Nonparametric estimation of age-specific reference percentile curves with radial smoothing

doi:10.1016/j.cct.2011.09.002

Contemporary Clinical Trials

Volume 33, Issue 1, January 2012, Pages 13-22

https://doi.org/10.1016/j.cct.2011.09.002 Get rights and content

Abstract

Reference percentile curves represent the covariate-dependent distribution of a quantitative measurement and are often used to summarize and monitor dynamic processes such as human growth. We propose a new nonparametric method based on a radial smoothing (RS) technique to estimate age-specific reference percentile curves assuming the underlying distribution is relatively close to normal. We compared the RS method with both the LMS and the generalized additive models for location, scale and shape (GAMLSS) methods using simulated data and found that our method has smaller estimation error than the two existing methods. We also applied the new method to analyze height growth data from children being followed in a clinical observational study of growth hormone treatment, and compared the growth curves between those with growth disorders and the general population.

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 {(t_i, y_i), i = 1, ⋯, n}, we can select {knot_j ; j = 1, ⋯, m} as the knot locations for the t_i's and fit the data using the following form $y_{i} = β_{0} + \sum_{j = 1}^{m} γ_{j} \cdot f (t_{i}; {knot}_{j}) + ε_{i}$ where β₀ is the intercept, $ε_{i} \overset{i n d .}{\sim} N (0, σ_{ε}^{2})$ , f(⋅) is the basis function, and γ_j is the coefficient of the basis function. It has been shown that, instead of fitting this model with γ_j as fixed effects, one can treat γ_j 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: $g (t) = A \{1 - \frac{1}{1 + {(t / D_{1})}^{C_{1}} + {(t / D_{2})}^{C_{2}} + {(t / D_{3})}^{C_{3}}}\}$ with parameters $\begin{array}{l} A = 175.0, \\ C_{1} = 0.5912, C_{2} = 3.630, C_{3} = 21.84, \\ D_{1} = 3.032, D_{2} = 9.982, D_{3} = 13.75. \end{array}$

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

References (19)

R.J. Kuczmarski et al.
2000 cdc growth charts for the united states: methods and development
Vital Health Stat
(2002)
E. Borghi et al.
Construction of the world health organization child growth standards: selection of methods for attained growth curves
Stat Med
(2006)
T.J. Cole et al.
Smoothing reference centile curves: the LMS method and penalized likelihood
Stat Med
(1992)
T.J. Cole
Lmschartmaker
CDC
Biv cutoffs documentation
R.A. Rigby et al.
The gamlss project: a flexible approach to statistical modelling
R.A. Rigby et al.
Using the Box–Cox t distribution in GAMLSS to model skewness and kurtosis
Statistical Modelling
(2006)
Y. Li et al.
Application of nonparametric quantile regression to body mass index percentile curves from survey data
Stat Med
(2010)
J.L. French et al.
Comment on ke and wang
J Am Stat Assoc
(2001)

There are more references available in the full text version of this article.

Cited by (7)

Continuous reference intervals for 21 biochemical and hematological analytes in healthy Chinese children and adolescents: The PRINCE study
2022, Clinical Biochemistry
Citation Excerpt :
Many methodological researches have been conducted on modelling continuous RIs in recent years [28–32]. Typical methods include non-parametric models such as quantile regression and smoothing [8,9,17,28,29], and semi-parametric models such as LMS and GAMLSS [10,30–32]. Non-parametric models fit continuous reference limits by polynomial functions of age regardless of the analytes’ distribution, and thus show advantages when there is no suitable transformation method to achieve normality.
Critical gaps have existed in pediatric reference intervals in China. In this study, we presented the sex and age distributions of 21 laboratory analytes from childhood to adolescence, and established the corresponding continuous reference intervals based on direct samples.
We used the data from the Pediatric Reference Intervals in China (PRINCE), which is a nation-wide cross-sectional study enrolling 15,150 healthy children and adolescents aged 0 – <20 years from 11 centers across China. Blood samples were collected and analyzed by trained staff following standard operating procedures. Biochemical tests were performed with Cobas C702 at the central laboratory, and hematological tests were performed with Sysmex XE, XN, or XS that satisfy the national standards at each participating center. Children younger than 3 months were excluded due to high neonatal variability and insufficient samples. Continuous reference intervals were calculated using the generalized additive models for location, shape, and scale, and were validated among another 387 healthy volunteers.
We provided pediatric continuous reference intervals for 21 commonly used biochemical and hematological analytes in China, and depicted the changes in analyte concentrations from 3 months to 20 years. The out-of-range values for all analytes were less than 10%, indicating a well applicability of the continuous reference intervals to the general pediatric population.
This is the first comprehensive report of continuous reference intervals based on healthy Chinese children, reflecting the complex dynamic trends of analytes from infancy to adulthood. Applying continuous reference intervals to clinical practice would not only improve the laboratory test result interpretation, but also help better clinical decision making.
Comparison of Pulmonary Function Decline in Steroid-Treated and Steroid-Naïve Patients with Duchenne Muscular Dystrophy
2019, Journal of Pediatrics
Citation Excerpt :
Decile curves were derived using a linear mixed model with radial smoothing. The within-patient covariance structure was the identity matrix; in other words, longitudinal data were treated as independent to generate the fitted percentile curves.25 Of 416 patients with DMD reviewed at the Cincinnati Children's Hospital Medical Center, 184 patients met criteria for more than 2 acceptable and repeatable spirometry tests.
To describe and compare the lung function decline in patients with Duchenne muscular dystrophy on glucocorticoid therapy in contrast with glucocorticoid-naïve patients, and to define the deciles of pulmonary decline in glucocorticoid-treated patients.
This retrospective study examined lung function of patients with Duchenne muscular dystrophy over 6 years of age followed between 2001 and 2015 at 2 centers—glucocorticoid-treated patients in Cincinnati, Ohio, and glucocorticoid-naïve patients in Paris, France. Forced vital capacity (FVC, FVC%), forced expiratory volume in 1 second, maximal inspiratory pressure, maximal expiratory pressure, and peak expiratory flow data were analyzed. Only FVC data were available for the French cohort.
There were 170 glucocorticoid-treated patients (92%), 5 patients (2.7%) with past glucocorticoid use, and 50 French glucocorticoid-naïve patients. The peak absolute FVC was higher and was achieved at earlier ages in glucocorticoid-treated compared with glucocorticoid-naïve patients (peak FVC, 2.4 ± 0.6 L vs 1.9 ± 0.7 L; P < .0001; ages 13.5 ± 3.0 years vs 14.3 ± 2.8 years; P = .03). The peak FVC% was also higher and was achieved at earlier ages in glucocorticoid-treated patients (peak FVC%, 105.1 ± 25.1% vs 56 ± 20.9%; P < .0001; ages 11.9 ± 2.9 years vs 13.6 ± 3.2 years; P = .002). Rates of decline for both groups varied with age. Maximal rates of decline were 5.0 ± 0.26% per year (12-20 years) for glucocorticoid-treated and 5.1 ± 0.39% per year for glucocorticoid-naïve patients (11-20 years; P = .2). Deciles of FVC% decline in glucocorticoid-treated patients show that patients experience accelerated decline at variable ages.
These data describe nonlinear rates of decline of pulmonary function in patients with Duchenne muscular dystrophy, with improved function in glucocorticoid-treated patients. FVC% deciles may be a useful tool for clinical and research use.
Development of next-generation reference interval models to establish reference intervals based on medical data: current status, algorithms and future consideration
2023, Critical Reviews in Clinical Laboratory Sciences
Current status and challenges in establishing reference intervals based on real-world data
2023, Critical Reviews in Clinical Laboratory Sciences
Evaluation of posterior fossa biometric measurements on fetal mri in the evaluation of dandy-walker continuum
2021, American Journal of Neuroradiology
Comparison of four algorithms on establishing continuous reference intervals for pediatric analytes with age-dependent trend
2020, BMC Medical Research Methodology

View all citing articles on Scopus

View full text

Nonparametric estimation of age-specific reference percentile curves with radial smoothing

Abstract

Introduction

Section snippets

Material and methods

Results

Discussion and conclusion

Acknowledgments

2000 cdc growth charts for the united states: methods and development

Vital Health Stat

Construction of the world health organization child growth standards: selection of methods for attained growth curves

Stat Med

Smoothing reference centile curves: the LMS method and penalized likelihood

Stat Med

Lmschartmaker

Biv cutoffs documentation

The gamlss project: a flexible approach to statistical modelling

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

Statistical Modelling

Application of nonparametric quantile regression to body mass index percentile curves from survey data

Stat Med

Comment on ke and wang

J Am Stat Assoc