Biology Reference
In-Depth Information
Minimum Deviance Method
The second method assesses the goodness-of-fit of a model to the observed covariance
matrix. This method, introduced by
Marquez (2008)
and subsequently modified to
improve the assessment of relative fit (
Parsons et al., 2012
), is implemented in Mint
(
Marquez, 2012
). As outlined briefly above, the expected covariance matrix is modeled by
placing modules into orthogonal subspaces, one per module. This is done by making as
many copies of the data as there are modules, and assigning a value of zero to the coordi-
nates that do not belong to the hypothesized module. For example, given the hypothesis
that the front and back are two modules, with landmarks 1, 2, 3, 4, 5, 6, 7 and 13 in the
front and landmarks 8, 9, 10, 11, 12, 14 and 15 in the back, we would make two copies of
the data and arrange them in the extended matrix of the Front/Back Model:
1234567 3
00000000
0000000
89 0 1 2 4 5
Front
=
Back Model
5
Each of the numbers represent the
x
and
y
(and, if present,
z
) coordinates for that
landmark. The values for the coordinates that belong to a module are taken from the data.
The values for the coordinates that do not belong to the model are fixed to 0, 0. Before
assessing the fit of the model, the matrix predicted under the model (e.g. Front/Back
Model) is superimposed. The procedure is somewhat more complicated when the hypoth-
esis predicts that some landmarks belong to two or more modules; the modules then par-
tially overlap each other hence the subspaces are not orthogonal to each other and the
variances of the overlapping landmarks must be allocated to multiple modules without
altering the overall value of the variance. As currently implemented in Mint, the variance
of the landmark's coordinates is equally partitioned across all the modules that contain
that landmark.
The fit of the model to the data can be assessed by several metrics (and Mint offers
three). We describe only one of them, the one that is fully standardized to allow for com-
paring results across data sets and also for assessing the relative fit of models that differ in
the number of modules. This goodness-of-fit statistic is
γ
:
T
γ
5
ðð
trace
S
2
S
0
Þð
S
2
S
0
Þ
Þ
(12.8)
where S and S
0
are the observed and expected (modeled) covariance matrices, respectively
(
Richtsmeier et al., 2005
). For example, in the case of the Front/Back model, S
0
is the
covariance matrix of the Front/Back Model. To make it comparable across data sets,
γ
is
scaled by its maximum value,
γ
max
, which is com-
puted by comparing the data to the null model of “no integration”. That null model is a
diagonal matrix that has the variances of the coordinates along the diagonal and zeros for
all the off-diagonal elements, which are the covariances. The second scaling step removes
the dependence of
γ
max
, by dividing
γ
for each model by
on the number of fixed parameters (the landmarks whose coordinates
are fixed to zero by model). This makes
γ
comparable across models that differ in the
number of fixed parameters. The scaling is done by regressing the value of
γ
γ
on the
