2. 1 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs) Olivier Guilbaud AstraZeneca R&D, Sweden

3. 2 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Outline Bonferroni Confidence Regions Elements of Proposed Simultaneous Confidence Regions The Simultaneous  Confidence Regions Illustration Extensions Nice Reduction Property

4. 3 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 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  i s and C i,  s

5. 4 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Can other simultaneous confidence regions based on marginal confidence regions be constructed that share these nice properties ? YES, and Bonferroni regions constitute a special case in a class of such regions.

6. 5 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements of Proposed Simultaneous Confidence Regions Estimated quantities (m specified): Marginal  -Confidence Regions (m specified): Target Regions of interest (m specified): Aim is to ”show”, if possible, that  i  R i (target assertion) Additional

7. 6 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Examples with  i       and Target region R i   for  i : Show, i.e., ”superiority” or ”non-inferiority” Show, i.e., ”inequality” Show, i.e., ”equivalence” using appropriate marginal confidence regions C i,  s for  i s No restriction to particular kinds/dimensions of  i s, R i s, or C i,  s

8. 7 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Raw p-value p i = p i (data) : p i = ( infimum of levels  ' for which the test rejects H i ) Marginal Confidence-Region  ' Test of H i :  i  R i to Show H i c :  i  R i : Connection with Holm’s (1979) Step-down MTP through

9. 8 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 (Brief Refresher) 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 > ! Sture Holm

10. 9 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Can be arbitrarily chosen. Can be chosen to sharpen inferences about  1, …,  m Given , introduce I Reject  ( set of 1  i  m of H i s rejected by Holm at multiple level  ) I Accept   1, 2, …, m   I Reject (i) : the 2 index sets (ii) : additional Estimated quantity & Marginal  Confidence Region

11. 10 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 The Simultaneous  Confidence Regions For 1  i  m+1, define : Main Result : Bonferroni |I Accept | -adjustment of marginal conf region for  i Reflects by how much/little one missed the Target assertion ”  i  R i ” Useful to sharpen inferences for 1  i  m

12. 11 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Nice Reduction Property What happens if all target regions R 1, …, R m are chosen to be empty ? Reduction to m ordinary Bonferroni  Conf Regions for  1, …,  m ! because : Only this is informative Empty

13. 12 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Illustration Estimated quantities in , , e.g. Differences of Trt-means Show, if possible, that each ! That is, each. Marginal 1-sided  -Confidence Regions ( t or W based t or W p-values for H 0 :  i  0 vs. H 0 c :  i > 0 ) Possible realization of  Conf Regions with m  5 : 0  Holm non-rejections (Bonf 2 -adjustm) Holm rejections (Target assertions) ”extra free information”

14. 13 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Illustration (cont.) (Reasonable if scales of are equal or sufficiently similar) Possible choice of additional and to sharpen inferences : Rectangular region Sharpening of  Conf Regions with m  5 if occurs : 0  0  instead of L min

15. 14 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Extension 1: Holm with Weights Holm’s (1979) MTP based on p 1, …, p m and given weights 1, …, m > 0 For 1  i  m+1 : Holm rejection index set

16. 15 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Extension 2: Hommel, Bretz & Maurer (2007) class of MTPs CTP with Bonferroni test of H I using certain w i (I)s, i  I, I  {1, …, m}. (Fixed-Seq MTP, Holm’s MTP, Gatekeeping MTPs, Fallback MTP, …) For 1  i  m+1, again : Weights w i (I) are such that for each non-empty I  {1, …, m} : H-B-M rejection index set

17. 16 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Final Comments Flexible (no restriction concerning kinds/dim of  i s, R i s and C i,  s) Multi-dimensional  i s and R i s can be combined with ”marginal” simultaneous CIs within  i s (e.g. Hsu-Berger CIs for  i -components): Generally valid (if marginal confidence regions are valid) Intuitively appealing Simple to implement Leads e.g. to Simultaneous 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 m=2 : Superiority vs. PlaceboNon-inferiority vs. Placebo

18. 17 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Selected References Aitchinson, J. (1964). Confidence-region Tests. Journal of the Royal Statistical Society, Ser. B, 26, 462-476. 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. 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).

19. 18 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Summary Bonferroni Confidence Regions Elements of Proposed Simultaneous Confidence Regions The Simultaneous  Confidence Regions Illustration Extensions Nice Reduction Property