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1 Systems Management Control Concepts It is the mark of a truly intelligent person to be moved by statistics. George Bernard Shaw

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2 R. A. Fisher Studied math and astronomy at Cambridge Conducted agricultural research in Britain Developed powerful methods of experimental design Sir Ronald A. Fisher 1890 - 1962

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Lady Tasting Tea At a summer tea party in Cambridge, England, a lady states that tea poured into milk has a different taste than milk poured into tea Most guests, including distinguished scientists, think this nonsense, but Ronald Fisher proposes a method to scientifically test the lady's claim In Fishers experiment, the lady correctly identified every cup! Salsburg, D., The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century, W.H. Freeman, 2001

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A Cure for Scurvy Scurvy once common among sailors Dr. James Lind, a Scottish surgeon in the British Royal Navy, designed experiments in 1754 –12 scurvy patients on the HMS Salisbury selected –Divided into six pairs and given remedies Cider, vitriol, seawater, garlic, vinegar, citrus –The pair on oranges and lemons fit for duty in six days First controlled clinical trial Origin of limey as term for British sailors Lind, J., A Treatise of the Scurvy, 1753

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5 Design of Experiments As we vary process parameters to reduce the loss function, it may be hard to determine what changes cause what effects Traditional method is to vary one parameter at a time –Inefficient –Provides no information about interactions

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6 Example Goal: improve some quality characteristic Factors: temperature (T) and pressure (P) Approach: conduct a series of experiments –First, the classic one factor at a time design –Baseline plus 2 trials TrialTPResult 1LL10 2LH30 3HL20

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7 One Factor at a Time Design The T-effect increase result by 10 The P-effect increase result by 20 Each estimate based on two trials No estimate of the interaction effect TrialTPResult 1LL10 2LH30 3HL20

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8 Factorial Design The T-effect (20+26)/2 – (10+30)/2 = 3 The P-effect (30+26)/2 – (20+10)/2 = 13 Each estimate based on four trials Next, interaction effect TrialTPR 1LL10 2LH30 3HL20 4HH26

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9 Factorial Design TxP Interaction Effect T-effect (P Low) 20 – 10 = 10 T-effect (P High) 26 – 30 = -4 By convention TxP effect is half the difference (10+4) / 2 = 7 Average of two T-effects equal to main effect of T [10 + (-4)] / 2 = 3 TrialTPR 1LL10 2LH30 3HL20 4HH26

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10 Factorial Design Interaction Effect TPR 1LL10 2LH30 3HL20 4HH26 Interaction: Effect of T changes depending on level of P Best result

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11 Factorial Design PxT Interaction Effect P-effect (T Low) 30 – 10 = 20 P-effect (T High) 26 – 20 = 6 Half the difference is the PxT effect (same as TxP) (20-6) / 2 = 7 Average of two P-effects equal to main effect of P [20 + 6] / 2 = 13 TrialTPR 1LL10 2LH30 3HL20 4HH26

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12 Factorial Design Lurking Variable Suppose Operator also affects the response T and O perfectly correlated Which variable caused the effect attributed to T? Control what you can Randomize trial order to break link with variables you cant control (or dont know about) TrialTPRO 1LL101 2LH301 3HL202 4HH262

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13 Factorial Design Summary With only four trials, we get –Main effect of T –Main effect of P –2-factor interaction effect Each estimate based on all four trails Estimating effects ANalysis Of Means (ANOM) Determining which effects are statistically significant ANalysis Of VAriances (ANOVA) TPR 1LL10 2LH30 3HL20 4HH26 With good design, analysis can be simple

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14 Factorial Design Same as earlier example Here with algebraic notation Recall that T-effect = 3 P-effect = 13 T x P effect = 7 Much easier to get these results with this notation TrialTPTxPR 1 +10 2 + 30 3+ 20 4+++26

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Calculating Effects in the 2 3 Design ABCABACBCABC +++++++ ++ + + + + + ++ ++ + + + + ++ + +++ 15

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The Fractional Factorial ABCABACBCABC +++++++ ++ + + + + + ++ 16 One-half fraction of a full 2 3 design: 2 3-1 design Note that effects B, AB and BC, ABC are confounded We always lose something with a fractional design

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Fractional Factorial Designs As the number of factors increases, the runs required for a full factorial may outgrow the experimental resources available Fractional factorial designs are used for screening experiments: to determine which factors are truly significant Goal: distinguish main effects and 2-factor interactions 17

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Alias structure for 2 6-2 design EffectAlias ABCECDFABDEF BACEDEFABCDF CABEADFBCDEF DADFBEFABCDE EABCBDFACDEF FACDBDEABCEF ABCEBCDFADEF ACBEDFABCDEF ADCFBCDEABEF AEBCCDEFABDF AFCDBCEFABDE BDEFACDEABCF BFDEABCDACEF 18

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Design Resolution Resolution III Designs (2 3-1 ) –No main effects are aliased with any other main effect, but main effects are aliased with 2-factor interactions Resolution IV Designs (2 4-1 ) –No main effect is aliased with any other main effect or 2-factor interaction, but 2-factor interactions are aliased with each other Resolution V Designs (2 5-1 ) –No main effect or 2-factor interaction is aliased with any other main effect or 2-factor interaction 19

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20 George Box Born in Britain Distinguished career at the University of Wisconsin Box, Hunter, and Hunter, Statistics for Experimenters Dr. John MacGregor one of Boxs Ph.D. students George E. P. Box 1919 -

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21 Douglas Montgomery Ph.D. VPI, 1969 G.T. faculty, 1969 - 84 ASU faculty, 1988 – present ACQC Shewhart Medal, 1997 Supervised Philip Coyles M.S. thesis, "An Adaptation of Bayesian Statistical Methods to the Determination of Optimal Sample Sizes for Operational Testing"

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How to use statistical techniques Find out as much as you can about the problem Get help from experts on the process Define objectives Dont invest more than one quarter of the experimental effort (budget) in a first design Box, G., Hunter, W., Hunter, S., Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, John Wiley and Sons, 1978 22

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How to use statistical techniques Use your non-statistical knowledge of the problem Keep the design and analysis as simple as possible Recognize the difference between practical and statistical significance Experiments are usually iterative Montgomery, D., Design and Analysis of Experiments, 2 nd edition, John Wiley and Sons, 1984 23

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