Interplay of Probability and Precision: Event Size Calculation for Hypothesis Testing, Confidence Interval and Prediction Interval of Poisson Rates

Wei-ming  Luh; Jiin-Huarng  Guo

Authors

Wei-ming Luh Institute of Education, National Cheng Kung University, Tainan, Taiwan
Jiin-Huarng Guo Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan

Keywords:

Conditional probability, coverage, precision, statistical power, R Shiny app, relative width

Abstract

The Poisson distribution is a key model for describing rare events, from machinery failures in engineering to disease onset in medicine. While prior research has examined the determination of event size for testing equality between Poisson distributions, comprehensive approaches that integrate both statistical power and precision remain limited. To address this gap, our study aims to achieve two goals. First, we present methods for calculating event sizes based on incidence rate ratios, incorporating rejection, validity, and confidence interval width to ensure both power and precision. Second, we extend the discussion to prediction intervals, establishing the original event sizes needed to achieve a specified probability of coverage. Simulation studies confirm that the proposed methods yield valid results for coverage and target probabilities. To support practical use, we developed three interactive R Shiny applications, making the methodology accessible for researchers and practitioners. Together, the empirical validation and user-friendly tools demonstrate the practical value of our framework.

References

Anderson SF, Kelley K. Sample size planning for replication studies: The devil is in the design. Psychol Methods. 2024; 29(5): 844-867.

Anderson SF, Maxwell SE. There’s more than one way to conduct a replication study: beyond statistical significance. Psychol Methods. 2016; 21(1), 1-12.

Beal SL. Sample size determination for confidence intervals on the population mean and on the difference between two population means. Biometrics. 1989; 45(3): 969-977.

Bradley RA, Schumann DEW. The comparison of the sensitivities of similar experiments: applications. Biometrics. 1957; 13(4): 496-510.

Calin-Jageman RJ, Cumming G. The new statistics for better science: Ask how much, how uncertain, and what else is known. Am Stat. 2019; 73(sup1): 271-280.

Cesana BM, Antonelli P. A new approach to sample size calculations for the power of testing and estimating population means of Gaussian distributed variables. Biomed Stat Clin Epidemiol. 2010; 4(2): 67-78.

Chang W, Cheng J, Allaire J, et al. shiny: Web application framework for R. R package version 1.7.4.9002. 2023 [cited 20 June 2025]. Available from: https://shiny.rstudio.com/

Chang C-H, Pal N, Lin J-J. A note on comparing several Poisson means. Commun Stat Simul Comput. 2010; 39(8): 1605-1627.

Chiolero A, Santschi V, Burnand B, Platt RW, Paradis G. Meta-analyses: with confidence or prediction intervals? Eur J Epidemiol. 2012; 27: 823-825.

Cinlar E. Introduction to Stochastic Processes. New Jersey: Prentice-Hall; 1975.

Cox DR. Some simple approximate tests for Poisson variates. Biometrika. 1953; 40(3-4): 354-360.

Cox DR, Lewis PA. The statistical analysis of series of events. London: Methuen; 1966.

Cumming G. Replication and p intervals: p values predict the future only vaguely, but confidence intervals do much better. Perspect Psychol Sci. 2008; 3(4): 286-300.

Faulkenberry GD. A method of obtaining prediction intervals. J Am Stat Assoc. 1973; 68(342): 433-435.

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

Gu K, Ng HK, Tang ML, Schucany WR. Testing the ratio of two Poisson rates. Biom J. 2008; 50(2): 283-298.

Hahn GJ, Meeker WQ. Statistical intervals: a guide for practitioners. Hoboken: John Wiley & Sons; 1991.

Hartnack S, Roos M. Teaching: confidence, prediction and tolerance intervals in scientific practice: a tutorial on binary variables. Emerg Themes Epidemiol. 2021; 18(1): 17. https://doi.org/10.1186/s12982-021-00108-1

Hsu LM. Unbalanced designs to maximize statistical power in psychotherapy efficacy studies. Psychother Res. 1994; 4(2): 95-106.

IntHout J, Ioannidis JP, Rovers MM, Goeman JJ. Plea for routinely presenting prediction intervals in meta-analysis. BMJ Open. 2016; 6(7): e010247. https://doi.org/10.1136/bmjopen-2015-010247

Jiroutek MR, Muller KE, Kupper LL, Stewart PW. A new method for choosing sample size for confidence interval-based inferences. Biometrics. 2003; 59(3): 580-590.

Kelley K, Rausch JR. Sample size planning for the standardized mean difference: accuracy in parameter estimation via narrow confidence intervals. Psychol Methods. 2006; 11(4): 363-385.

Kharrati-Kopaei M, Dorosti-Motlagh R. Confidence intervals for the ratio of two independent Poisson rates: parametric bootstrap, modified asymptotic, and approximate-estimate approaches. Stat Methods Med Res. 2020; 29(8): 2140-2150.

Kim T, Lieberman B, Luta G, Peña EA. Prediction regions for Poisson and over-dispersed Poisson regression models with applications in forecasting the number of deaths during the COVID-19 pandemic. Open Stat. 2021; 2(1): 81-112.

Krishnamoorthy K, Lv S. Highest posterior mass prediction intervals for binomial and Poisson distributions. Metrika. 2018; 81(7): 775-796.

Krishnamoorthy K, Peng J. Improved closed-form prediction intervals for binomial and Poisson distributions. J Stat Plan Inference. 2011; 141(5): 1709-1718.

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

Lai K, Kelley K. Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: sample size planning via narrow confidence intervals. Br J Math Stat Psychol. 2012; 65(2): 350-370.

Li HQ, Tang ML, Wong WK. Confidence intervals for ratio of two Poisson rates using the method of variance estimates recovery. Comput Stat. 2014; 29(3): 869-889.

Liu XS. Sample size and the width of the confidence interval for mean difference. Br J Math Stat Psychol. 2009; 62(Pt 2): 201-215.

Liu XS. Implications of statistical power for confidence intervals. Br J Math Stat Psychol. 2012; 65(3): 427-437.

Luh WM. Probabilistic thinking is the name of the game: integrating test and confidence intervals to plan sample sizes. Methodology. 2022; 18(2): 80-98.

Luh WM, Guo JH. Unequal allocation of sample/event sizes with considerations of sampling cost for testing equality, non-inferiority/superiority, and equivalence of two Poisson rates. Int J Biostat. 2022; 20(1): 143-156.

Maguire BA, Pearson ES, Wynn AHA. The time intervals between industrial accidents. Biometrika. 1952; 39(1-2): 168-180.

Meeker WQ, Hahn GJ. Sample sizes for prediction intervals. J Qual Tech. 1982; 14(4): 201-206.

Nelson LS. Comparison of Poisson means: the general case. J Qual Tech. 1987; 19(4): 173-179.

Nelson W. Confidence intervals for the ratio of two Poisson means and Poisson predictor intervals. IEEE Trans Reliab. 1970; R-19(2): 42-49.

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

Pan ZY, Kupper LL. Sample size determination for multiple comparison studies treating confidence interval width as random. Stat Med. 1999; 18(12): 1475-1488.

Patil P, Peng RD, Leek JT. What should researchers expect when they replicate studies? A statistical view of replicability in psychological science. Perspect Psychol Sci. 2016; 11(4): 539-544.

Peckham E, Brabyn S, Cook L, Devlin T, Dumville J, Torgerson DJ. The use of unequal randomisation in clinical trials—An update. Contemp Clin Trials. 2015; 45(Pt A): 113-122.

Preston S. Teaching prediction intervals. J Stat Educ. 2000; 8(3). https://doi.org/10.1080/10691898.

12131297

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

R Core Team. R: A language and environment for statistical computing [Internet]. Vienna, Austria: R Foundation for Statistical Computing; 2024 [cited 2 July 2024]. Available from: http://www.r-project.org

Ross SM. Introduction to probability models. 11th ed. San Diego: Academic Press; 2014.

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

Schumann DEW, Bradley RA. The comparison of the sensitivities of similar experiments: theory. Ann Math Stat. 1957; 28(4): 902-920.

Schumann DEW, Bradley RA. The comparison of the sensitivities of similar experiments: model II of the analysis of variance. Biometrics. 1959; 15(3): 405-416.

Shan G. Exact sample size determination for the ratio of two incident rates under the Poisson distribution. Comput Stat. 2016; 31(4): 1633-1644.

Shiue WK, Bain LJ. Experiment size and power comparisons for two-sample Poisson tests. JR Stat Soc Ser C Appl Stat. 1982; 31(2): 130-134.

Spence JR, Stanley DJ. Prediction interval: What to expect when you’re expecting . . . A replication. PLoS One. 2016; 11(9): e0162874. http://doi.org/10.1371/journal.pone.0162874

Spence JR, Stanley DJ. Tempered expectations: A tutorial for calculating and interpreting prediction intervals in the context of replications. Adv Methods Pract Psychol Sci. 2024; 7(1): 25152459231217932. http://doi.org/10.1177/25152459231217932

Thode HC. Power and sample size requirements for tests of differences between two Poisson rates. Statistician. 1997; 46(2): 227-230.

Wang Y, Kupper LL. Optimal sample sizes for estimating the difference in means between two normal populations treating confidence interval length as a random variable. Commun Stat Theory Methods. 1997; 26(3): 727-741.

Xiao M, Jiang T, Zhang H, Shan G. Exact one-sided confidence limit for the ratio of two Poisson rates. Stat Biopharm Res. 2017; 9(2): 180-185.