# Improved Centile Estimation by Transformation And/Or Adaptive Smoothing of the Explanatory Variable

## TL;DR

This paper introduces two methods to improve centile estimation in growth references by addressing high curvature in distribution parameters.

## Contribution

The novel contribution is the introduction of transformation and adaptive smoothing methods to reduce high curvature in centile estimation.

## Key findings

- Transforming the explanatory variable reduces high curvature in centile estimation.
- Adaptive smoothing allows the smoothing parameter to vary with the response variable, improving fit.
- Simulations and examples demonstrate smoother and better-fitting centiles using these methods.

## Abstract

A popular approach to growth reference centile estimation is the LMS (Lambda‐Mu‐Sigma) method, which assumes a parametric distribution for response variable Y and fits the location, scale and shape parameters of the distribution of Y as smooth functions of explanatory variable X. This article provides two methods, transformation and adaptive smoothing, for improving the centile estimation when there is high curvature (i.e., rapid change in slope) with respect to X in one or more of the Y distribution parameters. In general, high curvature is reduced (i.e., attenuated or dampened) by smoothing. In the first method, X is transformed to variable T to reduce this high curvature, and the Y distribution parameters are fitted as smooth functions of T. Three different transformations of X are described. In the second method, the Y distribution parameters are adaptively smoothed against X by allowing the smoothing parameter itself to vary continuously with Y. Simulations are used to compare the performance of the two methods. Three examples show how the process can lead to substantially smoother and better fitting centiles.