Biostatistics Advance Access published online on June 16, 2007
Biostatistics, doi:10.1093/biostatistics/kxm020
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Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty
Centre for Paediatric Epidemiology and Biostatistics, UCL Institute of Child Health, 30 Guilford Street, London WC1N 1EH, UK tim.cole{at}ich.ucl.ac.uk
Department of Social Medicine, University of Bristol, Bristol, UK
Statistical Laboratory, University of Cambridge, Cambridge, UK
Centre for Paediatric Epidemiology and Biostatistics, UCL Institute of Child Health, 30 Guilford Street, London WC1N 1EH, UK
* To whom correspondence should be addressed.
Higher moments of the frequency distribution of child height and weight change with age, particularly during puberty, though why is not known. Our aims were to confirm that height skewness and kurtosis change with age during puberty, to devise a model to explain why, and to test the model by analyzing the data longitudinally. Heights of 3245 Christ's Hospital School boys born during 1927–1956 were measured twice termly from 9 to 20 years (
). Treating the data as independent, the mean, standard deviation (SD), skewness, and kurtosis were calculated in 40 age groups and plotted as functions of age t. The data were also analyzed longitudinally using the nonlinear random-effects growth model 
, with 
the cross-sectional data, 
the individual mean curve, and 
and 
subject-specific random effects reflecting variability in age and height at peak height velocity (PHV). Mean height increased monotonically with age, while the SD, skewness, and kurtosis changed cyclically with, respectively, 1, 2, and 3 turning points. Surprisingly, their age curves corresponded closely in shape to the first, second, and third derivatives of the mean height curve. The growth model expanded as a Taylor series in predicted such a pattern, and the longitudinal analysis showed that adjusting for age at PHV on a multiplicative scale largely removed the trends in the higher moments. A nonlinear growth process where subjects grow at different rates, such as in puberty, generates cyclical changes in the higher moments of the frequency distribution.
Keywords: Growth; Height; Kurtosis; Random effects; Skewness
Received July 28, 2006; revised January 15, 2007; revised March 16, 2007; accepted for publication April 25, 2007.