Objective To construct growth curves for school-aged children and adolescents that accord with the WHO Child Growth Standards for preschool children and the body mass index (BMI) cut-offs for adults. Methods Data from the 1977 National Center for Health Statistics (NCHS)/WHO growth reference (1–24 years) were merged with data from the under-fives growth standards’ cross-sectional sample (18–71 months) to smooth the transition between the two samples. State-of-the-art statistical methods used to construct the WHO Child Growth Standards (0–5 years), i.e. the Box-Cox power exponential (BCPE) method with appropriate diagnostic tools for the selection of best models, were applied to this combined sample. Findings The merged data sets resulted in a smooth transition at 5 years for height-for-age, weight-for-age and BMI-for-age. For BMI-for-age across all centiles the magnitude of the difference between the two curves at age 5 years is mostly 0.0 kg/m² to 0.1 kg/m². At 19 years, the new BMI values at +1 standard deviation (SD) are 25.4 kg/m² for boys and 25.0 kg/m² for girls. These values are equivalent to the overweight cut-off for adults (> 25.0 kg/m²). Similarly, the +2 SD value (29.7 kg/m² for both sexes) compares closely with the cut-off for obesity (> 30.0 kg/m²). Conclusion The new curves are closely aligned with the WHO Child Growth Standards at 5 years, and the recommended adult cut-offs for overweight and obesity at 19 years. They fill the gap in growth curves and provide an appropriate reference for the 5 to 19 years age group. Bulletin of the World Health Organization 2007;85:660–667.

/pdf/development-of-a-who-growth-reference-for-school-aged-2ms904e97f.pdf

Development of a WHO growth reference for school-aged children and adolescents

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Modern Applied Statistics With S

WHO child growth standards: length/height-for-age, weight-for-age, weight-for-length, weight-for-height and body mass index-for-age - Methods and development

Aim: To describe the methods used to construct the WHO Child Growth Standards based on length/height, weight and age, and to present resulting growth charts. Methods: The WHO Child Growth Standards were derived from an international sample of healthy breastfed infants and young children raised in environments that do not constrain growth. Rigorous methods of data collection and standardized procedures across study sites yielded very high-quality data. The generation of the standards followed methodical, state-of-the-art statistical methodologies. The Box-Cox power exponential (BCPE) method, with curve smoothing by cubic splines, was used to construct the curves. The BCPE accommodates various kinds of distributions, from normal to skewed or kurtotic, as necessary. A set of diagnostic tools was used to detect possible biases in estimated percentiles or z-score curves. Results: There was wide variability in the degrees of freedom required for the cubic splines to achieve the best model. Except for length/height-for-age, which followed a normal distribution, all other standards needed to model skewness but not kurtosis. Length-for-age and height-for-age standards were constructed by fitting a unique model that reflected the 0.7-cm average difference between these two measurements. The concordance between smoothed percentile curves and empirical percentiles was excellent and free of bias. Percentiles and z-score curves for boys and girls aged 0–60 mo were generated for weight-for-age, length/height-for-age, weight-for-length/height (45 to 110 cm and 65 to 120 cm, respectively) and body mass index-for-age.

Conclusion: The WHO Child Growth Standards depict normal growth under optimal environmental conditions and can be used to assess children everywhere, regardless of ethnicity, socio-economic status and type of feeding.

WHO Child Growth Standards based on length/height, weight and age

Summary. A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9876.2005.00510.x

Generalized additive models for location, scale and shape

GAMLSS is a general framework for fitting regression type models where the distribution of the response variable does not have to belong to the exponential family and includes highly skew and kurtotic continuous and discrete distribution. GAMLSS allows all the parameters of the distribution of the response variable to be modelled as linear/non-linear or smooth functions of the explanatory variables. This paper starts by defining the statistical framework of GAMLSS, then describes the current implementation of GAMLSS in R and finally gives four different data examples to demonstrate how GAMLSS can be used for statistical modelling.

Generalized Additive Models for Location Scale and Shape (GAMLSS) in R

The Box–Cox power exponential (BCPE) distribution, developed in this paper, provides a model for a dependent variable Y exhibiting both skewness and kurtosis (leptokurtosis or platykurtosis). The distribution is defined by a power transformation Yν having a shifted and scaled (truncated) standard power exponential distribution with parameter τ. The distribution has four parameters and is denoted BCPE (µ,σ,ν,τ). The parameters, µ, σ, ν and τ, may be interpreted as relating to location (median), scale (approximate coefficient of variation), skewness (transformation to symmetry) and kurtosis (power exponential parameter), respectively.

Smooth centile curves are obtained by modelling each of the four parameters of the distribution as a smooth non-parametric function of an explanatory variable. A Fisher scoring algorithm is used to fit the non-parametric model by maximizing a penalized likelihood. The first and expected second and cross derivatives of the likelihood, with respect to µ, σ, ν and τ, required for the algorithm, are provided. The centiles of the BCPE distribution are easy to calculate, so it is highly suited to centile estimation.

This application of the BCPE distribution to smooth centile estimation provides a generalization of the LMS method of the centile estimation to data exhibiting kurtosis (as well as skewness) different from that of a normal distribution and is named here the LMSP method of centile estimation. The LMSP method of centile estimation is applied to modelling the body mass index of Dutch males against age. Copyright © 2004 John Wiley & Sons, Ltd.

Smooth centile curves for skew and kurtotic data modelled using the Box–Cox power exponential distribution

A class of generalized autoregressive moving average (GARMA) models is developed that extends the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data The dependent variable is assumed to have a conditional exponential family distribution given the past history of the process The model estimation is carried out using an iteratively reweighted least squares algorithm Properties of the model, including stationarity and marginal moments, are either derived explicitly or investigated using Monte Carlo simulation The relationship of the GARMA model to other models is shown, including the autoregressive models of Zeger and Qaqish, the moving average models of Li, and the reparameterized generalized autoregressive conditional heteroscedastic GARCH model (providing the formula for its fourth marginal moment not previously derived) The model is demonstrated by the application of the GARMA model with a negative binomial conditional distribution to 

Generalized Autoregressive Moving Average Models

The Box-Cox t (BCT) distribution is presented as a model for a dependent variable Y exhibiting both skewness and leptokurtosis. The distribution is defined by a power transformation Yv having a shifted and scaled (truncated) t distribution with degrees of freedom parameter τ. The distribution has four parameters and is denoted by BCT(μ, σ,ν, τ). The parameters μ, σ,ν and τ may be interpreted as relating to location (median), scale (centile-based coefficient of variation), skewness (power transformation to symmetry) and kurtosis (degrees of freedom), respectively. The generalized additive model for location, scale and shape (GAMLSS) is extended to allow each of the parameters of the distribution to be modelled as linear and/or non-linear parametric and/or smooth non-parametric functions of explanatory variables. A Fisher scoring algorithm is used to fit the model by maximizing a (penalized) likelihood. The first and expected second and cross derivatives of the likelihood with respect to μ, σ,ν and τ, requi...

Using the Box-Cox t distribution in GAMLSS to model skewness and kurtosis:

A model for the statistical analysis of the total amount of insurance paid out on a policy is developed and applied. The model simultaneously deals with the number of claims (zero or more) and the amount of each claim. The number of claims is from a Poisson-based discrete distribution. Individual claim sizes are from a continuous right skewed distribution. The resulting distribution of total claim size is a mixed discrete-continuous model, with positive probability of a zero claim. The means and dispersions of the claim frequency and claim size distribution are modeled in terms of risk factors. The model is applied to a car insurance data set.

D. Mikis Stasinopoulos

Papers

Generalized Additive Models for Location Scale and Shape (GAMLSS) in R

Smooth centile curves for skew and kurtotic data modelled using the Box–Cox power exponential distribution

Generalized Autoregressive Moving Average Models

Using the Box-Cox t distribution in GAMLSS to model skewness and kurtosis:

Mean and dispersion modelling for policy claims costs