Developmental mechanisms that canalize or compensate perturbations of organismal development (targeted or compensatory growth) are widely considered a prerequisite of individual health and the evolution of complex life, but little is known about the nature of these mechanisms. It is even unclear if and how a "target trajectory" of individual development is encoded in the organism's genetic-developmental system or, instead, emerges as an epiphenomenon. Here we develop a statistical model of developmental canalization based on an extended autoregressive model. We show that under certain assumptions the strength of canalization and the amount of canalized variance in a population can be estimated, or at least approximated, from longitudinal phenotypic measurements, even if the target trajectories are unobserved. We extend this model to multivariate measures and discuss reifications of the ensuing parameter matrix. We apply these approaches to longitudinal geometric morphometric data on human postnatal craniofacial size and shape as well as to the size of the frontal sinuses. Craniofacial size showed strong developmental canalization during the first 5 years of life, leading to a 50% reduction of cross-sectional size variance, followed by a continual increase in variance during puberty. Frontal sinus size, by contrast, did not show any signs of canalization. Total variance of craniofacial shape decreased slightly until about 5 years of age and increased thereafter. However, different features of craniofacial shape showed very different developmental dynamics. Whereas the relative dimensions of the nasopharynx showed strong canalization and a reduction of variance throughout postnatal development, facial orientation continually increased in variance. Some of the signals of canalization may owe to independent variation in developmental timing of cranial components, but our results indicate evolved, partly mechanically induced mechanisms of canalization that ensure properly sized upper airways and facial dimensions.

We studied developmental canalization in human cranial size and shape using a longitudinal sample of 26 untreated individuals (13 girls, 13 boys) from the Denver Growth Study, a longitudinal X-ray study carried out in the US between 1931 and 1966. On a total of 500 lateral radiographs, covering the age range from birth to early adulthood, 18 landmarks were digitized by Ekaterina Stansfield (see Figure 2 of the paper). Six semilandmarks were located on the external outline of the frontal bone, which were allowed to slide along the bone outline so as to minimize the bending energy of the data set around its Procrustes average. Additionally, the area of the lateral projection of the frontal sinuses was measured on every radiograph. Further information on the Denver Growth Study and the original X-rays are available at https://www.aaoflegacycollection.org/aaof_collection.html?id=UOKDenver. 

individuals.txt: Contains the sex and the number of measured radiographs for each individual (a total of 500).

age.txt: Contains the individual's age for all 500 measured radiographs.

sinus.txt: Contains the area of the projection of the frontal sinuses for all 500 radiographs.

landmarks.txt: A 500 x 36 array containing the x and y coordinates of all 18 landmarks for all 500 measured radiographs. The coordinates are in units of pixel size, which is 0.15625mm.