&Four DOE Class_90a 1

&5 二因子實驗設計 二因子無交互作用

&Four DOE Class_90a 2

二因子有交互作用

&Four DOE Class_90a 3

One-factor at a time 之方法

&Four DOE Class_90a 4

&Four DOE Class_90a 5

二因子實驗設計之模式

0 oneleast at :

, allfor ,0: 3.

0 oneleast at :

0: 2.

0 oneleast at :

0: 1.

:

1

0

1

210

1

210

ij

ij

j

b

i

a

H

jiH

H

H

H

H

TestingHypothesis

&Four DOE Class_90a 6

Data Sheet

&Four DOE Class_90a 7

ANOVA 表 – Two-Factor Factorial

&Four DOE Class_90a 8

a

i

b

j

ijsubtotals abn

y

n

ySS

1 1

2...

2.

&Four DOE Class_90a 9

Example

&Four DOE Class_90a 10

&Four DOE Class_90a 11

決策模式： 因為 F0(Primer Types) = 28.63 > F0.05,2,12 = 3.89

F0(Application Methods) = 61.38 > F0.05,1,12 = 4.75

所以此二因子對黏著力皆有顯著影響。 但 F0(Interaction) = 1.5 < F0.05,2,12 = 3.89 ，所以此二因子的交互作

用對黏著力無明顯之影響。

&Four DOE Class_90a 12

Another Example

&Four DOE Class_90a 13

ANOVA Results

Computer Output (Model Adequacy Checking)

&Four DOE Class_90a 14

Multiple Comparisons

&Four DOE Class_90a 15

The Three-factor ANOVA Model

&Four DOE Class_90a 16

The ANOVA TABLE

&Four DOE Class_90a 17

An Example

&Four DOE Class_90a 18

The Results

Computer Results

&Four DOE Class_90a 19

Fitting Response Curve

&Four DOE Class_90a 20

Example 5-5

Computer Results (Contour Plot and Response Surface)

&Four DOE Class_90a 21

&6 2k 因子階層設計 k 個因子，每個因子 2 個水準 (+,-) ，共 2k次實驗（當

n = 1 時）。

在因子數不多的狀況下，常用於實驗初期，來了解因子對反應變數之可能影響。

只能看出因子對反應變數之線性作用 (linear effect) ，無法預估高階曲面作用。

&Four DOE Class_90a 22

The 22 Factorial Design

&Four DOE Class_90a 23

The Calculation (I)

Effects

Contrast

baabn

ababn

AB

ababn

baabn

B

baabn

ababn

A

12

11

2

1

12

11

2

1

12

11

2

1

12

1

1

1

kA

AB

B

A

n

ContrastA

baabContrast

ababContrast

baabContrast

&Four DOE Class_90a 24

Sum of Square Errors

The Calculation (II)

ABBATE

ijkijkT

kAB

AB

kB

B

kA

A

SSSSSSSSSSn

yyyySS

n

ContrastSS

n

ContrastSS

n

Contrast

n

baabSS

4

2

2

24

1

222

2

2

22

&Four DOE Class_90a 25

The AVONA Table

&Four DOE Class_90a 26

The Regression Model Since the effect of AB is not significant, the

regression model would be

22110 xxy

The estimates of these coefficients are

2

2

2

^

^

1

^

0

B

A

Effect

Effect

y

Why?

&Four DOE Class_90a 27

The Residuals

Residuals at x1=1 and x2 = -1

Computer Output

&Four DOE Class_90a 28

23 因子階層設計

&Four DOE Class_90a 29

23 因子階層設計 _ 符號表

&Four DOE Class_90a 30

計算 各因子之 Contrast = (Sign) (Treatment Combination)

ContrastA = -(1)+a-b+ab-c+ac-bc+abc ContrastB = -(1)-a+b+ab-c-ac+bc+abc ContrastAB =

各因子之效用 Effect = Contrast / (n 2k-1)

各因子之 Sum of Square = Contrast / (n 2k) SSA = {-(1)+a-b+ab-c+ac-bc+abc}2 / 8n

&Four DOE Class_90a 31

Example for 23 Design

A: 速度 B: 切割深度 C: 切刀角度

&Four DOE Class_90a 32

計算 _Example 各因子之效用

各因子之 Sum of Square SSA = {-(1)+a-b+ab-c+ac-bc+abc}2 / 8n = (27)2 / (82) = 45.5625

&Four DOE Class_90a 33

ANOVA 表 _Example

&Four DOE Class_90a 34

Example for 24 Design

&Four DOE Class_90a 35

24 因子階層設計 _ 符號表

請完成

&Four DOE Class_90a 36

ANOVA 表 _Example

&Four DOE Class_90a 37

AD 交互作用與迴歸函數

2/

2/

2/

3

2

1

3210

之效用為

之效用為

之效用為

AD

D

A

ADDAy

&Four DOE Class_90a 38

2k Design with Center Points

增加預估曲線作用之能力

不破壞設計之平衡性 (Balanced Design)

只需增加少數幾個實驗

cF

cFcFquadraticpure nn

yynnSS

)(

&Four DOE Class_90a 39

Example

&Four DOE Class_90a 40

ANOVA 表 _Example

&Four DOE Class_90a 41

&7 2k 因子實驗之區隔化與混雜化 22 factorial design with Blocking

&Four DOE Class_90a 42

The ANOVA

&Four DOE Class_90a 43

Confounding( 混雜化 ) 受限於資源（時間、金錢、人力等），無法再每一個

區隔皆有完整的因子實驗。

Confounding is a design technique for arranging a complete factorial design in blocks, where the block size is smaller than the number of treatment combinations in one replicate.

It causes information about certain treatment effects (usually high-order interactions) to be indistinguishable from, or confounded with, blocks.

&Four DOE Class_90a 44

Confounding in 2 blocks

Effect AB confounded with block.

&Four DOE Class_90a 45

The 23 Design Confounded(I)

&Four DOE Class_90a 46

The 23 Design Confounded(II)

The Degrees of Freedom

&Four DOE Class_90a 47

The 23 Design Confounded(III) Replications => n = 4.

&Four DOE Class_90a 48

The 23 Design Confounded(IV)

Replications => n = 4.

&Four DOE Class_90a 49

An Example

&Four DOE Class_90a 50

The ANOVA

&Four DOE Class_90a 51

Confounding in 4 blocks

ADE and BCE are confounded, in addition, ABCD is also confounded. Why?

Suggested Blocking => pp. 298, Table 7-8.