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Issues of Simultaneous Tests for Non-Inferiority and Superiority

Issues of Simultaneous Tests for Non-Inferiority and Superiority. Tie-Hua Ng*, Ph. D. U.S. Food and Drug Admi n istration Ng@cber.fda.gov Presented at MCP 2002 August 5-7 , 200 2 Bethesda, Maryland _______

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Issues of Simultaneous Tests for Non-Inferiority and Superiority

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  1. Issues of Simultaneous Tests for Non-Inferiority and Superiority Tie-Hua Ng*, Ph. D. U.S. Food and Drug Administration Ng@cber.fda.gov Presented at MCP 2002 August 5-7, 2002 Bethesda, Maryland _______ * The views expressed in this presentation are not necessarily of the U.S. Food and Drug Administration.

  2. Simultaneous Tests for Non-Inferiority and Superiority • Multiplicity adjustment is not necessary • Intersection-union principle (IU) • Dunnett and Gent (1996) • Closed testing procedure (CTP) • Morikawa and Yoshida (1995) • Indisputable

  3. A Big Question Is Multiplicity Adjustment Necessary?

  4. Is Multiplicity Adjustment Necessary?

  5. Outline • Assumptions and Notations • Switching between Superiority and Non-Inferiority • Is Simultaneous Testing Acceptable? • Use of Confidence Interval in Hypothesis Testing --- Pitfall • Problems of Simultaneous Testing • Conclusion

  6. Assumptions/Notations • Normality and larger is better • T: Test/Experimental treatment (t) • S: Standard therapy/Active control (s) • : Non-InferiorityMargin (> 0) • For a given d (real number), define • Null: H0(d): T S - d • Alternative: H1(d): T >S - d • Non-Inferiority: d =  • Superiority: d = 0

  7. ° • T Boundary Non-Inferiority (d = ) H0(): T S - against H1(): T >S - H1() H0()  S Worse Better Mean Response

  8. Superiority (d = 0) H0 (0): T S against H1 (0): T >S H1(0) H0(0) ° • S T Worse Better Boundary Mean Response

  9. Switching between Superiority and Non-Inferiority CPMP (Committee for Proprietary Medicinal Products), European Agency for the Evaluation of Medicinal Products Points to Consider on Switching Between Superiority and Non-Inferiority, 2000. http://www.emea.eu.int/htms/human/ewp/ewpptc.htm

  10. Switching between Superiority and Non-Inferiority (2) • Non-Inferiority Trial • If H0() is rejected, proceed to test H0(0) • No multiplicity issue, closed testing procedure • Superiority Trial • Fail to reject H0(0), proceed to test H0() • No multiplicity issue • Post hoc specification of 

  11. Switching between Superiority and Non-Inferiority (3) • Non-inferiority Trial • Intention-to-treat (ITT) • Per protocol (PP) • Superiority Trial • Primary: Intention-to-treat (ITT) • Supportive: Per protocol (PP) • Assume ITT = PP

  12. Simultaneous Testing One-sided 100(1 - )% lower Confidence Interval for T - S Superiority Non-inferiority Neither - 0 Test is worse Test is better Mean Difference (T – S)

  13. Simultaneous Testing (2) • Multiplicity adjustment is not necessary • Dunnett and Gent (1996) • Intersection-Union (IU): Superiority: Both H0() andH0(0) are rejected • Morikawa and Yoshida (1995) • Closed Testing Procedure (CTP): Test H0(0) whenH0()H0(0) is rejected

  14. Simultaneous Testing (3) • Discussion Forum (October 1998) • London • PSI (Statisticians in Pharmaceutical Industry) • Is Simultaneous Testing of Equivalence [Non-Inferiority] and Superiority Acceptable? • Superiority trial: • Fail to reject H0 (0) • No equivalence/non-inferiority claim • Ok: Morikawa and Yoshida (1995) • Ref: Phillips et al (2000), DIJ

  15. Is Simultaneous TestingAcceptable?

  16. Use of Confidence Interval inHypothesis Testing H0(d): T S - d (at significance level ) One-sided 100(1-)% lower CI for T-S Reject H0(d)if and only if the CI excludes -d Reject H0(d) Do not reject H0(d) -d Test is worse Test is better Mean Difference (T – S)

  17. Use of Confidence Interval inHypothesis Testing (2) • If CI = (L, ), then H0(d) will be rejected for all -d < L. • A Tricky Question • Suppose CI = (-1.999, ), L = -1.999 • H0(2): T S - 2 is rejected (d=2) since -d < L • Can we conclude that T > S - 2? • Yes, if H0(2) is prespecified. • No, otherwise.

  18. Use of Confidence Interval inHypothesis Testing (3) Post hoc specification of H0(d) is a No No

  19. Simultaneous Testing: Problems H0(d1) and H0(d2), for d1 > d2 One-sided (1 - )100% lower CI for T - S Reject H0(d2) Reject H0(d1) Neither -d1 -d2 Test is worse Test is better Mean Difference (T – S)

  20. Simultaneous Testing: Problems (2) H0(d1), H0(d2) and H0(d3), for d1 > d2 > d3 One-sided (1 - )100% lower CI for T - S Reject H0(d3) Reject H0(d2) Reject H0(d1) None -d1 -d2 -d3 Test is worse Test is better Mean Difference (T – S)

  21. Simultaneous Testing: Problems (3) H0(d1), H0(d2),…, H0(dk), for d1 > d2 > … > dk One-sided (1 - )100% lower CI for T - S Reject H0(dk) . . . Reject H0(d2) … Reject H0(d1) None -d1 -d2 -d3 … -dk Test is worse Test is better Mean Difference (T – S)

  22. Simultaneous Testing: Problems (4) • Choose k large enough • Pr[-d1 < Lower limit < -dk] close to 1 • Max |dk - dk-1| < a given small number • Simultaneous testing of H0(di), i = 1,…, k  Post hoc specification of H0(d)

  23. Confirmatory (oneH0(d)) Simultaneous H0() and H0(0) Exploratory (manyH0(d)) 1 2 3 4 …………. k ………… Number of Nested hypotheses Simultaneous Testing: Problems (5)

  24. Simultaneous Testing: Problems (6) • What is wrong with IU and CTP? • Nothing • Pr[Rejecting at least one true null]   • What kind of problems?

  25. Simultaneous Testing: Problems (7) • Post hoc specification of H0(d) • Let -d0 = 100(1 - )% lower limit -  • Reject H0(d0), since -d0 < lower limit • Repeat the same trial independently • Pr[Rejecting H0(d0)] = 0.5 +

  26. Simultaneous Testing: Problems (8) • Simultaneous testing of many H0(d) • Repeat the same trial independently • Low probability of confirming the finding • 1st trial: Reject H0(dj) but not H0(dj+1) • 2nd trial: Pr[Rejecting H0(dj)] is low (e.g., 0.5+)

  27. Simultaneous Testing: Problems (9) • Simultaneous testing of H0() and H0(0)? • Confirm the finding •  = 2 • Known variance • Let   T - S • Significance level  = 0.025 • 80% power for H0() (at  = 0)

  28. Simultaneous Testing: Problems (10) f() = Pr[Rejecting H0() | ] f0() = Pr[Rejecting H0(0) | ]

  29. Simultaneous Testing: Problems (11) • Test one null hypothesis H0() • Suppose that H0() is rejected • Repeat the same trial independently • Pr[Rejecting H0() again] = f()

  30. Simultaneous Testing: Problems (12) • Test H0() and H0(0) simultaneously • Suppose that H0() or H0(0) is rejected • Repeat the same trial independently • Pr[Rejecting the same null hypothesis again] = [1 - w()]· f() + w() · f0() = f() - f0() [1 – f0()/f()], where w() = f0()/f()

  31. Simultaneous Testing: Problems (13) Simultaneous tests in the 2nd trial [1 - w()]· f() + w() · f0() where w() = f0()/f()

  32. Simultaneous Testing: Problems (14) • Ratio: 1 – [f0()/f()] [1 – f0()/f()] • Ratio may be as low as 0.75

  33. Conclusion • Many H0(d): Problematic • Not type I error rate • H0() and H0(0): Acceptable? • If “zero tolerance policy”: No • If 25% reduction cannot be tolerated: No • If 25% reduction can be tolerated: Yes

  34. Is Simultaneous Testing of H0()andH0(0) Acceptable?

  35. You be the judge

  36. References • Dunnett and Gent (1976), Statistics in Medicine, 15, 1729-1738. • Committee for Proprietary Medicinal Products (CPMP; 2002). Points to Consider on Switching Between Superiority and Non-Inferiority. http://www.emea.eu.int/htms/human/ewp/ewpptc.htm • Morikawa T, Yoshida M. (1995), Journal of Biopharmaceutical Statistics, 5:297-306. • Phillips et al., (2000), Drug Information Journal, 34:337-348. 

  37. God Bless America

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