Time-to-event outcome variables

Suppose we are planning a drug development program testing the superiority of an experimental treatment over a control treatment. Our drug development program consists of an exploratory phase II trial which is, in case of promising results, followed by a confirmatory phase III trial.

The drugdevelopR package enables us to optimally plan such programs using a utility-maximizing approach. To get a brief introduction, we presented a very basic example on how the package works in Introduction to planning phase II and phase III trials with drugdevelopR. In the introduction, the observed outcome variable “tumor growth” was normally distributed. However, the drugdevelopR package is not only restricted to normally distributed outcome variables but also binary distributed outcome variables and a time-to-event outcome variables. In this article we want explain how the setting with time-to-event variables works.

The example setting

Suppose we are developing a new tumor treatment, exper. The patient variable that we want to investigate is how long the patient survives without further progression of the tumor (progression-free survival). This is a time-to-event outcome variable.

Within our drug development program, we will compare our experimental treatment exper to the control treatment contro. The treatment effect measure is given by $$\theta = −\log(HR)$$, which is the negative logarithm of the hazard ratio $$HR$$, which in turn is the ratio of the hazard rates. The hazard ratio is a little difficult to understand. A hazard ratio of 0.75 would mean that the momentary rate of experiencing (the “hazard”) at any point in time is reduced by 75% in the experimental group.

Applying the package to the example

After installing the package according to the installation instructions, we can load it using the following code:

library(drugdevelopR)

Defining all necessary parameters

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

• hr1 is our hazard ratio. As already explained above, we assume that our experimental treatment exper leads to a hazard reduced by 75% compared to the control treatment contro. Therefore, we set hr1 = 0.75. For now, we will not model the hazard ratio on any prior distribution. Thus, we will set fixed = TRUE.
• d2min and d2max specify the minimal and maximal number of events 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 events. In addition, we will tell the program to search this region in steps of four participants at a time by setting stepd2 = 4.
• hrgomin and hrgomax specify the minimal and maximal threshold value for the go/no-go decision rule in terms of the negative logarithm of the hazard ratio. The package will search for the optimal threshold value within this region. For now, we want the program to search in the interval between 0.7 and 0.9 while going in steps of stephrgo = 0.01. Note that the lower bound of the decision rule set 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. Moreover, note that the interval specified above corresponds to the set $$\{-\log(0.9), ..., -\log(0.7)\}$$.
• xi2 and xi3 correspond to the event rates in phase II and phase III. After calculating the optimal number of events d2 and d3, the event rates are used to calculate the optimal sample sizes in phase II and III. We assume event rates of 0.7 in each phase, indicating that 70 events correspond to an optimal sample size of 100, respectively.
• c02 and c03 are fixed costs for phase II and phase III respectively. We will set the phase II costs to 100 and the phase III costs to 150 (in $$10^5$$$), i.e. we have fixed costs of 10 000 000$ in phase II and 15 000 000$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 0.75 in phase II and 0.1 in phase III. Again, these values are given in $$10^5$$$, i.e. we have per patient costs of 75 000$in phase II and 100 000$ 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 1000, a medium benefit of 2000, and a large benefit of 3000. The effect size categories directly correspond to the treatment effect: For example, if the treatment effect is between 1 and 0.95 (in terms of the risk ratio) we have a small treatment effect yielding an expected benefit of 100 000 000$.
• 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%.

As in the setting with normally distributed outcomes, the treatment effect may be fixed (as in this example) or may be distributed with respect to a prior distribution. Furthermore, all options to adapt the program to your specific needs are also available in this setting.

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_tte(), which calculates the optimal sample size and the optimal threshold value for a time-to-event outcome variable.

res <- optimal_tte(w = 0.3,                               # define parameters for prior
hr1 = 0.75, hr2 = 0.8, id1 = 210, id2 = 420,           # (https://web.imbi.uni-heidelberg.de/prior/)
d2min = 20, d2max = 400, stepd2 = 5,                   # define optimization set for d2
hrgomin = 0.7, hrgomax = 0.9, stephrgo = 0.01,         # define optimization set for HRgo
alpha = 0.025, beta = 0.1, xi2 = 0.7, xi3 = 0.7,        # drug development planning parameters
c2 = 0.75, c3 = 1, c02 = 100, c03 = 150,               # define fixed and variable costs
K = Inf, N = Inf, S = -Inf,                            # set constraints
steps1 = 1,  stepm1 = 0.95, stepl1 = 0.85,             # define boundary for  effect size categories
b1 = 1000, b2 = 2000, b3 = 3000,                       # define expected benefits
gamma = 0,                                             # assume different/same population structures
fixed = TRUE,                                          # choose if  effects are fixed or random
skipII = FALSE,                                        # more parameters
num_cl = 1)

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: 377.1
#>  Sample size:
#>    phase II: 236, phase III: 614, total: 850
#>  Expected number of events:
#>    phase II: 165, phase III: 430, total: 595
#>  Assumed event rate:
#>    phase II: 0.7, phase III: 0.7
#>  Probability to go to phase III: 0.81
#>  Total cost:
#>    phase II: 277, phase III: 736, cost constraint: Inf
#>  Fixed cost:
#>    phase II: 100, phase III: 150
#>  Variable cost per patient:
#>    phase II: 0.75, phase III: 1
#>  Effect size categories (expected gains):
#>   small: 1 (1000), medium: 0.95 (2000), large: 0.85 (3000)
#>  Success probability: 0.62
#>  Success probability by effect size:
#>    small: 0.09, medium: 0.29, large: 0.24
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.86 [HRgo]
#>  Assumed true effect: 0.75 [hr]
#>  Treatment effect offset between phase II and III: 0 [gamma]

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

• res$d2 is the optimal number of events for phase II and res$d3 the resulting number of events for phase III. We see that the optimal scenario requires 165 events in phase II and 430 events in phase III, which correspond to 236 participants in phase II and 614 in phase III.
• res$HRgo is the optimal threshold value for the go/no-go decision rule. We see that we need a hazard ratio of less than 0.86 in phase II in order to proceed to phase III. • res$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to 377.1, i.e. we have an expected utility of 37 710 000\$.

Where to go from here

In this article we introduced the setting, when the outcome is a time-to-event variable. For more information on how to use the package, see: