Optimal phase II/III drug development planning when discounting phase II results with normally distributed endpoint

The function optimal_bias_normal of the drugdevelopR package enables planning of phase II/III drug development programs with optimal sample size allocation and go/no-go decision rules including methods for discounting of phase II results for normally distributed endpoints (Preussler et. al, 2020). The discounting may be necessary as programs that proceed to phase III can be overoptimistic about the treatment effect (i.e. they are biased). The assumed true treatment effects can be assumed fixed or modelled by a prior distribution. The R Shiny application prior visualizes the prior distributions used in this package. Fast computing is enabled by parallel programming.

Usage

optimal_bias_normal(
  w,
  Delta1,
  Delta2,
  in1,
  in2,
  a,
  b,
  n2min,
  n2max,
  stepn2,
  kappamin,
  kappamax,
  stepkappa,
  adj = "both",
  lambdamin = NULL,
  lambdamax = NULL,
  steplambda = NULL,
  alphaCImin = NULL,
  alphaCImax = NULL,
  stepalphaCI = NULL,
  alpha,
  beta,
  c2,
  c3,
  c02,
  c03,
  K = Inf,
  N = Inf,
  S = -Inf,
  steps1 = 0,
  stepm1 = 0.5,
  stepl1 = 0.8,
  b1,
  b2,
  b3,
  fixed = FALSE,
  num_cl = 1
)

Arguments

w

weight for mixture prior distribution

Delta1

assumed true prior treatment effect measured as the standardized difference in means, see here for details

Delta2

assumed true prior treatment effect measured as the standardized difference in means, see here for details

in1

amount of information for Delta1 in terms of sample size, see here for details

in2

amount of information for Delta2 in terms of sample size, see here for details

a

lower boundary for the truncation of the prior distribution

b

upper boundary for the truncation of the prior distribution

n2min

minimal total sample size for phase II; must be an even number

n2max

maximal total sample size for phase II, must be an even number

stepn2

step size for the optimization over n2; must be an even number

kappamin

minimal threshold value kappa for the go/no-go decision rule

kappamax

maximal threshold value kappa for the go/no-go decision rule

stepkappa

step size for the optimization over the threshold value kappa

adj

choose type of adjustment: "multiplicative", "additive", "both" or "all". When using "both", res[1,] contains the results using the multiplicative method and res[2,] contains the results using the additive method. When using "all", there are also res[3,] and res[4,], containing the results of a multiplicative and an additive method which do not only adjust the treatment effect but also the threshold value for the decision rule.

lambdamin

minimal multiplicative adjustment parameter lambda (i.e. use estimate with a retention factor)

lambdamax

maximal multiplicative adjustment parameter lambda (i.e. use estimate with a retention factor)

steplambda

stepsize for the adjustment parameter lambda

alphaCImin

minimal additive adjustment parameter alphaCI (i.e. adjust the lower bound of the one-sided confidence interval)

alphaCImax

maximal additive adjustment parameter alphaCI (i.e. adjust the lower bound of the one-sided confidence interval)

stepalphaCI

stepsize for alphaCI

alpha

one-sided significance level

beta

type II error rate; i.e. 1 - beta is the power for calculation of the sample size for phase III

c2

variable per-patient cost for phase II in 10^5 $

c3

variable per-patient cost for phase III in 10^5 $

c02

fixed cost for phase II in 10^5 $

c03

fixed cost for phase III in 10^5 $

K

constraint on the costs of the program, default: Inf, e.g. no constraint

N

constraint on the total expected sample size of the program, default: Inf, e.g. no constraint

S

constraint on the expected probability of a successful program, default: -Inf, e.g. no constraint

steps1

lower boundary for effect size category "small", default: 0

stepm1

lower boundary for effect size category "medium" = upper boundary for effect size category "small" default: 0.5

stepl1

lower boundary for effect size category "large" = upper boundary for effect size category "medium", default: 0.8

b1

expected gain for effect size category "small" in 10^5 $

b2

expected gain for effect size category "medium" in 10^5 $

b3

expected gain for effect size category "large" in 10^5 $

fixed

choose if true treatment effects are fixed or following a prior distribution, if TRUE Delta1 is used as fixed effect

num_cl

number of clusters used for parallel computing, default: 1

Value

The output of the function is a data.frame object containing the optimization results:

Adj

optimal adjustment parameter (lambda or alphaCI according to Method)

u

maximal expected utility under the optimization constraints, i.e. the expected utility of the optimal sample size and threshold value

Kappa

optimal threshold value for the decision rule to go to phase III

n2

total sample size for phase II; rounded to the next even natural number

n3

total sample size for phase III; rounded to the next even natural number

n

total sample size in the program; n = n2 + n3

K

maximal costs of the program (i.e. the cost constraint, if it is set or the sum K2+K3 if no cost constraint is set)

pgo

probability to go to phase III

sProg

probability of a successful program

sProg1

probability of a successful program with "small" treatment effect in phase III

sProg2

probability of a successful program with "medium" treatment effect in phase III

sProg3

probability of a successful program with "large" treatment effect in phase III

K2

expected costs for phase II

K3

expected costs for phase III

and further input parameters. Taking cat(comment()) of the data frame lists the used optimization sequences, start and finish time of the optimization procedure. The attribute attr(,"trace") returns the utility values of all parameter combinations visited during optimization.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences.

Examples

# Activate progress bar (optional)
if (FALSE) progressr::handlers(global = TRUE) # \dontrun{}
# Optimize
# \donttest{
optimal_bias_normal(w=0.3,             # define parameters for prior
  Delta1 = 0.375, Delta2 = 0.625, in1=300, in2=600,    # (https://web.imbi.uni-heidelberg.de/prior/)
  a = 0.25, b = 0.75,
  n2min = 20, n2max = 100, stepn2 = 10,                # define optimization set for n2
  kappamin = 0.02, kappamax = 0.2, stepkappa = 0.02,   # define optimization set for kappa
  adj = "both",                                        # choose type of adjustment
  lambdamin = 0.2, lambdamax = 1, steplambda = 0.05,   # define optimization set for lambda
  alphaCImin = 0.025, alphaCImax = 0.5,
  stepalphaCI = 0.025,                                 # define optimization set for alphaCI
  alpha = 0.025, beta = 0.1,                           # drug development planning parameters
  c2 = 0.675, c3 = 0.72, c02 = 15, c03 = 20,           # fixed and variable costs for phase II/III
  K = Inf, N = Inf, S = -Inf,                          # set constraints
  steps1 = 0,                                          # define lower boundary for "small"
  stepm1 = 0.5,                                        # "medium"
  stepl1 = 0.8,                                        # and "large" effect size categories
  b1 = 3000, b2 = 8000, b3 = 10000,                    # define expected benefits
  fixed = TRUE,                                        # true treatment effects are fixed/random
  num_cl = 1)                                          # number of coresfor parallelized computing 
#> Optimization result with multiplicative adjustment of the treatment effect:
#>  Utility: 1854.94
#>  Bias adjustment parameter: 0.8
#>  Sample size:
#>    phase II: 100, phase III: 628, total: 728
#>  Probability to go to phase III: 0.9
#>  Total cost:
#>    phase II: 82, phase III: 470, 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: 0.8
#>  Joint probability of success and observed effect of size ... in phase III:
#>    small: 0.8, medium: 0, large: 0
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.12 [Kappa] 
#>  Assumed true effect: 0.375 [Delta] 
#> 
#> Optimization result with additive adjustment of the treatment effect:
#>  Utility: 1768.13
#>  Bias adjustment parameter: 0.5
#>  Sample size:
#>    phase II: 100, phase III: 460, total: 560
#>  Probability to go to phase III: 0.92
#>  Total cost:
#>    phase II: 82, phase III: 350, 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: 0.73
#>  Joint probability of success and observed effect of size ... in phase III:
#>    small: 0.73, medium: 0, large: 0
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.1 [Kappa] 
#>  Assumed true effect: 0.375 [Delta] 
#> 
  # }