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

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.