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Construction and Evaluation of Response Surface Designs Incorporating Bias from Model Misspecification

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### Construction and Evaluation of Response Surface Designs Incorporating Bias from Model Misspecification

Connie M. Borror, Arizona State University West

Christine M. Anderson-Cook, Los Alamos National Laboratory

Bradley Jones, JMP SAS Institute

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

- 2in general, are not fully estimable
- Assume 2~ N(0, )

Criteria

- Mean-squared error
- Expected squared bias (ESB):
- Expected MSE sum of PV and ESB

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

Cases

- I. Two-factor response surface design
- Assume a second-order model:
- True form of the model is cubic:

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

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.

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 II

FDS 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:

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.

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