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Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs)

Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs). Olivier Guilbaud AstraZeneca R&D, Sweden. Bonferroni Confidence Regions. Elements of Proposed Simultaneous Confidence Regions. The Simultaneous  Confidence Regions. Nice Reduction Property.

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Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs)

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  1. Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs) Olivier Guilbaud AstraZeneca R&D, Sweden

  2. Bonferroni Confidence Regions • Elements of Proposed Simultaneous Confidence Regions • The Simultaneous  Confidence Regions • Nice Reduction Property • Illustration • Extensions Outline

  3. Bonferroni Confidence Regions Estimated quantities (m specified): Marginal -Confidence Regions (m specified): Simultaneous ()-Confidence Regions: Bonferroni m -adjustment • Nice properties: • Flexible, Generally valid (if marginal regions are valid) • No restriction to particular kinds/dimensions of isand Ci,s

  4. Can other simultaneous confidence regions based on marginal confidence regions be constructed that share these nice properties ? and Bonferroni regions constitute a special case in a class of such regions . YES ,

  5. Estimated quantities (m specified): Marginal -Confidence Regions (m specified): Additional Target Regions of interest (m specified): Aim is to ”show”, if possible, that i Ri (target assertion) Elements of Proposed Simultaneous Confidence Regions

  6. Examples with i      and Target region Ri  for i : • Show , i.e. , ”superiority” or ”non-inferiority” • Show , i.e. , ”inequality” • Show , i.e. , ”equivalence” • using appropriate marginal confidence regions Ci,s for is Elements … (cont.) No restriction to particular kinds/dimensions of is, Ris,or Ci,s

  7. Connection with Holm’s (1979) Step-down MTP through Marginal Confidence-Region ' Test of Hi : i Rito ShowHic : i Ri: Raw p-value pi= pi(data) : • pi = ( infimum of levels ' for which the test rejects Hi ) Elements … (cont.)

  8. (Brief Refresher) Sture Holm Ordered p-values:p(1) p(2) …p(m) ; Corresp Hs: H(1), H(2), …,H(m) Bonferroni-Holm’s(1979) MTP with multiple-level  : Reject successively H(1), H(2), …, H(m), as long as p(1) /m , p(2) /(m-1) , …, p(m) /1 ; Stop at first > !

  9. Elements … (cont.) Given  ,introduce (i) : the 2 index sets • IReject ( set of 1  i  m of Hi s rejected by Holm at multiple level  ) • IAccept  1, 2, …, m   IReject (ii) : additional Estimated quantity & Marginal Confidence Region Can be arbitrarily chosen. Can be chosen to sharpen inferences about 1, …, m

  10. Reflects by how much/little one missed the Target assertion ”i Ri” For 1  i  m+1 , define : Useful to sharpen inferences for 1im Main Result : Bonferroni |IAccept| -adjustment of marginal conf region for i The Simultaneous  Confidence Regions

  11. Nice Reduction Property What happens if all target regionsR1, …, Rm are chosen to be empty ? Empty Only this isinformative because : Reduction to m ordinary BonferroniConf Regions for 1, …, m !

  12. Estimated quantities in  ,  , e.g. Differences of Trt-means Show, if possible, thateach ! That is, each . Marginal 1-sided -Confidence Regions ( t or W based t or W p-values for H0: i  0 vs. H0c: i > 0 ) Possible realization of  Conf Regions with m 5 : Holm non-rejections(Bonf 2 -adjustm) Holm rejections(Target assertions) 0  Illustration ”extra free information”

  13. Illustration (cont.) Possible choice of additional and to sharpen inferences : Rectangular region (Reasonable if scales of are equal or sufficiently similar) Sharpening of  Conf Regions with m 5if occurs : instead of 0  0  Lmin

  14. Holm rejection index set For 1  i  m+1 : Extension 1: Holm with Weights Holm’s (1979) MTP based on p1 , …, pm and given weights 1 , …, m > 0

  15. Weights wi(I) are such that for each non-empty I{1, …, m} : For 1  i  m+1 , again : H-B-M rejection index set Extension 2: Hommel, Bretz & Maurer (2007) class of MTPs CTP with Bonferroni test of HI using certainwi(I)s , i I , I {1, …, m} . (Fixed-Seq MTP, Holm’s MTP, Gatekeeping MTPs, Fallback MTP, …)

  16. Superiority vs. Placebo Non-inferiority vs. Placebo m=2 : Final Comments • Flexible (no restriction concerning kinds/dim of is, Risand Ci,s) • Generally valid (if marginal confidence regions are valid) • Intuitively appealing • Simple to implement • Multi-dimensional is and Ris can be combined with ”marginal” simultaneous CIs within i s (e.g. Hsu-Berger CIs for i -components): Leads e.g. toSimultaneous Confidence Regions corresponding to the Bonferroni-Holm MTP for m families of hypotheses (Bauer et al. 1998 appendix, Bauer et al. 2001)with Extra ”free” Information

  17. Selected References • Aitchinson, J. (1964). Confidence-region Tests. Journal of the Royal Statistical Society, Ser. B, 26, 462-476. • Bauer, P., Brannath, W., and Posch, M. (2001). Multiple Testing for Identifying Effective and Safe Treatments. Biometrical Journal 43, 605-616. • Bauer, P., Röhmel, J., Maurer, W., and Hothorn, L. (1998). Testing Strategies in Multi-dose Experiments Including Active Control. Statistics in Medicine 17, 2133-2146. • Holm, S. (1979). A Simple Sequentially Rejective Multiple Test Procedure. Scandinavian Journal of Statistics 6, 65-70. • Hommel, G., Bretz, F., and Maurer, W. (2007). Powerful Short-Cuts for Multiple Testing Procedures with Special Reference to Gatekeeping Strategies. Statistics in Medicine (in press). • Hsu, J. C. and Berger, R. L. (1999). Stepwise Confidence Intervals Without Multiplicity Adjustment for Dose-response and Toxicity Studies. Journal of the American Statistical Association 94, 468-482.

  18. Bonferroni Confidence Regions • Elements of Proposed Simultaneous Confidence Regions • The Simultaneous  Confidence Regions • Nice Reduction Property • Illustration • Extensions Summary

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