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

• Published in: Statistics in medicine Vol. 23; no. 19; pp. 3053 - 3076
• Main Authors: Rigby, Robert A., Stasinopoulos, D. Mikis
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 15.10.2004; Wiley Subscription Services, Inc
• Subjects: Adolescent; Adult; Algorithms; Body Height - physiology; Body Mass Index; Body Weight - physiology; Child; Child, Preschool; Cross-Sectional Studies; cubic smoothing splines; Data Interpretation, Statistical; Female; Humans; Infant; Likelihood Functions; Male; Medical research; Models, Statistical; Netherlands; reference curves; regression quantites; Statistics; Netherlands
• ISSN: 0277-6715; 1097-0258
• DOI: 10.1002/sim.1861

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.