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Residual Analysis for twoway ANOVA
The twoway model with
K
replicates, including inter
action, is
Y
ijk
=
μ
ij
+
²
ijk
=
μ
+
α
i
+
β
j
+
γ
ij
+
²
ijk
with
i
= 1
, . . . , I
,
j
= 1
, . . . , J
,
k
= 1
, . . . , K
.
In carrying out the
F
tests for interaction, and for the
main eFects of factors A and B, we have assumed that
²
ijk
are as sample from
N
(0
, σ
2
)
.
Among other things, this means that:
•
the distribution of the errors (and in particular, the
variance
σ
2
) does not diFer depending on the level
of factor A, the level of factor B, or the mean of
the response (
μ
ij
=
μ
+
α
i
+
β
j
+
γ
ij
)
•
the errors are a sample from a normal distribution
If these assumptions hold, then the pvalues for the
tests of interaction and main eFects are valid.
If the as
sumptions do not hold, then the pvalues may substan
tially over or underestimate the evidence against the null
hypotheses.
Residuals
are usually de±ned as the diFerence “data
prediction”.
In the twoway anova model with interaction, the pre
dicted value of
Y
ijk
is
ˆ
μ
ij
, and so the residuals are
r
ijk
=
Y
ijk

ˆ
μ
ij
=
Y
ijk

¯
Y
ij.
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View Full Document(Another way of writing the residual for the twoway model
with interaction is
r
ijk
=
Y
ijk

ˆ
μ

ˆ
α
i

ˆ
β
j

ˆ
γ
ij
.)
If the sample size is moderately large, the residuals should
be approximately equal to the errors
²
ijk
, and so we use
the residuals (which are known to us) in place of the errors
²
ijk
(which are unknown) to assess the plausibility of the
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 Spring '12
 daniel

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