On D-optimal Block Designs for Second-order Polynomial Models: An Algorithm
Ngozi Umelo-Ibemere *
Department of Statistics, School of Physical Sciences, Federal University of Technology, Owerri, Nigeria.
Polycarp Chigbu
Department of Statistics, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
Often, standard block designs such as Randomized complete block design, Balanced incomplete block designs are used in block experiments. However, these designs are not always feasible in experiments designed to meet some specific needs. In such cases, optimal block designs are constructed. This paper presents an algorithmic approach to constructing D-optimal designs for second-order polynomial models in the presence of fixed block effects. The procedure involves the selection of non-singular initial design points from the experimental space and arrangement of these design points in different blocks with pre-specified block sizes. A line search equation which includes the direction of search, starting point and step length is employed to locate the design point with maximum prediction variance referred to as an iterate. The iterate replaces the design point with minimum prediction variance thereby improving the initial design. The algorithm converges in few iterations. Numerical examples are given to illustrate the implementation of the algorithm for both equal and unequal block sizes.
Keywords: Block size, D-optimal design, exchange algorithm, fixed block effect, line search, prediction variance, second-order model