How to study complex shapes when the sample size cannot be increased

Statistical shape analysis is a cross-disciplinary field, allowing for applications in biology, geology, medicine and many other sciences, since it is characterized by flexible theory and techniques, potentially adaptable
to any appropriate configuration matrices.
The statistical community has shown an increased interest in shape analysis in the last decade and lots of efforts have been addressed to the development of powerful statistical methods for the comparison of shapes. Actually, traditional
inferential methods make use of congurations of landmarks optimally superimposed using a
least-squares procedure or analyze matrices of interlandmark distances.
For a detailed review, see Rohlf (2000). For example, in the two independent sample case,
a practical method for comparing the mean shapes in the two groups is to use the Procrustes tangent space coordinates, if data are concentrated, calculate the Mahalanobis distance and then the Hotelling s T2 test statistic.
Under the assumption of isotropy, another simple approach is to work with statistics based on
the squared Procrustes distance and then consider the Goodall s F test statistic.
Despite their widespread use, on the one hand it is well known that Hotelling s T2 test may not
be very powerful unless there are a large number of observations available, and on the other hand the underlying model required by Goodall s F test is very restrictive.
Hence, these methods are based on strong assumptions and often require large sample size while, in practice, researchers have to deal with few individuals and many landmarks, implying over-dimensioned spaces and loss of power.
In light of all these considerations, we propose an extension of the nonparametric combination (NPC) methodology to shape analysis. We illustrate how it is possible to obtain powerful tests in a nonparametric framework by increasing the number of informative variables while leaving the number of cases fixed. In particular,
the power of the suggested tests increases when the number of processed variables increases, provided that the induced noncentrality parameter increases, and this result holds even when the number of variables is larger than the permutation sample space (Brombin, Pesarin and Salmaso, 2010). Applications to real case
studies in biology and in rhinoseptoplasty surgery are shown.