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Unequal allocation of sample/event sizes with considerations of sampling cost for testing equality, non-inferiority/superiority, and equivalence of two Poisson rates

  • Wei-Ming Luh ORCID logo EMAIL logo and Jiin-Huarng Guo
Published/Copyright: December 30, 2022
The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 20 Issue 1

Abstract

For non-inferiority/superiority and equivalence tests of two Poisson rates, the determination of the required number of sample sizes has been studied but the studies for the number of events to be observed are very limited. To fill the gap, the present study first is aimed toward determining the number of events to be observed for testing non-inferiority/superiority and equivalence of two Poisson rates, respectively. Also, considering the cost for each event, the second purpose is to apply an exhaustive search to find the unequal but optimal allocation of events for each group such that the budget is minimal for a user-specified power level, or the statistical power is maximal for a user-specified budget. Four R Shiny apps were developed to obtain the number of events needed for each group. A simulation study showed the proposed approach to be valid in terms of Type I error and statistical power. A comparison of the proposed approach with extant methods from various disciplines was performed, and an illustrative example of comparing the adverse reactions to the COVID-19 vaccines was demonstrated. By applying the proposed approach, researchers also can estimate the most economical number of subjects or time intervals after determining the number of events.

Keywords: equivalence margin; optimal allocation; R Shiny app; sample size; sampling cost; superiority margin
Corresponding author: Weiming Luh, National Cheng Kung University, Tainan, 70101, Taiwan, E-mail:

Funding source: Ministry of Science and Technology, Taiwan

Award Identifier / Grant number: MOST 107-2410-H-006-061-My3

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by Ministry of Science and Technology, Taiwan under the grant no. MOST 107-2410-H-006-061-My30.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

1. Graham, PL, Mengersen, K, Morton, AP. Confidence limits for the ratio of two rates based on likelihood scores: non-iterative method. Stat Med 2003;22:2071–83. https://doi.org/10.1002/sim.1405.Search in Google Scholar PubMed

2. Krishnamoorthy, K, Thomson, J. A more powerful test for comparing two Poisson means. J Stat Plann Inference 2004;119:23–35. https://doi.org/10.1016/s0378-3758(02)00408-1.Search in Google Scholar

3. Liu, G. Sample size for epidemiologic studies. In: Gail, M, Benichou, J, editors. Encyclopedia of epidemiologic methods. Chichester, England: John Wiley & Sons; 2000:777–94 pp.Search in Google Scholar

4. Nelson, LS. Comparison of Poisson means: the general case. J Qual Technol 1987;19:173–9. https://doi.org/10.1080/00224065.1987.11979062.Search in Google Scholar

5. Kikuchi, T, Gittins, J. A behavioural Bayes approach to the determination of sample size for clinical trials considering efficacy and safety: imbalanced sample size in treatment groups. Stat Methods Med Res 2011;20:389–400. https://doi.org/10.1177/0962280209358131.Search in Google Scholar PubMed

6. Shan, G. Exact sample size determination for the ratio of two incidence rates under the Poisson distribution. Comput Stat 2016;31:1633–44. https://doi.org/10.1007/s00180-016-0654-6.Search in Google Scholar

7. Schulz, K, Grimes, D. Unequal group sizes in randomised trials: guarding against guessing. Lancet 2002;359:966–70. https://doi.org/10.1016/S0140-6736(02)08029-7.Search in Google Scholar PubMed

8. 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;45:113–22. https://doi.org/10.1016/j.cct.2015.05.017.Search in Google Scholar PubMed

9. Sverdlov, O, Ryeznik, Y. Implementing unequal randomization in clinical trials with heterogeneous treatment costs. Stat Med 2019;38:2905–27. https://doi.org/10.1002/sim.8160.Search in Google Scholar PubMed

10. Chandereng, T, Wei, X, Chappell, R. Imbalanced randomization in clinical trials. Stat Med 2020;39:2185–96. https://doi.org/10.1002/sim.8539.Search in Google Scholar PubMed

11. Fackle-Fornius, E, Nyquist, H. Optimal allocation to treatment groups under variance heterogeneity. Stat Sin 2015;25:537–49. https://doi.org/10.5705/ss.2012.042.Search in Google Scholar

12. Wong, WK, Zhu, W. Optimum treatment allocation rules under a variance heterogeneity model. Stat Med 2008;27:4581–95. https://doi.org/10.1002/sim.3318.Search in Google Scholar

13. Dumville, JC, Hahn, S, Miles, JN, Torgerson, DJ. The use of unequal randomisation ratios in clinical trials: a review. Contemp Clin Trials 2006;27:1–12. https://doi.org/10.1016/j.cct.2005.08.003.Search in Google Scholar

14. Hung, H, Wang, S, O’Neill, R. Consideration of regional difference in design and analysis of multi-regional trials. Pharmaceut Stat 2010;9:173–8. https://doi.org/10.1002/pst.440.Search in Google Scholar

15. Manju, M, Candel, M, Berger, M. Optimal and maximin sample sizes for multicentre cost-effectiveness trials. Stat Methods Med Res 2015;24:513–39. https://doi.org/10.1177/0962280215569293.Search in Google Scholar

16. Torgerson, D, Campbell, M. Unequal randomisation can improve the economic efficiency of clinical trials. J Health Serv Res Pol 1997;2:81–5. https://doi.org/10.1177/135581969700200205.Search in Google Scholar

17. Allison, DB, Allison, RL, Faith, MS, Paultre, F, Pi-Sunyer, FX. Power and money: designing statistically powerful studies while minimizing financial costs. Psychol Methods 1997;2:20–33. https://doi.org/10.1037/1082-989x.2.1.20.Search in Google Scholar

18. Nam, JM. Optimum sample sizes for the comparison of the control and treatment. Biometrics 1973;29:101–8. https://doi.org/10.2307/2529679.Search in Google Scholar

19. Schouten, HJ. Sample size formula with a continuous outcome for unequal group sizes and unequal variances. Stat Med 1999;18:87–91. https://doi.org/10.1002/(sici)1097-0258(19990115)18:1<87::aid-sim958>3.0.co;2-k.10.1002/(SICI)1097-0258(19990115)18:1<87::AID-SIM958>3.0.CO;2-KSearch in Google Scholar

20. Gu, K, Ng, HKT, Tang, ML, Schucany, WR. Testing the ratio of two Poisson rates. Biom J 2008;50:283–98. https://doi.org/10.1002/bimj.200710403.Search in Google Scholar

21. Wang, L, Fan, C. Sample size calculations for comparing two groups of count data. J Biopharm Stat 2019;29:115–27. https://doi.org/10.1080/10543406.2018.1489409.Search in Google Scholar

22. Chow, SC, Shao, J, Wang, H. Sample size calculations in clinical research, 2nd ed. New York, NY: Taylor & Francis; 2008:243–5 pp.10.1201/9781584889830Search in Google Scholar

23. Guo, JH, Luh, WM. Sample size calculations for testing equivalence of two exponential distributions with right censoring: allocation with costs. Methodology 2017;13:144–56. https://doi.org/10.1027/1614-2241/a000139.Search in Google Scholar

24. Guo, JH, Luh, WM. Testing two variances for non-inferiority/superiority and equivalence: using the exhaustion algorithm for sample size allocation with cost. Br J Math Stat Psychol 2020;73:316–32. https://doi.org/10.1111/bmsp.12172.Search in Google Scholar PubMed

25. Julious, SA. Sample sizes for clinical trials. Boca Raton, FL: Taylor & Francis; 2010:14 p.10.1201/9781584887409Search in Google Scholar

26. Julious, SA, Campbell, MJ. Tutorial in biostatistics: sample sizes for parallel group clinical trials with binary data. Stat Med 2012;31:2904–36. https://doi.org/10.1002/sim.5381.Search in Google Scholar PubMed

27. Liu, JP, Chow, SC. Sample size determination for the two one-sided tests procedure in bioequivalence. J Pharmacokinet Biopharm 1992;20:101–4. https://doi.org/10.1007/bf01143188.Search in Google Scholar PubMed

28. Luh, WM, Guo, JH. Sample size planning for the non-inferiority or equivalence of a linear contrast with cost considerations. Psychol Methods 2016;21:13–34. https://doi.org/10.1037/met0000039.Search in Google Scholar PubMed

29. Wellek, S. Testing statistical hypotheses of equivalence and noninferiority, 2nd ed. Boca Raton, FL: Chapman & Hall/CRC; 2010:119–218 pp.10.1201/EBK1439808184Search in Google Scholar

30. Lui, KJ. Sample size calculation for testing non-inferiority and equivalence under Poisson distribution. Stat Methodol 2005;2:37–48. https://doi.org/10.1016/j.stamet.2004.11.002.Search in Google Scholar

31. Wellek, S. On powerful exact nonrandomized tests for the Poisson two-sample setting. Stat Methods Med Res 2020;29:2538–53. https://doi.org/10.1177/0962280219900901.Search in Google Scholar PubMed

32. Stucke, K, Kieser, M. Sample size calculations for noninferiority trials with Poisson distributed count data. Biom J 2013;55:203–16. https://doi.org/10.1002/bimj.201200142.Search in Google Scholar PubMed

33. Maguire, B, Pearson, E, Wynn, A. The time intervals between industrial accidents. Biometrika 1952;39:168–80. https://doi.org/10.2307/2332475.Search in Google Scholar

34. Cinlar, E. Introduction to stochastic processes. NJ: Prentice-Hall; 1975:83 p.Search in Google Scholar

35. Ross, SM. Introduction to probability models, 11th ed Amsterdam: Elsevier; 2014:307 p.10.1016/B978-0-12-407948-9.00001-3Search in Google Scholar

36. Cox, DR, Lewis, PA. The statistical analysis of series of events. London: Methuen; 1966:229 p.10.1007/978-94-011-7801-3Search in Google Scholar

37. Desu, MM, Raghavarao, D. Sample size methodology. San Diego, CA: Academic; 1990:35 p.10.1016/B978-0-12-212165-4.50009-7Search in Google Scholar

38. Johnson, NL, Kotz, S. Distributions in statistics: continuous univariate distributions-2. New York, NY: John Wiley & Sons; 1970:81 p.Search in Google Scholar

39. Mace, AE. Sample-size determination. Huntington, NY: Robert Krieger; 1974:94–7 pp.Search in Google Scholar

40. Schuirmann, DJ. A comparison of the two one-sided test procedure and the power approach for assessing the equivalence of average bioavailability. J Pharmacokinet Biopharm 1987;15:657–80. https://doi.org/10.1007/bf01068419.Search in Google Scholar PubMed

41. Berger, RL, Hsu, JC. Bioequivalence trials, intersection-union tests and equivalence confidence sets. Stat Sci 1996;11:283–319. https://doi.org/10.1214/ss/1032280304.Search in Google Scholar

42. Guo, JH, Chen, HJ, Luh, WM. Sample size planning with the cost constraint for testing superiority and equivalence of two independent groups. Br J Math Stat Psychol 2011;64:439–61. https://doi.org/10.1348/000711010x512408.Search in Google Scholar

43. Cochran, WG. Sampling techniques, 3rd ed. New York, NY: John Wiley & Sons; 1977:97 p.Search in Google Scholar

44. Stuart, A. A simple presentation of optimum sampling results. J Roy Stat Soc 1954;B16:239–41. https://doi.org/10.1111/j.2517-6161.1954.tb00165.x.Search in Google Scholar

45. R Development Core Team. R: A language and environment for statistical computing. Vienna, Austria; 2021 [Online]. Available from: http://www.R-project.org/ [Accessed 6 Aug 2021].Search in Google Scholar

46. World Health Organization. Annex 9: guidelines on clinical evaluation of vaccines: regulatory expectations; 2017. Available from: https://cdn.who.int/media/docs/default-source/prequal/vaccines/who-trs-1004-web-annex-9.pdf?sfvrsn=9c8f4704_2&download=true [Accessed 6 Aug 2021].Search in Google Scholar

47. European Centre for Disease Prevention and Control. Suspected adverse reactions to COVID-19 vaccination and the safety of substances of human origin, A technical report; 2021. Available from: https://www.ecdc.europa.eu/en/publications-data/suspected-adverse-reactions-covid-19-vaccination-and-safety-substances-human [Accessed 6 Aug 2021].Search in Google Scholar

48. Cao, S. COVID-19 vaccine prices revealed from Pfizer, moderna, and AstraZeneca. Observer; 2020. Available from: https://observer.com/2020/11/covid19-vaccine-price-pfizer-moderna-astrazeneca-oxford/ [Accessed 6 Aug 2021].Search in Google Scholar

49. Chan, AW, Hróbjartsson, A, Jørgensen, K, Gøtzsche, PC, Altman, DG. Discrepancies in sample size calculations and data analyses reported in randomized trials: Comparison of publications with protocols. BMJ 2008;337:a2299. https://doi.org/10.1136/bmj.a2299.Search in Google Scholar PubMed PubMed Central

50. Kieser, M, Hauschke, D. Approximate sample sizes for testing hypotheses about the ratio and difference of two means. J Biopharm Stat 1999;9:641–50. https://doi.org/10.1081/bip-100101200.Search in Google Scholar PubMed

51. Wang, WB, Mehrotra, DV, Chan, ISF, Heyse, JF. Statistical considerations for noninferiority/equivalence trials in vaccine development. J Biopharm Stat 2006;16:429–41. https://doi.org/10.1080/10543400600719251.Search in Google Scholar PubMed

52. EMA-CHMP. Guideline on the choice of the non-inferiority margin; 2005. Available from: https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-choice-non-inferiority-margin_en.pdf [Accessed 6 Aug 2021].Search in Google Scholar

53. US-FDA. Scientific considerations in demonstrating biosimilarity to a reference product: guidance for industry; 2015. Available from: https://www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/UCM291128.pdf [Accessed 6 Aug 2021].Search in Google Scholar

54. Althunian, T, de Boer, A, Groenwold, RHH, Klungel, DH. Defining the noninferiority margin and analyzing noninferiority: an overview. Br J Clin Pharmacol 2017;83:1636–42. https://doi.org/10.1111/bcp.13280.Search in Google Scholar PubMed PubMed Central

55. Chow, SC, Song, F. On selection of margin in non-inferiority trails. J Biometrics Biostat 2016;7:301. https://doi.org/10.4172/2155-6180.1000301.Search in Google Scholar

56. US-FDA. Non-inferiority clinical trials to establish effectiveness: guidance for industry; 2016. Available from: https://www.fda.gov/downloads/Drugs/Guidances/UCM202140.pdf [Accessed 6 Aug 2021].Search in Google Scholar

57. CPMP. Points to consider on switching between superiority and non-inferiority. Br J Clin Pharmacol 2001;52:223–8. https://doi.org/10.1046/j.1365-2125.2001.01397-3.x.Search in Google Scholar

58. Zhu, H, Lakkis, H. Sample size calculation for comparing two negative binomial rates. Stat Med 2014;33:376–87. https://doi.org/10.1002/sim.5947.Search in Google Scholar PubMed

59. Lin, J, Lin, LA, Sankoh, S. A general overview of adaptive randomization design for clinical trials. J Biometrics Biostat 2016;7:294. https://doi.org/10.4172/2155-6180.1000294.Search in Google Scholar
Received: 2022-03-26
Accepted: 2022-12-09
Published Online: 2022-12-30

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Survival analysis using deep learning with medical imaging
  4. Using a population-based Kalman estimator to model the COVID-19 epidemic in France: estimating associations between disease transmission and non-pharmaceutical interventions
  5. Approximate reciprocal relationship between two cause-specific hazard ratios in COVID-19 data with mutually exclusive events
  6. Sensitivity of estimands in clinical trials with imperfect compliance
  7. Highly robust causal semiparametric U-statistic with applications in biomedical studies
  8. Hierarchical Bayesian bootstrap for heterogeneous treatment effect estimation
  9. Penalized logistic regression with prior information for microarray gene expression classification
  10. Bayesian learners in gradient boosting for linear mixed models
  11. Unequal allocation of sample/event sizes with considerations of sampling cost for testing equality, non-inferiority/superiority, and equivalence of two Poisson rates
  12. HiPerMAb: a tool for judging the potential of small sample size biomarker pilot studies
  13. Heterogeneity in meta-analysis: a comprehensive overview
  14. On stochastic dynamic modeling of incidence data
  15. Power of testing for exposure effects under incomplete mediation
  16. Exact correction factor for estimating the OR in the presence of sparse data with a zero cell in 2 × 2 tables
  17. Right-censored partially linear regression model with error in variables: application with carotid endarterectomy dataset
  18. Assessing HIV-infected patient retention in a program of differentiated care in sub-Saharan Africa: a G-estimation approach
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https://doi.org/10.1515/ijb-2022-0039
Keywords for this article
equivalence margin; optimal allocation; R Shiny app; sample size; sampling cost; superiority margin

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Survival analysis using deep learning with medical imaging
  4. Using a population-based Kalman estimator to model the COVID-19 epidemic in France: estimating associations between disease transmission and non-pharmaceutical interventions
  5. Approximate reciprocal relationship between two cause-specific hazard ratios in COVID-19 data with mutually exclusive events
  6. Sensitivity of estimands in clinical trials with imperfect compliance
  7. Highly robust causal semiparametric U-statistic with applications in biomedical studies
  8. Hierarchical Bayesian bootstrap for heterogeneous treatment effect estimation
  9. Penalized logistic regression with prior information for microarray gene expression classification
  10. Bayesian learners in gradient boosting for linear mixed models
  11. Unequal allocation of sample/event sizes with considerations of sampling cost for testing equality, non-inferiority/superiority, and equivalence of two Poisson rates
  12. HiPerMAb: a tool for judging the potential of small sample size biomarker pilot studies
  13. Heterogeneity in meta-analysis: a comprehensive overview
  14. On stochastic dynamic modeling of incidence data
  15. Power of testing for exposure effects under incomplete mediation
  16. Exact correction factor for estimating the OR in the presence of sparse data with a zero cell in 2 × 2 tables
  17. Right-censored partially linear regression model with error in variables: application with carotid endarterectomy dataset
  18. Assessing HIV-infected patient retention in a program of differentiated care in sub-Saharan Africa: a G-estimation approach
  19. Prediction-based variable selection for component-wise gradient boosting
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