Testing treatment combinations versus the corresponding monotherapies in clinical trials

1
Testing treatment combinations versus the
corresponding monotherapies in clinical trials

• Ekkekhard Glimm, Novartis Pharma AG
• 8th Tartu Conference on Multivariate Statistics
• Tartu, Estonia, 29 June 2007

2
Setting the scene (I)

• The problem
• Two monotherapies available for the
  treatment of a disease
• Question Does a combination /
  simultaneous administration of the treatments
  (combination) have a benefit?
• Might be
• synergism (? positive interaction between monos)
• a way to overcome dose limitations of monos

3
Setting the scene (II)

• Ultimate task is to find if the best combination
  therapydose is better than the best dose of any
  of the monos.
• Frequent problem in clinical trials (e.g.
  hypertension treatment)
• A lot of literature on the topic
• Laska and Meisner (1989)
• Sarkar, Snapinn and Wang (1995)
• Hung (2000)
• Chuang-Stein, Stryszak, Dmitrienko and Offen
  (2007)

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Setting the scene (III)

• Limited goal in this talk
• Only two monotherapies
• Optimal doses are known
• Let A, B be the monotherapies, AB their
  combination.
• Assume n individuals per treatment group with
  response

5
Min Test (I)

• Laska and Meisner (1989)

• Reject H0 if min(Z1, Z2)gtu1-? (with u1-?
  N(0,1)-quantile).
• Note Assumption of known?? is just for
  convenience, min-t-test is also
  possible. Same with equal ns.

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Min Test (II)

• Rejection probability of this test

This test is uniformly most powerful in the class
of monotone tests ( tests whose teststatistic
is a monotone function of Z1 and Z2).
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Min Test (III)

• The Min test is conservative
• Let ?AB ?B gt ?A and
• Then the null rejection probability is
• The least favorable constellation under H0 is
  d?8
• with
• But at nominal ?0.05!

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Laska and Meisner Min Test (IV)
Is there a way to alleviate this conservatism?
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Conditional tests

• Tests uniformly more powerful (UMP) than the Min
  test can be derived, if we adjust the critical
  value based on the observed difference

• In general such tests are of the form
• Reject if
• To be UMP than the Min test, a sufficient
  condition is

and keeping a is attainable.
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Sarkar et al. test

• Suggestion by Sarkar et al. (1995)
• Reject if
• k, d such that ?-level is kept.
• The null rejection prob. r0 can be written as a
  function of bivariate normal cdfs.
• The derivative can be written as a function
  of bivariate normal cdfs and pdfs.
• Using these two components, we can let the
  computer search for d corresponding to given k
  (or vice versa).

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What can be inferred about the derivative

• . As d ? from 0, ? for all d lt some d.
• For d?8 ? 0.
• There is either no or one d where
  . If there is, for ? lt ? and
  for ? gt ?, so r0 has a maximum in
  ?.

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Some remarks on computer implementation

• k is fixed.
• For given d, calculate at d 4.5 If
  this is lt0, decrease d.
• Stop if
• Idea If the conditionshold, ? 0. d 4.5
  is close enough to ?.
• This approach finds d within a few steps.

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Modified Suggestion linearized conditional test

• Reject if
• With this, it is also possible to write down the
  rejection prob r0 and its derivative
• Need to find k,c and d. To limit options, k0
  and c u1??/d were assumed, so just search for
  d.
• Same search algorithm as before.
• For non-linear c(V), I did not try to work out
  r0 (maybe possible for special functions).

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Rejection probability of conditional tests

• Rejection probability is highest at max d which
  has k ??
• Once k gt ??, power relatively quickly ? min test
  as d ?

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More Modifications

• With both variants, we may want to allow
• r0 approaches a value lt? as d?8.
• Resulting test is no longer UMP than min test,
  but
• we gain more power (in the vicinity of H0) for
  small d.
• Here, k, d, c are even easier to find
• The max r0 is at d where

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More Modifications Maximin test

• Idea Find c(V) such that r0(0)r0(8).
• This test maximizes the minimum rejection
  probability among all conditional tests.
• Results
• Sarkar test with k0 d0.08025, c?1.767 ?
  r0(0)r0(8)0.0386.
• Linearized test with k0 d0.2125, c8.539,
  c?1.81 ? r0(0)r0(8)0.0348.

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Rejection probability of maximin test (k0)
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A remark on the power of conditional tests

• Suppose
• These tests do not dominate each other.
• As ? and d increase, the tests with large k, d
  overtake tests with small k, d.
• Unfortunately, real gains coincide with low
  power Power gain over Min test (all at d
  0)
• ? 0.8 k0 10.4 Min test 8.7 (max
  absolute gain, 1.73)
• ? 2 k0 49.2 Min test 48.3
• ? 2.8 k1.3 79.9 Min test 79.6
• ? 3 k1.5 85.4 Min test 85.2

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A few remarks on generalizations

• Unequal ns No problem.
• The ? in the bivariate Normal distribution
  changes, so k, c and d change, but approach
  remains the same.
• Estimated s instead of known ? In principle,
  same approach.
• Rejection prob a sum of bivariate t- rather than
  normal cdfs.
• Basic idea for constructing a UMP conditional
  test works the same.
• k, c and d can be found by a grid search.
• gt 2 monos Again, in principle same approach,
  but gets messy
• more than one d to be considered.
• Generalization of rule if Vlt d to V1, ..., Vg
  not obvious.

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Contentious issues about conditional tests

• If we allow klt0, it is possible that we identify
  the combi as superior, although its observed
  average is lower than the better of the monos.
• ? This can be avoided by requiring k ? 0.
• Non-monotonicity It can happen thatrejects, but
  does not, although

(However, we should keep in mind The power
never only depends on
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Conclusions

• Conditional non-monotone tests are UMP than the
  Laska-Meisner min test.
• There is not that much to be gained
• The power depends on .
• ? Even for modest n, the region where the min
  tests r0mltlt ? is very small. E.g. n8, (?B ?
  ?A)/? 1 has r0m 0.0471.
• d is also very small. Only if the monotherapies
  arereally similar, this makes a difference.

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Conclusions

• Power profile is primarily driven by choice of k,
  irrespective of the variant of the conditional
  test.
• Gains over the Min test are in the wrong
  places
• They are where power is low (?10). Here, small
  values of k are best.
• At powers that matter to the pharma industry,
  biggest gains are achieved for large k, but are
  generally very small (ltlt1).
• k and d are easy to obtain with a relatively
  simple search algorithm on a computer.
• In practice, well rarely experience a difference
  from the Min test (with k0, P( Vltd
  ?0)2.6).

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Literature

• Laska, E.M. and Meisner, M. (1989) Testing
  whether an identified treatment is best.
  Biometrics 45, 1139-1151.
• Hung, H.M.J. (2000) Evaluation of a combination
  drug with multiple doses in unbalanced factorial
  design clinical trials. Statistics in Medicine
  19, 2079-2087.
• Sarkar, S.K., Snapinn, S., and Wang, W. (1995)
  On improving the min test for the analysis of
  combination drug trials. Journal of Statistical
  Computation and Simulation 51, 197-213.
• Chuang-Stein, C., Stryszak, P., Dmitrienko, A.,
  Offen, W. (2007) Challenge of multiple
  co-primary endpoints a new approach. Statistics
  in Medicine 26, 1181-1192.