Biology Reference
In-Depth Information
TABLE 9.17
Mutlivariate Analysis of Variance of the Alpine Chipmunk Jaw Shape Measured Twice on the
Right Side of the Jaw
Source
SS
df
MS
EMS
2
2
Ind
Ind
0.084781
93
0.00091162
σ
1
2
σ
ME
2
ME
0.003619
94
0.00003850
σ
ME
Total
0.088400
187
The expected mean squares (EMS) are equated to the MS and used to compute the Individual (Ind) and measurement error (ME)
variance components.
different results, and whether one is more prone to measurement error than another, and
whether the difference is large enough to matter given the magnitude of the biological var-
iation. This approach is especially valuable when you can choose your imaging and mea-
surement methods. Even when you cannot, it is still useful to do repeated measures of the
same specimens so that you can quantify measurement error.
Measurement error is often quantified as repeatability (R) using a ratio of two variance
components, that for the among-individual to the sum of the among-individual and
measurement error components. These components can be calculated from the MANOVA
table by equating the mean squares (MS) to the expected mean squares (EMS). The
EMS for the Individual term is
2
I
Ind
, where k is the number of replicate measure-
ments. To calculate the repeatability of alpine chipmunk jaw shape data, we would
use
Table 9.17
, which gives the MS and EMS for the right sides of the jaw, each mea-
sured twice. In this case, k
2
ME
σ
1
k
σ
2. So, to compute the value for Individual variance compo-
nent, we subtract the measurement error MS from the Individual MS and divide by
the number of replicates: (0.000912
5
0.0004365. We then compute the
ratio between that component and the total, with the total being
0.0000385)/2
2
5
2
ME
2
Ind
σ
1
σ
2
(0.0000385
0.0004365
0.00047525). We would then take the ratio between
σ
Ind
and
1
5
2
ME
2
σ
0.919. So the repeatability of shape is 0.92. We can
reduce measurement error by averaging the repeated measures of the same specimens.
1
σ
Ind
: 0.0004365/0.00047525
5
IMPLEMENTING GLM
Several software packages can be used to analyze GLM for shape. For permutation
MANOVAs or permutation tests of general linear models, one usually needs to specify for
each hypothesis:
How sums of squares and cross products should be calculated (i.e. Type I, II, III)
What the numerator and denominator of the F-tests should be
What factors (or labels) are exchangeable under the null hypothesis.
Some programs, such as adonis in the vegan package (Oksanen et al., 2011) in R
(
R_Development_Core _Team, 2011
), are highly automated so all that you do is to specify
the model. Others, such as DISTLM (
Anderson, 2004
) require you to input the design
matrices for the factor(s) of interest, as well as the design matrices for other terms
