Design and Analysis of Experiments

Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Two-Level Factorial Designs

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Outline Introduction The 22 Design The 23 Design The general 2k Design A single Replicate of the 2k design Additional Examples of Unreplicated 2k

Designs 2k Designs are Optimal Designs The additional of center Point to the 2k Design

Introduction Special case of general factorial designs k factors each with two levels Factors maybe qualitative or quantitative A complete replicate of such design is 2k

factorial design Assumed factors are fixed, the design are

completely randomized, and normality Used as factor screening experiments Response between levels is assumed linear

The 22 Design

Factor Treatment Combination

Replication

A B I II III IV

- - A low, B low 28 25 27 80

+ - A high, B low 36 32 32 100

- + A low, B high 18 19 23 60

+ + A high, B high 31 30 29 90

The 22 Design

“-” and “+” denote the low and high levels of a factor, respectively

Low and high are arbitrary terms

Geometrically, the four runs form the corners of a square

Factors can be quantitative or qualitative, although their treatment in the final model will be different

Estimate factor effects Formulate model

With replication, use full model With an unreplicated design, use normal probability plots

Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results

The 22 Design

12

12

12

(1)2 2[ (1)]

(1)2 2[ (1)]

(1)2 2[ (1) ]

A A

n

B B

n

n

A y y

ab a bn nab a b

B y y

ab b an nab b a

ab a bABn n

ab a b

ABBATE

i j

n

kijkT

AB

B

A

SSSSSSSSSSn

yySS

nbaabSS

nbaabSS

nbaabSS

2

1

2

1 1

2...2

2

2

2

4

4)]1([

4)]1([

4)]1([

The 22 Design

The 22 Design Standard order Yates’s order

Effects (1) a b ab

A -1 +1 -1 +1

B -1 -1 +1 +1

AB +1 -1 -1 +1

Effects A, B, AB are orthogonal contrasts with one degree of freedom

Thus 2k designs are orthogonal designs

The 22 Design ANOVA table

The 22 Design Algebraic sign for calculating effects in 22 design

The 22 Design Regression model

x1 and x2 are code variable in this case

Where con and catalyst are natural variables

22110 xxy

2/)(2/)(

2/)(2/)(

2

1

lowhigh

highlow

lowhigh

highlow

catalystcatalystcatalystcatalystcatalyst

x

conconconconcon

x


Factorial Fit: Yield versus Conc., Catalyst Estimated Effects and Coefficients for Yield (coded units)Term Effect Coef SE Coef T PConstant 27.500 0.5713 48.14 0.000Conc. 8.333 4.167 0.5713 7.29 0.000Catalyst -5.000 -2.500 0.5713 -4.38 0.002Conc.*Catalyst 1.667 0.833 0.5713 1.46 0.183

S = 1.97906 PRESS = 70.5R-Sq = 90.30% R-Sq(pred) = 78.17% R-Sq(adj) = 86.66%

Analysis of Variance for Yield (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 2 283.333 283.333 141.667 36.17 0.0002-Way Interactions 1 8.333 8.333 8.333 2.13 0.183Residual Error 8 31.333 31.333 3.917 Pure Error 8 31.333 31.333 3.917Total 11 323.000




Estimated Coefficients for Yield using data in uncoded units

Term CoefConstant 28.3333Conc. 0.333333Catalyst -11.6667Conc.*Catalyst 0.333333

Estimated Coefficients for Yield using data in uncoded units

Term CoefConstant 18.3333Conc. 0.833333Catalyst -5.00000

Regression model (without interaction)

The 22 Design Response surface

The 22 Design Response surface (note: the axis of catalyst is

reversed with the one from textbook)

The 23 Design 3 factors, each at two level. Eight combinations

The 23 Design Design matrix Or geometric notation

The 23 Design Algebraic sign

22

The 23 Design -- Properties of the Table

Except for column I, every column has an equal number of + and – signs

The sum of the product of signs in any two columns is zero

Multiplying any column by I leaves that column unchanged (identity element)

23

The 23 Design -- Properties of the Table

The product of any two columns yields a column in the table:

Orthogonal design Orthogonality is an important property shared by

all factorial designs

2

A B AB

AB BC AB C AC

The 23 Design -- example Nitride etch process Gap, gas flow, and RF power

The 23 Design -- example Nitride etch process Gap, gas flow, and RF power

The 23 Design -- example

Estimated Effects and Coefficients for Etch Rate (coded units)Term Effect Coef SE Coef T PConstant 776.06 11.87 65.41 0.000Gap -101.62 -50.81 11.87 -4.28 0.003Gas Flow 7.37 3.69 11.87 0.31 0.764Power 306.12 153.06 11.87 12.90 0.000Gap*Gas Flow -24.88 -12.44 11.87 -1.05 0.325Gap*Power -153.63 -76.81 11.87 -6.47 0.000Gas Flow*Power -2.12 -1.06 11.87 -0.09 0.931Gap*Gas Flow*Power 5.62 2.81 11.87 0.24 0.819

S = 47.4612 PRESS = 72082R-Sq = 96.61% R-Sq(pred) = 86.44% R-Sq(adj) = 93.64%

Analysis of Variance for Etch Rate (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 3 416378 416378 138793 61.62 0.0002-Way Interactions 3 96896 96896 32299 14.34 0.0013-Way Interactions 1 127 127 127 0.06 0.819Residual Error 8 18020 18020 2253 Pure Error 8 18021 18021 2253Total 15 531421

Full model


Factorial Fit: Etch Rate versus Gap, Power Estimated Effects and Coefficients for Etch Rate (coded units)Term Effect Coef SE Coef T PConstant 776.06 10.42 74.46 0.000Gap -101.62 -50.81 10.42 -4.88 0.000Power 306.12 153.06 10.42 14.69 0.000Gap*Power -153.63 -76.81 10.42 -7.37 0.000


Analysis of Variance for Etch Rate (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 2 416161 416161 208080 119.71 0.0002-Way Interactions 1 94403 94403 94403 54.31 0.000Residual Error 12 20858 20858 1738 Pure Error 12 20858 20858 1738Total 15 531421

Reduced model

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R2 and adjusted R2

R2 for prediction (based on PRESS)

52

5

25

5.106 10 0.96085.314 10

/ 20857.75 /121 1 0.9509/ 5.314 10 /15

Model

T

E EAdj

T T

SSR

SSSS dfRSS df

2Pred 5

37080.441 1 0.93025.314 10T

PRESSRSS

The 23 Design – example -- Model Summary Statistics for Reduced Model



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The Regression Model

32

Cube Plot of Ranges

What do the large ranges

when gap and power are at the high level tell

you?

33

The General 2k Factorial Design

There will be k main effects, and

two-factor interactions2

three-factor interactions3

1 factor interaction

k

k

k

34

The General 2k Factorial Design Statistical Analysis

35

The General 2k Factorial Design

Statistical Analysis

36

Unreplicated 2k Factorial Designs

These are 2k factorial designs with one observation at each corner of the “cube”

An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k

These designs are very widely used Risks…if there is only one observation at each

corner, is there a chance of unusual response observations spoiling the results?

Modeling “noise”?

37

If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

More aggressive spacing is usually best


38

Lack of replication causes potential problems in statistical testing Replication admits an estimate of “pure error” (a

better phrase is an internal estimate of error) With no replication, fitting the full model results

in zero degrees of freedom for error Potential solutions to this problem

Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels,

1959)


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A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin

The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate

Experiment was performed in a pilot plant

Unreplicated 2k Factorial Designs -- example

40


41


42

Unreplicated 2k Factorial Designs – example –full model

43

Unreplicated 2k Factorial Designs -- example –full model

44

Unreplicated 2k Factorial Designs -- example –full model

45

Unreplicated 2k Factorial Designs -- example –reduced model

Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units)Term Effect Coef SE Coef T PConstant 70.063 1.104 63.44 0.000Temperature 21.625 10.812 1.104 9.79 0.000Conc. 9.875 4.938 1.104 4.47 0.001Stir Rate 14.625 7.312 1.104 6.62 0.000Temperature*Conc. -18.125 -9.062 1.104 -8.21 0.000Temperature*Stir Rate 16.625 8.313 1.104 7.53 0.000


Analysis of Variance for Filtration (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 3 3116.19 3116.19 1038.73 53.23 0.0002-Way Interactions 2 2419.62 2419.62 1209.81 62.00 0.000Residual Error 10 195.12 195.12 19.51 Lack of Fit 2 15.62 15.62 7.81 0.35 0.716 Pure Error 8 179.50 179.50 22.44Total 15 5730.94

46


47


48


49

Unreplicated 2k Factorial Designs -- example –Design projection

Since factor B is negligible, the experiment can be interpreted as a 23 factorial design with factors A, C, D.

2 replicates

50


51


Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units)Term Effect Coef SE Coef T PConstant 70.063 1.184 59.16 0.000Temperature 21.625 10.812 1.184 9.13 0.000Conc. 9.875 4.938 1.184 4.17 0.003Stir Rate 14.625 7.312 1.184 6.18 0.000Temperature*Conc. -18.125 -9.062 1.184 -7.65 0.000Temperature*Stir Rate 16.625 8.313 1.184 7.02 0.000Conc.*Stir Rate -1.125 -0.562 1.184 -0.48 0.647Temperature*Conc.*Stir Rate -1.625 -0.813 1.184 -0.69 0.512

S = 4.73682 PRESS = 718R-Sq = 96.87% R-Sq(pred) = 87.47% R-Sq(adj) = 94.13%

Analysis of Variance for Filtration (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 3 3116.19 3116.19 1038.73 46.29 0.0002-Way Interactions 3 2424.69 2424.69 808.23 36.02 0.0003-Way Interactions 1 10.56 10.56 10.56 0.47 0.512Residual Error 8 179.50 179.50 22.44 Pure Error 8 179.50 179.50 22.44Total 15 5730.94

52

Dealing with Outliers Replace with an estimate Make the highest-order interaction zero In this case, estimate cd such that ABCD =

0 Analyze only the data you have Now the design isn’t orthogonal Consequences?

53

Duplicate Measurements on the Response

Four wafers are stacked in the furnace Four factors: temperature, time, gas flow, and

pressure. Response: thickness Treated as duplicate not replicate Use average as the response

54


55


Stat DOE Factorial Pre-process Response for Analyze

56


Stat DOE Factorial Analyze Factorial Design

57


Factorial Fit: average versus Temperature, Time, Pressure Estimated Effects and Coefficients for average (coded units)Term Effect Coef SE Coef T PConstant 399.188 1.049 380.48 0.000Temperature 43.125 21.562 1.049 20.55 0.000Time 18.125 9.062 1.049 8.64 0.000Pressure -10.375 -5.187 1.049 -4.94 0.001Temperature*Time 16.875 8.438 1.049 8.04 0.000Temperature*Pressure -10.625 -5.312 1.049 -5.06 0.000


Analysis of Variance for average (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 3 9183.7 9183.69 3061.23 173.81 0.0002-Way Interactions 2 1590.6 1590.62 795.31 45.16 0.000Residual Error 10 176.1 176.12 17.61 Lack of Fit 2 60.6 60.62 30.31 2.10 0.185 Pure Error 8 115.5 115.50 14.44Total 15 10950.4

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59


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The 2k design and design optimality

The model parameter estimates in a 2k design (and the effect estimates) are least squares estimates. For example, for a 22 design the model is

211222110 xxxxy

61

The four observations from a 22 design


412210

312210

212210

112210

)1)(1()1()1()1)(1()1()1()1)(1()1()1(

)1)(1()1()1()1(

abba

In matrix form: XY

62

The matrix is diagonal – consequences of an orthogonal design

X X

The regression coefficient estimates are exactly half of the ‘usual” effect estimates

The “usual” contrasts


YXXX '1' )(

1

0

14

2

12

ˆ

4 0 0 0 (1)0 4 0 0 (1)0 0 4 0 (1)0 0 0 4 (1)

(1)4ˆ (1) (

ˆ (1)1ˆ (1)4

(1)ˆ

a b aba ab bb ab a

a b ab

a b ab

a b ab a ab ba ab bb ab a

a b ab

-1β = (X X) X y

I

1)4

(1)4

(1)4

b ab a

a b ab

63


The matrix X’X has interesting and useful properties:

2 1

2

ˆ( ) (diagonal element of ( ) )

4

V

X X

|( ) | 256 X X

Minimum possible value for a four-run

designMaximum possible value for a four-run

design

Notice that these results depend on both the design that you have chosen and the model


The 22 design is called D-optimal design In fact, all 2k design is D-optimal design for

fitting first order model with interaction. Consider the variance of the predicted

response in the 22 design:


21 2

1 2 1 2

22 2 2 2

1 2 1 2 1 2

1 2

21 2

1 2

2

1 2

ˆ[ ( , )][1, , , ]

ˆ[ ( , )] (1 )4

The maximum prediction variance occurs when 1, 1ˆ[ ( , )]

The prediction variance when 0 is

ˆ[ ( , )]

V y x xx x x x

V y x x x x x x

x x

V y x xx x

V y x x

-1x (X X) xx

4What about prediction variance over the design space?average


1 12

1 2 1 21 1

1 12 2 2 2 2

1 2 1 2 1 21 1

2

1 ˆ[ ( , ) = area of design space = 2 4

1 1 (1 ) 4 4

49

I V y x x dx dx AA

x x x x dx dx

The 22 design is called G-optimal design In fact, all 2k design is G-optimal design for

fitting first order model with interaction.

Minimize the maximum prediction variance


The 22 design is called I-optimal design In fact, all 2k design is I-optimal design for

fitting first order model with interaction.

Smallest possible value of the average prediction variance


The Minitab provide the function on “Select Optimal Design” when you have a full factorial design and are trying to reduce the it to a partial design or “fractional design”.

It only provide the “D-optimal design” One needs to have a full factorial design first

and the choose the number of data points to be allowed to use.

69

These results give us some assurance that these designs are “good” designs in some general ways

Factorial designs typically share some (most) of these properties

There are excellent computer routines for finding optimal designs


70

Addition of Center Points to a 2k Designs

Based on the idea of replicating some of the runs in a factorial design

Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models:

01 1

20

1 1 1

First-order model (interaction)

Second-order model

k k k

i i ij i ji i j i

k k k k

i i ij i j ii ii i j i i

y x x x

y x x x x

Quadratic effects

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When adding center points, we assume that the k factors are quantitative.

Example on 22 design

72


Five point:(-,-),(-,+),(+,-),(+,+), and (0,0).

nF=4 and nC=4 Let be the average of the

four runs at the four factorial points and let be the average of nC run at the center point.

Fy

Cy

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If the difference of is small, the center points lie on or near the plane passing through factorial points and there is no quadratic effects.

The hypotheses are:

CF yy

01

11

: 0

: 0

k

iii

k

iii

H

H

74


2

Pure Quad( )F C F C

F C

n n y ySSn n

Test statistics:

with one degree of freedom

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Addition of Center Points to a 2k Designs -- example

In example 6.2, it is a 24 factorial. By adding center points x1=x2=x3=x4=0, four

additional responses (filtration rates) are : 73, 75, 66,69.

So =70.75 and =70.06.Cy Fy

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Term Effect Coef SE Coef T PConstant 70.063 1.008 69.52 0.000Temperature 21.625 10.812 1.008 10.73 0.002Pressure 3.125 1.562 1.008 1.55 0.219Conc. 9.875 4.937 1.008 4.90 0.016Stir Rate 14.625 7.312 1.008 7.26 0.005Temperature*Pressure 0.125 0.063 1.008 0.06 0.954Temperature*Conc. -18.125 -9.063 1.008 -8.99 0.003Temperature*Stir Rate 16.625 8.313 1.008 8.25 0.004Pressure*Conc. 2.375 1.188 1.008 1.18 0.324Pressure*Stir Rate -0.375 -0.187 1.008 -0.19 0.864Conc.*Stir Rate -1.125 -0.563 1.008 -0.56 0.616Temperature*Pressure*Conc. 1.875 0.937 1.008 0.93 0.421Temperature*Pressure*Stir Rate 4.125 2.063 1.008 2.05 0.133Temperature*Conc.*Stir Rate -1.625 -0.813 1.008 -0.81 0.479Pressure*Conc.*Stir Rate -2.625 -1.312 1.008 -1.30 0.284Temperature*Pressure*Conc.*Stir Rate 1.375 0.687 1.008 0.68 0.544Ct Pt 0.687 2.253 0.31 0.780

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Analysis of Variance for Filtration (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 4 3155.25 3155.25 788.813 48.54 0.0052-Way Interactions 6 2447.88 2447.88 407.979 25.11 0.0123-Way Interactions 4 120.25 120.25 30.062 1.85 0.3204-Way Interactions 1 7.56 7.56 7.562 0.47 0.544 Curvature 1 1.51 1.51 1.512 0.09 0.780Residual Error 3 48.75 48.75 16.250 Pure Error 3 48.75 48.75 16.250Total 19 5781.20

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If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model

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Use current operating conditions as the center point

Check for “abnormal” conditions during the time the experiment was conducted

Check for time trends Use center points as the first few runs when

there is little or no information available about the magnitude of error

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Center Points and Qualitative Factors

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Design and Analysis of Experiments

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