Deep-SITAR: A SITAR-Based Deep Learning Framework for Growth Curve Modeling via Autoencoders

TL;DR

This work addresses predicting growth trajectories for new individuals by extending the SITAR nonlinear mixed-effects model with a neural-network–based autoencoder, termed Deep-SITAR. The encoder predicts subject-specific random effects $u_i=(a_{1i}, b_{1i}, c_{1i})$, while the decoder employs a B-spline fit to reproduce each growth curve, enabling end-to-end learning via SGD. Across simulations based on Berkeley growth data, Deep-SITAR shows improved generalization with larger sample sizes and more flexible spline bases, though fit precision can be more variable than SITAR on training data; the approach provides a scalable, predictive framework that preserves the interpretable random-effects structure of SITAR. The work highlights the potential of hybrid deep-learning and mixed-effects methods for growth-curve analysis and provides open-source code for replication and further development.

Abstract

Several approaches have been developed to capture the complexity and nonlinearity of human growth. One widely used is the Super Imposition by Translation and Rotation (SITAR) model, which has become popular in studies of adolescent growth. SITAR is a shape-invariant mixed-effects model that represents the shared growth pattern of a population using a natural cubic spline mean curve while incorporating three subject-specific random effects -- timing, size, and growth intensity -- to account for variations among individuals. In this work, we introduce a supervised deep learning framework based on an autoencoder architecture that integrates a deep neural network (neural network) with a B-spline model to estimate the SITAR model. In this approach, the encoder estimates the random effects for each individual, while the decoder performs a fitting based on B-splines similar to the classic SITAR model. We refer to this method as the Deep-SITAR model. This innovative approach enables the prediction of the random effects of new individuals entering a population without requiring a full model re-estimation. As a result, Deep-SITAR offers a powerful approach to predicting growth trajectories, combining the flexibility and efficiency of deep learning with the interpretability of traditional mixed-effects models.

Figures (8)

• Figure 1: Example of growth curves with four fitted individuals using the LME. A: Fitted curves. B: Population curve. C: Individual deviation curves from the overall trend.
• Figure 2: SITAR features. The black dashed line represents the mean growth curve, while the colored curves illustrate the effects of the random effects. The red line indicates the height shift corresponding to $a_{1i}$, the green line corresponds to the age shift corresponding to $b_{1i}$, and the blue line represents the shrinking–stretching of the age scale corresponding to $c_{1i}$.
• Figure 3: Fully connected neural network scheme.
• Figure 4: Autoencoder architecture, where $\boldsymbol{u}$ is a latent variable.
• Figure 5: Deep-SITAR scheme where the output is an input estimation, and ${\boldsymbol{u}}_i=(a_{\theta}^{1i},b_{\theta}^{1i},c_{\theta}^{1i})$.