# Optimal phase II/III drug development planning with normally distributed endpoint

Source:`R/optimal_normal.R`

`optimal_normal.Rd`

The function `optimal_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 normally
distributed endpoints. The treatment effect is measured by the standardized
difference in means. The assumed true treatment effects can be assumed to be
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_normal(
w,
Delta1,
Delta2,
in1,
in2,
a,
b,
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,
gamma = 0,
fixed = FALSE,
skipII = 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

- 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 $

- gamma
to model different populations in phase II and III choose

`gamma != 0`

, default: 0, see here for details- fixed
choose if true treatment effects are fixed or following a prior distribution, if TRUE

`Delta1`

is used as fixed effect- skipII
choose if skipping phase II is an option, default: FALSE; if TRUE, the program calculates the expected utility for the case when phase II is skipped and compares it to the situation when phase II is not skipped. The results are then returned as a two-row data frame,

`res[1, ]`

being the results when including phase II and`res[2, ]`

when skipping phase II.`res[2, ]`

has an additional parameter,`res[2, ]$median_prior_Delta`

, which is the assumed effect size used for planning the phase III study when the phase II is skipped.- num_cl
number of clusters used for parallel computing, default: 1

## Value

The output of the function `optimal_normal`

is a data.frame containing the optimization results:

- 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

- 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

- 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 object lists the used optimization sequences, start and finish date of the optimization procedure.

## Examples

```
# Activate progress bar (optional)
if (FALSE) progressr::handlers(global = TRUE)
# Optimize
# \donttest{
optimal_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 = 4, # define optimization set for n2
kappamin = 0.02, kappamax = 0.2, stepkappa = 0.02, # define optimization set for kappa
alpha = 0.025, beta = 0.1, # drug development planning parameters
c2 = 0.675, c3 = 0.72, c02 = 15, c03 = 20, # fixed/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, # benefit for each effect size category
gamma = 0, # population structures in phase II/III
fixed = FALSE, # true treatment effects are fixed/random
skipII = FALSE, # skipping phase II
num_cl = 1) # number of cores for parallelized computing
#> Optimization result:
#> Utility: 2090.29
#> Sample size:
#> phase II: 100, phase III: 372, total: 472
#> Probability to go to phase III: 0.93
#> Total cost:
#> phase II: 82, phase III: 287, 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: 0.76
#> Success probability by effect size:
#> small: 0.72, medium: 0.03, large: 0
#> Significance level: 0.025
#> Targeted power: 0.9
#> Decision rule threshold: 0.1 [Kappa]
#> Parameters of the prior distribution:
#> Delta1: 0.375, Delta2: 0.625, in1: 300, in2: 600,
#> a: 0.25, b: 0.75, w: 0.3
#> Treatment effect offset between phase II and III: 0 [gamma]
# }
```