Post on 31-Mar-2015
Quantitative Methods
Interactions - getting more complex
Interactions - getting more complex
The factorial principle
Interactions - getting more complex
The factorial principle
Interactions - getting more complex
The factorial principle
3 3 3 3
6
6
Interactions - getting more complex
The factorial principle
Investigate interactions
Hidden replication
3 3 3 3
6
6
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Analysis of factorial experiments
BLOOMS=μ +
BED
1 α1
2 α2
3 −α1 −α2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
WATER
1 β1
2 β2
3 −β1 −β2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
SHADE
1 γ1
2 γ2
3 −γ1 −γ2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
WATER SHADE
1 1 δ11
1 2 δ12
1 3 −δ11−δ12
2 1 δ21
2 2 δ22
2 3 −δ21−δ22
3 1 −δ11−δ21
3 2 −δ12 −δ22
3 3 δ11+δ12 +δ21+δ22
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
+ε
(Model)
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Analysis of factorial experiments
BLOOMS=μ +
BED
1 α1
2 α2
3 −α1 −α2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
WATER
1 β1
2 β2
3 −β1 −β2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
SHADE
1 γ1
2 γ2
3 −γ1 −γ2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
WATER SHADE
1 1 δ11
1 2 δ12
1 3 −δ11−δ12
2 1 δ21
2 2 δ22
2 3 −δ21−δ22
3 1 −δ11−δ21
3 2 −δ12 −δ22
3 3 δ11+δ12 +δ21+δ22
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
+ε
(Model)
Interactions - getting more complex
Analysis of factorial experiments
Shade\ Water 1 2 3 Sum
1 δ11 δ21 0
2 δ12 δ22 0
3 0
Sum 0 0 0
Shade\ Water 1 2 3 Sum
1 δ11 δ21 −δ11 −δ21 0
2 δ12 δ22 −δ12 −δ22 0
3 0
Sum 0 0 0
Shade\ Water 1 2 3 Sum
1 δ11 δ21 −δ11−δ21 0
2 δ12 δ22 −δ12−δ22 0
3 −δ11−δ12 −δ21−δ22 0
Sum 0 0 0
Shade\ Water 1 2 3 Sum
1 δ11 δ21 −δ11−δ21 0
2 δ12 δ22 −δ12 −δ22 0
3 −δ11−δ12 −δ21−δ22 δ11+δ12 +δ21+δ22 0
Sum 0 0 0
Interactions - getting more complex
Analysis of factorial experiments
Shade\ Water 1 2 3 Sum
1 δ11 δ21 −δ11−δ21 0
2 δ12 δ22 −δ12 −δ22 0
3 −δ11−δ12 −δ21−δ22 δ11+δ12 +δ21+δ22 0
Sum 0 0 0
Interactions - getting more complex
Analysis of factorial experiments
Shade\ Water 1 2 3 Sum
1 δ11 δ21 −δ11−δ21 0
2 δ12 δ22 −δ12 −δ22 0
3 −δ11−δ12 −δ21−δ22 δ11+δ12 +δ21+δ22 0
Sum 0 0 0
Interactions - getting more complex
Analysis of factorial experiments
(Fitted value equation)
Interactions - getting more complex
Analysis of factorial experiments
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions with continuous variables
BACAFTER=BACBEF+TREATMT+BACBEF* TREATMT
BACAFTER=BACBEF|TREATMT
BACAFTER=μ +β ×BACBEF+
TREATMT
1 α1
2 α2
3 −α1 −α2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
TREATMT
1 γ1
2 γ2
3 −γ1 −γ2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
×BACBEF+ε
BACAFTER=(μ +α1)+(β +γ1)×BACBEF+ε
BACAFTER=(μ +α2)+(β +γ2)×BACBEF+ε
BACAFTER=(μ −α1 −α2)+(β −γ1 −γ2)×BACBEF+ε
(Model formula)
(Model)
Interactions - getting more complex
Interactions with continuous variables
FITTED
BACAFTER=−0.126+0.8894×BACBEF+
TREATMT
1 −0.946
2 −0.896
3 1.842
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
+
TREATMT
1 −0.0611
2 0.0167
3 0.0444
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
×BACBEF
BACAFTER=−1.072+0.8283×BACBEF
BACAFTER=−1.022+0.9061×BACBEF
BACAFTER=1.716+0.9338×BACBEF
(Fitted value equation)
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions with continuous variables
Interactions - getting more complex
Interactions with 2 continuous variables
Interactions - getting more complex
Interactions with 2 continuous variables
VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε
(Model)
Interactions - getting more complex
Interactions with 2 continuous variables
VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε
(Model)
VOLUME=μ +(β +δ×HEIGHT)×DIAMETER+γ ×HEIGHT+ε
VOLUME=μ +β ×DIAMETER+(γ +δ×DIAMETER)×HEIGHT +ε
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Interactions with 2 continuous variables
VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε
(Model)
VOLUME=μ +(β +δ×HEIGHT)×DIAMETER+γ ×HEIGHT+ε
VOLUME=μ +β ×DIAMETER+(γ +δ×DIAMETER)×HEIGHT +ε
Interactions - getting more complex
Interactions with 2 continuous variables
VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε
(Model)
VOLUME=μ +(β +δ×HEIGHT)×DIAMETER+γ ×HEIGHT+ε
VOLUME=μ +β ×DIAMETER+(γ +δ×DIAMETER)×HEIGHT +ε
FITTED
VOLUME=69.40−5.856×DIAMETER−1.2971×HEIGHT +0.13465×DIAMETER×HEIGHT
(Fitted value equation)
Interactions - getting more complex
Interactions in essence
Interactions - getting more complex
Interactions in essence
Simple additive/linear story, separate story for each x-variable? or more complicated story involving both variables?
Two x-variables interact if the effect of one x-variable on y depends on the level of the other.
Interactions - getting more complex
Last words…
• Factorial experimental designs are very useful• Interactions are about one x-variable affecting
how another affects y• Know how to construct the model• Know how to construct the fitted value equation• Marginality is important, but more on that later…
Checking the models I: independence
Read Chapter 8