Robust Second-Order Slope-Rotatable Designs with Maximum Directional Variance

Abstract

In response surface methodology, rotatability and slope-rotatability are natural and highly desirable properties for second-order regression models. In this article, we introduce the concept of robust slope-rotatable designs with equal maximum directional variance for second-order response surface models with correlated observations. This requires that the maximum variance of the estimated slope over all possible directions to be only a function of the distance of the point from the design origin, and independent of correlation parameter or parameters involved in the variance–covariance matrix of errors. It is derived that robust second-order rotatable designs of two factors are also robust slope-rotatable designs with equal maximum directional variance. It is also established that within the robust second-order symmetric balanced designs, robust rotatable designs are also robust slope-rotatable with equal maximum directional variance for more than two factors. We also investigate a class of robust second-order slope-rotatable designs with equal maximum directional variance for special correlation structures of errors.