Week 8 Flashcards
General formula for coding
General form of response function
Where β is a p vector of the model parameters and g(x) is a p vector of the model expansion of the point x
Second order response function
First order response model
Requirements for lack of fit check
At least one treatment has been replicated in the design (N > t)
Design has more treatments than fitted response function has parameters (t > p)
Lack of fit for first or second order response function
Under H0, both estimators are equal
LoF test stat follows F dist with t-p and N-t DoF
Reason for central composite design
Design requires each factor to be included at at least 3 distinct values to allow estimation of second order model
This is combinatorially prohibitive
CCD can be constructed and applied in stages
Components of CCD
Factorial component with n1 points (most commonly a 2 level full factorial resolution V fractional factorial design)
n2 centre points (0, 0, … , 0)
n3 = 2f axial points (+/- α, 0, …, 0), (0, +/- α, 0, …, 0), …, (0, …, +/- α)
5 reasons for CCD for response surface experiments
1) allows estimation of params of second order model
2) # of treatments doesn’t become too large as # of factors increases
3) design can be run sequentially starting with factorial component and some centre points to estimate first order model, then adding centre points and axial points to estimate second order
4) design allows LoF testing for first order model and (provided |α| > 1) for second order model
5) design points are spread reasonably evenly across region of the factors being explored
Rotatable designs
If α is chosen to be = n11/4
Prediction variance is then same at any point in a sphere of common distance around centre point
optimal response surface designs
As optimal designs in week 5
Alternative to CCD
May not provide same mix of prediction efficiency, rotatability, and ability to detect LoF
I optimaility
Sometimes called v optimality
Minimised average (scaled) prediction variance
First stage of sequential response optimisation
(Fractional) factorial design, with some additional centre points is used to collect data to fit a first order model(Typically not including interactions)
Partial deriv vector Is directional deriv and indicates direction in which new experiments should be conducted
Second stage of sequential response surface optimisation
LoF on first order (without interactions) model
Curvature in the estimated response surface (if is LoF) indicates possibility of an optimum response (max or min)
Location+nature of turning points in η^ can be investigated by once again exploring derivs:
If eigenvalues of B^ are all positive or negative, stationary point is a max or min respectively
If they’re mixed it’s a saddle point
