Quantitative Methods Interactions - getting more complex

Quantitative Methods

Interactions - getting more complex


The factorial principle





3 3 3 3

6

6



Investigate interactions

Hidden replication

3 3 3 3

6

6


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.


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)







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)



Shade\ Water 1 2 3 Sum

1 δ11 δ21 0

2 δ12 δ22 0

3 0

Sum 0 0 0


1 δ11 δ21 −δ11 −δ21 0

2 δ12 δ22 −δ12 −δ22 0

3 0

Sum 0 0 0


1 δ11 δ21 −δ11−δ21 0

2 δ12 δ22 −δ12−δ22 0

3 −δ11−δ12 −δ21−δ22 0

Sum 0 0 0


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




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




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



(Fitted value equation)




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)



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







Interactions with 2 continuous variables



VOLUME=μ +β ×DIAMETER+γ ×HEIGHT+δ×DIAMETER×HEIGHT +ε

(Model)




(Model)

VOLUME=μ +(β +δ×HEIGHT)×DIAMETER+γ ×HEIGHT+ε

VOLUME=μ +β ×DIAMETER+(γ +δ×DIAMETER)×HEIGHT +ε








(Model)






(Model)



FITTED

VOLUME=69.40−5.856×DIAMETER−1.2971×HEIGHT +0.13465×DIAMETER×HEIGHT









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

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Quantitative Methods Interactions - getting more complex