Superior Safety in Noninferiority Trials

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Superior Safety in Noninferiority Trials

• David R. Bristol
• To appear in Biometrical Journal, 2005

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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.

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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.

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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.

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Notation

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

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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.

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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.

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Notation

• Let Xij and Yij denote the efficacy and safety
  responses, respectively, for Subject j on
  Treatment i, i0,1, j1, ,n. It is assumed that
• (Xij,Yij)' BVN(µXi, µYi, s2X, s2Y, ?),
• where all parameters are unknown. Assume small
  values of efficacy and safety are preferable.

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Testing

• It is desired to show that µX1 lt µX0 ? and µY1 lt
  µY0, where the noninferiority margin ? is a
  specified positive number and is defined by
  clinical importance, often as a proportion of the
  average efficacy seen previously for Treatment 0.

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Testing

• This goal can be achieved by simultaneously
  testing
• H0X µX1 µX0? against H1X µX1 lt µX0 ?, and
• H0Y µY1 µY0 against H1Y µY1 lt µY0.

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Testing

• Let H0H0X U H0Y and let H1H1X nH1Y.
• It is desired to test H0 against H1.

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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 H0X and H0Y must be rejected.

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Test Procedures

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

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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.

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Notation

• Let
• and
• where and are (pooled) unbiased
  estimates of s2X and s2Y, respectively.
•  

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Rejection Rule(s)

• The rejection rule for efficacy is to
• Reject H0X µX1 µX0 ? in favor of
• H1X µX1 lt µX0 ? if ZX -za
• and the rejection rule for safety is to
• Reject H0Y µY1 µY0 in favor of
• H1Y µY1 lt µY0 if ZY -za,
• where za is the 100 (1-a)-th percentile of the
  standard normal distribution.

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Notation

• Let ?X µX1 -µX0 and ?Y µY1 - µY0. Then the
  problem is to simultaneously test
• H0X ?X ? against H1X ?Xlt ?
• and
• H0Y ?Y 0 against H1Y ?Y lt 0.

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Notation

• (ZX,ZY)'
• BVN((.5n)1/2(?X-?)/ sX,(.5n)1/2?Y/sY,1,1,?).
• (approx.)
• Tests could be based on linear combinations of ZX
  and ZY.
• Such tests will be inappropriate for the RM
  formulation.

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Max Test (Bivariate Approach)

• The simultaneous comparison is performed using a
  test based on WmaxZX,ZY.

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Max Test

• The rejection rule is
•  
• Reject H0 in favor of H1 if W C,
• where C is chosen such that
• P(Reject H0 ?X ? and ?Y 0)a.

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Max Test

• Let G(.,. ?) is the joint cdf of a bivariate
  normal distribution with zero means, unit
  variances, and correlation ?.
• Then
• P(Reject H0 ?X ? and ?Y 0) G(C, C ?).

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Max Test

• Given ?, C can be chosen such that
• G(C,C ?) a.
• However, ? is unknown. The critical value can be
  estimated by satisfying
• where r is an estimate of ?
• (pooled or average).

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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.

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Stepwise Approach

• Test H0X.
• If H0X is not rejected in favor of H1X, stop.
• If H0X is rejected in favor of H1X,
• (II) Test H0Y.
• If H0Y is not rejected in favor of H1Y, stop.
• If H0Y is rejected in favor of H1Y,
• Reject H0 in favor of H1.

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Stepwise Approach

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

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Simulation Results

•  The following results are based on 10,000 for
  each set of parameters, unit variances and n50
  subjects per treatment. Each test is conducted at
  the a0.05 level. The simulations were conducted
  with the same seed for comparison.

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Simulation Results

• Let PX PY 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 H0Y only if H0X is rejected,
  where both tests are performed at the 0.05 level.
  
• Maximum is test using W, with pooled or
  average estimate of correlation.

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Power Estimates ()
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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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