On the Difference in Cycling Pattern on Linear and Higher-Order Effect Designs

Abstract

D-optimality is a design criterion that seeks to maximize the determinant of the information matrix, or equivalently minimize the determinant of the inverse information matrix of the design. This design criterion results in maximizing the differential Shannon information content of the parameter estimates. Cycling, a phenomenal problem associated with the construction of optimal designs, impedes the rate of convergence to such desired optimum, whenever it occurs in a variance exchange process. Different polynomial functions may have varying effects on the pattern of convergence due to cycling. This paper seeks to determine the nature and extent to which the influence of cycling affects the pattern of convergence on Linear, Interactive, and Quadratic order effect designs. The variance exchange algorithmic search method was adopted based on the philosophy of numerically searching the design space for optimum designs. Two and three-variable response functions are used in the investigation of even and odd-sized point designs. Generated data from designs of sizes 10 and 11 were employed in the investigation. Numerical illustrations were given to ascertain the pattern of convergence on each of the degree polynomial designs. The computations and graphs were conducted in R version 4.1.1 (2021). The results show that cycling patterns differ with respect to the degree of the response function whether it is of even or odd-sized design, or has two or three variables. The result will enable researchers to find appropriate measures to accommodate the challenge posed by cycling.