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Optimal phase II/III drug development planning for programs with multiple normally distributed endpoints

Source: R/optimal_multiple_normal.R
optimal_multiple_normal.Rd

The function optimal_multiple_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 for two-arm trials with two normally distributed endpoints and one control group (Preussler et. al, 2019).

Usage

optimal_multiple_normal(
  Delta1,
  Delta2,
  in1,
  in2,
  sigma1,
  sigma2,
  n2min,
  n2max,
  stepn2,
  kappamin,
  kappamax,
  stepkappa,
  alpha,
  beta,
  c2,
  c3,
  c02,
  c03,
  K = Inf,
  N = Inf,
  S = -Inf,
  steps1 = 0,
  stepm1 = 0.5,
  stepl1 = 0.8,
  b1,
  b2,
  b3,
  rho,
  fixed,
  relaxed = FALSE,
  num_cl = 1
)

Arguments

Delta1

assumed true treatment effect for endpoint 1 measured as the difference in means

Delta2

assumed true treatment effect for endpoint 2 measured as the difference in means

in1

amount of information for Delta1 in terms of number of events

in2

amount of information for Delta2 in terms of number of events

sigma1

variance of endpoint 1

sigma2

variance of endpoint 2

n2min

minimal total sample size in phase II, must be divisible by 3

n2max

maximal total sample size in phase II, must be divisible by 3

stepn2

stepsize for the optimization over n2, must be divisible by 3

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

alpha

one-sided significance level/family-wise error rate

beta

type-II error rate for any pair, i.e. 1 - beta is the (any-pair) 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 $

rho

correlation between the two endpoints

fixed

assumed fixed treatment effect

relaxed

relaxed or strict decision rule

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:

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.

Details

For this setting, the drug development program is defined to be successful if it proceeds from phase II to phase III and all endpoints show a statistically significant treatment effect in phase III. For example, this situation is found in Alzheimer’s disease trials, where a drug should show significant results in improving cognition (cognitive endpoint) as well as in improving activities of daily living (functional endpoint).

The effect size categories small, medium and large are applied to both endpoints. In order to define an overall effect size from the two individual effect sizes, the function implements two different combination rules:

  • A strict rule (relaxed = FALSE) assigning a large overall effect in case both endpoints show an effect of large size, a small overall effect in case that at least one of the endpoints shows a small effect, and a medium overall effect otherwise, and

  • A relaxed rule (relaxed = TRUE) assigning a large overall effect if at least one of the endpoints shows a large effect, a small effect if both endpoints show a small effect, and a medium overall effect otherwise.

Fast computing is enabled by parallel programming.

Monte Carlo simulations are applied for calculating utility, event count and other operating characteristics in this setting. Hence, the results are affected by random uncertainty.

References

Meinhard Kieser, Marietta Kirchner, Eva Dölger, Heiko Götte (2018). Optimal planning of phase II/III programs for clinical trials with multiple endpoints

IQWiG (2016). Allgemeine Methoden. Version 5.0, 10.07.2016, Technical Report. Available at https://www.iqwig.de/ueber-uns/methoden/methodenpapier/, assessed last 15.05.19.

Examples

# Activate progress bar (optional)
if (FALSE) progressr::handlers(global = TRUE) # \dontrun{}
# Optimize
# \donttest{
set.seed(123) # This function relies on Monte Carlo integration
optimal_multiple_normal(Delta1 = 0.75,
  Delta2 = 0.80, in1=300, in2=600,                   # define assumed true HRs
  sigma1 = 8, sigma2= 12,                            # variances for both endpoints
  n2min = 30, n2max = 90, stepn2 = 10,               # define optimization set for n2
  kappamin = 0.05, kappamax = 0.2, stepkappa = 0.05, # define optimization set for HRgo
  alpha = 0.025, beta = 0.1,                         # planning parameters
  c2 = 0.75, c3 = 1, c02 = 100, c03 = 150,           # fixed/variable costs: 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 = 1000, b2 = 2000, b3 = 3000,                   # define expected benefit
  rho = 0.5, relaxed = TRUE,                         # strict or relaxed rule
  fixed = TRUE,                                      # treatment effect
  num_cl = 1)                                        # parallelized computing
#> Optimization result:
#>  Utility: -126.46
#>  Sample size:
#>    phase II: 30, phase III: 29, total: 59
#>  Probability to go to phase III: 0.37
#>  Total cost:
#>    phase II: 122, phase III: 84, 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: 0 (1000), medium: 0.5 (2000), large: 0.8 (3000)
#>  Success probability: 0.08
#>  Joint probability of success and observed effect of size ... in phase III:
#>    small: 0.08, medium: 0, large: 0
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.05 [Kappa] 
#>  Assumed true effects [Delta]: 
#>    endpoint 1: 0.75, endpoint 2: 0.8
#>  Correlation between endpoints: 0.5
  # }