# Optimal phase II/III drug development planning with time-to-event endpoint

Source:`R/optimal_tte.R`

`optimal_tte.Rd`

The function `optimal_tte`

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 time-to-event endpoints
(Kirchner et al., 2016). The assumed true treatment effects can be assumed to
be fixed or modelled by
a prior distribution. When assuming fixed true treatment effects, planning can
also be done with the user-friendly R Shiny app
basic.
The app prior visualizes
the prior distributions used in this package. Fast computing is enabled by
parallel programming.

## Usage

```
optimal_tte(
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,
steps1 = 1,
stepm1 = 0.95,
stepl1 = 0.85,
b1,
b2,
b3,
gamma = 0,
fixed = FALSE,
skipII = 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

- steps1
lower boundary for effect size category "small" in HR scale, default: 1

- stepm1
lower boundary for effect size category "medium" in HR scale = upper boundary for effect size category "small" in HR scale, default: 0.95

- stepl1
lower boundary for effect size category "large" in HR scale = upper boundary for effect size category "medium" in HR scale, default: 0.85

- b1
expected gain for effect size category "small"

- b2
expected gain for effect size category "medium"

- b3
expected gain for effect size category "large"

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

`gamma != 0`

, default: 0- fixed
choose if true treatment effects are fixed or random, if TRUE hr1 is used as a fixed effect and hr2 is ignored

- 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_HR`

, which is the assumed hazards ratio used for planning the phase III study when the phase II is skipped. It is calculated as the exponential function of the median of the prior function.- 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:

- 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

- 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 date of the optimization procedure.

## References

Kirchner, M., Kieser, M., Goette, H., & Schueler, A. (2016). Utility-based optimization of phase II/III programs. Statistics in Medicine, 35(2), 305-316.

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.

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_tte(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/variable costs for phase II/III
K = Inf, N = Inf, S = -Inf, # set constraints
steps1 = 1, # define lower boundary for "small"
stepm1 = 0.95, # "medium"
stepl1 = 0.85, # and "large" treatment effect size categories
b1 = 1000, b2 = 2000, b3 = 3000, # expected 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: 75.8
#> Sample size:
#> phase II: 122, phase III: 210, total: 332
#> Expected number of events:
#> phase II: 85, phase III: 147, total: 232
#> Assumed event rate:
#> phase II: 0.7, phase III: 0.7
#> Probability to go to phase III: 0.46
#> Total cost:
#> phase II: 192, phase III: 278, 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: 0.24
#> Success probability by effect size:
#> small: 0.05, medium: 0.09, large: 0.11
#> Significance level: 0.025
#> Targeted power: 0.9
#> Decision rule threshold: 0.8 [HRgo]
#> Parameters of the prior distribution:
#> hr1: 0.69, hr2: 0.88, id1: 210, id2: 420, w: 0.3
#> Treatment effect offset between phase II and III: 0 [gamma]
# }
```