Transcript Testing Treatment Combinations versus the Corresponding

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
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
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Setting the scene (II)
Ultimate task is to find if the best combination therapy
dose 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
xij ~ N (i ,  2 ), i  A, B, AB, j  1,, n
H 0 :  AB  max(  A ,  B )
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vs
A :  AB  max(  A , B )
Min Test (I)
Laska and Meisner (1989):
Z1 
n
2
 x AB  x A  / 
Z2 
n
2
 x AB  x B  / 
Reject H0 if min(Z1, Z2)>u1- (with u1- N(0,1)-quantile).
Note: Assumption of known  is just for convenience,
min-t-test is also possible. Same with equal n’s.
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Min Test (II)
Rejection probability of this test:
rm   0.5

n
2
2
 AB   A   u1 ,
where   .,. is the cdf of
n
2 2
 AB   B   u1 
 0  1
N 2   , 
 0  
 
 
1 
This test is uniformly most powerful
in the class of monotone tests (= tests whose test
statistic is a monotone function of Z1 and Z2).
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Min Test (III)
The Min test is „conservative“:
n
Let AB = B > A and  
  B   A  / 
2
Then the null rejection probability is
r0m     0.5   u1 ,u1 
The „least favorable constellation“ under H0 is δ→∞
with lim r0m     .
 
But r0m 0  0.0122 at nominal =0.05!
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Laska and Meisner Min Test (IV)
Min test rejection probability if  AB = B > A
0.05
0.045
rejection prob
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

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
V 
n
2
 x B  x A  / 
In general such tests are of the form:
Reject if min( Z1 , Z 2 )  cV 
To be UMP than the Min test, a sufficient condition is:
cV   u1 , cV  
 u1
V 
cV   u1 for some | V | and keeping  is attainable.
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Sarkar et al. test
Suggestion by Sarkar et al. (1995):
Reject if min( Z1 , Z 2 ) 
k , if V  d
u1 , if V  d
k, d such that -level is kept.
 The null rejection prob. r0 can be written as a function
of bivariate normal cdfs.
r0
 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
r0
r
(0)  0 . As δ  from 0, 0  for all δ < some δ+.


r0
 For δ→∞:
→ 0.

r
 There is either no or one δ* where 0  *  0 .


If there is, r0 ( )  0 for  < * and r0 ( )  0 for  > *,


so r0 has a maximum in *.
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Some remarks on computer implementation
 k is fixed.
r0
 For given d, calculate
at  = 4.5

If this is <0, decrease d.
r0
 10 6

 u1
Idea: If the conditions cV   u1 , cV  
V 
 Stop if 0 
hold, 0 
r0
 0.  = 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 min( Z1 , Z 2 ) 
k  c  V , if V  d
u1 , if V  d
 With this, it is also possible to write down the rejection
prob r0 and its derivative r0 .

 Need to find k,c and d. To limit options, k=0 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
0.05
d=0.02
d=0.03
d=0.033, k=0
d=0.04
d=0.1
linear, d=0.047, k=0
min test
rejection prob
0.04
0.03
0.02
0.01
0
0.5
1
1.5
2
2.5
3

 Rejection probability is highest at max d which has k = 
 Once k > , power relatively quickly  min test as d 
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More Modifications
With both variants, we may want to allow
cV  
 c  u1
V 
 r0 approaches a value < as δ→∞.
 Resulting test is no longer UMP than min test, but
 we gain more power (in the vicinity of H0) for small δ.
Here, k, d, c are even easier to find:
r0
 0.
The max r0 is at δ where

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More Modifications: „Maximin test“
Idea: Find c(|V|) such that r0(0)=r0(∞).
This test maximizes the minimum rejection probability
among all conditional tests.
Results:
Sarkar test with k=0: d=0.08025, c=1.767 
r0(0)=r0(∞)=0.0386.
Linearized test with k=0: d=0.2125, c=8.539, c=1.81 
r0(0)=r0(∞)=0.0348.
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Rejection probability of „maximin“ test (k=0)
0.05
rejection
rejection prob
prob
0.045
0.04
0.035
0.03
min test
d=0.033
d=0.033
minimax
maximin
linear
linear minimax
maximin
0.025
0.02
0.015
0.015
0.01
0.01
0
0
0.2
0.2 0.4
0.4 0.6
0.6 0.8
0.8
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1
1 1.2
1.2 1.4
1.4 1.6
1.6 1.8
1.8
delta
delta
2
2
A remark on the power of conditional tests
n
  AB   B  /  .
2
These tests do not dominate each other.
Suppose  
As  and δ 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 δ = 0):
-
 =0.8: k=0: 10.4%; Min test: 8.7% (max absolute gain, 1.73%)
 =2: k=0: 49.2%; Min test: 48.3%
 =2.8: k=1.3: 79.9%; Min test: 79.6%
 =3: k=1.5: 85.4%; Min test: 85.2%
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A few remarks on generalizations
 Unequal n‘s: 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.
 > 2 monos: Again, in principle same approach, but gets messy:
• more than one δ to be considered.
• Generalization of rule „if |V|< d“ to V1, ..., Vg not obvious.
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Contentious issues about conditional tests
 If we allow k<0, 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 that x AB , xB , x A 
rejects, but x AB , xB *, x A * does not, although
xB  xB *,
xA  xA *
(However, we should keep in mind:
The power never only depends on  AB  max(  A , B ).)
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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:
n
 The power depends on  
  B   A  /  .
2
 Even for modest n, the region where the min
test‘s r0m<<  is very small.
E.g. n=8, (B  A)/ =1 has r0m = 0.0471.
 d is also very small. Only if the monotherapies are
really 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 (<<1%).
 k and d are easy to obtain with a relatively simple search
algorithm on a computer.
 In practice, we‘ll rarely experience a difference from the
Min test (with k=0, P( |V|<d |=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.
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