Smoothing Spline Growth Curves With Covariates

TL;DR

This paper extends Wahba's interactive spline model to estimate growth curves with covariates using a non-parametric smoothing spline approach, with data-driven smoothness selection and insights into the risk estimation process.

Contribution

It adapts the spline model for growth curves with covariates, providing a data-driven method for smoothness selection and analyzing the risk estimate's properties.

Findings

Growth curve estimates are weakly dependent on knot placement when discretization error is small.

The smoothness parameter is optimally chosen by minimizing an empirical error estimate.

The risk estimate is a weighted goodness of fit, with a modified version replacing $\sigma^2$ by its unbiased estimate.

Abstract

We adapt the interactive spline model of Wahba to growth curves with covariates. The smoothing spline formulation permits a non-parametric representation of the growth curves. In the limit when the discretization error is small relative to the estimation error, the resulting growth curve estimates often depend only weakly on the number and locations of the knots. The smoothness parameter is determined from the data by minimizing an empirical estimate of the expected error. We show that the risk estimate of Craven and Wahba is a weighted goodness of fit estimate. A modified loss estimate is given, where $\sigma^2$ is replaced by its unbiased estimate.