Biology Reference
In-Depth Information
The expected mean square error (or the expected mean square residual) is then esti-
mated as
SS
error
df
error
MS
error
(9.17)
5
σ
e
2
. In contrast to the mean square (MS) term, which is
calculated from the data, the expected mean square (EMS) is calculated from the model. If
the model does describe the data, the MS and the EMS should be similar, differing only by
relatively minor random variation.
The
su
m of squares for the between groups term, or that contributed by the factor A
(when Y
whose expected value (EMS
error
)is
0) is
5
X
J
n
j
Y
2
SS
A
5
(9.18)
j
j
1
5
The mean square value is estimated as MS
A
5
SS
A
/(J
1) because the degrees of free-
2
dom are df
A
5
1). The expected mean square (EMS
A
) has two contributions, a pooled
variance of the error terms across the groups plus a component representing the squared
effects of the factors:
(J
2
X
J
2
j
n
j
α
2
e
EMS
A
5
σ
(9.19)
1
ð
J
1
Þ
2
j
1
5
The null hypothesis we want to test is that the factor A does not contribute to the value
of Y. Having constrained the mean of Y to be zero, the null hypothesis is therefore that the
α
j
values are all equal and are, in fact, all equal to zero. Under these conditions, the
expected mean square value of EMS
A
is simply
σ
e
2
. The F-ratio of the variance explained
by the model relative to the error or residual variance is then expected to be 1 should the
null hypothesis be true. Consequently, we can compute the F-ratio based on the data as:
MS
A
MS
error
5
SS
A
=
df
A
F
(9.20)
5
SS
error
=
df
error
which will follow an F-distribution with degrees of freedom df
A
, df
error.
Notice that the F-
ratio is the variance explained by the model divided by the unexplained variance, just as
it was when we applied F-tests to regression models, or used Goodall's F-test to compare
the mean shapes of two groups. Because the expected mean term in the numerator is equal
to the denominator plus one additional term, the F-value will be larger than one if the
additional term:
X
J
2
j
n
j
α
(9.21)
ð
J
1
Þ
2
j
1
5
is not zero. A larger F-value would indicate a lower probability that the null hypothesis is
true. Rejecting the null hypothesis allows us to interpret the MS
A
term as the variance
explained by the factor A, and to compare that to the unexplained variance, MS
error
.
