交互作用分析
若某因子不同水準間輸出的差異，會隨 其它因子水準的設定改變而改變，則這 些因子之間存在交互作用。

交互作用的處理方法
y Study control factor interactions to quantify their
effects.
觀察控制因子間的交互作用
y To minimize the likelihood of significant interactions
and avoid having to estimate them.
減少顯著的交互作用的發生
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Interactions defined
y Antisynergistic 反向交互作用
Factor A 水準一到水準二間的輸出差異，會隨著 B水準設定而改變，且其變化方向相反。

Response Factor B High
A , B 間有相 互影響
Factor B Low Low Factor A
High
Interactions defined
y Synergistic 正向交互作用
Factor A水準一到水準二間的輸出差異，不會隨 著 B 水準設定而改變。

Response Factor B High A , B 間不會
影響到水準 的決定
Factor B Low
High
Low Factor A
不平行的量愈 大表示相互的 作用愈大
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Measurement of Interactions
B1 B2
10 9 8 7 6 5 4 3 2 1
Run (L4) 1 2 3 4
A 1 1 2 2
B 1 2 1 2
C (A-B) 1 2 2 1
Y 7 5 10 8
1 Factor A
2
Delta 1 = A2B1 – A1B1 Delta 2 = A2B2 – A2B1 Delta 1 = Delta 2 表示因子 A 與 B之間 無交互作用
Measurement of Interactions
B1 B2
10 9 8 7 6 5 4 3 2 1
Run (L4) 1 2 3 4
A 1 1 2 2
B 1 2 1 2
C (A-B) 1 2 2 1
Y 9 5 6 8
1 Factor A
2
最佳水準組合將發生錯誤
Delta 1 = A2B1 – A1B1 -3 = 3 Delta 2 = A2B2 – A2B1 = Delta 1 ≠ Delta 2 且異號，表示因子 A 與 B之間有強交互作用
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Degrees of Freedom of Interactions
y 2-level factors
In the L4, one 2-level column (column 3), with DOF=(2-1)=1, is adequate to estimate the interaction between factors A and B.
High Factor B Low
Run (L4) 1 2 3 4
A 1 1 2 2
B 1 2 1 2
C (A-B) 1 2 2 1
Low
Factor A
High
Degrees of Freedom of Interactions
y 3-level factors
In the L9, one 3-level column (column 3,4), with DOF=(31)*2=4, is adequate to estimate the interaction between factors A and B.
High Factor B Medium Low
Ru n (L9)
A (1)
B (2)
C D (1-2) (1-2)
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2
1 2 3 3 1 2 2 3 1
Low
Medium
High Factor A
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Resolution (解析度)
y When all the two-way interactions are free of
confounding with other two-way interactions and main effects, this is referred to as a Resolution V array.
y When two-way interactions confound with two-way
interactions but not main effects, this is referred to as Resolution IV array.
交互作用 主因子效應
y When two-way interactions confound with main
effects, this is referred to as Resolution III array.
L8 (23), Resolution V
Run 1 2 3 4 5 6 7 8 1 A 1 1 1 1 2 2 2 2 2 B 1 1 2 2 1 1 2 2 3 1 1 2 2 2 2 1 1 1-2 4 C 1 2 1 2 1 2 1 2 5 1 2 1 2 2 1 2 1 1-4 6 1 2 2 1 1 2 2 1 2-4 7 1 2 2 1 2 1 1 2
Main effects and all two-way interaction are free of confounding called Resolution V.
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L8 (24), Resolution IV
Run 1 2 3 4 5 6 7 8 1 A 1 1 1 1 2 2 2 2 2 B 1 1 2 2 1 1 2 2 3 1 1 2 2 2 2 1 1 1-2 4-7 4 C 1 2 1 2 1 2 1 2 5 1 2 1 2 2 1 2 1 1-4 2-7 6 1 2 2 1 1 2 2 1 1-7 2-4 7 D 1 2 2 1 2 1 1 2
Two-way interactions confound with other two-way interaction, but not main effect, called Resolution IV.
L8 (27), Resolution III
Run 1 2 3 4 5 6 7 8 1 A 1 1 1 1 2 2 2 2 2-3 4-5 6-7 2 B 1 1 2 2 1 1 2 2 1-3 4-6 5-7 3 C 1 1 2 2 2 2 1 1 1-2 4-7 5-6 4 D 1 2 1 2 1 2 1 2 2-6 3-7 5 E 1 2 1 2 2 1 2 1 1-4 2-7 3-5 6 F 1 2 2 1 1 2 2 1 2-4 3-5 7 G 1 2 2 1 2 1 1 2 1-6 2-5 3-4
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L9 (32), Resolution V
Run (L9) 1 A 2 B 3 4
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2 1-2
1 2 3 3 1 2 2 3 1 1-2
OA 應用場合
y 五級解析度OA，可以用來精確的評估因子效應與
因子間的交互作用。

y 四級解析度OA，只能計算因子效應但無法計算因
子間的交互作用。

y 三級解析度OA，當交互作用可以完全忽略時，會
是個很有效率的實驗。

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分散交互作用(distributed interactions) OA
OA Total DOFs DOFs occupied by columns DOFs for grand average Remaining DOFs
L12 (211)
L18 (2337)
L36 L36 L54 (23313) (211312) (21325)
L32 (2149)
L50 (21511)
12 11 1 0
36 15 1 2
36 29 1 6
36 35 1 1
54 51 1 2
32 28 1 3
50 45 1 4
y They distribute the effect of an interaction between any pair
of factors over all the other columns, thus minimizing its confounding with any main effect.
加法模式預測誤差的原因
1. 因子選擇不當，有其它重要因子未考慮到。

2. 因子間存在交互作用，可以藉由因子交互作用分
析圖(Interaction plot)來判定。

3. 系統存在強烈的非線性關係，因子水準數選擇不
當所致，可以縮小水準值的範圍，來改善之。

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