Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 170

We stress that these data exploration techniques should not be
applied haphazardly. Statistical exploration for influential landmarks
can be important scientifically, but biological knowledge, focused
thinking, and common sense are necessary for understanding and
explaining the differences observed, as well as guarding against spuri-
ous results. Investigators using these techniques should understand
the underlying logic, and should always use their judgment when
interpreting and explaining the implications of the results. Simplistic
conclusions reporting localized increases and decreases in particular
linear distances are rarely informative by themselves. Plausible pro-
cesses that may underlie such changes should always be discussed. In
the best case, the results of data exploration should be formulated as
hypotheses to be tested using alternate, relevant data that may need
to be generated through other experiments, or the analysis of addi-
tional specimens.
4.11 A graphical tool for the detection of influential
landmarks
This method, proposed by Cole and Richtsmeier (1998), involves the use
of a two-dimensional scatter plot to summarize, explore, and interpret a
form difference matrix. Positions along the horizontal axis of this scatter
plot correspond to the landmarks numbered from 1 to K . Positions along
the vertical axis are real numbers corresponding to values of the off-
diagonal elements of the form difference matrix for each landmark pair.
Let us continue with data presented earlier ( Figure 4.8 ) . Rewrite
the form difference matrix for this comparison as a square, symmetric
matrix. The first row of this matrix contains the ratios of the distances
in the two objects that involve landmark 1 as one of the end points.
Similarly, the second row of the matrix contains the ratios of the dis-
tances in two objects that involve landmark 2 as one of the end points.
Now consider the scatter plot in Figure 4.9. The landmark numbers
1, 2, 3, and 4 are plotted on the horizontal axis. Above the point corre-
sponding to landmark 1, those values corresponding to the first row of
the form difference matrix are plotted. Above the point corresponding
to landmark 2, the ratios that correspond to the second row of the form
difference matrix are plotted. The same is done for those ratios that
correspond to the third and fourth row of the form difference matrix.
Remember that the form difference matrix is a square symmetric
matrix. This means that our scatter plot will contain two points for the