This R Markdown document provides an example for planning a trial with a binary endpoint using rpact. It also illustrates the use of ggplot2 for illustrating the characteristics of a sample size recalculation strategy. Another example for planning a trial with binary endpoints can be found here.

**First, load the rpact package**

`## [1] '3.1.1'`

Suppose a trial should be conducted in 3 stages where at the first stage 50%, at the second stage 75%, and at the final stage 100% of the information should be observed. Oâ€™Brien & Fleming boundaries should be used with one-sided \(\alpha = 0.025\) and non-binding futility bounds 0 and 0.5 for the first and the second stage, respectively, on the \(z\)-value scale.

The endpoints are binary (failure rates) and should be compared in a parallel group design, i.e., the null hypothesis to be tested is \(H_0:\pi_1 - \pi_2 = 0\,,\) which is tested against the alternative \(H_1: \pi_1 - \pi_2 < 0\,.\)

The necessary sample size to achieve 90% power if the failure rates are assumed to be \(\pi_1 = 0.40\) and \(\pi_2 = 0.60\) can be obtained as follows:

```
dGS <- getDesignGroupSequential(informationRates = c(0.5,0.75,1), alpha = 0.025, beta = 0.1,
futilityBounds = c(0,0.5))
r <- getSampleSizeRates(dGS, pi1 = 0.4, pi2 = 0.6)
```

The `summary()`

command creates a nice table for the study design parameters:

```
## Sample size calculation for a binary endpoint
##
## Sequential analysis with a maximum of 3 looks (group sequential design), overall
## significance level 2.5% (one-sided).
## The sample size was calculated for a two-sample test for rates
## (normal approximation),
## H0: pi(1) - pi(2) = 0, H1; treatment rate pi(1) = 0.4, control rate pi(2) = 0.6,
## power 90%.
##
## Stage 1 2 3
## Information rate 50% 75% 100%
## Efficacy boundary (z-value scale) 2.863 2.337 2.024
## Futility boundary (z-value scale) 0 0.500
## Overall power 0.2958 0.6998 0.9000
## Expected number of subjects 198.3
## Number of subjects 133.1 199.7 266.3
## Exit probability for futility 0.0100 0.0056
## Cumulative alpha spent 0.0021 0.0105 0.0250
## One-sided local significance level 0.0021 0.0097 0.0215
## Efficacy boundary (t) -0.248 -0.165 -0.124
## Futility boundary (t) 0.000 -0.035
## Overall exit probability (under H0) 0.5021 0.2275
## Overall exit probability (under H1) 0.3058 0.4095
## Exit probability for efficacy (under H0) 0.0021 0.0083
## Exit probability for efficacy (under H1) 0.2958 0.4040
## Exit probability for futility (under H0) 0.5000 0.2191
## Exit probability for futility (under H1) 0.0100 0.0056
##
## Legend:
## (t): treatment effect scale
```

Note that the calculation of the efficacy boundaries on the approximate
treatment effect scale is performed under the assumption that \(\pi_2 = 0.60\) is the
observed failure rate in the control group and states the *treatment difference
to be observed* in order to reach significance (or stop the trial due to
futility).

The optimum allocation ratio yields the smallest overall sample size and
depends on the choice of \(\pi_1\) and \(\pi_2\). It can be obtained by specifying
`allocationRatioPlanned = 0`

. In our case, the optimum allocation ratio is 1
but calculated numerically, therefore slightly unequal 1:

```
r <- getSampleSizeRates(dGS, pi1 = 0.4, pi2 = 0.6, allocationRatioPlanned = 0)
r$allocationRatioPlanned
```

`## [1] 0.9999976`

`## [1] 1`