# Optimal phase II/III drug development planning where several phase III trials are performed for time-to-event endpoints

Source:`R/optimal_multitrial.R`

`optimal_multitrial.Rd`

The function `optimal_multitrial`

of the drugdevelopR package enables planning of phase II/III drug development programs with time-to-event endpoints for programs with several phase III trials (Preussler et. al, 2019).
Its main output values are the optimal sample size allocation and optimal go/no-go decision rules.
The assumed true treatment effects can be assumed to be fixed (planning is then also possible via user friendly R Shiny App: multitrial) or can be 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_multitrial(
w,
hr1,
hr2,
id1,
id2,
d2min,
d2max,
stepd2,
hrgomin,
hrgomax,
stephrgo,
alpha,
beta,
xi2,
xi3,
c2,
c3,
c02,
c03,
K = Inf,
N = Inf,
S = -Inf,
b1,
b2,
b3,
case,
strategy = TRUE,
fixed = FALSE,
num_cl = 1
)
```

## Arguments

- w
weight for mixture prior distribution, see this Shiny application for the choice of weights

- hr1
first assumed true treatment effect on HR scale for prior distribution

- hr2
second assumed true treatment effect on HR scale for prior distribution

- id1
amount of information for

`hr1`

in terms of number of events- id2
amount of information for

`hr2`

in terms of number of events- d2min
minimal number of events for phase II

- d2max
maximal number of events for phase II

- stepd2
step size for the optimization over d2

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

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

- stephrgo
step size for the optimization over HRgo

- alpha
one-sided significance level

- beta
type II error rate; i.e.

`1 - beta`

is the power for calculation of the number of events for phase III by Schoenfeld's formula (Schoenfeld 1981)- xi2
assumed event rate for phase II, used for calculating the sample size of phase II via

`n2 = d2/xi2`

- xi3
event rate for phase III, used for calculating the sample size of phase III in analogy to

`xi2`

- 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

- b1
expected gain for effect size category "small"

- b2
expected gain for effect size category "medium"

- b3
expected gain for effect size category "large"

- case
choose case: "at least 1, 2 or 3 significant trials needed for approval"

- strategy
choose strategy: "conduct 1, 2, 3 or 4 trials in order to achieve the case's goal"; TRUE calculates all strategies of the selected

`case`

- fixed
choose if true treatment effects are fixed or random, if TRUE hr1 is used as a fixed effect and hr2 is ignored

- 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:

- Strategy
Strategy: "number of trials to be conducted in order to achieve the goal of the case"

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

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

- d2
optimal total number of events for phase II

- d3
total expected number of events for phase III; rounded to next natural number

- d
total expected number of events in the program; d = d2 + d3

- 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 (lower boundary in HR scale is set to 1, as proposed by IQWiG (2016))

- sProg2
probability of a successful program with "medium" treatment effect in phase III (lower boundary in HR scale is set to 0.95, as proposed by IQWiG (2016))

- sProg3
probability of a successful program with "large" treatment effect in phase III (lower boundary in HR scale is set to 0.85, as proposed by IQWiG (2016))

- 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 date of the optimization procedure.

## Effect sizes

In other settings, the definition of "small", "medium" and "large" effect
sizes can be user-specified using the input parameters `steps1`

, `stepm1`

and
`stepl1`

. Due to the complexity of the multitrial setting, this feature is
not included for this setting. Instead, the effect sizes were set to
to predefined values as explained under sProg1, sProg2 and sProg3 in the
*Value* section.

## References

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.

Preussler, S., Kieser, M., and Kirchner, M. (2019). Optimal sample size allocation and go/no-go decision rules for phase II/III programs where several phase III trials are performed. Biometrical Journal, 61(2), 357-378.

Schoenfeld, D. (1981). The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika, 68(1), 316-319.

## Examples

```
# Activate progress bar (optional)
if (FALSE) progressr::handlers(global = TRUE)
# Optimize
# \donttest{
optimal_multitrial(w = 0.3, # define parameters for prior
hr1 = 0.69, hr2 = 0.88, id1 = 210, id2 = 420, # (https://web.imbi.uni-heidelberg.de/prior/)
d2min = 20, d2max = 100, stepd2 = 5, # define optimization set for d2
hrgomin = 0.7, hrgomax = 0.9, stephrgo = 0.05, # 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, # fixed and variable costs for phase II/III
K = Inf, N = Inf, S = -Inf, # set constraints
b1 = 1000, b2 = 2000, b3 = 3000, # expected benefit for each effect size
case = 1, strategy = TRUE, # chose Case and Strategy
fixed = TRUE, # true treatment effects are fixed/random
num_cl = 1) # number of cores for parallelized computing
#> Optimization result with 1 significant trial(s) needed, strategy 1:
#> Utility: 859.71
#> Sample size:
#> phase II: 144, phase III: 446, total: 590
#> Expected number of events:
#> phase II: 100, phase III: 312, total: 412
#> Assumed event rate:
#> phase II: 0.7, phase III: 0.7
#> Probability to go to phase III: 0.85
#> Total cost:
#> phase II: 208, phase III: 574, 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: 0.67
#> Success probability by effect size:
#> small: 0.07, medium: 0.21, large: 0.38
#> Significance level: 0.025
#> Targeted power: 0.9
#> Decision rule threshold: 0.85 [HRgo]
#> Assumed true effect: 0.69 [hr]
#> Treatment effect offset between phase II and III: 0 [gamma]
#>
#> Optimization result with 1 significant trial(s) needed, strategy 2:
#> Utility: 714.39
#> Sample size:
#> phase II: 144, phase III: 628, total: 772
#> Expected number of events:
#> phase II: 100, phase III: 440, total: 540
#> Assumed event rate:
#> phase II: 0.7, phase III: 0.7
#> Probability to go to phase III: 0.77
#> Total cost:
#> phase II: 208, phase III: 859, 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: 0.68
#> Success probability by effect size:
#> small: 0.05, medium: 0.18, large: 0.46
#> Significance level: 0.025
#> Targeted power: 0.9
#> Decision rule threshold: 0.8 [HRgo]
#> Assumed true effect: 0.69 [hr]
#> Treatment effect offset between phase II and III: 0 [gamma]
#>
# }
```