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Chapter 13 Design of Experiments

Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

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Page 1: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

Chapter 13

Design of Experiments

Page 2: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

Introduction

• “Listening” or passive statistical tools: control charts.

• “Conversational” or active tools: Experimental design.– Planning of experiments– A sequence of experiments

Page 3: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.1 A Simple Example of Experimental Design Principles

• The objective is to compare 4 different brands of tires for tread wear using 16 tires (4 of each brand) and 4 cars in an experiment.

• Illogical Design:– Randomly assign the 16 tires to the four cars– Assign each car will have all 4 tires of a given brand (confounded

with differences between cars, drivers, and driving conditions)– Assign each car will have one tire of each brand

Wheel Position

Car

1 2 3 4

LF A B A B

RF B A B A

LR D C D C

RR C D C D

(poor design because brands A and B would be used only on the front of each car, and brands C and D would be used only on the rear positions. Brand effect would be confounded with the position effect.

Page 4: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.1 A Simple Example of Experimental Design Principles

• Logical Design:– Each brand is used once at each position, as well as once with

each car.Wheel

PositionCar

1 2 3 4

LF A B C D

RF B A D C

LR C D A B

RR D C B A

Page 5: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.2 Principles of Experimental Design

• The need to have processes in a state of statistical control when designed experiments are carried out.

• It is desirable to use experimental design and statistical process control methods together.

• General guidelines on the design of experiments:1. Recognition of and statement of the problem2. Choice of factors and levels3. Selection of the response variable(s)4. Choice of experimental design5. Conduction of the experiment6. Data analysis7. Conclusions and recommendations

• The levels of each factor used in an experimental run should be reset before the next experimental run.

Page 6: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.3 Statistical Concepts inExperimental Design: Example

• Assume that the objective is to determine the effect of two different levels of temperature on process yield, where the current temperature is 250F and the experimental setting is 300F.

• Assume that temperature is the only factor that is to be varied.

Page 7: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.3 Statistical Concepts inExperimental Design: Example

Day 250F 300FM 2.4 2.6Tu 2.7 2.4W 2.2 2.8Th 2.5 2.5F 2 2.2

M 2.5 2.7Tu 2.8 2.3W 2.9 3.1Th 2.4 2.9F 2.1 2.2

Page 8: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.3 Statistical Concepts inExperimental Design: Example

Observations:•Neither temperature setting is uniformly superior to the other over the entire test period.•The fact that the lines are fairly close together would suggest that increasing temperature may not have a perceptible effect on the process yield.•The yield at each temperature setting is the lowest on Friday of each week.•There is considerable variability within each temperature setting.

Page 9: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.4 t-Tests

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13.4.1 Exact t-Test

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Page 11: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.4.1 Exact t-TestExample

250F 300F

Mean 2.45 2.57

Variance 0.0872 0.0934

H0: 1=2

H1: 1<2

Page 12: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.4.1 Assumptions for Exact t-Test

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13.4.2 Approximate t-Test

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13.4.3 Confidence Intervals for Differences

Page 15: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5 Analysis of Variance (ANOVA)for One Factor

• Experimental Variable: Factor (e.g. Temperature)• Values of Experimental Variable: Levels (250, 300)• Output Variable: Effect (yield)• Distinguish “between” variation from “within” variation

Page 16: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5 Analysis of Variance (ANOVA)for One Factor: Example

Day 250F 300F SS(Within)M 2.4 2.6 0.0025 0.0009Tu 2.7 2.4 0.0625 0.0289W 2.2 2.8 0.0625 0.0529Th 2.5 2.5 0.0025 0.0049F 2.0 2.2 0.2025 0.1369M 2.5 2.7 0.0025 0.0169Tu 2.8 2.3 0.1225 0.0729W 2.9 3.1 0.2025 0.2809Th 2.4 2.9 0.0025 0.1089F 2.1 2.2 0.1225 0.1369

0.785 0.841 1.626Avg. 2.45 2.57 0.0036 0.0036 0.0072

Page 17: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5 Analysis of Variance (ANOVA)for One Factor: Example

Anova: Single Factor

SUMMARYGroups Count Sum Average Variance250F 10 24.5 2.45 0.087222222300F 10 25.7 2.57 0.093444444

ANOVASource of Variation SS df MS F P-value F critBetween Groups 0.072 1 0.0727 0.797 0.3838 4.4139Within Groups 1.626 18 0.0903

Total 1.698 19

Output from Excel

Page 18: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5 Analysis of Variance (ANOVA)for One Factor: Example

Output from Minitab

One-way ANOVA: Yield versus Temp

Source DF SS MS F PTemp 1 0.0720 0.0720 0.80 0.384Error 18 1.6260 0.0903Total 19 1.6980

S = 0.3006 R-Sq = 4.24% R-Sq(adj) = 0.00%

Individual 95% CIs For Mean Based on Pooled StDevLevel N Mean StDev +---------+---------+---------+---------250 10 2.4500 0.2953 (------------*-------------)300 10 2.5700 0.3057 (------------*-------------) +---------+---------+---------+--------- 2.25 2.40 2.55 2.70

Pooled StDev = 0.3006

Page 19: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5 Analysis of Variance (ANOVA)for One Factor

• The degrees of freedom for “Total” will always be the total number of data values minus one.

• The degrees of freedom for “Factor” will always be equal to the number of levels of the factor minus one.

• The degrees of freedom for “Within” will always be equal to (one less than the number of observations per level) multiplied by (the number of levels).

• The ratio of these mean squares is a random variable of an F distribution with numerator and denominator d.f.

• Assumptions of normality of the population and equality of the variances

Page 20: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.1 ANOVA for a Single Factorwith More than Two Levels

• Assume the process has three temperature settings, and data were collected over 6 weeks, with 2 weeks at each temperature setting.

Page 21: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

Day 250F 300F 350F

M 2.4 2.6 3.2

Tu 2.7 2.4 3.0

W 2.2 2.8 3.1

Th 2.5 2.5 2.8

F 2 2.2 2.5

M 2.5 2.7 2.9

Tu 2.8 2.3 3.1

W 2.9 3.1 3.4

Th 2.4 2.9 3.2

F 2.1 2.2 2.6

13.5.1 ANOVA for a Single Factorwith More than Two Levels: Example

Page 22: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.1 ANOVA for a Single Factorwith More than Two Levels: Example

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13.5.1 ANOVA for a Single Factorwith More than Two Levels

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Page 24: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.1 ANOVA for a Single Factorwith More than Two Levels

Page 25: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

Output from Excel

13.5.1 ANOVA for a Single Factorwith More than Two Levels: Example

Anova: Single Factor

SUMMARYGroups Count Sum Average Variance

250F 10 24.5 2.45 0.087222300F 10 25.7 2.57 0.093444350F 10 29.8 2.98 0.079556

ANOVASource of Variation SS df MS F P-value F crit

Between Groups 1.544667 2 0.772333 8.903928 0.001072 3.354131Within Groups 2.342 27 0.086741

Total 3.886667 29       

Page 26: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

Output from Minitab

One-way ANOVA: Yield versus Temp

Source DF SS MS F PTemp 2 1.5447 0.7723 8.90 0.001Error 27 2.3420 0.0867Total 29 3.8867

S = 0.2945 R-Sq = 39.74% R-Sq(adj) = 35.28%

Individual 95% CIs For Mean Based on Pooled StDevLevel N Mean StDev +---------+---------+---------+---------250 10 2.4500 0.2953 (-------*-------)300 10 2.5700 0.3057 (-------*------)350 10 2.9800 0.2821 (------*-------) +---------+---------+---------+--------- 2.25 2.50 2.75 3.00

Pooled StDev = 0.2945

13.5.1 ANOVA for a Single Factorwith More than Two Levels: Example

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13.5.2 Multiple Comparison Procedures13.5.3 Sample Size Determination

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Page 28: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.4 Additional Terms and Concepts in One-Factor ANOVA

• An experimental unit is the unit to which a treatment is applied (the days).

• If the temperature settings had been randomly assigned to the days, it would be a “completely randomized design.”

• Blocks: Extraneous factors that vary and have an effect on the response, but not interested.

• One should “block” on factors that could be expected to influence the response variable and randomize over factors that might be influential, but that could not be “blocked”.

Page 29: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

The cars were the blocks and the variation due to cars would be isolated. have one tire of each brand

Wheel Position

Car

1 2 3 4

LF A B A B

RF B A B A

LR D C D C

RR C D C D

13.5.4 Additional Terms and Concepts in One-Factor ANOVA

Randomized block design

Wheel Position

Car

1 2 3 4

LF A B C D

RF B A D C

LR C D A B

RR D C B A

The cars and wheel position were the blocks. Each brand is used once at each position, as well as once with each car.

Latin square design

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13.5.4 Additional Terms and Concepts in One-Factor ANOVA

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Page 31: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.4 Additional Terms and Concepts in One-Factor ANOVA

Page 32: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.5.4 Additional Terms and Concepts in One-Factor ANOVA

• The data in the temperature example were “balanced” in that there was the same number of obs for each level of the factor.

Page 33: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

• Regression and ANOVA both could be used as methods of analysis.

• Regression provides the tools for residual analysis, and the estimation of parameters.

• For fixed factors, ANOVA should be supplemented or supplanted.

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13.6 Regression Analysis of Data from Designed Experiments

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13.6 Regression Analysis of Data from Designed Experiments

Page 36: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

Page 37: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments: Example

Day 250F Res. Res^2 300F Res. Res^2 350F Res. Res^2M 2.4 -0.05 0.0025 2.6 0.03 0.0009 3.2 0.22 0.0484Tu 2.7 0.25 0.0625 2.4 -0.17 0.0289 3.0 0.02 0.0004W 2.2 -0.25 0.0625 2.8 0.23 0.0529 3.1 0.12 0.0144Th 2.5 0.05 0.0025 2.5 -0.07 0.0049 2.8 -0.18 0.0324F 2.0 -0.45 0.2025 2.2 -0.37 0.1369 2.5 -0.48 0.2304Sum -0.45 -0.35 -0.30M 2.5 0.05 0.0025 2.7 0.13 0.0169 2.9 -0.08 0.0064Tu 2.8 0.35 0.1225 2.3 -0.27 0.0729 3.1 0.12 0.0144W 2.9 0.45 0.2025 3.1 0.53 0.2809 3.4 0.42 0.1764Th 2.4 -0.05 0.0025 2.9 0.33 0.1089 3.2 0.22 0.0484F 2.1 -0.35 0.1225 2.2 -0.37 0.1369 2.6 -0.38 0.1444Sum 24.5 0.45 0.785 25.7 0.35 0.841 29.8 0.30 0.716 2.342Avg 2.45 2.57 2.98

Page 38: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

• The production is higher for the 2nd week at each temperature setting.

• The production is especially high during Wednesday of the week.

• The more ways we look at data, the more we are apt to discover.

Page 39: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

Page 40: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

Page 41: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

Page 42: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.6 Regression Analysis of Data from Designed Experiments

Page 43: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7 ANOVA for Two Factors

• Example now includes two factors: “weeks” and “temperature”.

• In a factorial design (or cross-classified design), each level of every factor is “crossed” with each level of every other factor. (If there are a levels of one factor and b levels of a second factor, there are ab combinations of factor levels.)

• In a nested factor design, one factor is “nested” within another factor.

Page 44: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7 ANOVA for Two Factors

Page 45: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.1 ANOVA with Two Factors:Factorial Designs

• Why not study each factor separately rather than simultaneously?– Interaction among factors

Page 46: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.1.1 Conditional Effects

• Factor effects are generally called main effects.• Conditional effects (simple effects): the effects of one

factor at each level of another factor.

Page 47: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.2 Effect Estimates

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13.7.2 Effect Estimates

Page 49: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.2 Effect Estimates

Page 50: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.3 ANOVA Table for Unreplicated Two-Factor Design

• When both factors are fixed, the main effects and the interaction are tested against the residual.

• When both factors are random, the main effects are tested against the interaction effect, and the interaction effect is tested against the residual.

• When one factor is fixed and the other random, the fixed factor is tested against the interaction, the random factor is tested against the residual, and the interaction is tested against the residual.

ANOVASource of Variation SS df MS F

T 0 1 0P 0 1 0TP (residual) 100 1 100Total 100 3    

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13.7.4 Yates’s Algorithm

A

Low High

B Low 10, 12, 16 8, 10, 13

High 14, 12, 15 12, 15, 16

Page 52: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

Page 53: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

A

Low High

B Low 10, 12, 16 8, 10, 13

High 14, 12, 15 12, 15, 16

Page 54: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

Page 55: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

Page 56: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

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13.7.4 Yates’s Algorithm

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13.7.4 Yates’s Algorithm

ANOVASource of Variation SS df MS F

A 2.08 1 2.08 <1B 18.75 1 18.75 3.36AB 6.75 1 6.75 1.21Residual 44.67 8 5.58Total 72.25 11    

Page 59: Chapter 13 Design of Experiments. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental

13.7.4 Yates’s Algorithm

Two-way ANOVA: Yield versus B, A

Source DF SS MS F PB 1 18.7500 18.7500 3.36 0.104A 1 2.0833 2.0833 0.37 0.558Interaction 1 6.7500 6.7500 1.21 0.304Error 8 44.6667 5.5833Total 11 72.2500

S = 2.363 R-Sq = 38.18% R-Sq(adj) = 14.99%