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# Test for non-additivity; 3 factor designs.pdf

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University of Alberta

Statistics

STAT368

Douglas Wiens

Winter

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117
16. Test for non-additivity; 3 factor designs
Sometimes we can make only one observation per cell
( = 1). Then all 1 ¯ = 0, so = 0 on
( 1) = 0 d.f. The interaction SS, which for
= 1 is
X ³ ´2
¯ ¯ + ¯ (*)
is what we should be using to estimate experimental
error.There is still however a way to test for inter-
actions, if we assume that they take a simple form:
( ) =
We carry out Tukeys one d.f. test for interaction,
which is an application of the usual reduction in SS
hypothesis testing principle. Our full model is
= + + + +
Under the null hypothesi0: = 0 of no interac-
tions, the reduced model is
= + + + 118
in which the minimum SS (i.e. ) is (*) above.
One computes
0 = ( 1)( 1) 1
( )
The di erence
=
is called the SS for non-additivity, and uses 1 d.f. to
estimate the one parameter . The ANOVA becomes
Source SS df MS
A 1 = 1
B 1 = 1
N 1 =
( 1)( 1) 1
Error 1 = ( )
Total 1
The error SS is . To obtain it one has to
minimize
³ h i´
X 2
+ + + 119
After a calculation it turns out that
n P ³ ´o 2
¯ ¯ ¯ + + ¯2
=
·
Then is obtained by subtraction: =
.
An R function to calculate this, and carry out the F-
test, is at R commands for Tukeys 1 df test on the
course web site.
Example. For the experiment at Example 5.2 of the
text there are= 3 levels of temperature and= 5
of pressure; response i= impurities in a chemical
product.
> h

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