CENTRAL COMPOSITE DESIGNS ROBUST TO THREE MISSING OBSERVATIONS

Abstract

In well-planed experimental work,situation may arise where some observations are lost or destroyed or unavailable due certain reasons that arc beyond the control of the experimenter. Unavailability of the observations destroys the orthogonality and the balance of the design and also affects the inference. The purpose of this study is to assess the consequences of missing any combination of m observations (three in our case) of factorial, axial and centre points. The intensity of the consequences depends upon the size and type of the design. Generally smaller designs are more affected by the missing observations. We emphasized on various types of Central Composite Designs (CCDs) which includes Cuboidal, Spherical, Orthogonal, Rotatable, Minimum Variance, Box and Draper Outlier Robust Designs with an intention to introduce CCDs robust to in missing observations. It is observed that different relations occur between different combinations of three missing observations of factorial, axial and centre points and the determinant of the reduced information matrix{X'Xr), the main contributor in the definition of the loss of missing observation. This loss also depends the distance of the axial point from the center ol'the experiment (a), number of factors ( k ) and the position of the missing point. A complete sensitivity analysis is conducted by comparing the losses against all possible combinations of missing observations for a variety of a and k values, 1 .0 < a < 3.0; 2 < k < 0. These losses fall in predetermined groups of combinations producing same losses with a predictable frequency. I'or each configuration designs robust to one, two and three missing observations arc developed under the mininiaxloss criterion and are termed as minimaxlossl . minimaxloss2 and minimaxloss3 respectively. The minimaxloss3 design for each k value are compared w-ith other CCD counterparts. If the loss of missing in observations approaches one, the design breaks down. To avoid this breakdown and as a precautionary measure certain influential points in the design are additionally replicated when there are higher chances of loosing them. The replication of factorial or axial points depends on the values of a and k. It not only refrain the design from breaking down but helps in improving the efficiency of the design by reducing the loss.