1. A new group-sequential phase II/III clinical trial design Nigel Stallard and Tim Friede Warwick Medical School, University of Warwick, UK n.stallard@warwick.ac.uk

2. 1 A new group-sequential phase II/III clinical trial design Outline 1.Seamless phase II/III design 2.Background 2.1 Notation 2.2 Standard group-sequential approach (k 1 = 1) 2.3 Selection of best treatment at first look (k 2 = … = k n = 1) 3. k 2, …, k n pre-specified case 3.1 Strong control of FWER 4. k 2, …, k n data dependent case 4.1 Error rate control 4.2 Simulation study 5. Conclusions

3. 2 Start T 0 T 1 T 2  T k 1 T 0 : Control Treatment T 1,…, T k 1 Experimental Treatments Superiority? Futility?  Interim 2 T 0 T (1)  T (k 2 ) Interim n etc. Aim: control FWER in strong sense Interim 1 T 0 T 1 T 2  T k 1 Superiority? Futility? Select treatments 1. Seamless phase II/III design

4. 3  i measures superiority of T i over T 0 Test H 0i :  i  0 vs. H Ai :  i > 0 Let Z ij be stagewise test statistic for H 0i at stage j S ij be cumulative test statistic for H 0i at stage j Number of treatments at each stage, k 1, …, k n Monitor S ij : reject H 0i at look j if S ij  u j Find boundaries with Pr(reject any true H 0i by look j )   *(j) for specified  *(1)  …   * (n) =  2. Background 2.1 Notation

5. 4 2.2 Standard group-sequential approach (k 1 = 1) (Jennison and Turnbull, 2000)  measures superiority of T 1 over T 0 Test H 0 :   0 vs. H A :  > 0 Obtain null distribution of S numerically using S 1 (first look) is normal S 1 ~ N (0, I 1 ) S j (subsequent looks) has S j – S j–1 = Z j ~ N (0, I j – I j–1 ) sum of truncated normals and normal increment density given by convolutions of normal densities Hence find boundaries to satisfy spending function

6. 5 Let Z 1 max = max{Z i1 } Under H 0 if I 11 = … = I k1 = : I 1 Obtain distribution of S 1 max= Z 1 max under global null hypothesis as in Dunnett test; density is given by 2.3 Selection of best treatment at first look (k 2 = … = k n = 1) (Stallard and Todd, 2003)

7. 6 Continue with best treatment, T (1), only Let S 1 max = Z 1 max Monitor S j max := S 1 max + …+ Z (1)j Increments in S 1 max are normal; under H 0(1) S j max – S j–1 max = Z  j ~ N (0, I j – I j–1 ) Density given via convolutions as in standard case Use distribution of S j max to give boundary to satisfy spending function for monitoring S j max Reject H 0(1) at look j if S j max  u j

8. 7 Let Z j max = max{Z ij } S j max be sum of Z j max Obtain distribution of S j max (given prespecified k 1, …, k n ) under global null hypothesis Find boundary to control type I error rate for monitoring S j max Use this boundary to monitor S ij i.e. reject H 0i at look j if S ij  u j 3. k 2, …, k n pre-specified case Test is conservative as S j max  st max{S ij } (sum of maxima is > maximum of sums)

9. 8 Consider test of H 0K :  i = 0  i  K  {1, …, k 1 }  control error rate for H 0K Hence control I error rate for H 0i in strong sense by CTP (Markus et al., 1976) Note: can select any treatments since S ij  st S j max (Jennison and Turnbull, 2006) 3.1 Strong control of FWER

10. 9 FWER is strongly controlled for pre-specified k 1, …, k n In practice may wish to have k 1, …, k n data-dependent Proposal: use u 1, …, u n as above 4. k 2, …, k n data dependent case Error rate can be inflated (neither weak nor strong control) Example: k 1 = 2, n = 2, I 1 = I 2 /2,  =  *(2) = 0.025,  *(1) = 0 Define conditional error functions probability of rejecting H 0 given stage 1 data (Z 1,1, Z 2,1 ) CE 1 (Z 1,1, Z 2,1 ) for k 2 = 1 (depends only on max{Z 1,1, Z 2,1 } ) CE 2 (Z 1,1, Z 2,1 ) for k 2 = 2 4.1 Error rate control

11. 10 k 2 = 1 CE 1 (Z 1,1, Z 2,1 )  CE 1 (Z 1,1, Z 2,1 )f(Z 1,1, Z 2,1 )dZ 1,1 dZ 2,1 =  Type I error rate = 0.025

12. 11 k 2 = 2 CE 2 (Z 1,1, Z 2,1 )  CE 2 (Z 1,1, Z 2,1 )f(Z 1,1, Z 2,1 )dZ 1,1 dZ 2,1   Type I error rate = 0.01654

13. 12 Data-dependent k 2 to maximise type I error rate k 2 = argmax{CE k (Z 1,1, Z 2,1 )} Error rate  max{CE k (Z 1,1, Z 2,1 )} f(Z 1,1, Z 2,1 )dZ 1,1 dZ 2,1 Exceeds  if CE 2 (Z 1,1, Z 2,1 )} > CE 1 (Z 1,1, Z 2,1 )} at any (Z 1,1, Z 2,1 ) Type I error rate = 0.02501

14. 13 Practical  treatment selection rule (Kelly et al., 2005) Drop treatment T i if S ij < max{S ij } –   I j  = 0.1

15. 14 Error rate does not exceed 0.025 for any  < 0.025

16. 15 k 1 = 2, n = 2, 32 patients per arm in each stage Drop treatment T i if S ij < max{S ij } –   I j  = 0  drop worst,  =   continue with both Estimate type I error rates pr(reject any H 0i ; H 0 ) Compare with other methods Estimate power pr(reject H 01 ; H 02 ) for range of  1 values pr(reject H 01 or H 02 ) for range of  1 values and  2 = 0.5 4.2 Simulation study

17. 16 Simulated type I error rates for range of  values Class. Dunnett (  ), Adap. Dunnett (  ), Comb. Test (  ), Gp-seq (  )

18. 17 Simulated power for range  1 values using both tmts (  =  )  2 = 0  2 = 0.5  1  1 Class. Dunnett (  ), Adap. Dunnett (  ), Comb. Test (  ), Gp-seq (  )

19. 18 Simulated power for range  1 values using best tmt (  = 0)  2 = 0  2 = 0.5  1  1 Class. Dunnett (  ), Adap. Dunnett (  ), Comb. Test (  ), Gp-seq (  )

20. 19 Simulated power for range  1 values using  = 1  2 = 0  2 = 0.5  1  1 Class. Dunnett (  ), Adap. Dunnett (  ), Comb. Test (  ), Gp-seq (  )

21. 20 5. Conclusions Group-sequential approach allows selection of >1 treatment extending Stallard and Todd (2003) method allows reduction of number of treatments over several stages does not allow further adaptations gives stopping boundaries in advance - can construct repeated c.i.’s (Jennison & Turnbull, 1989) strongly controls FWER for pre-specified k 1, …, k n appears to control FWER with selection rule simulated

22. 21 Choice of approach to maximise power depends on choice treatment selection rule true effectiveness of experimental treatments Single effective treatment, small  - group-sequential method can be (slightly) more powerful Several effective treatments, large  - adaptive Dunnett test can be more powerful

23. 22 References Stallard, N., Friede, T. Flexible group-sequential designs for clinical trials with treatment selection. Statistics in Medicine, 27, 6209-6227, 2008. Friede, T., Stallard, N. A comparison of methods for adaptive treatment selection. Biometrical Journal, 50, 767-781, 2008. Bauer P, Kieser M. Combining different phases in the development of medical treatments within a single trial. Stat. Med., 18, 1833-1848, 1999. Dunnett CW. A multiple comparison procedure for comparing several treatments with a control. JASA, 50, 1096-1121, 1955. Jennison C, Turnbull BW. Interim analyses: the repeated confidence interval approach. JRSS(B), 51, 305-361, 1989. Jennison C, Turnbull BW. Confirmatory seamless phase II/III clinical trials with hypothesis selection at interim: opportunities and limitations. Biom. J., 48, 650-655, 2006. Kelly PJ, Stallard N, Todd S. An adaptive group sequential design for phase II/III clinical trials that select a single treatments from several. J. Biopharm. Stat., 15, 641-658, 2005. Koenig F, Brannath W, Bretz F, Posch M. Adaptive Dunnett test for treatment selection. Stat. Med., 27, 1612-1625, 2008. Marcus R, Peritz E, Gabriel KR. On closed testing procedures with special reference to ordered analysis of variance. Biometrika, 63, 655-660, 1976. Stallard N, Todd S. Sequential designs for phase III clinical trials incorporating treatment selection. Stat. Med., 2003, 22, 689-703, 2003.