PoS Bayesian Framework for Pivotal Oncology Trials

Introduction

Here we describe the Bayesian framework that is used to estimate a phase 3 efficacy probability of success (PoS) based on work by Hampson et al. (2022). The framework consists of study and population level models that are described in the following sections.

Let $P$ be a progression-free survival (PFS), and we assume that the log hazard ratio (HR) of PFS for a phase III study is $\theta_{P3}$. The subscript 3 denotes the phase of the study (i.e., phase III). In addition, we assume that data on either the same endpoint $P$ or a different endpoint $D$ is observed in an earlier phase I/II or phase II study which preceded a pivotal trial ($\theta_{P2}$ or $\theta_{D2}$ respectively).

General study level model

The study level model for an observed logHR of PFS in the earlier study (phase I/II, phase II), $\hat{\theta}_{P2}$, is assumed to have the following form:

$$\begin{align} \hat{\theta}_{P2} \sim Normal(\theta_{P2}, \mathcal{I}^{-1}_{P2}) \label{eq:ph2_est}, \end{align}$$

where $\theta_{P2}$ represents a mean treatment effect on $P$ in the earlier study and $\mathcal{I}_{P2}$ is the Fisher information for $\theta_{P2}$. Using Bayesian framework, the prior for $\theta_{P2}$ is: $$\theta_{P2} \sim Normal (\mu_P, \tau_{P2}^2),$$ where $\mu_P$ is a population level parameter for the treatment effect on $P$. $\tau_{P2}$ characterizes the degree of heterogeneity in the treatment effect on $P$ across different earlier studies, which is assumed to follow a half-Normal distribution: $\tau_{P2} \sim HN(z^2_2)$.

Similar to the above, a phase III study level model for treatment effect on endpoint $P$, $\theta_{P3}$ has the following distribution:

$$\begin{align} \theta_{P3} \sim Normal (\mu_P, \tau_{P3}^2)\label{eq:p3} \end{align}$$

The population level parameter $\mu_P$ is shared across the phases of the clinical development. $\tau_{P3}$ is assumed to follow a half-Normal distribution: $\tau_{P3} \sim HN(z^2_3)$. The choice of $\tau_{P2},\ \tau_{P3}$ will follow Supplementary Materials E in Hampson et al. (2022).

Population level model

The population level treatment effect, $\mu_P$, is assumed to come from a mixture prior:

$$\begin{align} \mu_{P} \sim \omega Normal (\delta_{P}, \sigma_{P1}^2) + (1-\omega) Normal(0, \sigma_{P2}^2) , \end{align}$$

with the following components:

• $w$ is a probability that $\mu_P$ comes from the enthusiastic prior component. The value of $w$ is determined by the industry benchmark.

• $Normal(\delta_P, \sigma^2_{P1})$ is the enthusiastic component, i.e., a distribution which is centered at the target treatment effect $\delta_P$ (i.e., alternative hypothesis). $\sigma^2_{P1}$ is set as a solution to: $P(\mu_P \ge 0 | \omega = 1)=\gamma$, which is consistent with the interpretation of the enthusiastic (“alternative”) component, $\Leftrightarrow \sigma_{P1} = \frac{\delta_P}{\Phi^{-1}(\gamma)}$.

• $Normal(0, \sigma^2_{P2})$ is the skeptical component, i.e., a distribution which is centered at the null hypothesis. $\sigma^2_{P2}$ is set as a solution to: $P(\mu_P \le \delta_P | \omega = 0)=\gamma$, which is consistent with the interpretation of the skeptical (“null”) component, $\Leftrightarrow \sigma_{P2} = \frac{\delta_P}{\Phi^{-1}(\gamma)}$.

When $P$ denotes the logHR of PFS, $\gamma$ is the probability that the population level treatment effect is equal to or worse than the null (i.e., logHR is $\ge 0$) when the benchmarking data indicate an optimistic expectation of the treatment effect; or the probability that the population level treatment effect is equal to or better than the target effect in phase III (i.e., logHR is $\le \delta_P$) when the benchmarking data indicate we should have pessimistic expectation of the treatment effect. $\gamma$ should be set to a small number so that the probability of either lack treatment effect under the enthusiastic prior or substaintial treatment effect under the pessimistic prior is small.

Phase III efficacy PoS prediction

After fitting the models that are outlined above, we can generate a phase III efficacy PoS prediction based on the distribution of $\theta_{P3}$.

Let $J$ denotes the number of analyses considered in a group sequential design for a future phase III study. For instance, if $J = 2$, this means that a study has one interim analysis (IA) and one final analysis (FA). The distribution of the observed log HR for endpoint $P$ at the $j$-th analysis, $\hat{\theta}_{P3j}, j = 1, ..., J$ is as follows:

$$\begin{align} \hat{\boldsymbol{\theta}}_{P3} \sim Normal ({\theta}_{P3}\mathbf{1}_{J}, \mathbf{\Sigma}_{J \times J})\label{eq:thetahat_samp}, \end{align}$$

where $\theta_{P3}$ is the underlying true log hazard ratio for all $J$ analyses. $\mathbf{\Sigma}$ is the covariance matrix that encodes the Fisher’s information for $\hat{\theta}_{P3j}$:

$$\begin{align} \mathbf{\Sigma}_{ij} = \frac{\sigma_{unit}^2}{n_j}, ~~ \text{for all } i \leq j\label{eq:sigma}, \end{align}$$

with $n_j$ being the target number of events at the $j$-th analysis and $\sigma_{unit}^2 = \frac{1}{p_0(1-p_0)}$ where $p_0$ is the planned proportion of patients in the control group.

The predicted treatment effect $\hat{\boldsymbol{\theta}}_{P3}^{(l)}$ is generated $L$ times ($l = 1, \dots, L$) based on the Bayesian hierarchical model and the success at the $j$-th analysis is determined by a Frequentest efficacy boundary, $z_{P3j}$. Thus, the probability of stopping a phase III trial for efficacy at the first IA is estimated as: $$\hat{PoS}_{31} = \frac{1}{L} \sum_{l=1}^L I(\hat{\theta}_{P31}^{(l)} < z_{P31}),$$ and the probability of stopping a phase III trial for efficacy at the $j^{th}$ analysis is estimated as:

$$\hat{PoS}_{3j} = \frac{1}{L} \sum_{l=1}^L I(\hat{\theta}_{P3j}^{(l)} < z_{P3j}, \hat{\theta}_{P3i}^{(l)} \geq z_{P3i}, i = 1, \dots, j-1).$$ Finally, the overall PoS is: $\hat{PoS}_{3} = \sum_{j=1}^J \hat{PoS}_{3j}$.

Study level model when phase III primary endpoint is not available from earlier study(ies)

When an early study didn’t have a reliable PFS estimate, and only ORR is available from a randomized controlled phase II study, it can be used for PoS estimation instead. Let $\theta_{ORR,2}$ represents a log odds ratio (OR) of the treatment effect on ORR, the observed treatment effect, $\hat{\theta}_{ORR, 2}$, has the following distribution: $$\begin{align} \hat{\theta}_{ORR, 2} \sim Normal(\theta_{ORR, 2}, \mathcal{I}_{ORR, 2}^{-1}), \end{align}$$ where $\mathcal{I}_{ORR, 2}$ is the Fisher information associated with $\hat{\theta}_{ORR, 2}$. Further, let $\theta_{PFS, 2}$ be the treatment effect for PFS in Phase II. Motivated by the results in Blumenthal et al. (2015), we assume the following linear model between the PFS and ORR treatment effects: $$\begin{align} \theta_{ORR,2} \sim N(\beta_0 + \beta_1 \theta_{PFS,2}, \frac{\sigma_{WLS}^2}{N_{patients}}), \label{eq:ph23_pfs_orr_rel} \end{align}$$ where $N_{patients}$ is the number of patients in a given trial and the regression parameters are assigned the following priors: $$\begin{align*} \beta_0 \sim {Normal}(m_0, \nu_0) \\ \beta_1 \sim {Normal}(m_1, \nu_1). \end{align*}$$ The values of $m_0, m_1$, $\nu_0, \nu_1$ and $\sigma_{WLS}^2$ are determined from historical data, which is provided in the meta-analysis in Blumenthal et al. (2015). Specifically, $(m_0, m_1)$ are point estimates for the intercept and slope from a weighted liner simple (WLS) linear regression model of log(HR PFS) on log(OR ORR), while $(\nu_0, \nu_1)$ are their respective SEs, $\sigma_{WLS}^2$ is estimated based on WLS regression residual variance.

Based on the approximated correlation between $\theta_{ORR, 2}$ and $\theta_{PFS, 2}$, a distribution for $\theta_{PFS, 2}$ can be obtained and, therefore, a predicated efficacy PoS can be estimated as using models that are outlined above.

Blumenthal, Gideon M, Stella W Karuri, Hui Zhang, Lijun Zhang, Sean Khozin, Dickran Kazandjian, Shenghui Tang, Rajeshwari Sridhara, Patricia Keegan, and Richard Pazdur. 2015. “Overall Response Rate, Progression-Free Survival, and Overall Survival with Targeted and Standard Therapies in Advanced Non–Small-Cell Lung Cancer: US Food and Drug Administration Trial-Level and Patient-Level Analyses.” Journal of Clinical Oncology 33 (9): 1008.

Hampson, Lisa V, Björn Bornkamp, Björn Holzhauer, Joseph Kahn, Markus R Lange, Wen-Lin Luo, Giovanni Della Cioppa, Kelvin Stott, and Steffen Ballerstedt. 2022. “Improving the Assessment of the Probability of Success in Late Stage Drug Development.” Pharmaceutical Statistics 21 (2): 439–59.