Algebraic Foundations of Third Order Rotatability in Two Dimensions

Abstract

The mechanisms of some scientific phenomena are understood sufficiently well that useful mathematical models that flow directly from the physical mechanisms can be written down. Such models are not considered in this study. Response surface methodology in Schlaflian vectors and matrices representation for rotatability of experimental design points and optimal design theory in Kronecker product representation for measuring rotatability of experimental design points will be appropriate to the study of phenomena that are presently not sufficiently well understood to permit the mechanistic approach. These two techniques have three kinds of applications one approximate mapping of a surface within a limited region two choice of operating conditions to achieve desired specifications and three search for optimal conditions and are a generalization of factorial designs emphasizing the concept of rotatability. The 249 10th AIC Symposium 1: Peer Reviewed Papers problem of fitting a curve to the relationship between the concentration of a stimulus and the proportion of individuals responding transforming proportions to the corresponding normal deviates for data from psychological experiments is the precursor of these techniques. The concept of rotatability produced very strong reactions and the division between theoretical statisticians researching into the theory of optimal design and practical statisticians designing experiments for applied research workers is still very wide because the assumptions in the theory of optimal design have been restrictive with linear models assumed almost exclusively and the optimality criterion based on the generalized variance of the parameter estimates. This restrictiveness undoubtedly explains some of the reluctance of practical statisticians to try to produce “optimal” designs for practical problems. Development has come about mainly in answer to problems of determining optimum conditions in chemical investigations but the methods will be of value in other fields where experimentation is sequential and the error fairly small. The current endeavor is geared to be of use in deriving some new third order rotatable designs in higher dimensions from some of the available third order rotatable designs in lower dimensions. When these designs are used the results of the experiments performed according to the lower dimensional designs need not be discarded. Some of these designs may be performed sequentially in all factors and require a smaller number of points than most of the available third order rotatable designs. Algebra is used in the current effort and the results support existing moment and non-singularity conditions of third order rotatability hence “algebraic foundations” reference. Designs having a spherical variance insure that the estimated response has a constant variance at all points which are the same distance from the centre of the design. The unknown functional relationship may be represented by a Taylor series expansion of moderately low order within the region of interest. To get usable third order designs, we must combine at least two spherical sets of points with different positive radii as we have shown in these algebraic foundations of third order rotatability. The technique of fitting a response surface is one widely used to aid in the statistical analysis of experimental work in which the “yield” of a product depends, in some unknown fashion, on one or more controllable variables. Before the details of such an analysis can be carried out, experiments must be performed at predetermined levels of the controllable factors, i.e., an experimental design must be selected prior to experimentation. Rotatable designs permit a response surface to be fitted easily and provide spherical information contours where rotatability is coined from rotation in the multiplication of a vector and an orthogonal matrix when the original vector and the resulting vector have the same magnitude but face different directions from a common centre. In the real world 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 the tentative assumption of the validity of this linear model which we can justify on the basis of Taylor expansion, we can perform experiments, fit the model using regression techniques, and then apply standard statistical procedures to determine whether this model appears adequate. A particular selection of settings, or factor levels, at which observations are to be taken is called a design. Designs are usually selected to satify some desirable criteria chosen by the experimenter and errors can arise in one or more of the following ways one the true response may be observed with error two the functional relationship may not be the correct model three the observations on the independent variables may contain errors.