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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> and fits the location, scale and shape parameters of the distribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> as smooth functions of explanatory variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>X</mml:mi></mml:mrow> <mml:annotation>$$ X $$</mml:annotation></mml:semantics> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>X</mml:mi></mml:mrow> <mml:annotation>$$ X $$</mml:annotation></mml:semantics> </mml:math> in one or more of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> distribution parameters. In general, high curvature is reduced (i.e., attenuated or dampened) by smoothing. In the first method, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>X</mml:mi></mml:mrow> <mml:annotation>$$ X $$</mml:annotation></mml:semantics> </mml:math> is transformed to variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>T</mml:mi></mml:mrow> <mml:annotation>$$ T $$</mml:annotation></mml:semantics> </mml:math> to reduce this high curvature, and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> distribution parameters are fitted as smooth functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>T</mml:mi></mml:mrow> <mml:annotation>$$ T $$</mml:annotation></mml:semantics> </mml:math> . Three different transformations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>X</mml:mi></mml:mrow> <mml:annotation>$$ X $$</mml:annotation></mml:semantics> </mml:math> are described. In the second method, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> distribution parameters are adaptively smoothed against <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>X</mml:mi></mml:mrow> <mml:annotation>$$ X $$</mml:annotation></mml:semantics> </mml:math> by allowing the smoothing parameter itself to vary continuously with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Y</mml:mi></mml:mrow> <mml:annotation>$$ Y $$</mml:annotation></mml:semantics> </mml:math> . 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.