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 introduction we now want to investigate a scenario, where the results of phase II of a time-to-event outcome are discounted. The discounting may be necessary as programs that proceed to phase III can be overoptimistic about the treatment effect (i.e. they are biased). Hereby, we adjust the optimal number of events in phase III.
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. Therefore, we will
use the function optimal_bias, which calculates optimal
sample sizes and threshold decisions values for time-to-event outcomes
with bias adjustment.
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 is the ratio of the hazard rates. If we assume that unfavorable events as tumor progression occur only 75% as often as in the control group, we have a hazard ratio of 0.75.
After installing the package according to the installation instructions, we can load it using the following code:
library(drugdevelopR)
#> Lade nötiges Paket: doParallel
#> Lade nötiges Paket: foreach
#> Lade nötiges Paket: iterators
#> Lade nötiges Paket: parallelWe 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 (as in this example) or follows a prior distribution (see Fixed or Prior). Furthermore, most options to adapt the program to your specific needs are also available in this setting (see More parameters), however skipping phase II and choosing different treatment effects in phase II and III are not possible.
In addition to the parameters from the basic setting, some parameters are needed specifically for the bias adjustment:
adj is needed to select the type of bias
adjustment. There are four possible options for this parameter:
adj = "additive" selects the additive adjustment method
for the number of events in phase III. Here, the lower bound of the
one-sided confidence interval is adjusted according to Wang et al
(2006).adj = "multiplicative" selects the multiplicative
adjustment method for the number of events in phase III. Here, an
estimate with a retention factor is used according to Kirby et
al. (2012).adj = "both" returns the results of both adjustment
methods, i.e. the additive and the multiplicative one.adj = "all" returns the results of both adjustment
methods and in addition, also returns the results for both methods when
not only the number of events is adjusted but also the threshold value
for the decision rule.alphaCImax to 0.5. We want to investigate the optimization
region in the interval of [0.1, 0.5] with step size 0.05. Thus, we set
the lower bound to alphaCImin to 0.1 and the step size to
stepalphaCI = 0.05.lambdamax to 1. We want to investigate the optimization
region in the interval of [0.7, 1] with step size 0.05. Thus, we set the
lower bound to lambdamin to 0.1 and the step size to
steplambda = 0.05.Now that we have defined all parameters needed for our example, we are ready to feed them to the package.
res <- optimal_bias(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
adj = "both", # choose both adjustment methods
lambdamin = 0.7, lambdamax = 1, steplambda = 0.05, # optimization set for multiplicative adjustment
alphaCImin = 0.1, alphaCImax = 0.5, stepalphaCI = 0.05, # optimization set for additive adjustment
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
fixed = TRUE, # choose if effects are fixed or random
num_cl = 1)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 multiplicative adjustment of the treatment effect:
#> Utility: 378.47
#> Bias adjustment parameter: 0.95
#> Sample size:
#> phase II: 236, phase III: 624, total: 860
#> Expected number of events:
#> phase II: 165, phase III: 436, total: 601
#> Assumed event rate:
#> phase II: 0.7, phase III: 0.7
#> Probability to go to phase III: 0.79
#> Total cost:
#> phase II: 277, phase III: 742, 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: 0.63
#> Success probability by effect size:
#> small: 0.09, medium: 0.3, large: 0.24
#> Significance level: 0.025
#> Targeted power: 0.9
#> Decision rule threshold: 0.85 [HRgo]
#> Assumed true effect: 0.75 [hr]
#>
#> Optimization result with additive adjustment of the treatment effect:
#> Utility: 377.1
#> Bias adjustment parameter: 0.5
#> 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]The program returns output values for both adjustment methods. The most important ones for the multiplicative method are:
res[1,]$Adj is the optimal multiplicative adjustment
parameter. In this setting, our optimal value is 0.95, indicating that a
slight bias adjustment leads to a higher utility.res[1,]$d2 is the optimal number of events for phase II
and res$d3 the resulting number of events for phase III. We
see that the optimal scenario requires 165 events in phase II and 436
events in phase III, which correspond to 236 participants in phase II
and 624 in phase III.res[1,]$HRgo is the optimal threshold value for the
go/no-go decision rule. We see that we need a hazard ratio of less than
0.85 in phase II in order to proceed to phase III.res[1,]$u is the expected utility of the program for
the optimal sample size and threshold value. In our case it amounts to
378.47, i.e. we have an expected utility of 37 847 000$.For the additive method we get:
res[2,]$Adj is the optimal additive adjustment
parameter. In this setting, our optimal value is 0.5, indicating that no
bias adjustment leads to the highest utility. In this case, our results
match the results of the basic setting as can be verified here.res[2,]$d2 is the optimal number of events for phase II
and res$d3 the resulting number of events for phase III. We
see that the optimal scenario requires 165 events in phase II and 430
events in phase III, which correspond to 236 participants in phase II
and 614 in phase III.res[2,]$HRgo is the optimal threshold value for the
go/no-go decision rule. We see that we need a hazard ratio of less than
0.86 in phase II in order to proceed to phase III.res[2,]$u is the expected utility of the program for
the optimal sample size and threshold value. In our case it amounts to
377.1, i.e. we have an expected utility of 37 710 000$.In this article we presented an example how methods to discount the
results of phase II can be included for the purpose of bias adjustment.
Note that this example is not restricted to time-to-event endpoints but
can also be applied to binary and normally distributed endpoints using
the functions optimal_bias_binary and
optimal_bias_normal. For more information on how to use the
package, see:
Kirby, S., Burke, J., Chuang-Stein, C., and Sin, C. (2012). Discounting phase 2 results when planning phase 3 clinical trials. Pharmaceutical Statistics, 11(5):373–385.
Wang, S.-J., Hung, H. J., and O’Neill, R. T. (2006). Adapting the sample size planning of a phase III trial based on phase II data. Pharmaceutical Statistics, 5(2):85–97.