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. 2 | Testing treatment combinations | Ekkehard Glimm | 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

3. 3 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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)

4. 4 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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 vs

5. 5 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Min Test (I) Laska and Meisner (1989): Reject H 0 if min(Z 1, Z 2 )>u 1-  (with u 1-  N(0,1)-quantile). Note: Assumption of known  is just for convenience, min-t-test is also possible. Same with equal n’s.

6. 6 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Min Test (II) Rejection probability of this test: where is the cdf of This test is uniformly most powerful in the class of monotone tests (= tests whose test statistic is a monotone function of Z 1 and Z 2 ).

7. 7 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Min Test (III) The Min test is „conservative“: Let  AB =  B >  A and Then the null rejection probability is The „least favorable constellation“ under H 0 is δ→ ∞ with But at nominal  =0.05!

8. 8 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Laska and Meisner Min Test (IV) Is there a way to alleviate this conservatism?

9. 9 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 „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  is attainable.

10. 10 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Sarkar et al. test Suggestion by Sarkar et al. (1995): Reject if k, d such that  -level is kept.  The null rejection prob. r 0 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).

11. 11 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 What can be inferred about the derivative . As δ  from 0,  for all δ < some δ +.  For δ→ ∞: → 0.  There is either no or one δ* where. If there is, for   *, so r 0 has a maximum in  *.

12. 12 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Some remarks on computer implementation  k is fixed.  For given d, calculate at  = 4.5 If this is <0, decrease d.  Stop if Idea: If the conditions hold,  0.  = 4.5 is „close enough“ to . This approach finds d within a few steps.

13. 13 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Modified Suggestion: linearized conditional test Reject if  With this, it is also possible to write down the rejection prob r 0 and its derivative  Need to find k,c and d. To limit options, k=0 and c= u 1  /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 r 0 (maybe possible for special functions).

14. 14 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Rejection probability of conditional tests  Rejection probability is highest at max d which has k =   Once k > , power relatively quickly  min test as d 

15. 15 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 More Modifications With both variants, we may want to allow  r 0 approaches a value <  as δ→ ∞.  Resulting test is no longer UMP than min test, but  we gain more power ( in the vicinity of H 0 ) for small δ. Here, k, d, c are even easier to find: The max r 0 is at δ where

16. 16 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 More Modifications: „Maximin test“ Idea: Find c(|V|) such that r 0 (0)=r 0 (∞). This test maximizes the minimum rejection probability among all conditional tests. Results: Sarkar test with k=0: d=0.08025, c  =1.767  r 0 (0)=r 0 (∞)=0.0386. Linearized test with k=0: d=0.2125, c=8.539, c  =1.81  r 0 (0)=r 0 (∞)=0.0348.

17. 17 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 Rejection probability of „maximin“ test (k=0)

18. 18 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 A remark on the power of conditional tests Suppose These tests do not dominate each other. 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%

19. 19 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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 V 1,..., V g not obvious.

20. 20 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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 rejects, but does not, although (However, we should keep in mind: The power never only depends on

21. 21 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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 test‘s r 0m <<  is very small. E.g. n=8, (  B   A )/  =1 has r 0m = 0.0471.  d is also very small. Only if the monotherapies are really similar, this makes a difference.

22. 22 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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%).

23. 23 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007 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.