Biology Reference
In-Depth Information
TABLE 9.7 Sums of Squares, Degrees of Freedom and Mean Squares for a Two-Way, Balanced, Nested
Univariate ANOVA
Source
SS
df
MS
NJ P
I
i 5 1 ð
2
A
SS A 5
y i 2
y
Þ
I
2
1
SS A /(I
2
1)
N P
I
i 5 1 P
J
j 5 1 ð
2
B(A)
SS B ð A Þ 5
y j ð i Þ 2
y i Þ
I ( J
2
1)
SS B(A) /I(J
2
1)
SS error 5 P
1 P
1 P
J
N
I
2
Residual
1 ð
y ijk 2
y j ð i Þ Þ
IJ(N
1)
SS error /IJ(N
1)
2
2
k
5
i
5
j
5
5 P
k 5 1 P
1 P
N
I
J
2
1 ð
Þ
Total
SS total
y ijk
2
y
IJN
2
1
i
5
j
5
There are I levels of factor A, and J levels of factor B , which are nested within A, with N specimens per cell. The expressions of
the form y i indicate means over all cells with level i (or j, or the combination ij), while y is the overall (grand) mean of the data.
TABLE 9.8
F-Ratios and Variance Components for a Two Factor, Nested, Balanced Design
Source
A Fixed, B Random
A,B Fixed
Component
A
MS A /MS B(A)
MS A /MS error
(MS A 2
MS B(A) )/NJ
B(A)
MS B(A) /MS error
MS B(A) /MS error
(MS B(A) 2
MS error )/N
Error
MS error
TABLE 9.9 A Two-Way, Nested Analysis of Variance
Source
SS
df
MS
F
P
Diet
0.01694
2
0.00847
4.68
0.0132
Litter (Diet)
0.10135
56
0.00181
4.203
0.000
Error
0.09774
227
0.00043
Preliminary results for the impact of dietary consistency on deer mouse jaw size. One factor; ”litter” is nested within the
other,“diet”, because all members of the same litter ate the same food.
A (partial) table showing how to form F-ratios for a three factor ANOVA is shown in
Table 9.10 . For a more complete description of three factor ANOVAs see Quinn and
Keogh (2002) or Snedcor and Cochran (1980) .
Models with Covariates
When one or more discrete factors and one or more continuous covariates are also of
interest, the general approach is to identify the form of the dependence of Y (shape) on the
covariate, and then to remove the variance in Y explained by the covariate, assuming no
interaction between the covariate and factor(s). Having removed the variance explained by
the covariate, one can then proceed to test the significance of the factors, as discussed
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