Multilevel Models of Age-Related Changes in Facial Shape in Adolescents

Abstract

Here we study the effects of age on facial shape in adolescents by using a method called multilevel principal components analysis (mPCA). An associated multilevel multivariate probability distribution is derived and expressions for the (conditional) probability of age-group membership are presented. This formalism is explored via Monte Carlo (MC) simulated data in the first dataset; where age is taken to increase the overall scale of a three-dimensional facial shape represented by 21 landmark points and all other “subjective” variations are related to the width of the face. Eigenvalue plots make sense and modes of variation correctly identify these two main factors at appropriate levels of the mPCA model. Component scores for both single-level PCA and mPCA show a strong trend with age. Conditional probabilities are shown to predict membership by age group and the Pearson correlation coefficient between actual and predicted group membership is r = 0.99. The effects of outliers added to the MC training data are reduced by the use of robust covariance matrix estimation and robust averaging of matrices. These methods are applied to another dataset containing 12 GPA-scaled (3D) landmark points for 195 shapes from 27 white, male schoolchildren aged 11 to 16 years old. 21% of variation in the shapes for this dataset was accounted for by age. Mode 1 at level 1 (age) via mPCA appears to capture an increase in face height with age, which is consistent with reported pubertal changes in children. Component scores for both single-level PCA and mPCA again show a distinct trend with age. Conditional probabilities are again shown to reflect membership by age group and the Pearson correlation coefficient is given by r = 0.63 in this case. These analyses are an excellent first test of the ability of multilevel statistical methods to model age-related changes in facial shape in adolescents.