Detection of non-affine shape outliers for matched-pair shape data

ABSTRACT Cleft lip/palate (CLP) is a relatively common birth defect so dis- figuring that nowadays it is almost always corrected surgically as early as possible. The postnatal surgical correction does not, however, result in a normally growing upper jaw, but instead, owing to scar tissue, one that grows abnormally. It is important to decide if a clinical treatment group is homogeneous. The example involves data from digitally processed lateral X-ray films of 48 boys who have complete unilateral CLP but no other malformation. 22 landmarks were represented by their Procrustes shape coordinates, principal components of matched- pair differences were examined, and the distribution of the 48 shape changes was studied for outliers in the affine and non-affine subspaces of the full Procrustes shape and form space. To separate outliers from inliers we use bagplots. There are no outliers apparent in the affine subspace. In the non-affine subspaces, we found no outliers in the subspace of bending patterns at large scale but four out- liers in the subspace of local changes at small scale. Almost the same outliers were found by form-space PCA. These latter are associated with possible creases of the corresponding thin-plate splines. In those cases we can use the same spline formalism to relax the outlying form to an inlier by optimal relaxation along the curve d´ecolletage that weighs bending energy against Procrustes distance and stop relaxation on the fence. These maneuvers suggest a possibly novel and interesting fusion of the Procrustes-spline toolkit with outlier detection. They also have practical implications for craniofacial management of CLP follow-up as well as suggestive implications for outlier detection in applied craniometrics and anthropometrics more generally.