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Chapter 13
Design of Experiments
Introduction
• “Listening” or passive statistical tools: control charts.
• “Conversational” or active tools: Experimental design.– Planning of experiments– A sequence of experiments
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.
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
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.
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.
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
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.
13.4 t-Tests
(13.1)
13.4.1 Exact t-Test
(13.2)
13.4.1 Exact t-TestExample
250F 300F
Mean 2.45 2.57
Variance 0.0872 0.0934
H0: 1=2
H1: 1<2
13.4.1 Assumptions for Exact t-Test
13.4.2 Approximate t-Test
(13.3)
13.4.3 Confidence Intervals for Differences
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
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
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
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
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
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.
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
13.5.1 ANOVA for a Single Factorwith More than Two Levels: Example
13.5.1 ANOVA for a Single Factorwith More than Two Levels
(13.4)
13.5.1 ANOVA for a Single Factorwith More than Two Levels
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
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
13.5.2 Multiple Comparison Procedures13.5.3 Sample Size Determination
(13.5)
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”.
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
13.5.4 Additional Terms and Concepts in One-Factor ANOVA
(13.6)
(13.7)
13.5.4 Additional Terms and Concepts in One-Factor ANOVA
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.
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.
13.6 Regression Analysis of Data from Designed Experiments
(13.8)
13.6 Regression Analysis of Data from Designed Experiments
13.6 Regression Analysis of Data from Designed Experiments
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
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.
13.6 Regression Analysis of Data from Designed Experiments
13.6 Regression Analysis of Data from Designed Experiments
13.6 Regression Analysis of Data from Designed Experiments
13.6 Regression Analysis of Data from Designed Experiments
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.
13.7 ANOVA for Two Factors
13.7.1 ANOVA with Two Factors:Factorial Designs
• Why not study each factor separately rather than simultaneously?– Interaction among factors
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.
13.7.2 Effect Estimates
13.7.2 Effect Estimates
13.7.2 Effect Estimates
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
13.7.4 Yates’s Algorithm
A
Low High
B Low 10, 12, 16 8, 10, 13
High 14, 12, 15 12, 15, 16
13.7.4 Yates’s Algorithm
13.7.4 Yates’s Algorithm
A
Low High
B Low 10, 12, 16 8, 10, 13
High 14, 12, 15 12, 15, 16
13.7.4 Yates’s Algorithm
13.7.4 Yates’s Algorithm
13.7.4 Yates’s Algorithm
13.7.4 Yates’s Algorithm
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
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%
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