested in determining the relationship between a response and several independent variables. The independent variables may be controlled by the experimenter or observed without control. Suppose, further, that these independent variables represent all the factors that contribute to the response, and that the exact relationship between the response and the independent variables is the response function and, geometrically, it defines a surface called the response surface.
In the real world, however, we rarely know the exact relationship, or all the variables which affect that relationship. One way of proceeding then is to graduate, or approximate to, the true relationship by a polynomial function, linear in some unknown parameters to be estimated and of some selected order in the independent variables. Under tentative assumption of the validity of this linear model, which we can justify on the basis of Taylor expansion of the response function, we can perform experiments, fit the model using regression techniques, and then apply standard statistical procedures to determine whether this model appears adequate. Since in practice we do not know all of the factors which affect the response, we usually select a subset of the independent variables which
we believe might have significant effects. This selection may be made on the basis of prior knowledge, or a preliminary experiment may be performed to screen the important independent variables out of a larger set of possible independent variables.
Polynomial models of order higher than two are rarely fitted, in practice. This is partially because of the difficulty of interpreting the form of the fitted surface, which in any case, produces predictions whose standard errors are greater than those from the lower order fit, and partly because the region of interest is usually chosen small enough for a first or second order model to be a reasonable choice. Exceptions occur in meteorology, where quite high order polynomials have been fitted, but there are only two or three variables commonly used. When a second order polynomial is not adequate, and often even when it is, the possibility of making a simplifying transformation in response or in one or more of the variables would usually be explored before reluctantly proceeding to higher order, because more parsimonious representations involving fewer terms are generally more desirable. This has limited research in second order polynomials whence the gap in the mathematical world in respect of third order polynomials particularly the development of the mathematical formula for sequential construction in the factors leading to the current endeavour.
Once an experimenter has chosen a polynomial model of suitable order, the problem arises as how best to choose the settings for the independent variables over which he has control. 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 methods or criteria chosen by the experimenter. These methods or criteria include the rotatability method or criterion and the method or criterion of minimizing the mean square error of estimation over a given region in the factor space. The present endeavor represents an attempt to meet, in part, this need in third order polynomial models using the rotatability method or criterion along with the cyclical group of order six to generate the point sets. The criterion or method of rotatability says that the variances of estimates of the response made from the least squares estimates of the Taylor series are constant on circles, spheres or hyperspheres about the center of the design. Thus, a rotatable design, that is, a design which meets this criterion or method, could be rotated through any angle around its center and the variances of responses estimated from it would be unchanged while the 9
cyclical group of order six generating point sets provides the set of points on which the criterion or method of rotatability is applied.
Specifically, the problem considered is that sequential choice of combinations of levels of independent variables which will enable the experimenter to approximate a functional relationship by fitting a Taylor series expansion through terms of order three by the method of least squares and will also follow the criterion or method of rotatability in six factors. Such a sequential choice of combinations of levels of the independent variables will be called a third order rotatable design in six dimensions or factors. The objective is to have eventually the mathematical formulation of third order rotatable designs in a finite number of factors as is the case for second order rotatable designs. Already in the literature we have third order rotatable designs in five dimensions or factors but there is no mathematical formula of their sequential construction like we have for second order rotatable designs. This is necessary because when such sequential designs are used the results of the experiments performed according to the five dimensional designs need not be discarded when appending the sixth factor. In soil science for instance continuous cultivation of crops may exhaust previously available mineral elements necessitating sequential appendage of the mineral elements which become deficient in the soil in time among other examples
In our endeavor we were able to append the sixth factor. However, the other aspects of the problem for further study would include the practical field application after the estimation of the free or arbitrary parameters employing the general equivalence theorem which states that the minimization of the generalized variance of the estimates of the coefficients which are linear in the polynomial models is equivalent to the minimization of the maximum variance of the estimated response to identify specific optimum designs of order three and the mathematical formulation for third order rotatability. The moment and the non singularity conditions are byproducts of the criterion or method of rotatability which the cyclical group of order six generated points satisfy.

Keywords: Third order; rotatable designs; five dimensions; six dimensions; sequential.

Journal of Agriculture, Science & Technology Vol. 9 (1) 2007: pp. 78-87