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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) 1. Initial design to estimate linear main effects 2. Exploration along path of steepest ascent 3. Repeat step 1 in new optimal location – If main effects are still dominant, repeat step 2; if not, go to step 4 4. Augment to complete a 2nd-order design 5. 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- 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) 2 5-2 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 2 nd and 3 rd testosterone esters – Time for process, t IV – 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 R 2 > 99% Bad news: Improvement for resolution and rate point in opposite directions – Authors recommend a compromise: – Lower x 1 (pH) and x 5 (buffer) to increase rate – Lower x 2 (SHS%) and x 3 (Acet.) and increase x 4 (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 JMP’s 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 b , the angle between least squares estimator b and true coefficient vector Pivotal quantity: Upper confidence bound for sin 2 b Assuming b < 90 o,

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95% Confidence Cone for Paths of Steepest Ascent for Resolution & Rate 95% Confidence Cones for Paths of Steepest Ascent – Resolution (Y 1 ): b < 14.4 o – Rate (Y 2 ): b < 32.7 o These confidence cones do not overlap, since the angle between b Resolution and b Rate is 141.5 o ! 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 – b j : vector of least squares estimates for j th response – T j : corresponding vector of t statistics for j th response Convex combinations for two responses – For 0≤c≤1: x C =(1-c)T 1 + cT 2

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Efficient Frontier for Two Responses Let x N denote a vector that is not a convex combination of T 1 and T 2 There exists a convex combination x C, with |x C | = |x N |, such that x C ‘b j ≥ x N ‘b j (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 x’x=7.49 Grid lines match Predicted Y @ design center One quadrant shows gain in both Y j ’s

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Efficient Frontier for Resolution and Rate No change in Rate (Y 2 ): x C =(1-c 1 )T 1 + c 1 T 2 c 1 =0.63 x c = [-0.48, -0.10, -2.68, -0.10, -0.22] @ x c ’x c =7.49 Resolution = 1.76 Rate =.084

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Efficient Frontier for Resolution and Rate No change in Resolution (Y 1 ): x C =(1-c 2 )T 1 + c 2 T 2 c 2 =0.7355 x c = [-1.39, 0.46, -1.85, -1.22, -0.65] @ x c ’x c =7.49 Resolution =.86 Rate =.11

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Improving Both Responses If T 1 ’T 2 > 0, all convex combinations of T 1 and T 2 increase the predicted Y for both responses If T 1 ’T 2 < 0, all x c with c 1 < c < c 2 increase the predicted Y for both responses For our example,.63 < c <.735 increase predicted Resolution and Rate

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Efficient Frontier @ x’x=5 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 E[Y(x)], given x Lower confidence limit for change in E[Y(x)], given x where

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Efficient Frontier @ x’x=7.49 with 90% Lower Confidence Bound for E(Y)- 0

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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 t 2,.10 = 1.886, the upper bound for θ xb is 86.9 o for Resolution, and 83.3 o for Rate For simultaneous (in x) confidence region, replace t df, with (kF k,df, ) 1/2 or [(k-1)F k-1,df, ] 1/2

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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 b < 90 o, 100(1- )% confidence cone for steepest ascent – Assuming b < 90 o, 100(1- )% 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 = 0 + 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 shrinks, while the paths of improvement region expands toward half of R k

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Common Paths of Improvement Using predicted values, convex combinations x C =(1-c 2 )T 1 + c 2 T 2 yield improvement in both responses for c 1 < c < c 2 – For our example,.63 < c <.735 Using lower confidence bounds, a smaller set of directions yield “certain” improvement in both responses – For our example using t 2,.10 = 1.886, we are sure of improvement for.651 < c <.727

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Extensions to J > 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 H o : = 180 o ? – An approximate F test based on the difference in SSE for regression of Y 2 on X and regression of Y 2 on predicted Y 1. – For our example, F = 25.45 vs. F 4, 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?

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