Chapter 4 Phase II/III design

Adaptive Multiple Arm Multiple Stage (MAMS)

Adaptive MAMS are generalizations opf 2-arm group sequential designs:

  • Multiple treatment arms compared to a common control
  • Multiple looks at accumulating data
  • Early stopping for efficacy or futility
  • Treatments may be dropped at each interim look
  • Sample size re-estimation permitted at each interim look
  • Strong control of Family Wise Error Rate (FWER)$

Example: SOCRATES Reduced Trial (Gheorghiade et al. 2015)

FWER = probability of making one or more false claims

\[\begin{array}{||l|l||||} \hline \hline \text { Null Hypotheses } & \text { Type of Incorrect Conclusion } \\ \hline \hline H^{(1,2,3)}: \delta_1=\delta_2=\delta_3=0 & \begin{array}{l} \text { The selected treatment is } \\ \text { declared superior to placebo } \end{array} \\ \hline H^{(1,2)}: \delta_1=\delta_2=0, \delta_3>0 & \begin{array}{l} \text { Treatment } 1 \text { or } 2 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline H^{(1,3)}: \delta_1=\delta_3=0, \delta_2>0 & \begin{array}{l} \text { Treatment } 1 \text { or } 3 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline H^{(2,3)}: \delta_2=\delta_3=0, \delta_1>0 & \begin{array}{l} \text { Treatment } 2 \text { or } 3 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline H^{(1)}: \delta_1=0, \delta_2>0, \delta_3>0 & \begin{array}{l} \text { Treatment } 1 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline H^{(2)}: \delta_2=0, \delta_1>0, \delta_3>0 & \begin{array}{l} \text { Treatment } 2 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline H^{(3)}: \delta_3=0, \delta_1>0, \delta_2>0 & \begin{array}{l} \text { Treatment } 3 \text { is selected and } \\ \text { is declared superior to placebo } \end{array} \\ \hline \end{array}\]

Goal: test \(H^{(i)}: \delta_i=0, i=1,2,3\), with strong FWER control

  1. Form the Closed Set of all elementary and intersection hypotheses \[ \begin{gathered} H^{(1)}, H^{(2)}, H^{(3)} \\ H^{(1,2)}=H^{(1)} \cap H^{(2)}, H^{(1,3)}=H^{(1)} \cap H^{(3)}, H^{(2,3)}=H^{(2)} \cap H^{(3)} \\ H^{(1,2,3)}=H^{(1)} \cap H^{(2)} \cap H^{(3)} \end{gathered} \]
  2. To reject any elementary hypothesis at level \(\alpha\), must also reject every intersection hypothesis containing it at its local level- \(\alpha\)

Method 1: Stage-Wise MAMS

  • Let \(\left\{p_j^{(1)}, p_j^{(2)}, p_j^{(3)}\right\}\) be unadjusted \(p\)-values based only on the incremental data at stages (looks) \(j=1,2,3,4\)
  • Bonferroni adjusted p-value for \(H^{(123)}\) at stage \(j\) \[ p_j^{(123)}=3 \min \left\{p_j^{(1)}, p_j^{(2)}, p_j^{(2)}\right\} \]
  • Simes adjusted \(p\)-values for \(H^{(123)}\) at stage \(j\) \[ p_j^{(1,2,3)}=\min \left\{3 p_j^{(1)}, 1.5 p_j^{(2)}, p_j^{(3)}\right\} \]
  • Dunnett adjusted p-value at stage \(j\) \[ p_j^{(1,2,3)}=P\left(\cup_{i=1}^3 P_i \leq p_i\right) \]

Level-\(\alpha\) Test of \(H^{(1,2,3)}\): Look 3 Reject if \(\lambda_{13} \Phi^{-1}\left(1-p_1^{(1,2,3)}\right)+\lambda_{23} \Phi^{-1}\left(1-p_2^{(1,2,3)}\right)+\lambda_{33} \Phi^{-1}\left(1-p_3^{(1,2,3)}\right) \geq 2.4\)

Repeat this process to test \[ H^{(12)}, H^{(1,3)}, H^{(2,3)}, H^{(1)}, H^{(2)}, H^{(3)} \] - Reject \(H^{(1)}\) under closed testing if \(H^{(123)}, H^{(12)}, H^{(13)}\) and \(H^{(1)}\) are all rejected at level \(\alpha\) - Reject \(H^{(2)}\) under closed testing if \(H^{(123)}, H^{(12)}, H^{(23)}\) and \(H^{(2)}\) are all rejected at level \(\alpha\) - Reject \(H^{(3)}\) under closed testing if \(H^{(123)}, H^{(13)}, H^{(23)}\) and \(H^{(3)}\) are all rejected at level \(\alpha\)

Summary:

  • Can drop treatment arms at each stage
  • Can change the number and spacing of future stages
  • Can alter the sample size of future stages
  • Can change the \(\alpha\)-spending function for future stages

Stage Wise MAMS is flexible, easy to implement and applicable under a range of distributional assumptions

Method 2: Cumulative MAMS

Exploits asymptotic normality correlations structure

  • Cumulative Wald statistic for each dose \(i\) vs placebo, at look \(j\) \[ Z_{i j}=\frac{\hat{\delta}_{i j}}{\operatorname{se}\left(\hat{\delta}_{i j}\right)} \]

  • Construct multiplicity adjusted level- \(\alpha\) group sequential efficacy boundaries \(\left\{u_j, j=1, \ldots, 4\right\}\) \[ P_0\left(\bigcup_{j=1}^4 \max \left\{Z_{1 j}, Z_{2 j}, Z_{3 j}\right\} \geq u_j\right)=\alpha \]

  • FWER is automatically controlled if no adaptations

  • But what if we have the following adaptations?

    • Drop dose and re-allocate its remaining subjects
    • Re-estimate sample size of future stages
    • Change the number and spacing of future stages
    • Change error spending function for the future stages

  • Must recompute boundaries by conditional error rate (CER) method if adapt

Compute CER as if did not drop dose: \(CER = P_{\underline{0}}\left\{\max \left(Z_{13}, Z_{23}, Z_{33}\right) \geq 2.7 \text { or } \max \left(Z_{14}, Z_{24}, Z_{34}\right) \geq 2.4 \mid\left(z_{12}, z_{22}, z_{32}\right)\right\}\)

Recompute boundaries \(\left(b_3^*, b_4^*\right)\) so that \(P_0\left\{\max \left(Z_{23}^*, Z_{33}^*\right) \geq b_3^* \text { or } \max \left(Z_{24}^*, Z_{34}^*\right) \geq b_4^* \mid\left(z_{22}, z_{32}\right)\right\}= CER\)

Key differences between the two methods:

\[\begin{array}{|l|l|} \hline \hline \text { Stage Wise MAMS } & \text { Cumulative MAMS } \\ \hline \text { Inverse normal p-value Combination } & \text { Cumulative Wald statistic } \\ \text { Track a single statistic } & \text { Track one statistic for each dose } \\ \text { Compute two-arm boundaries } & \text { Compute multi-arm boundaries } \\ \text { Valid for any general setting } & \text { Valid for asymptotically normal setting } \\ \text { If adapt, use pre-specified weights } & \text { If adapt, use CER for FWER control } \\ \hline \end{array}\]

Final Comments:

  • Generally speaking Cumulative MAMS would be preferred to Stage Wise MAMS because of greater power
  • But Stage Wise MAMS controls FWER even for non-normal data and small sample sizes; hence might be preferable for rare disease trials
  • Decision rules for dropping arms play an important role in power comparisons and need further investigation
  • MAMS methods can be extended to investigating multiple populations and multiple endpoints with FWER
  • FWER control required on a case by case basis

Graphical testing for group sequential design

Basket trial

A basket trial evaluates a single therapy in multiple diseases or disease subtypes

Heng Zhou’s papers: (Zhou et al. 2019), (Wu et al. 2021), (Jing et al. 2022)

Umberlla trial

An umbrella trial evaluates multiple therapies simultaneously for a single disease

Platform trial

A platform trial evaluates multiple therapies for a single disease in a perpetual manner, with therapies allowed to enter or leave the platform over time

References

Gheorghiade, Mihai, Stephen J Greene, Javed Butler, Gerasimos Filippatos, Carolyn SP Lam, Aldo P Maggioni, Piotr Ponikowski, et al. 2015. “Effect of Vericiguat, a Soluble Guanylate Cyclase Stimulator, on Natriuretic Peptide Levels in Patients with Worsening Chronic Heart Failure and Reduced Ejection Fraction: The SOCRATES-REDUCED Randomized Trial.” Jama 314 (21): 2251–62.
Jing, Naimin, Fang Liu, Heng Zhou, Cong Chen, et al. 2022. “An Optimal Two-Stage Exploratory Basket Trial Design with Aggregated Futility Analysis.” Contemporary Clinical Trials 116: 106741.
Wu, Xiaoqiang, Cai Wu, Fang Liu, Heng Zhou, and Cong Chen. 2021. “A Generalized Framework of Optimal Two-Stage Designs for Exploratory Basket Trials.” Statistics in Biopharmaceutical Research 13 (3): 286–94.
Zhou, Heng, Fang Liu, Cai Wu, Eric H Rubin, Vincent L Giranda, and Cong Chen. 2019. “Optimal Two-Stage Designs for Exploratory Basket Trials.” Contemporary Clinical Trials 85: 105807.