Advertisement

Critical Boundary Refinement in a Group Sequential Trial When the Primary Endpoint Data Accumulate Faster Than the Secondary Endpoint

  • Jiangtao GouEmail author
  • Oliver Y. Chén
Chapter
Part of the ICSA Book Series in Statistics book series (ICSABSS)

Abstract

We propose a generalized framework for critical boundary refinement when conducting hierarchical hypothesis test in a clinical trial involving multiple interim stages. When the hypothesis test follows the stagewise hierarchical rule or the partially hierarchical rule, we provide an improvement on the secondary boundary. This refinement boosts the power to reject the secondary hypothesis significantly. For a trial using a stage-wise hierarchical rule, we deliver a feasible region of information fractions under which an α-level boundary can be directly used in testing the secondary hypothesis at each interim stage. For a trial using a partially hierarchical rule, we recommend using the refined O’Brien-Fleming boundary for both the primary and the secondary endpoint. To evaluate the efficacy of the framework, we present the theoretical underpinning for the boundary refinement, and prove the uniform monotonicity of as well as the upper bound for the type I error rate. The framework has particular advantage when the primary endpoint data can be assessed earlier than the secondary endpoint data. Finally, we extend the framework to include an adaptive update on the refined boundary when the attained sample sizes are different from what they are originally planned.

Notes

Acknowledgements

We thank Ajit C. Tamhane and Dong Xi for comments that greatly improved the manuscript. This work was partially supported by the Professional Staff Congress-City University of New York (PSC-CUNY) research grant, Cycle 48 (2017–2018). This research article extended the framework that was present at the 2017 ICSA Applied Statistics Symposium, Session 148, Recent Developments in Theory and Application of Multiple Comparison Methods, A gatekeeping test on a primary and a secondary endpoint in a group sequential design, by Dr. Ajit C. Tamhane. It was also present at the 2017 ICSA Applied Statistics Symposium, Session 121, Multiplicity in Clinical Trials, A gatekeeping test in a group sequential design with multiple interim looks, by Dr. Jiangtao Gou. The authors thank editor Dr. Lanju Zhang and an anonymous referee for suggestions that improved this paper.

Conflict of Interest

The authors have declared no conflict of interest.

References

  1. Amir, E., Seruga, B., Kwong, R., Tannock, I. F., Ocaña, A.: Poor correlation between progression-free and overall survival in modern clinical trials: are composite endpoints the answer? Eur. J. Cancer 48, 385–388 (2012)CrossRefGoogle Scholar
  2. Baselga, J., Campone, M., Piccart, M., Burris III, H. A., Rugo, H. S., Sahmoud, T., Noguchi, S., Gnant, M., Pritchard, K. I., Lebrun, F., et al.: Everolimus in postmenopausal hormone-receptor–positive advanced breast cancer. N. Engl. J. Med. 366, 520–529 (2012)CrossRefGoogle Scholar
  3. Bretz, F., Maurer, W., Brannath, W., Posch, M.: A graphical approach to sequentially rejective multiple test procedures. Stat. Med. 28, 586–604 (2009)MathSciNetCrossRefGoogle Scholar
  4. Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K.: Graphical approaches for multiple comparison procedures using weighted bonferroni, simes, or parametric tests. Biom. J. 53, 894–913 (2011)MathSciNetCrossRefGoogle Scholar
  5. Burman, C.-F., Sonesson, C., Guilbaud, O.: A recycling framework for the construction of bonferroni-based multiple tests. Stat. Med. 28, 739–761 (2009)MathSciNetCrossRefGoogle Scholar
  6. Dmitrienko, A., Tamhane, A.C.: Gatekeeping procedures with clinical trial applications. Pharm. Stat. 6, 171–180 (2007)CrossRefGoogle Scholar
  7. Dmitrienko, A., Tamhane, A.C., Bretz, F.: Multiple Testing Problems in Pharmaceutical Statistics. Taylor & Francis, Boca Raton (2009)CrossRefGoogle Scholar
  8. Dunnett, C.W., Tamhane, A.C.: A step-up multiple test procedure. J. Am. Stat. Assoc. 87, 162–170 (1992)MathSciNetCrossRefGoogle Scholar
  9. Finner, H., Dickhaus, T., Roters, M.: On the false discovery rate and an asymptotically optimal rejection curve. Ann. Stat. 37, 596–618 (2009)MathSciNetCrossRefGoogle Scholar
  10. Fiteni, F., Westeel, V., Pivot, X., Borg, C., Vernerey, D., and Bonnetain, F.: Endpoints in cancer clinical trials. J. Visc. Surg. 151, 17–22 (2014)CrossRefGoogle Scholar
  11. Glimm, E., Maurer, W., Bretz, F.: Hierarchical testing of multiple endpoints in group-sequential trials. Stat. Med. 29, 219–228 (2010)MathSciNetGoogle Scholar
  12. Gou, J., Tamhane, A.C. : On generalized Simes critical constants. Biom. J. 56, 1035–1054 (2014)MathSciNetCrossRefGoogle Scholar
  13. Gou, J., Tamhane, A.C.: A flexible choice of critical constants for the improved hybrid Hochberg–Hommel procedure. Sankhya B 80, 85–97 (2018a)MathSciNetCrossRefGoogle Scholar
  14. Gou, J., Tamhane, A.C.: Hochberg procedure under negative dependence. Stat. Sin. 28, 339–362 (2018b)MathSciNetzbMATHGoogle Scholar
  15. Gou, J., Tamhane, A.C., Xi, D., Rom, D.: A class of improved hybrid Hochberg–Hommel type step-up multiple test procedures. Biometrika 101, 899–911 (2014)MathSciNetCrossRefGoogle Scholar
  16. Gou, J., Xi, D.: Hierarchical testing of a primary and a secondary endpoint in a group sequential design with different information times. Stat. Biopharm. Res. (2019). https://doi.org/10.1080/19466315.2018.1546613
  17. Hochberg, Y.: A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75, 800–802 (1988)MathSciNetCrossRefGoogle Scholar
  18. Hochberg, Y., Tamhane, A.C.: Multiple Comparison Procedures. Wiley, New York (1987)CrossRefGoogle Scholar
  19. Holm, S.: A simple sequentially rejective multiple test procedure. Scand. J. Stat. 6, 65–70 (1979)MathSciNetzbMATHGoogle Scholar
  20. Hommel, G.: A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika 75, 383–386 (1988)CrossRefGoogle Scholar
  21. Hung, H.M.J., Wang, S.-J., O’Neill, R.: Statistical considerations for testing multiple endpoints in group sequential or adaptive clinical trials. J. Biopharm. Stat. 17, 1201–1210 (2007)MathSciNetCrossRefGoogle Scholar
  22. Jennison, C., Turnbull, B.W.: Group sequential tests for bivariate response: interim analyses of clinical trials with both efficacy and safety endpoints. Biometrics 49, 741–752 (1993)MathSciNetCrossRefGoogle Scholar
  23. Jennison, C., Turnbull, B.W.: Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC, New York (2000)zbMATHGoogle Scholar
  24. Lan, K.K.G., DeMets, D.L.: Discrete sequential boundaries for clinical trials. Biometrika 70, 659–663 (1983)MathSciNetCrossRefGoogle Scholar
  25. Lan, K.K.G., DeMets, D.L.: Group sequential procedures: calendar versus information time. Stat. Med. 8, 1191–1198 (1989)CrossRefGoogle Scholar
  26. Marcus, R., Peritz, E., Gabriel, K.R.: On closed testing procedures with special reference to ordered analysis of variance. Biometrika 63, 655–660 (1976)MathSciNetCrossRefGoogle Scholar
  27. Maurer, W., Bretz, F.: Multiple testing in group sequential trials using graphical approaches. Stat. Biopharm. Res. 5, 311–320 (2013)CrossRefGoogle Scholar
  28. Michiels, S., Saad, E.D., Buyse, M.: Progression-free survival as a surrogate for overall survival in clinical trials of targeted therapy in advanced solid tumors. Drugs 77, 713–719 (2017)CrossRefGoogle Scholar
  29. O’Brien, P.C., Fleming, T.R.: A multiple testing procedure for clinical trials. Biometrics 35, 549–556 (1979)CrossRefGoogle Scholar
  30. Plackett, R.L.: A reduction formula for normal multivariate integrals. Biometrika 41, 351–360 (1954)MathSciNetCrossRefGoogle Scholar
  31. Pocock, S.J.: Group sequential methods in the design and analysis of clinical trials. Biometrika 64, 191–199 (1977)CrossRefGoogle Scholar
  32. Rom, D.M.: A sequentially rejective test procedure based on a modified Bonferroni inequality. Biometrika 77, 663–665 (1990)MathSciNetCrossRefGoogle Scholar
  33. Sarkar, S.K.: Generalizing Simes’ test and Hochberg’s stepup procedure. Ann. Stat. 36, 337–363 (2008)MathSciNetCrossRefGoogle Scholar
  34. Simes, R.J.: An improved Bonferroni procedure for multiple tests of significance. Biometrika 73, 751–754 (1986)MathSciNetCrossRefGoogle Scholar
  35. Slepian, D.: The one-sided barrier problem for gaussian noise. Bell Syst. Tech. J. 41, 463–501 (1962)MathSciNetCrossRefGoogle Scholar
  36. Tamhane, A.C., Gou, J.: Advances in p-value based multiple test procedures. J. Biopharm. Stat. 28, 10–27 (2018)CrossRefGoogle Scholar
  37. Tamhane, A.C., Gou, J., Jennison, C., Mehta, C.R., Curto, T.: A gatekeeping procedure to test a primary and a secondary endpoint in a group sequential design with multiple interim looks. Biometrics 74, 40–48 (2018)MathSciNetCrossRefGoogle Scholar
  38. Tamhane, A.C., Mehta, C.R., Liu, L.: Testing a primary and a secondary endpoint in a group sequential design. Biometrics 66, 1174–1184 (2010)MathSciNetCrossRefGoogle Scholar
  39. Tamhane, A.C., Wu, Y., Mehta, C.R.: Adaptive extensions of a two-stage group sequential procedure for testing primary and secondary endpoints (I): unknown correlation between the endpoints. Stat. Med. 31, 2027–2040 (2012a)MathSciNetCrossRefGoogle Scholar
  40. Tamhane, A.C., Wu, Y., Mehta, C.R.: Adaptive extensions of a two-stage group sequential procedure for testing primary and secondary endpoints (II): sample size re-estimation. Stat. Med. 31, 2041–2054 (2012b)MathSciNetCrossRefGoogle Scholar
  41. Tang, D.-I., Geller, N.L.: Closed testing procedures for group sequential clinical trials with multiple endpoints. Biometrics 55, 1188–1192 (1999)CrossRefGoogle Scholar
  42. Xi, D., Tamhane, A.C.: Allocating recycled significance levels in group sequential procedures for multiple endpoints. Biom. J. 57, 90–107 (2015)MathSciNetCrossRefGoogle Scholar
  43. Ye, Y., Li, A., Liu, L., Yao, B.: A group sequential holm procedure with multiple primary endpoints. Stat. Med. 32, 1112–1124 (2013)MathSciNetCrossRefGoogle Scholar
  44. Zhang, F., Gou, J.: A p-value model for theoretical power analysis and its applications in multiple testing procedures. BMC Med. Res. Methodol. 16, 135 (2016)CrossRefGoogle Scholar
  45. Zhang, F., Gou, J.: Control of false positive rates in clusterwise fMRI inferences. J. Appl. Stat. (2019a). https://doi.org/10.1080/02664763.2019.1573883 MathSciNetCrossRefGoogle Scholar
  46. Zhang, F., Gou, J.: Refined critical boundary with enhanced statistical power for non-directional two-sided tests in group sequential designs with multiple endpoints. (2019b) (submitted)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsHunter College of CUNYNew YorkUSA
  2. 2.Department of Mathematics and StatisticsVillanova UniversityVillanovaUSA
  3. 3.Institute for Biomedical EngineeringUniversity of OxfordOxfordUK

Personalised recommendations