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. Specifically, it calculates
The decision rule is the optimal threshold for deciding whether we should proceed to phase III based on the results of phase II. Optimization is performed with respect to a utility function which takes the program’s cost, its success probability and its expected benefit when successfully launching the new drug on the market into account.
But now, let’s get started with a hands-on example.
Suppose we are developing a new tumor treatment, exper. The patient variable that we want to investigate is the difference in tumor width between the one-year visit and baseline. This is a normally distributed outcome variable.
Within our drug development program, we will compare our experimental treatment exper to the control treatment contro. Building on available data, we expect that the mean tumor difference amounts to \(20±1 \mathrm{mm}\) for contro and \(15±1 \mathrm{mm}\) for exper. Furthermore, the variance is assumed to be 8. The treatment effect is given as standardized difference in mean (\(\Delta=\frac{\mu_{contro} - \mu_{exper}}{\sigma}\)). Thus, we obtain a standardized treatment difference of \(\Delta_1 = 0.625\).
After installing the package according to the installation instructions, we can load it using the following code:
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.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%.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 NULLAfter 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$.This tutorial only covered drugdevelopR’s basic setting. Luckily, the package’s functionality extends to a multitude of different settings. If the example above doesn’t suit your needs, you can adapt it to your specific setting: