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God I hate stats.
In a
factorial experiment
, each possible combinations of the levels
of the factors in each replicate of the experiment is investigated.
An experiment with two factors (A, B) is represented by a
flat sheet in 3D surface plot.
Factorial experiments are the only way to discover
interactions between variables
.
A
2
k
factorial design provides the smallest # of runs for which k
factors can be studied in a complete factorial design.
2
2
Design (A,B factors):
=

.
a results for all a high
b,ab are similar
Main effect of A:
=
+
= +
 +
=
+
 
A yA yA
a ab2n b 12n 12na ab b 1
Main effect of B:
=
+
= +
 +
=
+
 
B yB yB
b ab2n a 12n 12nb ab a 1
AB interaction:
==
+( )
 +
=
+  
AB
ab 1 2n a b2n 12nab 1 a b
For the above 3 equations, quantities in brackets are
contrasts.
Sums of squares
formulas:
= +
 
SSA a ab b 124n
= +
 
SSB b ab a 124n
=
+( ) 
SSAB ab 1 a b24n
SS
T
calculated “as usual” with 4n1 DF, SS
E
with 4(n1) DF with
subtraction.
=
+ +
+ +
SST yi2 1 a b ab24n
Formulas for TwoLevel Factorial with k factors
Each at Two Levels and N total Trials
V(Effect)=

σ2n2k 2
Coefficient = effect/2
=
=

seeEffect 2σ2N2 σ2n2k 2
=
=(
)

seCoefficient 122σ2N2
12 σ2n2k 2
=
=
t ratio effectseeffect coefficientsecoefficient
=
σ2 MSE
=SS
E
/(4n4)
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 Spring '06
 andersonrowland

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