Some design needs: Projection Properties

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Some design needs Projection Properties Model
uncertainty

• Jan Engel (Centre for Quantitative Methods,
  Eindhoven, The Netherlands)

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Contents

• Examples 1 2
• Practical considerations
• Projection properties of designs
• Discussion needs and questions
• References

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1. Example 1
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1. Example 1

• Design 18 runs from Webb (1971).
• Blocked 2/3 replicate of the full 33 27
  factorial.
• Linear by linear interactions can be estimated.
• Factor Day (2 level block) is not significant
• Now 2/3 replicate of the full 33 design.
• Question opportunities for estimation of
  interactions?

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1. Example 2

• Screening Experiment IC design 0.13 µm MOS

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1. Example 2
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2. Practical considerations

• Conclusions from the Examples
• Step 1 Estimate main effects from screening
  design
• Step 2 Estimate 2-factor interactions from
  reduced design
• Stepwise experimentation is regular approach
• Question can we do with a single experiment?
• Some practical design needs
• Small designs
• Robust against model uncertainty
• Reasonable design for series of models
  (contrary to optimum design for single model)

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2. Practical considerations

• Back to the Examples
• Example 2 PB with 7 factors in 12 runs
• Design options in 2n-k series resolution be at
  least V to estimate main effects and
  interactions
• 27-2 32 runs Res IV , resolution too low
• 27-1 64 runs Res VII, experiment size
  non-feasible
• Benefit from sparse effects only some out of 7
  factors are active (Box 20, Pareto principle).
• Use empty columns in the design matrix to test
  for interactions of the active factors
• Projection
  Properties of Designs

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3. Projection properties of designs

• Design projects to subspaces that are relevant.
• Interesting 2-level designs
• 2.1 All main effects be tested.
• 2.2 In case of k active factors (k 2,3) all
  active factor interactions be tested.
• Projection properties, results obtained,
  e.g.
• 2n-k - designs, Resolution R design projects to
  full R-1 factorial (e.g. Myers Montgomery,
  1995, p. 140)
• Plackett-Burman designs, Example 2 (Box
  Tyssedal, 1996 Miller Sitter, 2001)
• 3n-k designs, project to second order designs
  (Cheng Wu, 2001, Xu et al., 2004) .

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4. Discussion

• Needs
• More Projection Properties for existing design
• 3 level designs their projection properties.
• Mixed designs (combined 2 and 3 level designs)
  their projection properties.
• Questions unanswered
• Projection instead of stepwise experimenting, or
  additional to?
• Relationship with supersaturated designs?
• Consequences of conditional testing?
  Null hypotheses on main
  effects accepted, then test for interactions
  significance level? power?

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5. References

• Box, G.E.P. Tyssedal, J. (1996). Projective
  properties of certain orthogonal designs.
  Biometrika, 83, 950-955.
• Cheng, S-W Wu, C.F.J. (2001). Factor screening
  and response surface exploration. Statistica
  Sinica, 11, 553-604.
• Miller, A. Sitter, R.R. (2001). Using the
  folded-over 12-run Plackett-Burman design to
  consider interactions. Technometrics, 43, 44-55.
• Myers, R.H. Montgomery, D.C. (1995). Response
  Surface Methodology. New York Wiley.
• Xu, H., Cheng, S-W Wu, C.F.J. (2004). Optimal
  projective three-level designs for factor
  screening and interaction detection.
  Technometrics, 46, 280-292.
• Webb, S.R. (1971). Small incomplete factorial
  experiment designs for two- and three-level
  factors. Technometrics, 13, 243-256.