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Multitrial - methods for programs with several phase III trials

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. Contrary to the basic setting, where only one phase III trial was planned, we now want to show how to plan several phase III trials using the package.

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 disease (progression-free survival). This is a time-to-event outcome variable. Therefore, we will use the function optimal_multitrial, which calculates optimal sample sizes and threshold decisions values for time-to-event outcomes when several phase III trials are performed.

Within our drug development program, we will compare our experimental treatment exper to the control treatment contro. The treatment effect measure is given by $\theta = −\log(HR)$, where the hazard ratio $HR$ is the ratio of the hazard rates between the two groups. If we assume that the hazard of experiencing a progression is reduced by 75% in the treatment group compared to the control group, we have a hazard ratio of 0.75.

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)

We insert the same input values as in the example for time-to-event endpoints. As in the basic setting, the treatment effect may be fixed or follow a prior distribution. Furthermore, some options to adapt the program to your specific needs are also available in this setting, however skipping phase II, setting predefined benefit categories and choosing different treatment effects in phase II and III are not possible.

The package implements a framework developed for phase II/III drug development programs where several phase III trials are performed. This is of particular relevance as regulatory agencies often require statistical significance in two or more phase III trials. Different cases, defined by the number of significant trials needed for approval, are possible. For each case, different strategies are possible. They are defined by the number of phase III trials to be conducted in order to reach the goal of the case. For the success of the drug development program, it is necessary that the treatment effects of all phase III trials point in the same direction. For example, if we select case 3 and strategy 4, we require four phase III trials, where three need to be significant at level $\alpha$ and the treatment effect of the fourth must point in the same direction.

The following cases and possible strategies are implemented in the package.

Case	Possible strategies for this case
1	1, 2
2	1 (with significance level of $\alpha^2$)* , 2, 3, 23*
3	1 (with significance level of $\alpha^3$)*, 3, 4

Note that setting case = 1 and strategy = 1 would lead to the same results as in the basic setting. There are two special strategies which we marked with a * in the table:

When using strategy 23 (the “two+one” strategy) in case 2, two phase III trials are conducted. If only one shows a significant positive treatment effect (and the other trial’s treatment effect points at least in the same direction), a third trial is conducted.
Setting strategy = 1 in cases 2 and 3 leads to a situation where one large phase III trial is conducted with adjusted significance level, i.e. $\alpha_{new} = \alpha^2$ in case 2 and $\alpha_{new} = \alpha^3$ in case 3. This feature is implemented as there is discussion that a larger trial with adjusted significance level may also may serve evidence on efficacy (see Koch (2005)).

So, in addition to the parameters from the basic setting, we provide the following parameters:

The parameter case allows us to chose the number of trials in phase III that have to show a significant positive treatment effect for the whole program to be deemed successful. In the following example we set the parameter case = 3, indicating that at least two trials in phase III have to show a significant treatment effect in phase III.
The parameter strategy defines the number of trials that are conducted in phase III. The strategies 1, 2, 3, and 4 (and 23) are implemented. Setting strategy = TRUE returns the results of the optimization for all implemented strategies for a specific case. Note that not all strategies are implemented for each case (see table above). For the following example we set strategy = TRUE.

Note that for the following example, some input parameters were changed compared to the basic setting in order to reduce computation time.

res <- optimal_multitrial(w = 0.3,                        # define parameters for prior
   hr1 = 0.7, hr2 = 0.8, id1 = 210, id2 = 420,            # (https://web.imbi.uni-heidelberg.de/prior/)
   d2min = 20, d2max = 200, stepd2 = 5,                   # define optimization set for d2
   hrgomin = 0.7, hrgomax = 0.9, stephrgo = 0.02,         # 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
   b1 = 1000, b2 = 2000, b3 = 3000,                       # define expected benefits
   fixed = TRUE,                                          # choose if  effects are fixed or random
   case = 3, strategy = TRUE                              # choose case and strategy
   num_cl = 3)

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 with 3 significant trial(s) needed, strategy 1:
#>  Utility: -231.07
#>  Sample size:
#>    phase II: 30, phase III: 368, total: 398
#>  Expected number of events:
#>    phase II: 20, phase III: 258, total: 278
#>  Assumed event rate:
#>    phase II: 0.7, phase III: 0.7
#>  Probability to go to phase III: 0.57
#>  Total cost:
#>    phase II: 122, phase III: 454, 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: 0.19
#>  Success probability by effect size:
#>    small: 0.06, medium: 0.11, large: 0.02
#>  Significance level: 1.5625e-05
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.76 [HRgo] 
#>  Assumed true effect: 0.7 [hr] 
#>  Treatment effect offset between phase II and III: 0 [gamma] 
#> 
#> Optimization result with 3 significant trial(s) needed, strategy 3:
#>  Utility: -411.1
#>  Sample size:
#>    phase II: 30, phase III: 258, total: 288
#>  Expected number of events:
#>    phase II: 20, phase III: 180, total: 200
#>  Assumed event rate:
#>    phase II: 0.7, phase III: 0.7
#>  Probability to go to phase III: 0.5
#>  Total cost:
#>    phase II: 122, phase III: 483, 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: 0.08
#>  Success probability by effect size:
#>    small: 0, medium: 0.05, large: 0.03
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.7 [HRgo] 
#>  Assumed true effect: 0.7 [hr] 
#>  Treatment effect offset between phase II and III: 0 [gamma] 
#> 
#> Optimization result with 3 significant trial(s) needed, strategy 4:
#>  Utility: -436.79
#>  Sample size:
#>    phase II: 58, phase III: 416, total: 474
#>  Expected number of events:
#>    phase II: 40, phase III: 292, total: 332
#>  Assumed event rate:
#>    phase II: 0.7, phase III: 0.7
#>  Probability to go to phase III: 0.5
#>  Total cost:
#>    phase II: 144, phase III: 716, 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: 0.2
#>  Success probability by effect size:
#>    small: 0, medium: 0.17, large: 0.02
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.7 [HRgo] 
#>  Assumed true effect: 0.7 [hr] 
#>  Treatment effect offset between phase II and III: 0 [gamma]

The program returns a data frame where the output for all implemented strategies is listed.

For strategy 1 we get:

res[1,]$d2 is the optimal number of events for phase II and res[1,]$d3 the resulting number of events for phase III. We see that the optimal scenario requires 20 events in phase II and 258 events in phase III, which correspond to 30 participants in phase II and 368 in phase III.
res[1,]$HRgo is the optimal threshold value for the go/no-go decision rule. We see that we need a hazard ratio of less than 0.76 in phase II in order to proceed to phase III.
res[1,]$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to -231.07, i.e. we have an expected utility of -23 107 000$.
Note that the input parameter alpha is returned having a value of 0.000016, which corresponds to the adjusted significance level $\alpha^3$.

The results for strategy 3 are:

res[2,]$d2 is the optimal number of events for phase II and res[2,]$d3 the resulting number of events for phase III. We see that the optimal scenario requires 20 events in phase II and 180 events in phase III, which leads to 30 participants in phase II and 258 in phase III. This corresponds to three trials in phase III with each trial having 86 participants.
res[2,]$HRgo is the optimal threshold value for the go/no-go decision rule. We see that we need a hazard ratio of less than 0.7 in phase II in order to proceed to phase III.
res[2,]$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to -411.1, i.e. we have an expected utility of -41 110 000$.

Strategy 4 yields the following results:

res[3,]$d2 is the optimal number of events for phase II and res[3,]$d3 the resulting number of events for phase III. We see that the optimal scenario requires 40 events in phase II and 292 events in phase III, which leads to 58 participants in phase II and 416 in phase III. This corresponds to four trials in phase III with each trial having 104 participants.
res[3,]$HRgo is the optimal threshold value for the go/no-go decision rule. We see that we need a hazard ratio of less than 0.70 in phase II in order to proceed to phase III.
res[3,]$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to -436.79, i.e. we have an expected utility of -43 679 000$.

Where to go from here

In this article we presented an example where several phase III trials are conducted. Note that this example is not restricted to time-to-event endpoints but can also be applied to binary and normally distributed endpoints by using the functions optimal_multitrial_binary and optimal_multitrial_normal. For more information on how to use the package, see:

Introduction to drugdevelopR: See how the package works in a basic example.
Different outcomes: Apply it to binary endpoints and time-to-event endpoints.
Interpreting the rest of the output: Obtain further details on your drug development program.
Fixed or prior: Model the assumed treatment effect on a prior distribution.
More parameters: Define custom effect size categories. Put constraints on the optimization by defining maximum costs, the total expected sample size of the program or the minimum expected probability of a successful program. Define an expected difference in treatment effect between phase II and III. Skip phase II.
Complex drug development programs: Adapt to situations with biased effect estimators, multi-arm trials, or multiple endpoints.
Parallel computing: Be faster at calculating the optimum by using parallel computing.

References

Koch, G. G. (2005). Statistical consideration of the strategy for demonstrating clinical evidence of effectiveness one larger vs two smaller pivotal studies by z. shun, e. chi, s. durrleman and l. fisher, statistics in medicine 2005; 24: 1619–1637. Statistics in Medicine, 24(11):1639–1646.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.

Case	Possible strategies for this case
1	1, 2
2	1 (with significance level of \(\alpha^2\))* , 2, 3, 23*
3	1 (with significance level of \(\alpha^3\))*, 3, 4