# A new group-sequential phase II/III clinical trial design Nigel Stallard and Tim Friede Warwick Medical School, University of Warwick, UK

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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

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

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

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

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

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)

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

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)

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

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

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

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

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

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

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

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

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

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 (  )

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 (  )

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

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

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

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.

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