11 Designs for the First Order Model “Orthogonal first order designs” minimize the variance of If the off diagonal elements of X´X are all zero then the first order design is orthogonal Examples of orthogonal first order designs  2 k factorial designs  2 k- p fractional factorial designs in which main effects are not aliased with each other  Simplex designs : Regularly sided figure with k+1 vertices in k dimensions => economical (The number of trials is the minimum for defining a direction of improvement. ) Figure The simplex design for (a) k = 2 variables (b)k = 3 variables

22 Designs for the Second Order Model The central composite design: the most widely used design for fitting the second-order model Box-Behnken Design Selection of a second-order design is an interesting problem There are numerous excellent second-order designs available

33 Central Composite Design It consists of  2 k factorial designs (or 2 k- p fractional factorial designs with resolution V) with n F runs  2k axial runs  n c center runs CCD is used as a sequential experimentation. - After fitting a first-order model with 2 k design, if the model shows lack of fit, then the axial runs added to allow the quadratic term to be incorporated. Two parameters  and n c must be specified where  is the distance of the axial runs from the design center Generally, 3~5 central runs (n c ) are recommended. Choose  which makes CCD “rotatable”

44 A second order model must provide good predictions throughout the region of interest One way to define good is to require that the model have a reasonably consistent and stable variance of the predicted response at x The variance of the predicted response at x Rotatable means that the variance of the predicted response at x is the same at all x that are the same distance from the design center A design with this property will leave the variance of the predicted response unchanged when the design is rotated about the center A CCD is rotatable when  =(n f ) 1/4 Rotatability

5 Figure Central composite designs for k = 2 and k = 3.

Figure Contours of constant standard deviation of predicted response for the rotatable CCD

77 Box Behnken Design The Box-Behnken design is an independent quadratic design in that it does not contain an embedded factorial or fractional factorial design In this design the treatment combinations are at the midpoints of edges of the process space and at the center Efficent in terms of the number of runs and rotatable (or near rotatable) 3 levels of each factor. Are required

8 For three factors, the CCD contains 14+n c while the BBD contains 12+n c BBD is used when one is not interested in predicting response at the extremes