quantitative methods interactions - getting more complex
TRANSCRIPT
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Quantitative Methods
Interactions - getting more complex
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Interactions - getting more complex
The factorial principle
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Interactions - getting more complex
The factorial principle
![Page 4: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/4.jpg)
Interactions - getting more complex
The factorial principle
3 3 3 3
6
6
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Interactions - getting more complex
The factorial principle
Investigate interactions
Hidden replication
3 3 3 3
6
6
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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.
![Page 7: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/7.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 8: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/8.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 9: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/9.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 10: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/10.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 11: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/11.jpg)
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.
![Page 12: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/12.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 13: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/13.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 14: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/14.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 15: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/15.jpg)
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.
![Page 16: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/16.jpg)
Interactions - getting more complex
Analysis of factorial experiments
![Page 17: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/17.jpg)
Interactions - getting more complex
Analysis of factorial experiments
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Interactions - getting more complex
Analysis of factorial experiments
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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)
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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.
![Page 21: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/21.jpg)
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)
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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
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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
![Page 24: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/24.jpg)
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
![Page 25: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/25.jpg)
Interactions - getting more complex
Analysis of factorial experiments
(Fitted value equation)
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Interactions - getting more complex
Analysis of factorial experiments
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Interactions - getting more complex
Interactions with continuous variables
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Interactions - getting more complex
Interactions with continuous variables
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Interactions - getting more complex
Interactions with continuous variables
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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.
![Page 31: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/31.jpg)
Interactions - getting more complex
Interactions with continuous variables
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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)
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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)
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Interactions - getting more complex
Interactions with continuous variables
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Interactions - getting more complex
Interactions with continuous variables
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Interactions - getting more complex
Interactions with 2 continuous variables
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Interactions - getting more complex
Interactions with 2 continuous variables
VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε
(Model)
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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
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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.
![Page 40: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/40.jpg)
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
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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)
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Interactions - getting more complex
Interactions in essence
![Page 43: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/43.jpg)
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.
![Page 44: Quantitative Methods Interactions - getting more complex](https://reader036.vdocuments.us/reader036/viewer/2022062417/5519cf8d550346443e8b49eb/html5/thumbnails/44.jpg)
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