Defining all necessary parameters

In order to apply the package to the setting from our example, we need to specify the following parameters:

• Delta1 is the assumed true treatment effect, defined as the standardized difference in means between the treatment arm and the control arm. We will use the value \(\Delta_1 = 0.625\) as defined above. For now, we will assume that the treatment effects are fixed and independent of any prior distribution. Thus, we will set fixed = TRUE.
• n2min and n2max specify the minimal and maximal number of participants for the phase II trial. The package will search for the optimal sample size within this region. For now, we want the program to search for the optimal sample size in the interval between 20 and 400 participants. In addition, we will tell the program to search this region in steps of four participants at a time by setting stepn2 = 4.
• kappamin and kappamax specify the minimal and maximal threshold value for the go/no-go decision rule. The package will search for the optimal threshold value within this region. For now, we want the program to search in the interval between 0.02 and 0.2 while going in steps of stepkappa = 0.02. Note that the lower bound of the decision rule set for represents the smallest size of treatment effect observed in phase II allowing to go to phase III, so it can be used to model the minimally clinically relevant effect size.
• c02 and c03 are fixed costs for phase II and phase III respectively. We will set the phase II costs to 15 and the phase III costs to 20 (in \(10^5\)$), i.e. we have fixed costs of 1 500 000$ in phase II and 2 000 000$ in phase III. Note that the currency of the input values does not matter, so an input value for c02 of 15 could also be interpreted as fixed costs of 1 500 000€ if necessary.
• c2 and c3 are the costs in phase II and phase III per patient. We will set them to be 0.675 in phase II and 0.72 in phase III. Again, these values are given in \(10^5\)$, i.e. we have per patient costs of 67 500$ in phase II and 72 000$ in phase III.
• b1, b2 and b3 are the expected small, medium and large benefit categories for successfully launching the treatment on the market for each effect size category in \(10^5\)$. We will define a small benefit of 3000, a medium benefit of 8000, and a large benefit of 10000. The effect size categories directly correspond to the treatment effect, i.e. if the treatment effect is between 0 and 0.5 (in standardized differences of mean) we have a small treatment effect, hence yielding expected benefits of the drug development program of 300 000 000$.
• alpha is the specified one-sided significance level. We will set alpha = 0.025.
• 1 - beta is the minimal power that we require for our drug development program. We will set beta = 0.1, meaning that we require a power of 90%.
• Further parameters: In more advanced settings, we would need to pre-specify even more parameters. For now, we can set all remaining parameters to NULL. Their use is explained in the vignette on parameters and in the vigentte on prior distributions.

Now that we have defined all parameters needed for our example, we are ready to feed them to the package. We will use the function optimal_normal(), which calculates the optimal sample size and the optimal threshold value for a normally distributed outcome variable.

 res <- optimal_normal(Delta1 = 0.625, fixed = TRUE, # treatment effect
                       n2min = 20, n2max = 400, # sample size region
                       stepn2 = 4, # sample size step size
                       kappamin = 0.02, kappamax = 0.2, # threshold region
                       stepkappa = 0.02, # threshold step size
                       c2 = 0.675, c3 = 0.72, # maximal total trial costs
                       c02 = 15, c03 = 20, # maximal per-patient costs
                       b1 = 3000, b2 = 8000, b3 = 10000, # gains for patients
                       alpha = 0.025, # one-sided significance level
                       beta = 0.1, # 1 - power
                       Delta2 = NULL, w = NULL, in1 = NULL, in2 = NULL, 
                       a = NULL,b = NULL) # setting all unneeded parameters to NULL

Interpreting the output

After setting all these input parameters and running the function, let’s take a look at the output of the program.

res
#> Optimization result:
#>  Utility: 2946.07
#>  Sample size:
#>    phase II: 92, phase III: 192, total: 284
#>  Probability to go to phase III: 1
#>  Total cost:
#>    phase II: 77, phase III: 158, cost constraint: Inf
#>  Fixed cost:
#>    phase II: 15, phase III: 20
#>  Variable cost per patient:
#>    phase II: 0.675, phase III: 0.72
#>  Effect size categories (expected gains):
#>   small: 0 (3000), medium: 0.5 (8000), large: 0.8 (10000)
#>  Success probability: 0.85
#>  Success probability by effect size:
#>    small: 0.72, medium: 0.12, large: 0
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.06 [Kappa] 
#>  Assumed true effect: 0.625 [Delta] 
#>  Treatment effect offset between phase II and III: 0 [gamma]

The program returns a total of thirteen values and the input values. For now, we will only look at the most important ones:

• res$n2 is the optimal sample size for phase II and res$n3 the resulting sample size for phase III. We see that the optimal scenario requires 92 participants in phase II and 192 participants in phase III.
• res$Kappa is the optimal threshold value for the go/no-go decision rule. We see that we need a treatment effect of more than 0.06 in phase II in order to proceed to phase III.
• res$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to 2946.07, i.e. we have an expected utility of 294 607 000$.