Bayesian Assurance of Consecutive Studies

Introduction

The intent of this example is to demonstrate the computation of Bayesian assurance, or probability of success, through the integration of R with Cytel products using a series of examples. These examples begin with a 2-arm, normal outcome, fixed sample trial assuming a non-standard prior for computation of assurance. The examples progress to a more complex setting of computing assurance for a sequence of a phase 2 trial with normal outcomes followed by a phase 3 trial where the outcome is time-to-event.

Once CyneRgy is installed, you can load this example in RStudio with the following commands:

CyneRgy::RunExample( "AssuranceConsecutiveStudies" )

Running the command above will load the RStudio project in RStudio.

East Workbook: AssuranceConsecutiveStudies.cywx

RStudio Project File: AssuranceConsecutiveStudies.Rproj

In the R directory of this example you will find the R files used in the examples.

The examples will all contain two treatments, Standard of Care (S) and Experimental (E) and are follows:

Fixed sample design using a mixture of normal distributions for computation of assurance
Expand example 1 to a group sequential design with an interim for futility based on a Bayesian predictive probability
Fixed design with a time-event-event outcome, this example provides the basis and a comparison for the phase 2 followed by a phase 3 example considered last
Two consecutive studies, phase 2 with a normal endpoint followed by a phase 3 with a normal endpoint
Two consecutive studies, phase 2 with a normal endpoint followed by a phase 3 with a time-to-event outcome

Example 1

Study Design

Fixed sample, with normally distributed outcomes Y.

Sample size: 80 patients per arm
Assume standard deviation is known: $\sigma = 1.9$

Denote the minimum acceptable value by MAV. For patients receiving treatment $j$ = S or E, we assume the outcomes Y $\sim N( \mu_j, \sigma^2)$ where, for simplicity, $\mu_j$ is unknown and $\sigma$ is the known, fixed quantity. We assume a priori $\mu_j \sim$ Normal( $\theta_j$ , $\tau_j^2$ = $1000^2$ ), for $j=$ S or E. At the end of the study the following is computed:

$\rho = Pr( \mu_E > \mu_S + MAV | data ) \text{ and}$

$\text{If } \rho > P_U \implies \text{ Go }$ $\text{ If } \rho \leq P_U \implies \text{ No-Go }$

For decision making we assume $P_U = 0.8$ . For assurance, a mixture of normal distributions is assumed. The assurance prior is specified in terms of the prior weight, mean, and variance for each component of the mixture. For simplicity, we assume mixture of two normal distributions as follows:

Weight: 25% on $N( 0, 0.05 )$
Weight: 75% on $N( 0.7, 0.3^2)$

R Integration

In order to evaluate the design above, one can develop an R function for analysis that can be called from East during simulation. By replacing only the analysis function with an R function, one can obtain the frequentist operating characteristics of the Bayesian design using East. In addition, by replacing how the patient data is simulated one can obtain the Bayesian assurance. Specifically, in the simulation when the patient data is simulated an R function will first sample the assurance prior then sample the patient data. The resulting power of this simulation will be the Bayesian assurance assuming the 2 component prior given above.

There is often a need for examination of posterior distribution of both observed and true treatment differences given a Go decision. These posterior distributions can be useful for planning the next phase of study and understanding potential risks. Obtaining the posterior distributions is described in the next section and they are applied to the phase 2 followed by phase 3 in later sections.

Required R Functions

The two functions that are needed to evaluate this design and obtain the Bayesian assurance are the analysis function, AnayzeUsingBayesianNormals, and patient simulation function, SimulatePatientOutcomeNormalAssurance.

To help understand the AnayzeUsingBayesianNormals one must first derive the posterior distributions.

After observing $n$ patients on treatment $j =$ S or E, the posterior distribution of $\mu_j$ is:

$\mu_j | \bar{y} \sim N( \theta_j^*, \tau_j^{2*} )$ where $\theta_j^* = \frac{\frac{\theta_j}{\tau_j^2} + \frac{n}{\sigma^2}\bar{y}}{\frac{1}{\tau_j^2}+ \frac{1}{\sigma^2}} \text{ and } \frac{1}{\tau_j^{2*}} = \frac{1}{\tau_j^2} +\frac{n}{\sigma^2}$

Integration with East

Using the East workbook named Assurance.cywx with East version $>=$ 6.5.4 the Example 1 simulation can be used to obtain the results found in the next section. After editing the simulation, on the User Define R Functions the Generate Response and Compute Test Statistic are both replaced with R code. The Generate Response utilizes the SimulatePatientOutcomeNormalAssurance found in the file named R/SimulatePatientOutcomeNormalAssurance.R and has variables as shown below. For the Compute Test Statistic the function name AnalyzeUsingBayesianNormals found in R/AnalyzeUsingBayesianNormals.R with input shown below.

Input for generate response in East

Input for test statistitic input in East

Integration with East Horizon Explore

The setup in East Horizon Explore is very similar to East. East Horizon Explore utilizes the same R functions. In the Responses tab, the Generate Response utilizes the SimulatePatientOutcomeNormalAssurance found in the file named R/SimulatePatientOutcomeNormalAssurance.R and has variables as shown below. For the Compute Test Statistic the function name AnalyzeUsingBayesianNormals found in R/AnalyzeUsingBayesianNormals.R with input shown below.

Input for generate response in East Horizon Explore

Input for test statistic in East Horizon Explore

Results

The probability of a Go is 15.3% and the probability of a No-Go is 84.7%.