&sixadvanced statistics_doe11 實驗目的 : &13 doe 簡介對 y 影響最大的變數為...

&Six Advanced Statistics_DOE1 1

實驗目的 :

&13 DOE 簡介

對 y 影響最大的變數為何？

如何設定 x1, x2, …, xp使 y 值趨近最佳值？

如何設定 x1, x2, …, xp使 y 值得變異最小？

如何設定 x1, x2, …, xp

使不可控制因素 z1, z2, …, zp 之影響最小？


An Example： Play Golf Objective: Lower score without much practicing. Response Variable: Score (per round) Possible Factors:

The type of driver used (oversized or regular-sized)

The type of ball used (balata or three-piece) Walking or riding in a golf cart Beverage Type (water or beer) Time (in the morning or afternoon) Weather (cool or hot, windy or calm) The type of golf shoe spike (metal or soft)


一般實驗進行方式 Best-guess approach

No Good, Guess Again Switching the levels of one (perhaps two) factors for

the next test based on the outcome of the current test

Good Enough, Stop! On-factor-at-a-

time Selecting a baseline starting point Varying each factor over its range with the other

factors held constant at the baseline level Interactions ruin everything


最佳因子水準組合為？ Driver: Regular Mode of travel: Ride Beverage: Water

But what if…………………

Results of the one-factor-a-time strategy for the golf experiment


The two-factor factorial design for the golf experiment (I)


The two-factor factorial design for the golf experiment (II)

Ball Effect = ？ Ball-Driver Interaction Effect = ？


Other Designs for the Golf Experiment

Four-factor factorial design

Three-factor factorial design


Other Designs for the Golf Experiment

Four-factor fractional factorial design


實驗計劃法 (DOE) 在一個或連串的試驗中刻意地改變製程輸入參數值 ,

以便觀察並找出影響製程輸出變數之因素 .

應用 : 改進製程產出率降低製程變異 , 改善產品品質降低研發時間降低總體成本評估各種可行之設定值評估各替代原料確定影響產品特性之因素


Example:Optimizing a Process


基本原則複製 (Replication)

隨機化 (Randomization)

區隔化 (Blocking)

增進實驗之精確度

估計自然誤差中央極限定理

“Averaging out” the effects from uncontrollable variables


DOE 之程序問題之認知與陳述選擇因子與其水準選擇反應變數選擇適當之實驗設計執行實驗資料分析結論與建議

Follow-up run and confirmation test Iterative No more than 25% of available resources should be invested in t

he first experiment


Notes

使用統計以外之專業知識實驗之設計與分析應愈簡單愈好實驗之統計分析結果與現實上之差異

成本技術時間

實驗通常是遞迴式的前幾次實驗通常只是學習經驗而已


二因子實驗設計二因子無交互作用


二因子有交互作用


One-factor at a time 之方法


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


Data Sheet


ANOVA 表 – Two-Factor Factorial


a

i

b

j

ijsubtotals abn

y

n

ySS

1 1

2...

2.


Example


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。


Another Example


ANOVA Results

Computer Output (Model Adequacy Checking)


Multiple Comparisons


&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。


The 22 Factorial Design


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


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


The AVONA Table


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?


The Residuals

Residuals at x1=1 and x2 = -1

Computer Output


23 因子階層設計


23 因子階層設計 _ 符號表


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


Example for 23 Design

A: 速度 B: 切割深度 C: 切刀角度


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


ANOVA 表 _Example


Example for 24 Design


24 因子階層設計 _ 符號表

請完成


ANOVA 表 _Example


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


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

)(


Example


ANOVA 表 _Example


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


The ANOVA


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.


Confounding in 2 blocks

Effect AB confounded with block.


The 23 Design Confounded(I)


The 23 Design Confounded(II)

The Degrees of Freedom


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


The 23 Design Confounded(IV)

Replications => n = 4.


An Example


The ANOVA


Confounding in 4 blocks

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

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

&sixadvanced statistics_doe11 實驗目的 : &13 doe 簡介 對 y 影響最大的變數為...

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&sixadvanced statistics_doe11 實驗目的 : &13 doe 簡介對 y 影響最大的變數為...