1 Superior Safety in Noninferiority Trials David R. Bristol To appear in Biometrical Journal, 2005

2 Abstract Noninferiority of a new treatment to a reference treatment with respect to efficacy is usually associated with the superiority of the new treatment to the reference treatment with respect to other aspects not associated with efficacy.

3 Abstract When the superiority of the new treatment to the reference treatment is with respect to a specified safety variable, the between-treatment comparisons with respect to safety may also be performed. Here techniques are discussed for the simultaneous consideration of both aspects.

4 Background ICH (1998) guidelines E-9 and E-10 discuss noninferiority trials, but only with respect to the efficacy comparison. The efficacy problem has been discussed by several authors. Bristol (1999) provides a review.

5 Notation Treatment 0 = Reference treatment, (efficacious with an associated adverse effect on a specified safety variable) Treatment 1 = New treatment.

6 GOAL Show that Treatment 1 is superior to Treatment 0 with respect to the specified safety variable and noninferior with respect to a specified efficacy variable.

7 Study Design A randomized parallel-group study is to be conducted to compare Treatment 0 and Treatment 1, with n subjects / group. A placebo group could be included in this design for completeness and sensitivity testing, but its inclusion will not have a direct impact on the primary analysis, which is discussed here.

8 Notation Let X ij and Y ij denote the efficacy and safety responses, respectively, for Subject j on Treatment i, i=0,1, j=1, …,n. It is assumed that (X ij,Y ij )' ~BVN(μ Xi, μ Yi, σ 2 X, σ 2 Y, ρ), where all parameters are unknown. Assume small values of efficacy and safety are preferable.

9 Testing It is desired to show that μ X1 < μ X0 +Δ and μ Y1 < μ Y0, where the noninferiority margin Δ is a specified positive number and is defined by clinical importance, often as a proportion of the average efficacy seen previously for Treatment 0.

10 Testing This goal can be achieved by simultaneously testing H 0X : μ X1 μ X0 +Δ against H 1X : μ X1 < μ X0 +Δ, and H 0Y : μ Y1 μ Y0 against H 1Y : μ Y1 < μ Y0.

11 Testing Let H 0 =H 0X U H 0Y and let H 1 =H 1X H 1Y. It is desired to test H 0 against H 1.

12 Testing The noninferiority (NI) aspect differs from that seen in most NI problems, as the response is bivariate. The reverse multiplicity (RM) aspect pertains to the all-pairs multiple comparisons problem, where both H 0X and H 0Y must be rejected.

13 Test Procedures Univariate approach composite score or a global statistic: OBrien (1984) Pocock, Geller, Tsiatis (1987) And many others

14 Test Procedures The multiplicity problem is solved by reducing the dimensionality of the response variable used for the comparison. This approach suffers from the possible impact of one variable on the new response variable. Thus, this approach should not be considered for this problem. However, it is briefly discussed for completeness.

15 Notation Let and where and are (pooled) unbiased estimates of σ 2 X and σ 2 Y, respectively.

16 Rejection Rule(s) The rejection rule for efficacy is to Reject H 0X : μ X1 μ X0 +Δ in favor of H 1X : μ X1 < μ X0 +Δ if Z X -z α and the rejection rule for safety is to Reject H 0Y : μ Y1 μ Y0 in favor of H 1Y : μ Y1 < μ Y0 if Z Y -z α, where z α is the 100 (1-α)-th percentile of the standard normal distribution.

17 Notation Let Δ X = μ X1 -μ X0 and Δ Y = μ Y1 - μ Y0. Then the problem is to simultaneously test H 0X : Δ X Δ against H 1X : Δ X < Δ and H 0Y : Δ Y 0 against H 1Y : Δ Y < 0.

18 Notation (Z X,Z Y )' ~ BVN((.5n) 1/2 (Δ X -Δ)/ σ X,(.5n) 1/2 Δ Y /σ Y,1,1,ρ). (approx.) Tests could be based on linear combinations of Z X and Z Y. Such tests will be inappropriate for the RM formulation.

19 Max Test (Bivariate Approach) The simultaneous comparison is performed using a test based on W=max{Z X,Z Y }.

20 Max Test The rejection rule is Reject H 0 in favor of H 1 if W C, where C is chosen such that P(Reject H 0 | Δ X =Δ and Δ Y = 0)=α.

21 Max Test Let G(.,.| ρ) is the joint cdf of a bivariate normal distribution with zero means, unit variances, and correlation ρ. Then P(Reject H 0 | Δ X =Δ and Δ Y = 0) =G(C, C | ρ).

22 Max Test Given ρ, C can be chosen such that G(C,C| ρ)= α. However, ρ is unknown. The critical value can be estimated by satisfying where r is an estimate of ρ (pooled or average).

23 Stepwise Approach Stepwise approaches to the multiple endpoints problem were considered by Lehmacher, Wassmer, and Reitmer (1991) and several others. However, because of the RM formulation, these results are not directly applicable. A stepwise procedure could be used here.

24 Stepwise Approach (I)Test H 0X. If H 0X is not rejected in favor of H 1X, stop. If H 0X is rejected in favor of H 1X, (II) Test H 0Y. If H 0Y is not rejected in favor of H 1Y, stop. If H 0Y is rejected in favor of H 1Y, Reject H 0 in favor of H 1.

25 Stepwise Approach The choice of level for each test has an important impact on the overall level, and using an α-level test for each of the univariate tests results in the overall level being much less than α. The properties of this testing procedure are examined below using simulations.

26 Simulation Results The following results are based on 10,000 for each set of parameters, unit variances and n=50 subjects per treatment. Each test is conducted at the α=0.05 level. The simulations were conducted with the same seed for comparison.

27 Simulation Results Let P X & P Y be the estimated power for the univariate tests based on X and Y respectively. Let P denote the estimated power of the stepwise procedure of testing H 0Y only if H 0X is rejected, where both tests are performed at the 0.05 level. Maximum is test using W, with pooled or average estimate of correlation.

28 Power Estimates (%)

29 Discussion and Summary Noninferiority trials are often conducted when the new treatment has an advantage, other than efficacy, over the reference treatment. To simultaneously test superiority with respect to safety and noninferiority with respect to efficacy, the single-stage testing approach based on maximum is easy to use and easy to interpret.

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