chapter 13 design of experiments

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

Car1 2 3 4

LF A B A BRF B A B ALR D C D CRR 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 DRF B A D CLR C D A BRR 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.


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


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

• The t statistic is of the general form

where is the parameter to be estimated is the sample statistic (estimator of ) is the estimator of the std. deviation of

• Let = the true average yield using 250F = the true average yield using 300F =

• Then (if and are known)

(13.1)

13.4.1 Exact t-Test

• The exact t-test is of the form

where is the square root of the estimate of the (assumed) common variance ()

• reduces to a simple average of and when • Degrees of freedom =

(13.2)

13.4.1 Exact t-TestExample

𝑠𝑝2=

(𝑛1−1 ) 𝑠12+(𝑛2−1 )𝑠22

(𝑛1+𝑛2−2 )=9 (.0872 )+9(.0934 )

18=.0903

250F 300FMean 2.45 2.57

Variance 0.0872 0.0934

Prob(t<-.893)19=.1916

H0: 1=2

H1: 1<2

13.4.1 Assumptions for Exact t-Test

• should be checked. (This assumption is not crucial when n1=n2.)

• The two samples are independent.• The observations are independent within each sample.

13.4.2 Approximate t-Test

• If n1 and n2 differ considerably and is unknown, an approximate t-test is used

where the degrees of freedom is calculated as (13.3)

13.4.3 Confidence Intervals for Differences

• 100(1-)% Confidence Bound

• 100(1-)% Two-sided Confidence Interval

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


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


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 350FM 2.4 2.6 3.2Tu 2.7 2.4 3.0W 2.2 2.8 3.1Th 2.5 2.5 2.8F 2 2.2 2.5

M 2.5 2.7 2.9Tu 2.8 2.3 3.1W 2.9 3.1 3.4Th 2.4 2.9 3.2F 2.1 2.2 2.6

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


• Sum of squares for factor (Temp.)

where represents the total of the obs for the ith level, is the number of levels of the factor, represents the number of obs for the ith level, denotes the grand total of all obs.N is the number of total obs.

• For the example

(13.4)


• Total sum of squares

where represents ith obs.N is the number of total obs.

• For the example

Output from Excel


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 MinitabOne-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


13.5.2 Multiple Comparison Procedures13.5.3 Sample Size Determination

where represents number of levels of a factor is the std dev of the obs. denotes the minimum pairwise difference that one wishes to detect with probability 0.90

(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

Car1 2 3 4

LF A B A BRF B A B ALR D C D CRR C D C D


Randomized block design

Wheel Position

Car1 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


• Regression model for One-factor ANOVA:

where j denotes the jth level of the single factor represents the ith obs for the jth level represents the effect of the jth level is a constant represents the error term

• If the effects were all the same,

• F-test determines whether the appropriate model is (13.6) or (13.7)

(13.6)

(13.7)


• Factors are generally classified as fixed (250F, 300 F, 350 F) or random (any number from a population)

• Data in one-factor ANOVA are analyzed in the same way regardless of whether the factor is fixed or random, but the interpretation does differ.

• is a constant if the factor is fixed, and a random variable if the factor is random.

• The error term is NID(0, 2) in both cases.• are assumed to be normally distributed in both cases• are not independent in the random-factor case.


• 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.


• The least squares estimator in regression analysis resulted from minimizing the sum of squared errors.

so that

• Assumption: the levels of the factor are fixed, balanced data.

(13.8)


• The effect can be thought as a deviation from the overall mean .

where is the expected value of the response variable for the jth level of the factor is the average of components

So • This restriction on the components allows and each to

be estimated using least squares.


• Minimizing produces and where denotes the average of all obs

is the avg of the obs for the jth factor level• Then,

• The residuals are defined as

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


• 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.

13.7 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

• Model for nested factor design

where (temperature settings) (the week) (replicate factor, days)

indicates j factor (week) is nested within factor (temperature)

indicates that the replicate factor is nested within each ( , ) combination𝑖 𝑗

• The nested factor design is also called “hierarchical design” and is used for estimating components of variance.

13.7.1 ANOVA with Two Factors:Factorial Designs

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

T1 T20

5

10

15

20

25

30

35

P1P2

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

•Temperature effect: (Effect of changing Temp from T1 to T2 at P1 and P2.

• Pressure Effect:

T1 T20

5

10

15

20

25

P1P2


•Interaction effect:

T1 T20

5

10

15

20

25

30

35

P1P2


•Temperature effect: (Effect of changing Temp from T1 to T2 at P1 and P2.

• Pressure Effect:

• Interaction Effect• T=P=0, TP=-10

T1 T20

5

10

15

20

25

P1P2

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

• For any design, where is the number of factors and 2 is the number of levels of each factor, any treatment combination can be represented by the presence or absence of each of lowercase letters, where presence would denote the high level, and absence the low level.

• For example, if = (A high, B high); = (A high, B low); = (A low, B high); = (A low, B low)

ALow High

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

High 14, 12, 15 12, 15, 16


• The procedure is initiated by writing down the treatment combinations in standard order:– 1 is always written first– The other combinations are listed relative to the natural

ordering, including combinations of letters

• The procedure can be employed using either the totals or averages for each treatment combination.


Treatment Combination Total (1) (2) SS

38

31

41

43

ALow High

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

High 14, 12, 15 12, 15, 16


• The columns designated by (1) and (2) are columns in which addition and subtraction are performed for each ordered pair of numbers. (In general, there will be such columns for factors.)

• Specifically, the number in each pair are first added, and then the first number in each pair is subtracted from the second number.



38 69=38+31

31 84=41+43

41 -7=31-38

43 2=43-41


38 69 153=69+84

31 84 -5=-7+2

41 -7 15=84-69

43 2 9=2-(-7)


• The process is continued on each new column that is created until the number of such columns is equal to the number of factors.

• The last column that is created by these operations is used to compute the sum of squares for each effect.

• Specifically, each number (except the first) is squared and divided by the number of replicates times .



38 69 153

31 84 -5 (-5)2/(3*22)=2.08 (A)

41 -7 15 (15)2/(3*22)=18.75 (B)

43 2 9 (9)2/(3*22)=6.75 (AB)


• The first number in the last column is actually the sum of all of the obs. (

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

𝐹 1,8 ,.95=5.32


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%

chapter 13 design of experiments

Documents

illogical design

experimental setting

experimental run

temperature setting

current temperature

increasing temperature

brand effect

state of statistical