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A new variance estimator for parameters of semiparametric generalized additive models

  • W. Dana FlandersEmail author
  • Mitch Klein
  • Paige Tolbert
Article

Abstract

Generalized additive models (GAMs) have become popular in the air pollution epidemiology literature. Two problems, recently surfaced, concern implementation of these semiparametric models. The first problem, easily corrected, was laxity of the default convergence criteria. The other, noted independently by Klein, Flanders, and Tolbert, and Ramsay, Burnett, and Krewski concerned variance estimates produced by commercially available software. In simulations, they were as much as 50% too small. We derive an expression for a variance estimator for the parametric component of generalized additive models that can include up to three smoothing splines, and show how the standard error (SE) estimated by this method differs from the corresponding SE estimated with error in a study of air pollution and emergency room admissions for cardiorespiratory disease. The derivation is based on asymptotic linearity. Using Monte Carlo experiments, we evaluated performance of the estimator in finite samples. The estimator performed well in Monte Carlo experiments, in the situations considered. However, more work is needed to address performance in additional situations. Using data from our study of air pollution and cardiovascular disease, the standard error estimated using the new method was about 10% to 20% larger than the biased, commercially available standard error estimate.

Key Words

Epidemiologic methods Generalized additive models Semiparametric models Variance 

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References

  1. Borja-Aburta, V.H., Castillejos, M., Gold, D. R., Bierzwinski, S. and Loomis, D. (1998), “Mortality and Ambient Fine Particles in Southwest Mexico City, 1993–1995,” Environmental Health Perspectives, 106, 849–855.CrossRefGoogle Scholar
  2. Buja, A., Hastie, T., Tibshirani, R. (1998), “Linear Smoothers and Additive Models,” The Annals of Statistics, 17, 453–510.CrossRefMathSciNetGoogle Scholar
  3. Burnett, R.T., Smith-Doiron M., Stieb, D., Cakmak, S., and Brook, J., (1999), “Effects of Particulate and Gaseous Air Pollution on Cardiorespiratory Hospitalizations,” Archives of Environmental Health, 54, 130–139.CrossRefGoogle Scholar
  4. Conceicao, G.M.S., Miraglia, S.G.E.K., Kishi, H.S., Saldiva, P.N.H., and Singer, J.M. (2001), “Air Pollution and Child Mortality: A Time-Series Study in Sao Paolo, Brazil,” Environmental Health Perspectives, 109, 347–350.CrossRefGoogle Scholar
  5. Dominici, F., McDermott, A., Zeger, S.L., and Samet, J.M. (2002), “On the Use of Generalized Additive Models in Time-Series Studies of Air Pollution and Health,” American Journal of Epidemiology, 156, 193–203.CrossRefGoogle Scholar
  6. Dominici, F., McDermott, A., and Hastie, T. (2002), “Semiparametric Regression in Time Series Analyses of Air Pollution and Mortality: Generalized Additive and Generalized Linear Models,” Presentation on Variance of GAM Estimators, Environmental Protection Agency Workshop on GAM-Related Statistical Issues in PM Epidemiology, November 4–6, 2002, Durham, NC.Google Scholar
  7. Dominici, F., McDermott, A., and Hastie, T. (2003), “Improved Semi-Parametric Time Series Models of Air Pollution and Mortality” [on-line], http: //www.biostat.jhsph.edu/∼fdominic/jasa.R2.pdf.Google Scholar
  8. Flanders, W.D., Klein, M., and Tolbert, P. (2002), “A New Variance Estimator for Parameters of Semi-parametric Generalized Additive Models. A Report to the U.S. Environmental Protection Agency,” Based on a Presentation at the Environmental Protection Agency Workshop on GAM-Related Statistical Issues in PM Epidemiology, November 4–6, 2002, Durham, NC.Google Scholar
  9. Flanders, W.D., Klein, M., and Tolbert, P. (2003), “A New Variance Estimator for Parameters of Semi-parametric Generalized Additive Models,” Technical Report, Rollins School of Public Health, Emory University. Department of Biostatistics, Atlanta, GA.Google Scholar
  10. Hastie, T.J., and Tibshirani, R.J. (1990), Generalized Additive Models, Monographs on Statistics and Applied Probability 43, New York: Chapman & Hall.zbMATHGoogle Scholar
  11. Health Effects Institute (2003), “Revised Analyses of Time Series Studies of Air Pollution and Health,” Special Report, Boston, MA: Health Effects Institute Boston, MA.Google Scholar
  12. Katsouyanni, K., Touloumi, G., Samoli, E., Gryparis, A., Monopolis, Y., LeTertre, A., Boumghar, A., Rossi, G., Zmirou, D., Ballester, F., Anderson, H.R., Wojtyniak, B., Paldy, A., Braunstein, R., Pekkanen, J., Schindler, C., and Schwartz, J. (2002), “Different Convergence Parameters Applied to the S-Plus GAM Function,” Epidemiology, 13, 742–743.CrossRefGoogle Scholar
  13. Klein, M., and Flanders, W. D., and Tolbert, P. E. (2002), “Variances may be Underestimated Using Available Software for Generalized Additive Models,” American Journal of Epidemiology, 155, s106.Google Scholar
  14. Lumley, T., and Sheppard, L., (2003), “Time Series Analyses of Air Pollution and Health: Straining at Gnats and Swallowing Camels,” Epidemiology, 14, 13–14.CrossRefGoogle Scholar
  15. McCullagh, P., and Nelder, J.A. (1989), Generalized Additive Models, New York: Chapman and Hall, pp. 327–329.Google Scholar
  16. Michelozzi, P., Forastiere, F., Fusco, D., Perucci, C.A., Ostro, B., Ancona, C., and Palotti, G. (1998), “Air Pollution and Daily Mortality in Rome, Italy,” Occupational and Environmental Medicine, 44, 605–610.CrossRefGoogle Scholar
  17. Moolgavkar, S. (2000), “Air Pollution and Hospital Admissions for Diseases of the Circulatory System in Three U.S. Metropolitan Areas,” Journal of Air Waste Management Association, 50, 1199–1206.Google Scholar
  18. Pope, C.A., Hill, R.W., and Villegas, G.M. (1999), “Particulate Air Pollution and Daily Mortality on Utah’s Wasatch Front,” Environmental Health Perspectives, 107, 567–573.CrossRefGoogle Scholar
  19. Ramsay, T., Burnett, R., and Krewski, D. (2003), “The Effect of Concurvity in Generalized Additive Models Linking Mortality to Ambient Air Pollution,” Epidemiology, 14, 18–23.CrossRefGoogle Scholar
  20. Samet, J.M., Dominici, F., Curriero, F., Coursac, I., and Zeger, S.L. (2000), “Fine Particulate Air Pollution and Mortality in 20 U.S. Cities: 1987–1994,” New England Journal of Medicine, 343, 1742–1757.CrossRefGoogle Scholar
  21. SAS Institute (2001), The SAS system for Windows, Release 8.02, TS Level 02M0, Cary, NC: SAS Institute.Google Scholar
  22. Schwartz, J. (1994a), “The Use of Generalized Additive Models in Epidemiology,” XVIIth International Biometric Conference, Hamilton, Ontario, Canada, August 8–12, 1994. Proceedings, Volume 1: Invited papers.Google Scholar
  23. — (1994b), “Air Pollution and Hospital Admissions for the Elderly in Birmingham, Alabama,” American Journal of Epidemiology, 139, 589–598.Google Scholar
  24. Tolbert, P.E., Klein, M., Metzger, K.B., Peel, J., Flanders, W.D., Todd, K., Mulholland, J.A., Ryan, P.B., and Frumkin, H. (2000), “Interim Results of the Study of Particulates and Health in Atlanta (SOPHIA),” Journal of Exposure Analysis and Environmental Epidemiology, 20, 446–460.CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2005

Authors and Affiliations

  • W. Dana Flanders
    • 1
    Email author
  • Mitch Klein
    • 1
    • 2
  • Paige Tolbert
    • 1
    • 2
  1. 1.Rollins School of Public Health, Department of EpidemiologyEmory UniversityAtlanta
  2. 2.Department of Environmental and Occupational HealthEmory University, Rollins School of Public HealthAtlanta

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