Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 14 Sequential Experimentation, Screening Designs, Fold-Over Designs.

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Presentation transcript:

Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 14 Sequential Experimentation, Screening Designs, Fold-Over Designs

Don’t Risk all Experimental Resources on a Single Comprehensive Experiment Usually many inert factors, few dominant ones Unexpected effects may be found early Experiment could be terminated early with substantial cost savings Comprehensive evaluation of a few dominant factors is usually more informative than little information on many factors

Conduct a Screening Experiment to Identify Dominant Factors Augment the Screening Experiment to Identify Strong Two-Factor Interactions Conduct a R V Experiment with the Dominant Factors Comprehensive Experiment with a Few Factors and Multiple Levels or Design to Quantitatively Characterize the Response Surface Figure 7.7 A simple strategy for a sequence of experiments.

Sequential Experimentation Large experiments Design so that key fractions can be run in sequence Key fractions : Resolution III, IV, or V Analyze each sequence of data as it is completed Based on the results of the analysis Continue experiment Terminate Redesign with dominant/new factors

Acid Plant Corrosion Study Factor Raw Material Feed Rate3,000pph6,000pph Gas Temperature100 o C220 o C Scrubber Water5%20% Reactor Bed Acid20%30% Exit Temperture300 o C360 o C Reactant Distribution PointEastWest Coded Level Plant must cease commercial production during experimentation -- test runs must be minimized MGH Table 7.1

Screening Experiments Very few test runs Ability to assess main effects only Generally leads to a comprehensive evaluation of a few dominant factors Potential for bias Highly effective for isolating vital few strong effects should be used ONLY under the proper circumstances Highly effective for isolating vital few strong effects should be used ONLY under the proper circumstances

Plackett-Burman Screening Designs Any number of factors, each having 2 levels Interactions nonexistent or negligible Relative to main effects Number of test runs is a multiple of 4 At least 6 more test runs than factors should be used

Construction Determine the number of factors (k) to be included in the design Determine the experiments size : at least k + 6 6df for error Select the design generator from Table 7A2 Generate the rows of the design Design generator is the first row Move all levels in the previous row one position to the left; move the first level of the previous row to the last position Continue the previous step until n - 1 rows are filled The last row has all levels equal to -1

Construction (con’t) Randomize Randomly Assign Factors to Columns; Delete Unassigned Columns Randomly Permute the Rows

Acid Plant Corrosion Study Factor Raw Material Feed Rate3,000pph6,000pph Gas Temperature100 o C220 o C Scrubber Water5%20% Reactor Bed Acid20%30% Exit Temperture300 o C360 o C Reactant Distribution PointEastWest Coded Level Seeking identification of dominant main effects

Plackett-Burman Design : Corrosion Study k = 6 Factors n = 12 (Minimum Recommended) Design Generator

Plackett-Burman Design : Corrosion Study k = 6 Factors n = 12 (Minimum Recommended)

Plackett-Burman Design : Corrosion Study k = 6 Factors n = 12 (Minimum Recommended)

Plackett-Burman Design : Corrosion Study k = 6 Factors n = 12 (Minimum Recommended)

Plackett-Burman Design : Corrosion Study Randomly assign factors to columns

Plackett-Burman Design : Corrosion Study Eliminate unassigned columns randomly permute rows

Plackett-Burman Design : Corrosion Study Resolution III

Human Performance Testing Response Eye Focus Time (ms) Predictors (A) Acuity or Sharpness of Vision (B) Distance from Eye to Target (C) Target Shape (D) Illumination Level (E) Target Size (F) Target Density (G) Subject 2 Levels Each Only a few effects anticipated, no interactions Only a few effects anticipated, no interactions

Design Considerations Complete factorial : repeats = repeats Very few effects expected, no interactions Solution Fractional Factorial R III Solution Fractional Factorial R III

Human Performance Testing Design: Defining Equation I = ABD = ACE = BCF = ABCG Added Factors D = AB, E = AC, F = BC, G = ABC Implicit Equations = 11 n = 8

Human Performance Testing Complete Defining Relation I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG Implicit Contrasts

Human Performance Testing Complete Defining Relation I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG Implicit Contrasts Main-Effect Aliases A = BD = CE = FG B = AD = CF = EG C = AE = BF = DG D = AB = CG = EF E = AC = BG = DF F = BC = AG = DE G = CD = BE = AF Assuming No 3fi

Human Performance Testing Complete Defining Relation I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG Implicit Contrasts Main-Effect Aliases A + BD + CE + FG B + AD + CF + EG C + AE + BF + DG D + AB + CG + EF E + AC + BG + DF F + BC + AG + DE G + CD + BE + AF Alternative Interpretation of Aliasing: Each Measured Effect is the Sum of Four Effects Alternative Interpretation of Aliasing: Each Measured Effect is the Sum of Four Effects

Human Performance Testing

Conclusions A, B, and D are the primary factors that affect eye focus times

Human Performance Testing Conclusions A, B, and D are the primary factors that affect eye focus times Key Main-Effect Aliases A = BD B = AD D = AB Could the primary effects be only two factors and their interaction ?

Human Performance Testing Fold-Over Design: Defining Equation I = -ABD = -ACE = -BCF = -ABCG Added Factors -D = AB, -E = AC, -F = BC, -G = ABC Reverse the signs on all levels of all factors in the design Reverse the signs on all levels of all factors in the design

Human Performance Testing: Fold-Over Design

Combined Effects: Original DesignA + BD + CE + FG Fold-Over DesignA - BD - CE - FG AverageA Difference/2 BD + CE + FG Conclusion: Reversing ALL signs in a second fraction unaliases ALL main effects from two-factor interactions (still assumes higher-order interactions are negligible)

Human Performance Testing: Fold-Over Design Conclusions ? n = 16 Original Design: D = AB

Fold-Over Designs Reverse the signs on one or more factors Run a second fraction with the sign reversals Use the confounding pattern of the original and the fold-over design to determine the alias structure Averages Half-Differences

Human Performance Testing Complete Defining Relation Reversing the Signs on B I = -ABD = ACE = -BCF = -ABCG = -BCDE = ACDF = CDG = -ABEF = -BEG = AFG = DEF = ADEG = CEFG = -BDFG = -ABCDEFG Main-Effect Aliases A - BD + CE + FG -B + AD + CF + EG C + AE - BF + DG D - AB + CG + EF E + AC - BG + DF F - BC + AG + DE G + CD - BE + AF

Human Performance Testing: Fold-Over Design Combined Effects: Original DesignA + BD + CE + FG Fold-Over DesignA - BD + CE + FG AverageA + CE + FG Difference/2 BD Conclusion: Reversing the signs on ONE factor in a second fraction unaliases its main effect and ALL its two-factor interactions Original DesignB + AD + CF + EG Fold-Over Design -B + AD + CF + EG Average AD + CE + FG Difference/2B Similar With All Main Effects Except B