Biology Reference
In-Depth Information
data and for data consisting of landmarks plus semilandmarks, but for the remainder of
this discussion we will mention only the two-dimensional landmark case. The adjustments
are straightforward except that, in the case of landmark-only data, the dimensionality of the
data equals the number of partial warps (including the two uniform components), which
will not be the case for data that include semilandmarks as well as landmarks. That distinc-
tion, however, is not important when using permutation or bootstrap methods based on
Procrustes distances, as we will see later. Regression models can be framed in terms of par-
tial warp scores or principal component scores or coordinates of landmarks because all the
mathematics involved is linear so a rotation of the data will not alter the answers, so long as
the mathematics is done correctly.
To regress shape on an independent (scalar) variable, we regress the shape data on the
independent variable. For example, suppose we have P partial warp and uniform compo-
nents, which we can write as a row vector {Y
1
, Y
2
, Y
3
,
Y
P
}. Then the (linear) model for
...
the regression of that vector on a scalar (X) is:
f
Y
1
;
Y
2
;
Y
3
;
...
Y
P
g
5
f
m
1
;
m
2
;
m
3
;
...
m
P
g
X
1
f
b
1
;
b
2
;
b
3
;
...
b
P
g
1
fε
; ε
; ε
;
...
ε
g
(8.13)
1
2
3
P
where {m
1
, m
2
, m
3
,
...
ε
P
} are vectors of slope and
intercept coefficients and residuals, respectively. Although this expression looks far more
complicated than the one for a bivariate regression, it actually is not. In fact, we can deter-
mine the ith component of the slope and intercept terms using the same m
i
and b
i
values
that minimize the residuals in the corresponding bivariate model. Each observation Y is
now a vector, as are the slope, intercept and each of the errors.
Estimating slope and intercept coefficients is no more complex in the multivariate case
than it was in the bivariate case. But in one important respect, the analysis actually is
more complex
checking the assumption of linearity. There are at least two ways to check
this assumption for multivariate data, although neither is ideal. One is to look at the rela-
tionship between each individual component of shape and the independent variable, such
as by regressing each partial warp on size. If one or more exhibits a strong and highly
non-linear relationship, such as shown in
Figure 8.1A
, then it is unlikely that shape and
size are linearly related. This method for checking linearity is not ideal because it falls
back on inspecting multiple bivariate regressions when it is multivariate linearity that
really matters. Another approach is to estimate the Procrustes distance between each spec-
imen and the shape at the lowest value on the independent variable. Regressing that dis-
tance on the independent variable may show if that relationship is non-linear (as in
Figure 8.1B
). If it is not, it is unlikely that shape and size are linearly related. This method
is again not ideal, because the Procrustes distance measures only the magnitude of the dif-
ference between each specimen and the reference, not its direction. Two specimens that
differ a great deal from each other in shape may be equally distant from the reference.
Despite the deficiencies of these two less than ideal methods, we can use them to check
whether it is unlikely that shape is linearly related to size. The results shown in
Figure 8.1
both indicate a non-linear relationship of shape and centroid size, and both suggest that
shape might be linearly related to the log of centroid size. That linear relationship to the
log of centroid size is suggested by the shape of the curves because they depict a very
rapid change in shape relative to size over the smaller values of size. So we can try a log
m
P
}, {b
1
, b
2
, b
3
,
b
P
} and {
ε
1
,
ε
2
,
ε
3
,
...
...
