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Issues of Simultaneous Tests for Non-Inferiority and Superiority Tie-Hua Ng*, Ph. D. U.S. Food and Drug Administration 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.

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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

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3 A Big Question Is Multiplicity Adjustment Necessary?

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4 IsMultiplicityAdjustmentNecessary?

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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

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6 Assumptions/Notations Normality and larger is better T: Test/Experimental treatment ( t ) S: Standard therapy/Active control ( s ) : Non-Inferiority Margin (> 0) For a given d (real number), define –Null: H 0 (d): T S - d –Alternative: H 1 (d): T > S - d Non-Inferiority: d = Superiority: d = 0

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7 Non-Inferiority (d = ) H 0 ( ): T S - against H 1 ( ): T > S - H0()H0() H1()H1() ° T Boundary WorseBetter Mean Response S

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8 Superiority (d = 0) H 0 (0): T S against H 1 (0): T > S H 0 (0) H 1 (0) ° T Boundary WorseBetter Mean Response S

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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,

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

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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

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

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13 Simultaneous Testing (2) Multiplicity adjustment is not necessary –Dunnett and Gent (1996) Intersection-Union (IU): Superiority: Both H 0 ( ) and H 0 (0) are rejected –Morikawa and Yoshida (1995) Closed Testing Procedure (CTP): Test H 0 (0) when H 0 ( ) H 0 (0) is rejected

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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 H 0 (0) No equivalence/non-inferiority claim –Ok: Morikawa and Yoshida (1995) Ref: Phillips et al (2000), DIJ

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15 Is Simultaneous Testing Acceptable?

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16 Use of Confidence Interval in Hypothesis Testing H 0 (d): T S - d (at significance level ) One-sided 100(1- )% lower CI for T-S Reject H 0 (d) if and only if the CI excludes -d Test is worseTest is better Mean Difference (T – S) -d Reject H 0 (d) Do not reject H 0 (d)

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17 Use of Confidence Interval in Hypothesis Testing (2) If CI = (L, ), then H 0 (d) will be rejected for all -d < L. A Tricky Question –Suppose CI = (-1.999, ), L = H 0 (2): T S - 2 is rejected (d=2) since -d < L Can we conclude that T > S - 2? Yes, if H 0 (2) is prespecified. No, otherwise.

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18 Use of Confidence Interval in Hypothesis Testing (3) Post hoc specification of H 0 (d) is a No

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19 Simultaneous Testing: Problems H 0 (d 1 ) and H 0 (d 2 ), for d 1 > d 2 One-sided (1 - )100% lower CI for T - S Test is worse Test is better Mean Difference (T – S) -d 2 -d 1 Reject H 0 (d 2 ) Reject H 0 (d 1 ) Neither

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20 Simultaneous Testing: Problems (2) H 0 (d 1 ), H 0 (d 2 ) and H 0 (d 3 ), for d 1 > d 2 > d 3 One-sided (1 - )100% lower CI for T - S Test is worse Test is better Mean Difference (T – S) -d 3 -d 1 Reject H 0 (d 3 ) Reject H 0 (d 2 ) None -d 2 Reject H 0 (d 1 )

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21 Simultaneous Testing: Problems (3) H 0 (d 1 ), H 0 (d 2 ),…, H 0 (d k ), for d 1 > d 2 > … > d k One-sided (1 - )100% lower CI for T - S Test is worse Test is better Mean Difference (T – S) -d k -d 1 Reject H 0 (d k ) Reject H 0 (d 2 ) None -d 2 Reject H 0 (d 1 ) … … d 3

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22 Simultaneous Testing: Problems (4) Choose k large enough Pr[-d 1 < Lower limit < -d k ] close to 1 Max |d k - d k-1 | < a given small number Simultaneous testing of H 0 (d i ), i = 1,…, k Post hoc specification of H 0 (d)

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23 Simultaneous Testing: Problems (5) Number of Nested hypotheses Exploratory (many H 0 (d)) Confirmatory (one H 0 (d)) …………. k ………… Simultaneous H 0 ( ) and H 0 (0)

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

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25 Simultaneous Testing: Problems (7) Post hoc specification of H 0 (d) Let -d 0 = 100(1 - )% lower limit - Reject H 0 (d 0 ), since -d 0 < lower limit Repeat the same trial independently Pr[Rejecting H 0 (d 0 )] = 0.5 +

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26 Simultaneous Testing: Problems (8) Simultaneous testing of many H 0 (d) –Repeat the same trial independently –Low probability of confirming the finding 1 st trial: Reject H 0 (d j ) but not H 0 (d j+1 ) 2 nd trial: Pr[Rejecting H 0 (d j )] is low (e.g., 0.5+)

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27 Simultaneous Testing: Problems (9) Simultaneous testing of H 0 ( ) and H 0 (0)? Confirm the finding = 2 Known variance Let T - S Significance level = 80% power for H 0 ( ) (at = 0)

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28 Simultaneous Testing: Problems (10) f ( ) = Pr[Rejecting H 0 ( ) | ] f 0 ( ) = Pr[Rejecting H 0 (0) | ]

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29 Simultaneous Testing: Problems (11) Test one null hypothesis H 0 ( ) Suppose that H 0 ( ) is rejected Repeat the same trial independently Pr[Rejecting H 0 ( ) again] = f ( )

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

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31 Simultaneous Testing: Problems (13) [1 - w( )] · f ( ) + w( ) · f 0 ( ) where w( ) = f 0 ( )/f ( ) Simultaneous tests in the 2 nd trial

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32 Simultaneous Testing: Problems (14) Ratio: 1 – [f 0 ( )/f ( )] [1 – f 0 ( )/f ( )] Ratio may be as low as 0.75

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

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34 Is Simultaneous Testing of H 0 ( ) and H 0 (0) Acceptable?

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35 You be the judge

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36 References Dunnett and Gent (1976), Statistics in Medicine, 15, Committee for Proprietary Medicinal Products (CPMP; 2002). Points to Consider on Switching Between Superiority and Non-Inferiority. Morikawa T, Yoshida M. (1995), Journal of Biopharmaceutical Statistics, 5: Phillips et al., (2000), Drug Information Journal, 34:

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