Rethinking Steepest Ascent for Multiple Response Applications

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Rethinking Steepest Ascent for Multiple Response
Applications

• Robert W. Mee
• Jihua Xiao
• University of Tennessee

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Outline

• Overview of RSM Strategy
• Steepest Ascent for an Example
• Efficient Frontier Plots
• Paths of Improvement (POI) Regions

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Sequential RSM Strategy

• Box and Wilson (JRSS-B, 1951)
• Initial design to estimate linear main effects
• Exploration along path of steepest ascent
• Repeat step 1 in new optimal location
• If main effects are still dominant, repeat step
  2 if not, go to step 4
• Augment to complete a 2nd-order design
• Optimization based on fitted second-order model

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Multiple Responses RSM Literature

• Del Castillo (JQT 1996), "Multiresponse
  Optimization
• Construct confidence cones for path of steepest
  ascent (i.e., maximum improvement) for each
  response
• Use very large 1-a for responses of secondary
  importance, e.g. 99-99.9 confidence
• Use 95-99 confidence for more critical
  responses
• Identify directions x falling inside every
  confidence cone
• If no such x exists, choose a convex combination
  of the paths of steepest ascent, giving greater
  weight for responses that are well estimated
• Constrain the solution to reside inside the
  confidence cones for the most critical responses.

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Multiple Responses RSM Literature

• Desirability Functions (Derringer and Suich, JQT
  1980)
• Score each response with a function between 0 and
  1.
• The geometric mean of the scores is the overall
  desirability
• Recent enhancements use score functions that are
  smooth (i.e., differentiable).

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An Example with Multiple Responses

• Vindevogel and Sandra (Analytical Chem. 1991)
• 25-2 fractional factorial design using micellar
  electrokinetic chromatography
• Higher surfactant levels required to separate two
  of four esters, but this increases the analysis
  time
• Response variables include
• Resolution for separation of 2nd and 3rd
  testosterone esters
• Time for process, tIV
• Four other responses of lesser importance

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Reaction Time vs. Reaction Rate

• Rate 1 / Time

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Fitted First-Order Models for Resolution and
Reaction Rate

• Good news Both models have R2 gt 99
• Bad news Improvement for resolution and rate
  point in opposite directions
• Authors recommend a compromise
• Lower x1 (pH) and x5 (buffer) to increase rate
• Lower x2 (SHS) and x3 (Acet.) and increase x4
  (surfactant) to increase resolution.

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What about Modeling Desirability?

• First-order model for Desirability

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What we just tried was a bad idea!

• Even when first-order models fit each response
  well, the desirability function for two or more
  responses will require a more complicated model
• Following an initial two-level design, one cannot
  model desirability directly.
• It is better to maximize desirability based on
  predicted response values from simple models for
  each response

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Maximizing Predicted Desirability for the
Vindevogel and Sandra Example

• JMPs default finds the maximum within a
  hypercube
• This does not identify a useful path for
  exploration

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Software Should Maximize Desirability Within a
Hypersphere
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Confidence Cone for Path of Steepest Ascent (Box
and Draper)

• Define ?bb, the angle between least squares
  estimator b and true coefficient vector b
• Pivotal quantity
• Upper confidence bound for sin2?bb
• Assuming ?bb lt 90o,

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95 Confidence Cone for Paths of Steepest Ascent
for Resolution Rate

• 95 Confidence Cones for Paths of Steepest Ascent
• Resolution (Y1) ?bb lt 14.4o
• Rate (Y2) ?bb lt 32.7o
• These confidence cones do not overlap, since the
  angle between bResolution and bRate is 141.5o!
• What compromise is best?

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Efficient Frontier Notation

• J larger-the-better response variables
• First-order model in k factors for each response
• Notation
• bj vector of least squares estimates for jth
  response
• Tj corresponding vector of t statistics for jth
  response
• Convex combinations for two responses
• For 0c1 xC(1-c)T1 cT2

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Efficient Frontier for Two Responses

• Let xN denote a vector that is not a convex
  combination of T1 and T2
• There exists a convex combination xC, with xC
  xN, such that xCbj xNbj (j 1,2)
• Proof by contradiction. I.e., suppose not. Then
  
• So one only need consider convex combinations of
  the paths of steepest ascent.

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Efficient Frontier for Resolution and Rate

• Predicted Resolution and Rate for xx7.49
• Grid lines match Predicted Y _at_ design center
• One quadrant shows gain in both Yjs

Gain!
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Efficient Frontier for Resolution and Rate

• No change in Rate (Y2) xC(1-c1)T1 c1T2
• c10.63
• xc -0.48, -0.10, -2.68, -0.10, -0.22 _at_
  xcxc7.49
• Resolution 1.76
• Rate .084

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Efficient Frontier for Resolution and Rate

• No change in Resolution (Y1) xC(1-c2)T1 c2T2
• c20.7355
• xc -1.39, 0.46, -1.85, -1.22, -0.65 _at_
  xcxc7.49
• Resolution .86
• Rate .11

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Improving Both Responses

• If T1T2 gt 0, all convex combinations of T1 and
  T2 increase the predicted Y for both responses
• If T1T2 lt 0, all xc with c1 lt c lt c2 increase
  the predicted Y for both responses
• For our example, .63 lt c lt .735 increase
  predicted Resolution and Rate

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Efficient Frontier _at_ xx5 versus Factorial
Points

• Factorial pts. 8 directions, none on the
  efficient frontier
• What about sampling error?

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Attaching Confidence to Improvement

• Lower confidence limit for EY(x), given x
• Lower confidence limit for change in EY(x),
  given x
• where

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Efficient Frontier _at_ xx7.49 with 90
Lower Confidence Bound for E(Y)-b0
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Paths of Improvement (POI) Region

• POI Region
• The POI Region is a cone about the path of
  steepest ascent, containing all x such that the
  angle
• Using t2,.10 1.886, the upper bound for ?xb is
  86.9o for Resolution, and 83.3o for Rate
• For simultaneous (in x) confidence region,
  replace tdf,a with (kFk,df,a)1/2 or
  (k-1)Fk-1,df,a1/2

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Paths of Improvement vs. Path of Steepest Ascent

• Path of Steepest Ascent b is perpendicular to
  contours for predicted Y
• The path of steepest ascent is not scale
  invariant
• Contours are invariant to the scaling of the
  factors
• Paths of improvement contours are complementary
  to the confidence cone for steepest ascent path
• Assuming ?bb lt 90o, 100(1-a) confidence cone for
  steepest ascent
• Assuming ?bb lt 90o, 100(1-a) confidence cone for
  paths of improvement

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Scale Dependence for Path of Steepest Ascent

• If the experiment uses a small range for one
  factor, steepest ascent will neglect that factor
• Suppose Y b0 X1 X2
• Experiment 1
• X1 -2,2
• X2 -1,1
• Path of S.A. 4,1
• Experiment 2
• X1 -1,1
• X2 -2,2
• Path of S.A. 1,4
• Contour 1,-1 for both

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Complementary Regions
As precision improves, the confidence cone for b
shrinks, while the paths of improvement region
expands toward half of Rk
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Common Paths of Improvement

• Using predicted values, convex combinations
  xC(1-c2)T1 c2T2 yield improvement in both
  responses for c1 lt c lt c2
• For our example, .63 lt c lt .735
• Using lower confidence bounds, a smaller set of
  directions yield certain improvement in both
  responses
• For our example using t2,.10 1.886, we are sure
  of improvement for .651 lt c lt .727

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Extensions to J gt 2 Responses

• The efficient frontier for more than two
  responses is the set of directions x that are a
  convex combination of all J vectors of steepest
  ascent
• If some directions of steepest ascent are
  interior to this set, they are not binding
• Overlaying contour plots can show the predicted
  responses for each direction x on the efficient
  frontier.

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Is Simultaneous Improvement Really Possible?

• Can we reject Ho ?b1b2 180o?
• An approximate F test based on the difference in
  SSE for regression of Y2 on X and regression of
  Y2 on predicted Y1.
• For our example, F 25.45 vs. F4, 2 (p .04)
• Can we construct an upper confidence bound for
  this angle?
• No solution at present
• The larger this angle, the further one must
  extrapolate in these k factors to achieve gain in
  both responses.

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Questions?