4. design of experiments (doe) (the 2k factorial...

Hae-Jin ChoiSchool of Mechanical Engineering,

Chung-Ang University

4. Design of Experiments (DOE)(The 2k Factorial Designs)

1Complex Sys. Des.

Example: Golfing

2

— How to improve my score in Golfing?— Practice!!!

— Other than that?— Type of driver used (oversized or regular sized)— Type of ball (2 piece or 3 piece)— Walking or riding cart— Drinking water or beer— Etc…

What combination of the factors is the best for me?

Complex Sys. Des.

How to find my best condition?

3

— One-factor-at-a time strategy

Any Problem??

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Interaction effect between the factors

4

— Interaction effect between type of driver and beverage

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Factorial Design of Experiments

5

Two factors with 2 level for each factor

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Factorial Design of Experiments

6

Three factors

Four factors

Any Problem??Complex Sys. Des.

Fractional Factorial Design of Experiments

7

16 experiments -> 8 experiments

Question for the semesterHow to effectively reduce the number of experiments?How to analyze the results of experiments?

Complex Sys. Des.

Introduction to 2k Factorial Designs

— Special case of the general factorial design; k factors, all at two levels

— The two levels are usually called low and high (they could be either quantitative or qualitative)

— Very widely used in industrial experimentation— Form a basic “building block” for other very useful

experimental designs— Useful for factor screening

8Complex Sys. Des.

Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

9Complex Sys. Des.

The Simplest Case: The 22

“-” 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

10Complex Sys. Des.

Notation of the 2k Designs

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— A special notation is used to represent the runs. In general, a run is represented by a series of lower case letters. If a letter is present, then the corresponding factor is set at the high level in that run; if it is absent, the factor is run at its low level. For example, run aindicates that factor A is at the high level and factor B is at the low level. The run with both factors at the low level is represented by (1).

— This notation is used throughout the 2k design series. For example, the run in a 24 with A and C at the high level and B and D at the low level is denoted by ac.

Estimation of Factor Effects

12

12

12

(1)2 2[ (1)]

(1)2 2[ (1)]

(1)2 2[ (1) ]

A A

n

B B

n

n

A y yab a b

n nab a b

B y yab b a

n nab b a

ab a bABn n

ab a b

+ -

+ -

= -

+ += -

= + - -

= -

+ += -

= + - -

+ += -

= + - -

The letters (1), a, b, and ab also represent the totals of all nobservations taken at these design points.

12Complex Sys. Des. Orthogonal Design

Contrasts in the 22

— Recall contrasts

— Effect = Contrast/2— Sum of Square of Contrasts

.1

= ya

i ii

C c=å

2

.1

2

1

c y =

1

a

i ii

c a

ii

SSc

n

=

=

æ ö÷ç ÷ç ÷ç ÷è øå

å

1 [ (1)]

A A B A B A B A BC y y y y

ab a bn

+ + + - - + - -= + - -

= + - -

[ ]

( )

2

2

2

1 [ (1)] (1)1 4(4)

4 /

A

ab a b ab a bnSSn

nContrast

n

é ù+ - -ê ú + - -ë û= =

=

13Complex Sys. Des.

Sum of Squares of the 22 Designs

— The analysis of variance is completed by computing the total sum of squares SST (with 4n-1 degrees of freedom) as usual, and obtaining the error sum of squares SSE [with 4(n-1) degrees of freedom] by subtraction.

SS a ab bn

SS b ab an

SS ab a bn

A

B

AB

=+ - -

=+ - -

=+ - -

[ ( )]

[ ( )]

[ ( ) ]

14

1414

2

2

2

14Complex Sys. Des.

ANOVA of the Chemical Processing

The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

15Complex Sys. Des.

Regression Model


— Regression model for 2k Designs

— Where x1 is coded variable of Factor A and x2 is coded variable of Factor B— Low lever = -1 and High level = +1

— Relationship between natural and coded variables

1 1 2 2 3 1 2oy x x x xb b b b e= + + + +

( )1( ) / 2

/ 2A A Ax

A A

+ -

+ -

- +=

-

Regression Model for Chemical Processing


— Since interaction effect is very small, the regression model employed is

— where x1 is coded variable of the reactant concentration and x2 is coded variable of the amount of catalyst

1 1 2 2oy x xb b b e= + + +

( )1

( ) / 2/ 2

(25 15) / 2 20(25 15) / 2 5

high low

high low

Conc Conc Concx

Conc Conc

Conc Conc

- +=

-

- + -= =

-

21.5

0.5Catalystx -

=

Regression Model for Chemical Processing


— Estimating of the regression model, using least square method

— We will return to least square method in response surface method

— Regression model with coded factors is

— where 27.5 is grand average of all observation, is one-half of the corresponding factor effect estimates

— Regression model with uncoded factors

0 1 2, ,b b b

1 28.33 5.00ˆ 27.5

2 2y x x

æ ö æ ö-÷ ÷ç ç= + +÷ ÷ç ç÷ ÷ç çè ø è ø

8.33 20 5.00 1.5ˆ 27.52 5 2 0.5

18.33 0.8333 5.00

Conc Catalysty

Conc Catalyst

æ öæ ö æ öæ ö- - -÷ ÷ ÷ ÷ç ç ç ç= + +÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè øè ø è øè ø

= + -

1 2ˆ ˆ,b b

Residual Analysis of Chemical Processing


— Residual

For example

ˆy ye= -

1 28 25.835e = -8.33 5.00ˆ 27.5 ( 1) ( 1)

2 2y

æ ö æ ö-÷ ÷ç ç= + - + -÷ ÷ç ç÷ ÷ç çè ø è ø

Review of Analysis Procedure— Estimate factor effects

— Main effects, interaction effects— Formulate model

— 22 design example — Statistical testing (ANOVA)— Refine the model

— Chemical processing example— Regression model estimation

— By Least Square Method — Analyze residuals (graphical)

— Normal probability plot of residuals— Interpret results

1 1 2 2 3 1 2oy x x x xb b b b e= + + + +

1 1 2 2oy x xb b b e= + + +

1 1 2 2ˆ ˆ ˆˆ oy x xb b b= + +

20Complex Sys. Des.

The 23 Factorial Design

21Complex Sys. Des.

Factor Effect of the 23 Designs


— 3 factors, each at two levels — 8 factor-level combinations— 3 main effects: A,B,C— 3 two-factor interactions:

AB, AC,BC— 1 three-factor interaction:

ABC



— Main effect of A

— Main effect of B

— Main effect of C

[ ]1 (1)4

A a ab ac abc b c bcn

= + + + - - - -

[ ]1 (1)4

B b ab bc abc a c acn

= + + + - - - -

[ ]1 (1)4

C c ac bc abc a b abn

= + + + - - - -



— Interaction effect of AB

— The same approach can be applied for the interaction effect of BC and AC

1 ( ) ( )2

1 1( ) [ (1)] [ ]2 21 1( ) [ ] [ ]

2 2

1 [ (1) ]4

high low

low

high

AB AB C AB C

where

AB C ab a bn n

AB C abc c ac bcn n

Therefore

AB ab abc c b a bc acn

é ù= +ê úë û

= + - +

= + - +

= + + + - - - -



— Interaction effect of ABC is defined as the average difference between the AB interaction at the two different level of C

— How to memorize the sign of coefficients?

[ ]

( ( ) ( )

[ ]

1 ( ) ( )21 1 1 1 1 ) ( ) - (1) 2 2 2 2n 21 - - + - + + -(1)

4

ABC AB C high AB C low

abc c ac ab ab a bn n n

abc bc ac c ab b an

= -

ìé ù é ùïïê ú ê ú= + - + + - +íïê ú ê úë û ë ûïî

=


26Complex Sys. Des.

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)— 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 ABAB BC AB C AC´ =

´ = =

27Complex Sys. Des.

Effects, Sum of Squares, and Contrast


— The 23 Designs— Effect = Contrast/4— Sum of squares = n(Contrast)2/8

— Contrast for factor A

— Main effect of factor A

— Sum of Square of factor A

[ ]1 (1)AContrast a ab ac abc b c bcn

= + + + - - - -

[ ]1/ 4 (1)4AA Contrast a ab ac abc b c bc

n= = + + + - - - -

[ ]22 1( ) / 8 (1)8A ASS n Contrast a ab ac abc b c bcn

= = + + + - - - -

Plasma Etching Process— A 23 factorial design was used to

develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the electrodes, the gas flow (C2F6 is used as the reactant gas), and the RF power applied to the cathode. Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride (Å/m)

A = gap, B = Flow, C = Power, y = Etch Rate

29Complex Sys. Des.

Plasma Etching Process


Plasma Etching ProcessWafer

Gap Gas flow Power

Etch rate

ANOVA Summary – Full Model

Important effects by A, C, AC,

31Complex Sys. Des.

The Regression Model with Reduced Factors

32Complex Sys. Des.

The Regression Model with Reduced Factors

33Complex Sys. Des.

Cube Plot of Ranges

What do the large ranges when gap and power are at the high level tell

you?

34Complex Sys. Des.

The General 2k Factorial Design

35Complex Sys. Des.

1

22

( )2

k

k

ContrastEffect

n ContrastSS

-=

=

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?

36Complex Sys. Des.

Spacing of Factor Levels in the Unreplicated 2kFactorial Designs

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

37Complex Sys. Des.

Unreplicated 2k Factorial Designs— 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)

38Complex Sys. Des.

Example of an Unreplicated 2k Design— A chemical product is produced in a pressure vessel. A factorial

experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product .

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

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

— Experiment was performed in a pilot plant

39Complex Sys. Des.

The Resin Plant Experiment

40Complex Sys. Des.

Contrast Constants for the 24 Design

41Complex Sys. Des.

Estimates of the Effects

42Complex Sys. Des.

ANOVA Summary for the Model as a 23 in Factors A, C, and D

43Complex Sys. Des.

The Regression Model

44Complex Sys. Des.

Experiments with the larger number of factors— The system is usually dominated by the main effects and

low-order interactions. Higher interactions are usually negligible.

— When the number of factors is larger than 3 or 4, a common practice is to run only a single replicate design and then pool the higher order interactions as an estimate of error.

— Normal probability plot of the effects may be useful— If none of the effects is significant, then the estimates will behave like a

random sample drawn from a normal distribution with zero mean, and the plotted effects will lie approximately along a straight line.

— Those effects that do not plot on the line are significant factors.

45Complex Sys. Des.

4. design of experiments (doe) (the 2k factorial...

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