The Annals of Statistics

Conditional growth charts

Ying Wei and Xuming He

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Growth charts are often more informative when they are customized per subject, taking into account prior measurements and possibly other covariates of the subject. We study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions. The model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers, and it is also flexible for including covariate information. We propose a rank score test for large sample inference on covariates, and develop a new model assessment tool for longitudinal growth data. Our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis.

Article information

Ann. Statist. Volume 34, Number 5 (2006), 2069-2097.

First available in Project Euclid: 23 January 2007

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Mathematical Reviews number (MathSciNet)

Primary: 62F35: Robustness and adaptive procedures
Secondary: 62J20: Diagnostics 62P10: Applications to biology and medical sciences

Centile reference curves longitudinal data quantile regression semiparametric


Wei, Ying; He, Xuming. Conditional growth charts. Ann. Statist. 34 (2006), no. 5, 2069--2097. doi:10.1214/009053606000000623.

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  • Altman, D. G. (1993). Construction of age-related reference centiles using absolute residuals. Statistics in Medicine 12 917--924.
  • Cai, Z. and Xu, X. (2004). Nonparametric quantile regression for dynamic smooth coefficient models. Preprint.
  • Chen, H. (1991). Polynomial splines and nonparametric regression. J. Nonparametr. Statist. 1 143--156.
  • Chiang, C.-T., Rice, J. A. and Wu, C. O. (2001). Smoothing spline estimation for varying coefficient models with repeatedly measured dependence variables. J. Amer. Statist. Assoc. 96 605--619.
  • Cole, T. J. (1988). Fitting smoothed centile curves to reference data. J. Roy. Statist. Soc. Ser. A 151 385--418.
  • Cole, T. J. (1994). Growth charts for both cross-sectional and longitudinal data. Statistics in Medicine 13 2477--2492.
  • Cole, T. J., Freeman, J. V. and Preece, M. A. (1995). Body mass index reference curves for the UK, 1990. Archives of Disease in Childhood 73 25--29.
  • Cole, T. J., Freeman, J. V. and Preece, M. A. (1998). British 1990 growth reference centiles for weight, height, body mass index and head circumference fitted by maximum penalized likelihood. Statistics in Medicine 17 407--429.
  • Cole, T. J. and Green, P. J. (1992). Smoothing reference centile curves: The LMS method and penalized likelihood. Statistics in Medicine 11 1305--1319.
  • Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 1031--1057.
  • Fan, J. and Li, R. (2004). New estimation and model selection procedures for semiparametric modelling in longitudinal data analysis. J. Amer. Statist. Assoc. 99 710--723.
  • Gutenbrunner, C. and Jurěcková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305--330.
  • Gutenbrunner, C., Jurěcková, J., Koenker, R. and Portnoy, S. (1993). Tests of linear hypotheses based on regression rank scores. J. Nonparametr. Statist. 2 307--331.
  • Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
  • Hamill, P. V., Drizd, T. A., Johnson, C. L., Reed, R. B., Roche, A. F. and Moore, W. M. (1979). Physical growth: National Center for Health Statistics percentiles. American J. Clinical Nutrition 32 607--629.
  • He, X. (1997). Quantile curves without crossing. Amer. Statist. 51 186--192.
  • He, X. and Shao, Q. (2000). On parameters of increasing dimensions. J. Multivariate Anal. 73 120--135.
  • He, X. and Shi, P. (1994). Convergence rate of B-spline estimators of nonparametric conditional quantile functions. J. Nonparametr. Statist. 3 299--308.
  • He, X. and Shi, P. (1996). Bivariate tensor-product B-splines in a partly linear model. J. Multivariate Anal. 58 162--181.
  • He, X. and Zhu, L.-X. (2003). A lack-of-fit test for quantile regression. J. Amer. Statist. Assoc. 98 1013--1022.
  • He, X., Zhu, Z. and Fung, W. (2002). Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika 89 579--590.
  • Healy, M. J. R., Rasbash, J. and Yang, M. (1988). Distribution-free estimation of age-related centiles. Ann. Human Biology 15 17--22.
  • Lin, D. Y. and Ying, Z. (2001). Semiparametric and nonparametric regression analysis of longitudinal data (with discussion). J. Amer. Statist. Assoc. 96 103--126.
  • Lin, X. and Carroll, R. J. (2001). Semiparametric regression for clustered data using generalized estimating equations. J. Amer. Statist. Assoc. 96 1045--1056.
  • Kim, M. (2003). Quantile regression in a varying coefficient model. Ph.D. dissertation, Univ. Illinois at Urbana-Champaign.
  • Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33--50.
  • Koenker, R. and Xiao, Z. (2002). Inference on the quantile regression process. Econometrica 70 1583--1612.
  • Koul, H. L. and Saleh, A. K. Md. E. (1995). Autoregression quantiles and related rank-score processes. Ann. Statist. 23 670--689.
  • Koziol, J. A., Ho, N. J., Felitti, V. J. and Beutler, E. (2001). Reference centiles for serum ferritin and percentage of transferrin saturation, with application to mutations of the HFE gene. Clinical Chemistry 47 1804--1810.
  • Lau, Y. L., Jones, B. M., Ng, K. W. and Yeung, C. Y. (1993). Percentile ranges for serum IgG subclass concentrations in healthy Chinese children. Clinical and Experimental Immunology 91 337--341.
  • Pere, A. (2000). Comparison of two methods of transforming height and weight to Normality. Ann. Human Biology 27 35--45.
  • Rigby, R. A. and Stasinopoulos, D. M. (2000). Construction of reference centiles using mean and dispersion additive models. The Statistician 49 41--50.
  • Royston, P. (1995). Calculation of unconditional and conditional reference intervals for foetal size and growth from longitudinal measurements. Statistics in Medicine 14 1417--1436.
  • Royston, P. and Matthews, J. N. S. (1991). Estimation of reference ranges from normal samples. Statistics in Medicine 10 691--695.
  • Scheike, T. H. and Zhang, M. (1998). Cumulative regression function tests for regression function models for longitudinal data. Ann. Statist. 26 1328--1355.
  • Scheike, T. H., Zhang, M. and Juul, A. (1999). Comparing reference charts. Biometrical J. 41 679--687.
  • Stettler, N., Zemel, B. S., Kumanyika, S. and Stallings, V. (2002). Infant weight gain and childhood overweight status in a multicenter cohort study. Pediatrics 109 194--199.
  • Thompson, M. L. and Fatti, L. P. (1997). Construction of multivariate centile charts for longitudinal measurements. Statistics in Medicine 16 333--345.
  • Thompson, M. L. and Theron, G. B. (1990). Maximum likelihood estimation of reference centiles. Statistics in Medicine 9 539--548.
  • Wei, Y. (2004). Longitudinal growth charts based on semi-parametric quantile regression. Ph.D. dissertation, Univ. of Illinois at Urbana-Champaign.
  • Wei, Y., Pere, A., Koenker, R. and He, X. (2006). Quantile regression methods for reference growth charts. Statistics in Medicine 25 1369--1382.
  • Wright, E. M. and Royston, P. (1997). Simplified estimation of age-specific reference intervals for skewed data. Statistics in Medicine 16 2785--2803.
  • Wright, E. M. and Royston, P. (1997). A comparison of statistical methods for age-related reference intervals. J. Roy. Statist. Soc. Ser. A 160 47--69.
  • Xu, X. (2005). Semiparametric quantile dynamic time series models and their applications. Ph.D. dissertation, Univ. North Carolina at Charlotte.