Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty
Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty
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DOI：
10.1093/biostatistics/kxm020
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发表时间：
2008-01-01
期刊：
影响因子：
2.1
通讯作者：
Pan, Huiqi
中科院分区：
文献类型：
--
作者：
Cole, Tim J.;Cortina-Borja, Mario;Pan, Huiqi
儿童身高和体重频率分布的高阶矩随年龄变化，特别是在青春期，尽管原因尚不清楚。我们的目的是确认，在青春期身高偏度和峰度随年龄的变化，设计一个模型来解释为什么，并通过纵向分析数据来测试模型。对1927-1956年出生的3245名基督医院男生的身高进行了两次定期测量，从9岁到20岁（n = 129508）。将数据作为独立数据处理，计算40个年龄组的平均值、标准差（SD）、偏度和峰度，并绘制为年龄t的函数。还使用非线性随机效应生长模型H（t）= h（t-t）+ α对数据进行了纵向分析，其中H（t）为横断面数据，h（t）为个体平均曲线，α和α为受试者特异性随机效应，反映了峰值身高增速（PHV）时年龄和身高的变异性。平均身高随年龄单调增加，而SD，偏度和峰度周期性变化，分别为1，2和3个转折点。令人惊讶的是，他们的年龄曲线在形状上与平均身高曲线的一阶、二阶和三阶导数密切对应。增长模型扩展为泰勒级数e预测这样的模式，纵向分析表明，调整年龄在PHV的乘法规模在很大程度上消除了趋势的较高时刻。受试者以不同速率生长的非线性生长过程（如青春期）会在频率分布的高阶矩中产生周期性变化。
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 (n = 129 508). 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 H( t) = h( t - epsilon) + alpha, with H( t) the cross-sectional data, h( t) the individual mean curve, and epsilon and alpha 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 e 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.