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&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
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0: 1.
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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
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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.