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Binary outcome variables

Johannes Cepicka

Source: vignettes/Binary_outcomes.Rmd
Binary_outcomes.Rmd

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 binary distributed variables works.

The example setting

Suppose we are developing a new treatment to prevent strokes, exper. The patient variable that we want to investigate is if the patient experienced a stroke (unfavorable outcome with value 1) or not (favourable outcome with value 0) over a pre-defined observation period. This is a binary 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 \(\rho = −\log(RR)\), which is the negative logarithm of the risk ratio (relative risk) \(RR = \frac{p_1}{p_0}\). Here, \(p_1\) is the failure probability of the experimental treatment (i.e. \(\mathbb{P}(X_i = 1|\text{experimental})\), the probability that patient \(i\) has a stroke) and analogously \(p_0\) is the failure probability of the control treatment.

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)
#> Loading required package: doParallel
#> Loading required package: foreach
#> Loading required package: iterators
#> Loading required package: parallel

Defining all necessary parameters

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

  • p11 is the assumed true probability that the unfavorable outcome occurs for our experimental treatment exper. p0 is the assumed true probability that the unfavorable outcome occurs for our control treatment contro. Let’s assume that 30% of patients in the experimental group have a stroke and 50% in the control group. Therefore, we set p11 = 0.3 and p0 = 0.5. For now, we will assume that the probabilities 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.
  • rrgomin and rrgomax specify the minimal and maximal threshold value for the go/no-go decision rule in terms of the negative logarithm of the risk 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 steprrgo = 0.01. Note that the lower bound of the decision rule 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 minimal clinically relevant effect size. Moreover, note that the interval specified above corresponds to the set \(\{-\log(0.9), ..., -\log(0.7)\}\).
  • 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 per-patient costs in phase II and phase III. 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 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_binary(), which calculates the optimal sample size and the optimal threshold value for a binary distributed outcome variable.

res <- optimal_binary(p0 = 0.5, p11 =  0.3,               # probabilities of the unfavorable outcome
   n2min = 20, n2max = 400, stepn2 = 4,                   # define optimization set for n2
   rrgomin = 0.7, rrgomax = 0.9, steprrgo = 0.01,         # define optimization set for RRgo
   alpha = 0.025, beta = 0.1,                              # drug development planning parameters
   c2 = 0.75, c3 = 1, c02 = 100, c03 = 150,               # define fixed and variable costs for phase II and III,
   K = Inf, N = Inf, S = -Inf,                            # constraints
   steps1 = 1, stepm1 = 0.95,  stepl1 = 0.85,             # treatment effect size categories as proposed by IQWiG (2016)
   b1 = 1000, b2 = 2000, b3 = 3000,                       # expected benefit categories
   w = 0.3, p12 = 0.5, in1 = 30, in2 = 60,                # prior (https://web.imbi.uni-heidelberg.de/prior/)
   gamma = 0,                                             # population structures in phase II and III
   fixed = TRUE,                                          # choose if true treatment effects are fixed or random
   skipII = FALSE,                                        # choose if skipping phase II would be an option
   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: 1399.56
#>  Sample size:
#>    phase II: 260, phase III: 346, total: 606
#>  Probability to go to phase III: 0.99
#>  Total cost:
#>    phase II: 295, phase III: 494, 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: 0.83
#>  Success probability by effect size:
#>    small: 0.06, medium: 0.18, large: 0.59
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.86 [RRgo] 
#>  Assumed true effect:
#>   rate in the control: 0.5, rate in the treatment group: 0.3
#>  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 260 participants in phase II and 346 participants in phase III.
  • res$RRgo is the optimal threshold value for the go/no-go decision rule. We see that we need a risk ratio of less than 0.86 in phase II in order to proceed to phase III.
  • res$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to 1399.56, i.e. we have an expected utility of 139 956 000$.

Where to go from here

In this article we introduced the setting, when the outcome variable is binary distributed. For more information on how to use the package, see: