Connie M. Borror, Arizona State University West
Christine M. Anderson-Cook, Los Alamos National Laboratory
Bradley Jones, JMP SAS Institute
Construction and Evaluation of Response Surface Designs Incorporating Bias from
Model Misspecification
Motivation Response surface design evaluation (and
creation) assuming a particular model Single number efficiencies Prediction variance performance Mean-squared error
Model misspecification? What effect does this have on prediction and
optimization?
Motivation Examine effect of model misspecification
Expected squared bias Prediction variance Expected mean squared error Using fraction of design space (FDS) plots and
box plots Evaluate designs based on the contribution of
ESB relative to PV.
Scenario Cuboidal regions True form of the model is of higher order than
the model being fit. Examine
Response surface models when the true form is cubic
Screening experiment when the true form is full second order.
Model Specifications The model to be fit is
Y = X11 + ε X1 = n × p design matrix for the assumed form of
the model The true form of the model is
Y = X11+ X22 + ε X2 = n × q design matrix pertaining to those
parameters (2) not present in the model to be fit (assumed model).
Model Specifications 2 in general, are not fully estimable
Assume 2 ~ N(0, ) 2β
1
2
2
2
2
2
0
0q
βΣ
Criteria Mean-squared error
Expected squared bias (ESB):
Expected MSE sum of PV and ESB
AwzwAzβ 2
2ESB
121 1 2 2ˆ[ ( )]MSE y x w X 'X w β w A z z w A β
Fraction of Design Space (FDS) Plots Zahran, Anderson-Cook,
and Myers (2003) scaled prediction variance values are plotted versus the fraction of the design space that has SPV at or below the given value
Adapt this to plot ESB and EMSE as well as PV.
We use FDS plots and box plots to assess the designs
0 0.25 0.5 0.75 1
Fraction of Design Space
3
4
5
6
7
8
9
SPV
CCD-1CR
Hexagon-1CRCCD-3CR
100% G-eff
Cases I. Two-factor response surface design
Assume a second-order model:
True form of the model is cubic:
20 i i ij i j ii iy x x x x
2 2 30 i i ij i j ii i ij i j iii iy x x x x x x x
Case I Designs Central Composite Design (CCD) Quadratic I-optimal (Q I-opt) Quadratic D-optimal (Q D-opt) Cubic I-optimal (C I-opt) Cubic D-optimal (C D-opt) Cubic Bayes I-optimal (C Bayes I-opt) Cubic Bayes D-optimal (C Bayes D-opt)
Case I CCD
22 1 assume weIf B
Case I Designs CCD (ESB and EMSE performance as bias increases)
Case I Designs PV for all designs
Case I Designs ESB for all designs
Case I Designs EMSE for all designs
Case I Designs FDS for EMSE for all designs
Case II Four factor response surface design
Assume a second-order model:
True form of the model is cubic:
20 additional terms as we move from the second-order model to cubic.
20 i i ij i j ii iy x x x x
2 2 30 i i ij i j ii i ij i j iii iy x x x x x x x
Case II Designs Six possible designs, with n = 27 runs
Central Composite Design (CCD) Box Behnken Design (BBD) Quadratic I-optimal (Q I-Opt) Quadratic D-optimal (Q D-Opt) Cubic Bayes I-optimal (C Bayes I-Opt) Cubic Bayes D-optimal (C Bayes D-Opt)
Note: Cubic I- and D-Optimal not possible with available size of design
Case II PV for all designs
Case II EMSE for all designs
Case IIFDS plot of EMSE for Four Factors
Case III Eight-factor Screening Design
Assume a first-order model:
True form of the model is full second-order:
0 i iy x
20 i i ij i j ii i
i jy x x x x
Case III Designs 28-4 fractional factorial design with 4 center
runs D-optimal (for first order) Bayes I-optimal (for second order) Bayes D-optimal (for second order)
Case III Designs The difference in the number of terms from
the assumed to the true form of the models increases from 8 to 44. We would expect bias to quickly dominate
EMSE.
Case III PV for all designs
Case III ESB for all designs
Case III EMSE for all designs
Design Notes For the two-factor case:
The I-optimal and CCD were equivalent. They performed the best based on minimizing the
maximum EMSE They performed the best based on prediction
variance
Design Notes For the four factor case,
the BBD was best based on EMSE criteria (in particular, the 95th percentile, median, mean) when size of the coefficients of missing terms are
moderate to large The I-optimal design was competitive for this case
only if small amounts of bias were present.
As the number of missing cubic terms increases, the BBD was best for EMSE.
Design Notes I-optimal designs were highly competitive
over 95% of the design region; not with respect to the maximum PV, ESB, and EMSE.
Cubic Bayesian designs did not perform well.
Design Notes In the screening design example:
The D-optimal designs best if the assumed model is correct, but break down quickly if quadratic terms are in the model Much more pronounced than in the response surface design
cases. Quadratic Bayesian I-optimal design was best based on
mean, median, and 95th percentile of EMSE The 28-4 fractional factorial design was best with respect
to the maximum EMSE. The 28-4 design was best for both PV and ESB when the
PV and ESB contribution to the model were balanced.
Conclusions Appropriate design can strongly depend on the assumption
that we know the true form of the underlying model If we select designs carefully it is often possible to select a
model that predicts well in the design space, and provide some protection against missing model terms.
The ESB approach to assessing the effect of missing terms provides is advantageous: do not have to specify coefficient values for the true underlying
model, Instead, the relative size of the missing terms can be calibrated
relative to the variance of the observations.
Conclusions Size of the bias variance relative to observational
error needed to balance contributions from PV and ESB is highly dependent on the number of missing terms from the assumed model.
As the number of missing terms increases, the ability of designs to cope with the bias decreases substantially different designs are able to handle this increasing bias
differently.