The Annals of Applied Statistics

A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development

Hai Liu and Wanzhu Tu

Full-text: Open access


This research examines the simultaneous influences of height and weight on longitudinally measured systolic and diastolic blood pressure in children. Previous studies have shown that both height and weight are positively associated with blood pressure. In children, however, the concurrent increases of height and weight have made it all but impossible to discern the effect of height from that of weight. To better understand these influences, we propose to examine the joint effect of height and weight on blood pressure. Bivariate thin plate spline surfaces are used to accommodate the potentially nonlinear effects as well as the interaction between height and weight. Moreover, we consider a joint model for paired blood pressure measures, that is, systolic and diastolic blood pressure, to account for the underlying correlation between the two measures within the same individual. The bivariate spline surfaces are allowed to vary across different groups of interest. We have developed related model fitting and inference procedures. The proposed method is used to analyze data from a real clinical investigation.

Article information

Ann. Appl. Stat. Volume 6, Number 4 (2012), 1861-1882.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bootstrap factor-by-surface interaction mixed effects model paired outcomes penalized estimation thin plate spline


Liu, Hai; Tu, Wanzhu. A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development. Ann. Appl. Stat. 6 (2012), no. 4, 1861--1882. doi:10.1214/12-AOAS567.

Export citation


  • Anderson, S. E. and Whitaker, R. C. (2009). Prevalence of obesity among US preschool children in different racial and ethnic groups. Archives of Pediatrics & Adolescent Medicine 163 344–348.
  • Baker, J. L., Olsen, L. W. and Sorensen, T. (2007). Childhood body-mass index and the risk of coronary heart disease in adulthood. New England Journal of Medicine 357 2329–2337.
  • Brady, T. M., Fivush, B., Parekh, R. S. and Flynn, J. T. (2010). Racial differences among children with primary hypertension. Pediatrics 126 931–937.
  • Brezger, A., Fahrmeir, L. and Hennerfeind, A. (2007). Adaptive Gaussian Markov random fields with applications in human brain mapping. J. Roy. Statist. Soc. Ser. C 56 327–345.
  • Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74 829–836.
  • Crainiceanu, C. M., Diggle, P. J. and Rowlingson, B. (2008). Bivariate binomial spatial modeling of Loa loa prevalence in tropical Africa. J. Amer. Statist. Assoc. 103 21–37.
  • Crainiceanu, C. M. and Ruppert, D. (2004). Likelihood ratio tests in linear mixed models with one variance component. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 165–185.
  • Davidson, R. and Flachaire, E. (2008). The wild bootstrap, tamed at last. J. Econometrics 146 162–169.
  • Davy, K. P. and Hall, J. E. (2004). Obesity and hypertension: Two epidemics or one? American Journal of Physiology—Regulatory Integrative and Comparative Physiology 286 R803–R813.
  • Dean, C. B., Nathoo, F. and Nielsen, J. D. (2007). Spatial and mixture models for recurrent event processes. Environmetrics 18 713–725.
  • Dubin, J. A. and Müller, H.-G. (2005). Dynamical correlation for multivariate longitudinal data. J. Amer. Statist. Assoc. 100 872–881.
  • Falkner, B. (2010). Hypertension in children and adolescents: Epidemiology and natural history. Pediatr. Nephrol. 25 1219–1224.
  • Falkner, B. and Gidding, S. (2011). Childhood obesity and blood pressure: Back to the future? Hypertension 58 754–755.
  • Falkner, B., Gidding, S. S., Ramirez-Garnica, G., Wiltrout, S. A., West, D. and Rappaport, E. B. (2006). The relationship of body mass index and blood pressure in primary care pediatric patients. J. Pediatr. 148 195–200.
  • Fujita, Y., Kouda, K., Nakamura, H., Nishio, N., Takeuchi, H. and Iki, M. (2010). Relationship between height and blood pressure in Japanese schoolchildren. Pediatr. Int. 52 689–693.
  • Ghosh, P. and Hanson, T. (2010). A semiparametric Bayesian approach to multivariate longitudinal data. Aust. N. Z. J. Stat. 52 275–288.
  • Ghosh, P. and Tu, W. (2009). Assessing sexual attitudes and behaviors of young women: A joint model with nonlinear time effects, time varying covariates, and dropouts. J. Amer. Statist. Assoc. 104 474–485.
  • Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Monographs on Statistics and Applied Probability 58. Chapman & Hall, London.
  • Guillas, S. and Lai, M.-J. (2010). Bivariate splines for spatial functional regression models. J. Nonparametr. Stat. 22 477–497.
  • Guo, X. and Carlin, B. P. (2004). Separate and joint modeling of longitudinal and event time data using standard computer packages. Amer. Statist. 58 16–24.
  • Hall, J. E. (2003). The kidney, hypertension, and obesity. Hypertension 41 625–633.
  • Hall, J. E., da Silva, A. A., do Carmo, J. M., Dubinion, J., Hamza, S., Munusamy, S., Smith, G. and Stec, D. E. (2010). Obesity-induced hypertension: Role of sympathetic nervous system, leptin, and melanocortins. J. Biol. Chem. 285 17271–17276.
  • He, X., Fung, W. K. and Zhu, Z. (2005). Robust estimation in generalized partial linear models for clustered data. J. Amer. Statist. Assoc. 100 1176–1184.
  • Huang, Z., Willett, W. C., Manson, J. E., Rosner, B., Stampfer, M. J., Speizer, F. E. and Colditz, G. A. (1998). Body weight, weight change, and risk for hypertension in women. Annals of Internal Medicine 128 81–88.
  • Humphreys, M. H. (2011). The brain splits obesity and hypertension. Nat. Med. 17 782–783.
  • Jones, R. H. (1993). Longitudinal Data with Serial Correlation: A State-Space Approach. Monographs on Statistics and Applied Probability 47. Chapman & Hall, London.
  • Kohn, R., Ansley, C. F. and Tharm, D. (1991). The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. J. Amer. Statist. Assoc. 86 1042–1050.
  • Lauer, R. M. and Clarke, W. R. (1989). Childhood risk factors for high adult blood pressure: The muscatine study. Pediatrics 84 633–641.
  • Lever, A. F. and Harrap, S. B. (1992). Essential hypertension: A disorder of growth with origins in childhood? J. Hypertens. 10 101–120.
  • Levin, A., Morad, Y., Grotto, I., Ravid, M. and Bar-Dayan, Y. (2010). Weight disorders and associated morbidity among young adults in Israel 1990–2003. Pediatr. Int. 52 347–352.
  • Lin, X. and Carroll, R. J. (2006). Semiparametric estimation in general repeated measures problems. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 69–88.
  • Lin, X. and Zhang, D. (1999). Inference in generalized additive mixed models by using smoothing splines. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 381–400.
  • Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. Ann. Statist. 16 1696–1708.
  • Liu, H. and Tu, W. (2012). Supplement to “A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development.” DOI:10.1214/12-AOAS567SUPP.
  • Mammen, E. (1993). Bootstrap and wild bootstrap for high-dimensional linear models. Ann. Statist. 21 255–285.
  • Masuo, K., Mikami, H., Ogihara, T. and Tuck, M. L. (2000). Weight gain-induced blood pressure elevation. Hypertension 35 1135–1140.
  • Morris, J. S. (2002). The BLUPs are not “best” when it comes to bootstrapping. Statist. Probab. Lett. 56 425–430.
  • Morris, J. S., Wang, N., Lupton, J. R., Chapkin, R. S., Turner, N. D., Hong, M. Y. and Carroll, R. J. (2001). Parametric and nonparametric methods for understanding the relationship between carcinogen-induced DNA adduct levels in distal and proximal regions of the colon. J. Amer. Statist. Assoc. 96 816–826.
  • Pickering, T. G., Hall, J. E., Appel, L. J., Falkner, B. E., Graves, J., Hill, M. N., Jones, D. W., Kurtz, T., Sheps, S. G. and Roccella, E. J. (2005). Recommendations for blood pressure measurement in humans and experimental animals: Part 1: Blood pressure measurement in humans: A statement for professionals from the Subcommittee of Professional and Public Education of the American Heart Association Council on High Blood Pressure Research. Hypertension 45 142–161.
  • Pinheiro, J. C. and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York.
  • Purkayastha, S., Zhang, G. and Cai, D. (2011). Uncoupling the mechanisms of obesity and hypertension by targeting hypothalamic IKK-beta and NF-kappa B. Nat. Med. 17 883–887.
  • Roca-Pardiñas, J., Cadarso-Suárez, C., Tahoces, P. G. and Lado, M. J. (2008). Assessing continuous bivariate effects among different groups through nonparametric regression models: An application to breast cancer detection. Comput. Statist. Data Anal. 52 1958–1970.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics 12. Cambridge Univ. Press, Cambridge.
  • Sain, S. R., Jagtap, S., Mearns, L. and Nychka, D. (2006). A multivariate spatial model for soil water profiles. J. Agric. Biol. Environ. Stat. 11 462–480.
  • Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605–610.
  • Shankar, R. R., Eckert, G. J., Saha, C., Tu, W. and Pratt, J. H. (2005). The change in blood pressure during pubertal growth. Journal of Clinical Endocrinology & Metabolism 90 163–167.
  • Steinberger, J., Daniels, S. R., Eckel, R. H., Hayman, L., Lustig, R. H., McCrindle, B. and Mietus-Snyder, M. L. (2009). Progress and challenges in metabolic syndrome in children and adolescents: A scientific statement from the American Heart Association atherosclerosis, hypertension, and obesity in the young Committee of the Council on cardiovascular disease in the young; Council on cardiovascular nursing; and Council on nutrition, physical activity, and metabolism. Circulation 119 628–647.
  • Stram, D. O. and Lee, J. W. (1994). Variance-components testing in the longitudinal mixed effects model. Biometrics 50 1171–1177.
  • Stray-Pedersen, M., Helsing, R. M., Gibbons, L., Cormick, G., Holmen, T. L., Vik, T. and Belizán, J. M. (2009). Weight status and hypertension among adolescent girls in Argentina and Norway: Data from the ENNyS and HUNT studies. BMC Public Health 9 398.
  • Tu, W., Eckert, G. J., Saha, C. and Pratt, J. H. (2009). Synchronization of adolescent blood pressure and pubertal somatic growth. Journal of Clinical Endocrinology & Metabolism 94 5019–5022.
  • Tu, W., Eckert, G. J., DiMeglio, L. A., Yu, Z., Jung, J. and Pratt, J. H. (2011). Intensified effect of adiposity on blood pressure in overweight and obese children. Hypertension 58 818–824.
  • Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Statist. 13 1378–1402.
  • Wood, S. N. (2003). Thin plate regression splines. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 95–114.
  • Wood, S. N. (2006). Generalized Additive Models: An Introduction with $R$. Chapman & Hall/CRC, Boca Raton, FL.
  • Wood, S. N. (2010). mgcv: GAMs with GCV/AIC/REML smoothness estimation and GAMMs by PQL. R package version 1.7-2.
  • Zhang, D. and Lin, X. (2003). Hypothesis testing in semiparametric additive mixed models. Biostatistics 4 57–74.

Supplemental materials

  • Supplementary material: Detailed model-fitting algorithm and model diagnostics. We provide the computational details of the model-fitting algorithm with sample R code and an R function to visualize the predicted bivariate surfaces. Some model diagnostics plots are also provided.