Estimating Growth in Height from Limited Longitudinal Growth Data Using Full-Curves Training Dataset: A Comparison of Two Procedures of Curve Optimization—Functional Principal Component Analysis and SITAR

Abstract

A variety of models are available for the estimation of parameters of the human growth curve. Several have been widely and successfully used with longitudinal data that are reasonably complete. On the other hand, the modeling of data for a limited number of observation points is problematic and requires the interpolation of the interval between points and often an extrapolation of the growth trajectory beyond the range of empirical limits (prediction). This study tested a new approach for fitting a relatively limited number of longitudinal data using the normal variation of human empirical growth curves. First, functional principal components analysis was done for curve phase and amplitude using complete and dense data sets for a reference sample (Brno Growth Study). Subsequently, artificial curves were generated with a combination of 12 of the principal components and applied for fitting to the newly analyzed data with the Levenberg–Marquardt optimization algorithm. The approach was tested on seven 5-points/year longitudinal data samples of adolescents extracted from the reference sample. The samples differed in their distance from the mean age at peak velocity for the sample and were tested by a permutation leave-one-out approach. The results indicated the potential of this method for growth modeling as a user-friendly application for practical applications in pediatrics, auxology and youth sport.

GROWTH: Functional Data Analysis of the human height postnatal growth in an on-line application

“…When the aim of the fitting and interpolation is also to obtain estimates of growth milestones (APV, VPV, etc., computed based on inflection points of the model curve), the method also performs reliably and comparably to the SITAR method as long as the computed milestone is within the range of empirically measured and entered values. This can be seen in the test section of our model publication (Králík et al 2021).…”

“…To ensure the biological plausibility of the fitted model, the scores of both the phase and amplitude parameters were constrained to +/-3 standard deviations during optimization. For further details of the estimation method see Králík et al (2021).…”

“…However, we ultimately moved to pure B-splines and the FDA method because the individual-specific B-spline provides a more accurate fit of the individual growth trajectory than the average spline used by the SITAR method. This does not imply a large difference in the estimates of major growth milestones (Králík et al 2021), but rather the advantage of being able to account for the details of individual child growth on a shorter time scale, which was one of the goals in developing this auxological application.…”

“…The above approach was also tested in our project but was deemed to be less suitable due to its shortcomings. We incorporated GROWTH: Functional Data Analysis of the human height postnatal growth in an on-line application the approach with two independent FPCA models (one for phase and one for amplitude) into our method (Králík et al 2021) and it is part of the presented GROWTH web application.…”

“…When the purpose of the model is to predict long-term growth over a range of many years beyond the specified range of empirical values (i.e., to extrapolate), the principle of the model calculation implies that the estimate will certainly be more approximate and more influenced by the tendency to the average curve. Permutation test approach of our model on samples of five measurements (with a span of 1 year between successive measurements) systematically selected from empirical data (of the same testing population sample-BGS by means of leave-one-out method) at varying distances from the APV showed that the FPCA model provides reasonable estimates even when extrapolated outside the empirically specified age range (i.e., predicting future growth or estimating past growth), but the resulting estimates have overall lower variance than the reference (true) APV values secured on the curves (Králík et al 2021). Reducing the variance of the estimates during extrapolation beyond limited empirical data is a general feature of growth models based on any optimization and is certainly better than the opposite extreme (increasing the variance of the estimates and large variability of the curves).…”