Additive versus multiplicative models in ecologic regression

  • W. Douglas Thompson
  • Daniel Wartenberg
Original Paper


Much research in environmental epidemiology relies on aggregate-level information on exposure to potentially toxic substances and on relevant covariates. We compare the use of additive (linear) and multiplicative (log-linear) regression models for the analysis of such data. We illustrate how both additive and multiplicative models can be fit to aggregate-level data sets in which disease incidence is the dependent variable, and contrast these results with similar models fitted to individual-level data. We find (1) that for aggregate-level data, multiplicative models are more likely than additive models to introduce bias into the estimation of rates, an effect not found with individual-level data; and (2) that under many circumstances multiplicative models reduce the precision of the estimates, an effect also not found in individual-level models. For both additive and multiplicative models of aggregate-level data, we find that, in the presence of covariates, narrow confidence interval are obtained only when two or more antecedent factors are strongly related to the measured covariate and/or the exposure of primary substantive interest. We conclude that the equivalency of fitting additive versus multiplicative models in studies with individual-level binary data does not carry over to studies that analyze aggregate-level information. For aggregate data, we strongly recommend use of additive models.


  1. Bjork J, Stromberg U (2002) Effects of systematic exposure assessment errors in partially ecologic case-control studies. Int J Epidemiol 31:154–160CrossRefGoogle Scholar
  2. Bjork J, Stromberg U (2005) Model specification and unmeasured confounders in partially ecologic analyses based on group proportions of exposed. Scand J Work Environ Health 31:184–190Google Scholar
  3. Dockery DW, Pope CA, Xu X, Spengler JD, Ware JH, Fay ME, Ferris BG Jr, Speizer FE (1993) An associaton between air pollution and mortality in six U.S. cities. New Eng J Med 329:1753–1759CrossRefGoogle Scholar
  4. Gail M, Simon R (1985) Testing for qualitative interactions between treatment effects and patient subsets. Biometrics 41:361–372CrossRefGoogle Scholar
  5. Greenland S (2001) Ecologic versus individual-level sources of bias in ecologic estimates of contextual health effects. Int J Epidemiol 30:1343–1350CrossRefGoogle Scholar
  6. Greenland S (2002) A review of multilevel theory for ecologic analyses. Stat Med 21:389–395CrossRefGoogle Scholar
  7. Greenland S, Poole C (1988) Invariants and noninvariants in the concept of interdependent effects. Scand J Work Environ Health 14:125–129Google Scholar
  8. Greenland S, Robins J (1994) Invited commentary: ecologic studies—biases, misconceptions, and counterexamples. Am J Epidemiol 139:747–760Google Scholar
  9. Koepsell TD, Weiss NS (2003) Epidemiologic methods: studying the occurrence of illness. Oxford University Press, New York, pp 281–307Google Scholar
  10. Kunzli N, Tager IB (1997) The semi-individual study in air pollution epidemiology: a valid design as compared to ecologic studies. Environ Health Perspect 105:1078–1983CrossRefGoogle Scholar
  11. Monson RR (1990) Occupational Epidemiology. CRC, Boca RatonGoogle Scholar
  12. Morgenstern H (1998) Ecologic studies. In: Rothman KJ, Morgenstern H (eds) Modern epidemiology, 2nd edn. Lippincott-Raven, Philadelphia, pp 459–480Google Scholar
  13. Parkin DM, Khlat M (1996) Studies of cancer in migrants: rationale and methodology. Eur J Cancer 32A:776–771Google Scholar
  14. Peto R (1982) Statistical aspects of clinical trials. In: Harnan KE (ed) Treatment of cancer. Chapman and Hall, London, pp 867–871Google Scholar
  15. Pope CA III, Thun MJ, Namboodiri MM, Dockery DW, Evans JS, Speizer FE, Heath CW (1995) Particulate air pollution as a predictor of mortality in a prospective study of U.S. adults. Am J Resp Crit Care Med 151:669–674Google Scholar
  16. Rothman KJ, Greenland S, Walker AM (1980) Concepts of interaction. Am J Epidemiol 112:467–470Google Scholar
  17. Samet JM, Dominici F, Curriero FC, Coursac I, Zeger SL (2000) Fine particulate air pollution and mortality in 20 U.S. cities, 1987–1994. New Eng J Med 343:1742–1749CrossRefGoogle Scholar
  18. SAS Institute, Inc. (2004) SAS/STAT® 9.1 user’s guide. SAS Institute, Inc., Cary, NCGoogle Scholar
  19. Thompson WD (1991) Effect modification and limits of biological inference from epidemiologic data. J Clin Epidemiol 44:221–232CrossRefGoogle Scholar
  20. Webster T (2002) Commentary: does the spectre of ecologic bias haunt epidemiology? Int J Epidemiol 31:161–162CrossRefGoogle Scholar
  21. Weed DL, Selmon M, Sinks T (1988) Links between categories of interaction. Am J Epidemiol 127:17–27Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • W. Douglas Thompson
    • 1
  • Daniel Wartenberg
    • 2
  1. 1.Department of Applied Medical SciencesUniversity of Southern MainePortlandUSA
  2. 2.Department of Environmental and Occupational MedicineUMDNJ–Robert Wood Johnson Medical SchoolPiscatawayUSA

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