Nonparametric estimation of age-specific reference percentile curves with radial smoothing
Introduction
Anthropometric measurements such as height, weight, body mass index (BMI) have become available through large-scale clinical, observational, and survey studies, e.g. the National Health and Nutrition Examination Survey (NHANES) in the USA [1]. Availability of anthropometric data at the population level makes it both feasible and necessary to estimate age-specific reference percentile curves. Age-specific reference percentile curves can be defined as a series of curves that describe the distribution of selected body measurements or characteristics at different ages. The human growth charts for length, height, weight, and head circumference are probably the best-known examples of age-specific reference percentile curves and have been used extensively in both clinical and research settings, see [1], [2]. The growth charts not only provide an overall impression for tracking the growth of infants, children and adolescents, but also summarize population-level data for clinical diagnostics purposes. The growth charts can also be used to calculate the standardized z-scores (also known as standard deviation scores in clinical literature), which are usually calculated as the difference between the individual's value and the mean value of the reference population, divided by the population standard deviation. The z-score is widely used in both clinical practice and medical research to quantify relationship between an individual's measurements and those of the reference population.
The LMS method developed in [3] is a well-established and widely-used technique for constructing age-related reference percentile curves. The method generalizes the Box–Cox transformation and maximizes the Box–Cox log-likelihood function with three penalized curvature terms for the L(degree of skewness)-curve, M(central tendency)-curve and S(dispersion)-curve. The Fisher scoring method and cubic splines are used for estimation in the LMS method and variations of the LMS method exist [1]. The US Centers for Disease Control and Prevention (CDC) used the LMS method to create the widely-adopted 2000 CDC growth charts based on the NHANES data, the readers may refer to [1]. The 3rd, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 97th percentile curves are usually presented.
The LMS method shares some limitations of power transformation with the original Box–Cox transformation. The smoothing parameters in the penalty terms are empirically chosen and users of publicly-available software packages such as LMSChartMaker available from [4] may run into computational difficulties. As an example, the LMS method has problems such as calculating roots of negative numbers because of the nature of power transformation, see [5].
In addition to the LMS method, about 30 existing methods have been reviewed and compared in [2] before the World Health Organization (WHO) chose to use the generalized additive models for location, scale and shape (GAMLSS) technique developed in [6] for its recently published growth standards. It was shown that the Box–Cox t (BCT) distribution in GAMLSS provides a generalization of the Box–Cox normal (BCN) distribution model in the LMS method [7]. Nonparametric methods for estimating percentile curves such as the double-kernel-based method developed in [8] have also been studied and applied to real-world data. Although there are numerous existing methods, including the most recently developed GAMLSS method, for constructing growth charts, accurate, flexible and robust methods are still nevertheless needed for estimating growth curves from real-world data since even small improvements in percentile estimation is important (especially in the tail percentiles) and at time points at the boundary because these tail percentiles are often used to diagnose serious medical conditions (e.g. idiopathic short stature).
Our aim in this paper is to present a new nonparametric approach for estimation of age-specific reference percentile curves with a focus on growth charts when the underlying distribution is close to normal, using the proposed radial smoothing (RS) method; and to compare the RS method with both the LMS and the GAMLSS methods. Section 2 describes the development of the RS method, Section 3 presents simulation and comparison results, and demonstrates the application of the RS method to growth data from children followed in a clinical observational study. Section 4 provides a summary and discussion of the results.
Section snippets
Material and methods
Assuming that the age-specific growth data is comprised of independent data points {(ti, yi), i = 1, ⋯, n}, we can select {knotj ; j = 1, ⋯, m} as the knot locations for the ti's and fit the data using the following formwhere β0 is the intercept, , f(⋅) is the basis function, and γj is the coefficient of the basis function. It has been shown that, instead of fitting this model with γj as fixed effects, one can treat γj as random effects instead [9], [10]. This
Results
To assess the performance of our RS method by comparing it with the existing LMS and GAMLSS methods, we designed two simulation studies including (1) equally spaced percentile curves where at any time point, the percentiles are symmetric around the median, and (2) unequally spaced percentile curves. The mean growth curve of a hypothetical population from [19] is used for both cases:with parameters
Discussion and conclusion
In this article, we propose a new nonparametric method to construct age-specific reference percentile curves based on the radial smoothing technique when the underlying distribution is close to normal. We have shown in the simulation studies in which the underlying distributions are either normal or close to normal that the RS method outperforms the two widely used methods, i.e. the LMS and GAMLSS methods. We also used three simple error measurements (i.e. square root of mean squared error,
Acknowledgments
The authors would like to thank Drs. Roy Tamura, Charmian Quigley and Mayme Wong for review and comments, Dr. Mikis Stasinopoulos for help with GAMLSS package, and Yuqin Li for programming assistance. The authors are also grateful to the two anonymous reviewers whose comments help to improve the paper. The first author presented some of the results in UCLA IPAM (Institute for Pure & Applied Mathematics) Cells and Materials Reunion Conference II December, 2008 and would like to thank IPAM for
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