Biology Reference
In-Depth Information
which have slopes dependent on
A
or
B
, but not both. These models might then, in turn,
be compared to the common slope model:
Y
A
B
A
B
MX
5
1
1
3
1
1ε
(9.58)
using basically the same approach to forming F-tests.
To exemplify a model with multiple factors and a covariate, we add a covariate, size, to our
model for alpine chipmunk jaw shape. As shown in
Table 9.16
, all three factors have a signifi-
cant impact on jaw shape. These results are based on the Type 1 (sequential) sums of squares,
with size entered first, then shape, and then region because we are primarily interested in the
impact of region. The results can be read as saying that size has a significant impact on shape,
that sex has a significant impact controlling for size, and that region has a significant impact
controlling for size and sex. An interesting pattern can be seen in
Figure 9.2
: all three factors
have a substantial effect on the angular process, differing in where and how they affect it.
ANALYZING MEASUREMENT ERROR
Measurement error contributes to the unexplained variation in the data and, by increas-
ing the noise, measurement error makes it more difficult to pick out the influence of the
factors of interest, especially when their effects are subtle. Measurement errors are gener-
ally thought of as being either systematic or random (see
Arnqvist and Martensson, 1998
).
Systematic errors are consistent biases in a measurement meaning that all measures are
incorrect to a consistent degree or extent. One of us (HDS) was thrilled to find a set of
inexpensive plastic rulers which were roughly 3 to 5% shorter than claimed by the mark-
ings on them, perhaps because of shrinkage of the plastic, or poor mold-making. All mea-
surements made with these rulers were uniformly short by a fixed factor, one specific to a
particular
ruler, which made them invaluable in an introductory physics lab on
TABLE 9.16
A Three-Factor Multivariate Analysis of Covariance: The Impact of Size, Sex and Region on
Alpine Chipmunk Jaw Shape (Using Sequential Sums of Squares)
R
2
Source
SS
df
MS
F
P
Size
0.001497
1
0.001497
3.6151
0.02712
0.001
Sex
0.001587
1
0.0015865
3.8312
0.02874
0.001
Region
0.005194
1
0.0051943
12.5434
0.09411
0.001
Size
Sex
0.000362
1
0.0003625
0.8753
0.00657
0.549
3
Size
Region
0.000597
1
0.0005972
1.4423
0.01082
0.128
3
Sex
Region
0.000655
1
0.000655
1.5816
0.01187
0.081
3
Size
Sex
Region
0.000167
1
0.0001666
0.4023
0.00302
0.990
3
3
Residuals
0.045138
109
0.0004141
0.81776
Total
0.055197
116
1
