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

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.