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Sequential tests and estimates after overrunning based on p-value combination

W. J. Hall and Keyue Ding

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Often in sequential trials additional data become available after a stopping boundary has been reached. A method of incorporating such information from overrunning is developed, based on the “adding weighted Zs” method of combining p-values. This yields a combined p-value for the primary test and a median-unbiased estimate and confidence bounds for the parameter under test. When the amount of overrunning information is proportional to the amount available upon terminating the sequential test, exact inference methods are provided; otherwise, approximate methods are given and evaluated. The context is that of observing a Brownian motion with drift, with either linear stopping boundaries in continuous time or discrete-time group-sequential boundaries. The method is compared with other available methods and is exemplified with data from two sequential clinical trials.

Chapter information

Bertrand Clarke and Subhashis Ghosal, eds., Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 33-45

First available in Project Euclid: 28 April 2008

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Mathematical Reviews number (MathSciNet)

Primary: 62L10: Sequential analysis
Secondary: 62P10: Applications to biology and medical sciences

delayed observations deletion method double sampling lagged data meta analysis ML ordering sequential clinical trial

Copyright © 2008, Institute of Mathematical Statistics


Hall, W. J.; Ding, Keyue. Sequential tests and estimates after overrunning based on p -value combination. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 33--45, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000039.

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  • [1] Anderson, T. W. (1964). Sequential analysis with delayed observations. J. Amer. Statist. Assoc. 59 1006–1015.
  • [2] Brannath, W., Posch, M. and Bauer, P. (2002). Recursive combination tests. J. Amer. Statist. Assoc. 97 236–244.
  • [3] Hall, W. J. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequential Analysis 16 345–352. Addendum in Sequential Analysis 17 123–124.
  • [4] Hall, W. J. and Liu, A. (2002). Sequential tests and estimators after overrunning based on maximum-likelihood ordering. Biometrika 89 699–707.
  • [5] Jennison, C. (1999). Group sequential software at website: mascj/book/programs/general.
  • [6] Jennison, C. and Turnbull, B. W. (2000). Group Sequential Methods with Applications to Clinical Trials. Chapman & Hall/CRC, Boca Raton, FL.
  • [7] Liptak, T. (1958). On the combination of independent tests. Magyar Tud. Akad. Mat. Kutato Int. Közl. 3 171–197.
  • [8] Moss, A. J., Hall, W. J., Cannom, D. S., Daubert, J. P., Higgins, M. D., Klein, H., Levine, J. H., Saksena, S., Waldo, A. L., Wilber, D., Brown, M. W., Heo, M.; for the Multicenter Automatic Defibrillator Implantation Trial Investigators (1996). Improved survival with an implanted defibrillator in patients with coronary disease at high risk for ventricular arrhythmia. New England Journal of Medicine 335 1933–1940.
  • [9] Moss, A. J., Zareba, W., Hall, W. J., Klein, H., Wilber, D. J., Cannom, D. S., Daubert, J. P., Higgins, S. L., Brown, M. W., Andrews, M. L.; for the Multicenter Automatic Defibrillator Implantation Trial-II Investigators (2002). Prophylactic implantation of a defibrillator in patients with myocardial infarction and reduced ejection fraction. New England J. Medicine 346 877–883.
  • [10] Mosteller, F. M. and Bush, R. R. (1954). Selected quantitative techniques. In Handbook of Social Psychology I. Theory and Methods (G. Lindzey, ed.). Addison-Wesley, Cambridge, MA.
  • [11] MPS Research Unit (2000). PEST: Planning and Evaluation of Sequential Trials, Version 4: Operating Manual. University of Reading, Reading, UK.
  • [12] Oosterhoff, J. (1969). Combination of One-Sided Statistical Tests. The Mathematical Centre, Amsterdam.
  • [13] Rosenthal, R. (1978). Combining results of independent studies. Psych. Bull. 85 185–193.
  • [14] Sooriyarachchi, M. R., Whitehead, J., Matsushita, T., Bolland, K., and Whitehead, A. (2003). Incorporating data received after a sequential trial has stopped into the final analysis: Implementation and comparison of methods. Biometrics 59 701–709.
  • [15] Stouffer, S. A., Suchman, E. A., DeVinner, L. C., Star, R. M., Williams, R. M. (1949). The American Soldier: Adjustment During Army Life I. Princeton Univ. Press, Princeton, NJ.
  • [16] Whitehead, J. (1992). Overrunning and underrunning in sequential clinical trials. Controlled Clinical Trials 13 106–121.
  • [17] Whitehead, J. (1997). The Design and Analysis of Sequential Clinical Trials, 2nd ed. revised. Wiley, New York.