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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.

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slide1

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
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?
motivation3
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
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
Model Specifications
  • The model to be fit is

Y = X11 + ε

    • X1 = n × p design matrix for the assumed form of the model
  • The true form of the model is

Y = X11+ X22 + ε

    • X2 = n × q design matrix pertaining to those parameters (2) not present in the model to be fit (assumed model).
model specifications6
Model Specifications
  • 2in general, are not fully estimable
  • Assume 2~ N(0, )
criteria
Criteria
  • Mean-squared error
  • Expected squared bias (ESB):
  • Expected MSE sum of PV and ESB
fraction of design space fds plots
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
Cases
  • I. Two-factor response surface design
    • Assume a second-order model:
    • True form of the model is cubic:
case i designs
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 designs12
Case I Designs
  • CCD (ESB and EMSE performance as bias increases)
case i designs13
Case I Designs
  • PV for all designs
case i designs14
Case I Designs
  • ESB for all designs
case i designs15
Case I Designs
  • EMSE for all designs
case i designs16
Case I Designs
  • FDS for EMSE for all designs
case ii
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
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 ii19
Case II
  • PV for all designs
case ii20
Case II
  • EMSE for all designs
case ii21
Case II

FDS plot of EMSE for Four Factors

case iii
Case III
  • Eight-factor Screening Design
    • Assume a first-order model:
    • True form of the model is full second-order:
case iii designs
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 designs24
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 iii25
Case III
  • PV for all designs
case iii26
Case III
  • ESB for all designs
case iii27
Case III
  • EMSE for all designs
design notes
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 notes29
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 notes30
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 notes31
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
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.
conclusions33
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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