Multiple%20Testing%20Procedures

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Simultaneous Confidence Regions corresponding
to Holms Step-down MTP (and other CTPs)
Olivier Guilbaud AstraZeneca RD, Sweden
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Outline
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• Nice properties
• Flexible, Generally valid (if marginal regions
  are valid)
• No restriction to particular kinds/dimensions of
  ?i s and Ci,? s

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YES ,
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Elements of Proposed Simultaneous Confidence
Regions
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Elements (cont.)
No restriction to particular kinds/dimensions of
?i s, Ri s, or Ci,? s
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Elements (cont.)
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Ordered p-values p(1) ? p(2) ? ? p(m)
Corresp Hs H(1) , H(2) , , H(m)
Bonferroni-Holms (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 gt !
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The Simultaneous ??? Confidence Regions
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Reduction to m ordinary Bonferroni ??? Conf
Regions for ?1 , , ?m !
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Illustration
extra free information
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Extension 1 Holm with Weights
Holms (1979) MTP based on p1 , , pm and given
weights ?1 , , ? m gt 0
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Extension 2 Hommel, Bretz Maurer (2007) class
of MTPs
CTP with Bonferroni test of HI using certain
wi(I)s , i? I , I ?1, , m . (Fixed-Seq MTP,
Holms MTP, Gatekeeping MTPs, Fallback MTP, )
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Final Comments

• Flexible (no restriction concerning kinds/dim of
  ?i s, Ri s and Ci,? s)

• Generally valid (if marginal confidence regions
  are valid)

• Intuitively appealing

• Simple to implement

• Multi-dimensional ?i s and Ri s can be combined
  with marginal simultaneous CIs within ?i s
  (e.g. Hsu-Berger CIs for ?i -components)

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
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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.

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Summary