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Closed Testing and the Partitioning Principle

Closed Testing and the Partitioning Principle. Jason C. Hsu The Ohio State University MCP 2002 August 2002 Bethesda, Maryland. Principles of Test-Construction. Union-Intersection Testing UIT S. N. Roy Intersection-Union Testing IUT Roger Berger (1982) Technometrics Closed testing

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Closed Testing and the Partitioning Principle

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  1. Closed Testing and the Partitioning Principle Jason C. Hsu The Ohio State University MCP 2002August 2002Bethesda, Maryland

  2. Principles of Test-Construction • Union-Intersection Testing UIT S. N. Roy • Intersection-Union Testing IUT Roger Berger (1982) Technometrics • Closed testing Marcus, Peritz, Gabriel (1976) Biometrika • Partitioning Stefansson, Kim, and Hsu (1984) Statistical Decision Theory and Related Topics, Berger & Gupta eds., Springer-Verlag. Finner and Strassberger (2002)Annals of Statistics • Equivariant confidence set Tukey (1953) Scheffe (195?) Dunnett (1955)

  3. Partitioning confidence sets • Multiple Comparison with the Best Gunnar Stefansson & Hsu • 1-sided stepdown method (sample-determined steps) = Naik/Marcus-Peritz-Gabriel closed test Hsu • Multiple Comparison with the Sample Best Woochul Kim & Hsu & Stefansson • Bioequivalence Ruberg & Hsu & G. Hwang & Liu & Casella & Brown • 1-sided stepdown method (pre-determined steps) Roger Berger & Hsu

  4. Partitioning • Formulate hypotheses H0i: i* for iI •  iIi* = entire parameter space • {i*: iI} partitions the parameter space • Test each H0i*: i *, iI, at  • Infer   i if H0i* is rejected • Pivot in each i a confidence set Ci for  •  iICi is a 100(1)% confidence set for 

  5. Partitioning • Formulate hypotheses H0i: i for i  I •  iJ I = entire parameter space • For each J  I, letJ * = iJi( jJj)c • Test each H0J*: J *, J  I, at  • {J*: J  I} partitions the parameter space • Infer   J if H0J* is rejected • Pivot in each J a confidence set CJ for  •  J  J CJ is a 100(1)% confidence set for 

  6. MCB confidence intervals i  maxji j [(Yi  maxji Yj W), (Yi  maxji Yj+ W)+], i = 1, 2, … , k • Upper bounds imply subset selection • Lower bounds imply indifference zone selection

  7. Multiple Comparison with the Best • H01: Treatment 1 is the best • H02: Treatment 2 is the best • H03: Treatment 3 is the best • … • Test each at  using 1-sided Dunnett’s • Collate the results

  8. Union-Intersection Testing UIT • Form Ha:  Hai(an “or” thing) • Test H0:  H0i, the complement of Ha • If reject, infer at least one H0i false • Else, infer nothing

  9. Closed Testing • Formulate hypotheses H0i: i for iI • For each JI, let J = iJi • Form closed family of null hypotheses{H0J: J: JI} • Test each H0J at  • Infer   iJi if all H0J’ with J  J’ rejected • Infer   i if all H0J’ with i  J’ rejected

  10. Oneway model Yir = i + ir, i = 0, 1, 2, … , k, r = 1, … , ni ir are i.i.d. Normal(0, 2) Dose i “efficacious” if i > 1 +  ICH E10 (2000) • Superiority if   0 • Non-inferiority if  < 0 • Equivalence is 2-sided • Non-inferiority is 1-sided

  11. Closed testing null hypotheses(sample-determined steps) • H02: Dose 2 not efficacious • H03: Dose 3 not efficacious • H01: Doses 2 and 3 not efficacious • Test each at  • Collate the results

  12. Partitioning null hypotheses(sample-determined steps) • H01: Doses 2 and 3 not efficacious • H02: Dose 2 not efficaciousbut dose 3 is • H03: Dose 3 not efficaciousbut dose 2 is • Test each at  • Collate the results

  13. Partitioning implies closed testing Partitioning implies closed testing because • A size  test for H0i is a size  test forH0i • Reject H01: Doses 2 and 3 not efficacious implies either dose 2 or dose 3 efficacious • Reject H02: Dose 2 not efficaciousbut dose 3 efficacious implies it is not the case dose 3 is efficacious but notdose 2 • Reject H01 and H02 thus implies dose 2 efficacious

  14. Intersection-Union Testing IUT • Form Ha:  Hai (an “and” thing) • Test H0:  H0i, the complement of Ha • If reject, infer all H0i false • Else, infer nothing

  15. PK concentration in blood plasma curve

  16. Bioequivalence defined Bioequivalence: clinical equivalence between • Brand name drug • Generic drug Bioequivalence parameters • AUC = Area Under the Curve • Cmax = maximum Concentration • Tmax = Time to maximum concentratin

  17. Average bioequivalence Notation  = expected value of brand name drug 2 = expected value of generic drug Average bioequivalence means .8 < /2 < 1.25 for AUC and .8 < /2 < 1.25 for Cmax

  18. Bioequivalence in practice If log of observations are normal with means  and2 and equal variances, then average bioequivalence becomes log(.8) <  2 < log(1.25) for AUC and log(.8) <  2 < log(1.25) for Cmax

  19. Partitioning Partition the parameter space as • H0<:  2 < log(0.8) • H0>:  2 > log(1.25) • Ha: log(.8) <  2 < log(1.25) Test H0< and H0> each at . Infer log(.8) <  2 < log(1.25) if both H0< and H0> rejected. Controls P{incorrect decision} at .

  20. Dose-Response (Phase II)

  21. Anti-psychotic drug efficacy trial Arvanitis et al. (1997 Biological Psychiatry) CGI = Clinical Global Impression

  22. Minimum Effective Dose (MED) Minimu Effective Dose = MED = smallest i so that i > 1 + for all j, i  j  k Want an upper confidence bound MED+so that P{MED < MED+}  100(1)%

  23. Closed testing inference • Infer nothing if H01 is accepted • Infer at least one of doses 2 and 3 effective if H01 is rejected • Infer dose 2 effective if, in addition to H01,H02 is rejected • Infer dose 3 effective if, in addition to H01,H03 is rejected

  24. Closed testing method(sample-determined steps) • Start from H01 to H02 and H03 • Stepdown from smallest p-value to largest p-value • Stop as soon as one fails to reject • Multiplicity adjustment decreases from k to k  1 to k  2  to 2 from step 1 to 2 to 3 … to step k  1

  25. Tests of equalities(pre-determined steps) • H0k: 1 = 2 =  = kHak: 1 = 2 =  < k • H0(k1): 1 = 2 =  = k1Ha(k1): 1 = 2 =  < k1 • H0(k2): 1 = 2 =  = k2Ha(k2): 1 = 2 =  < k2 •  • H02: 1 = 2Ha2: 1 < 2

  26. Closed testing of equalities • Null hypotheses are nested • Closure of family remains H0k H02 • Test each H0i at  • Stepdown from dose k to dose k1 to  to dose 2 • Stop as soon as one fails to reject • Multiplicity adjustment not needed

  27. H0k: 1 =  = k  H02: 1 = 2 H0i H0i H0k: 1  k  H02:1  2 H0i H0i Testing equalities is easy

  28. Partitioning null hypotheses (for pre-determined steps) • H0k: Dose k not efficacious • H0(k-1):Dose k efficacious but dose k1 not efficacious • H0(k-1):Doses k and k1 efficacious but dose k2 not efficacious •  • H02: Doses k 3 efficacious but dose 2 not efficacious • Test each at  • Collate the results

  29. Partitioning inference • Infer nothing if H0k is accepted • Infer dose k effective if H0k is rejected • Infer dose k1 effective if, in addition to H0k, H0(k-1) is rejected • Infer dose k2 effective if, in addition to H0k and H0(k-1) , H03 is rejected • 

  30. Partitioning method (for pre-determined steps) • Stepdown from dose k to dose k1 to  to dose 2 • Stop as soon as one fails to reject • Multiplicity adjustment not needed • Any pre-determined sequence of doses can be used • Confidence set given in Hsu and Berger (1999 JASA)

  31. Pairwise t tests for partitioning Size  tests for H0k H02 are also size  test forH0k H02 • H0k: Dose k not efficacious • H0(k-1): Dose k1 not efficacious • H0(k-2): Dose k2 not efficacious •  • H02: Dose 2 not efficacious • Test each with a size- 2-sample 1-sided t-test

  32. H0k: 1 =  = k  H02: 1 = 2 H0i H0i H0k: 1  k  H02:1  2 H0i H0i Testing equalities is easy

  33. Could reject for the wrong reason H0 Ha neither

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