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Confidence intervals for ratio of two Poisson rates using the method of variance estimates recovery

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Abstract

Inference based on ratio of two independent Poisson rates is common in epidemiological studies. We study the performance of a variety of unconditional method of variance estimates recovery (MOVER) methods of combining separate confidence intervals for two single Poisson rates to form a confidence interval for their ratio. We consider confidence intervals derived from (1) the Fieller’s theorem, (2) the logarithmic transformation with the delta method and (3) the substitution method. We evaluate the performance of 13 such types of confidence intervals by comparing their empirical coverage probabilities, empirical confidence widths, ratios of mesial non-coverage probability and total non-coverage probabilities. Our simulation results suggest that the MOVER Rao score confidence intervals based on the Fieller’s theorem and the substitution method are preferable. We provide two applications to construct confidence intervals for the ratio of two Poisson rates in a breast cancer study and in a study that examines coronary heart diseases incidences among post menopausal women treated with or without hormones.

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References

  • Altman DG, Machin D, Bryant TN et al (2000) Statistics with confidence, 2nd edn. BMJ Books, Bristol

    Google Scholar 

  • Bartlett MS (1953) Approximate confidence intervals II. More than one unknown parameter. Biometrika 40(3):306–317

    MATH  MathSciNet  Google Scholar 

  • Brown LD, Cai TT, Dasgupta A (2003) Interval estimation in exponential families. Statistica Sinica 13(1):19–49

    MATH  MathSciNet  Google Scholar 

  • Byrne J, Kabaila P (2005) Comparison of Poisson confidence intervals. Commun Stat Theory Methods 34(3):545–556

    Article  MATH  MathSciNet  Google Scholar 

  • Donner A, Zou GY (2002) Interval estimation for a difference between intraclass kappa statistics. Biometrics 58(1):209–215

    Article  MathSciNet  Google Scholar 

  • Donner A, Zou GY (2010) Closed-form confidence intervals for function of the normal mean and standard deviation. Stat Methods Med Res 8:1–12

    Google Scholar 

  • Fleiss JL, Levin B, Paik MC (2003) Statistical methods for rates and proportions. Wiley, New Jersey

    Book  MATH  Google Scholar 

  • Graham PL, Mengersen K, Morton AP (2003) Confidence limits for the ratio of two rates based on likelihood scores: non-iterative method. Stat Med 22(12):2071–2083

    Article  Google Scholar 

  • Gu KX, Ng HKT, Tang ML et al (2008) Testing the ratio of two Poisson rates. Biometr J 50(2):283–298

    Article  MathSciNet  Google Scholar 

  • Haight FA (1967) Handbook of the Poisson distribution. Wiley, New York

    MATH  Google Scholar 

  • Krishnamoorthy K, Thomson J (2004) A more powerful test for comparing two Poisson means. J Stat Plan Inference 119(1):23–35

    Article  MATH  MathSciNet  Google Scholar 

  • Li Y, Koval JJ, Donner A, Zou GY (2010) Interval estimation for the area under the receiver operating characteristic curve when data are subject to error. Stat Med 29(24):2521–2531

    Article  MathSciNet  Google Scholar 

  • Newcombe RG (1998) Two-sided confidence intervals for the single proportion: comparison of seven methods. Stat Med 17(8):857–872

    Article  Google Scholar 

  • Newcombe RG (2012) Confidence intervals for proportions and related measures of effect size. Chapman & Hall/CRC Biostatistics Series

  • Ng HKT, Gu K, Tang ML (2007) A comparative study of tests for the difference of the two Poisson means. Comput Stat Data Anal 51(6):3085–3099

    Article  MATH  MathSciNet  Google Scholar 

  • Ng HKT, Tang ML (2005) Testing the equality of two Poisson means using the rate ratio. Stat Med 24(6): 955–965

    Article  MathSciNet  Google Scholar 

  • Price RM, Bonett DG (2000) Estimating the ratio of two Poisson rates. Comput Stat Data Anal 34(3): 345–356

    Article  MATH  MathSciNet  Google Scholar 

  • Rothman KJ, Greenland S (1998) Modern epidemiology, 2nd edn. Lippincott-Raven, Philadelphia

    Google Scholar 

  • Sahai H, Khurshid A (1993) Confidence intervals for the ratio of two Poisson means. Math Sci 18(1):43–50

    MATH  MathSciNet  Google Scholar 

  • Singer J (2010) Letter to the editor. Construction of confidence limits about effect measures: a general approach, by Zou GY, Donner A (eds) Statistics in Medicine 2008; 27: 1693–1702. Statistics in Medicine 29(16):1757–1759

  • Stamey J, Hamilton C (2006) A note on confidence intervals for a linear function of Poisson rates. Commun Stat Simul Comput 35(4):849–856

    Article  MATH  MathSciNet  Google Scholar 

  • Stampfer MJ, Willett WC (1985) A prospective study of postmenopausal estrogen therapy and coronary heart disease. N Engl J Med 313(17):1044–1049

    Article  Google Scholar 

  • Stapleton JH (1995) Linear statistical models. Wiley, New York

    Book  MATH  Google Scholar 

  • Swift MB (2009) Comparison of confidence intervals for a Poisson mean-further considerations. Commun Stat Theory Methods 38(5):748–759

    Article  MATH  MathSciNet  Google Scholar 

  • Tang ML, Ling MH, Tian GL (2009) Exact and approximate unconditional confidence intervals for proportion difference in the presence of incomplete data. Stat Med 28(4):625–641

    Article  MathSciNet  Google Scholar 

  • Tang ML, Ng HKT (2004) Comment on: confidence limits for the ratio of two rates based on likelihood scores: non-iterative method. Stat Med 23(4):685–691

    Article  Google Scholar 

  • Zou GY (2008) On the estimation of additive interaction by use of the four-by-two table and beyond. Am J Epidemiol 168(2):212–224

    Article  Google Scholar 

  • Zou GY, Donner A (2008) Construction of confidence limits about effect measures: a general approach. Stat Med 27(10):1693–1702

    Article  MathSciNet  Google Scholar 

  • Zou GY, Donner A (2010) A generalization of Fiellers theorem for ratios of non-normal variables and some practical applications. In: Schuster H, Metzger W (eds) Biometrics: methods, applications and analyses. Nova Publications, London, pp 197–216

    Google Scholar 

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Acknowledgments

Weng-Kee Wong worked on the manuscript when he was a visiting fellow and scientific advisor for a six-month design workshop at The Sir Isaac Newton Institute at Cambridge, England. He would like to thank the Institute for the support during his repeated visits from August to December in 2011. The research was fully supported by the Natural Science Foundation of China (11201412) and grants from the Science Foundation of Yunnan Province(2011FB016). Tang’s research was fully supported by two grants from the Hong Kong Baptist University (Project Nos FRG2/11-12/013 and FRG2/12-13/073). We further thank Qin Wu from Hong Kong Baptist University for help with the graphics in this paper and also to the entire editorial team for their very helpful comments on our work.

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Li, HQ., Tang, ML. & Wong, WK. Confidence intervals for ratio of two Poisson rates using the method of variance estimates recovery. Comput Stat 29, 869–889 (2014). https://doi.org/10.1007/s00180-013-0467-9

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