```
library(rpact)
packageVersion("rpact")
```

# Planning a Trial with Binary Endpoints with rpact

# Designing a trial with binary endpoints

**First, load the rpact package**

`[1] '3.4.0'`

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. O’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 \(H_1: \pi_1 - \pi_2 < 0\,.\)

## Sample size calculation

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:

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

The `summary()`

command creates a nice table for the study design parameters:

`kable(summary(r))`

**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 |

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

**Design plan parameters and output for rates**

**Design parameters**

*Information rates*: 0.500, 0.750, 1.000*Critical values*: 2.863, 2.337, 2.024*Futility bounds (non-binding)*: 0.000, 0.500*Cumulative alpha spending*: 0.002101, 0.010451, 0.025000*Local one-sided significance levels*: 0.002101, 0.009711, 0.021475*Significance level*: 0.0250*Type II error rate*: 0.1000*Test*: one-sided

**User defined parameters**

*Assumed treatment rate*: 0.400*Assumed control rate*: 0.600

**Default parameters**

*Risk ratio*: FALSE*Theta H0*: 0*Normal approximation*: TRUE*Treatment groups*: 2*Planned allocation ratio*: 1

**Sample size and output**

*Direction upper*: FALSE*Reject per stage [1]*: 0.2958*Reject per stage [2]*: 0.4040*Reject per stage [3]*: 0.2002*Overall futility stop*: 0.01556*Futility stop per stage [1]*: 0.010003*Futility stop per stage [2]*: 0.005556*Early stop*: 0.7154*Maximum number of subjects*: 266.3*Maximum number of subjects (1)*: 133.1*Maximum number of subjects (2)*: 133.1*Number of subjects [1]*: 133.1*Number of subjects [2]*: 199.7*Number of subjects [3]*: 266.3*Expected number of subjects under H0*: 184.3*Expected number of subjects under H0/H1*: 228.8*Expected number of subjects under H1*: 198.3*Critical values (treatment effect scale) [1]*: -0.248*Critical values (treatment effect scale) [2]*: -0.165*Critical values (treatment effect scale) [3]*: -0.124*Futility bounds (treatment effect scale) [1]*: 0.0000*Futility bounds (treatment effect scale) [2]*: -0.0349*Futility bounds (one-sided p-value scale) [1]*: 0.5000*Futility bounds (one-sided p-value scale) [2]*: 0.3085

**Legend**

*(i)*: values of treatment arm i*[k]*: values at stage k

Note that the calculation of the efficacy boundaries on the 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).

## Optimum allocation ratio

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, due to \(\pi_1 = 1 - \pi_2\), the optimum allocation ratio is 1 but calculated numerically, therefore slightly unequal 1:

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

`[1] 0.9999976`

`round(r$allocationRatioPlanned, 5)`

`[1] 1`

## Boundary plots

The decision boundaries can be illustrated on different scales.

On the \(z\)-value scale:

`plot(r, type = 1)`

On the effect size scale:

`plot(r, type = 2)`

On the \(p\)-value scale:

`plot(r, type = 3)`

## Power assessment

Suppose that \(N = 280\) subjects were planned for the study. The power if the failure rate in the active treatment group is \(\pi_1 = 0.40\) or \(\pi_1 = 0.50\) can be achieved as follows:

```
<- getPowerRates(dGS,
power maxNumberOfSubjects = 280,
pi1 = c(0.4, 0.5), pi2 = 0.6, directionUpper = FALSE
)$overallReject power
```

`[1] 0.914045 0.377853`

Note that `directionUpper = FALSE`

is used because the study is powered for alternatives \(\pi_1 - \pi_2\) being smaller than 0.

The power for \(\pi_1 = 0.50\) (37.8%) is much reduced as compared to the case \(\pi_1 = 0.40\) (where it exceeds 90%):

## Graphical illustration

We also can graphically illustrate the power, the expected sample size, and the early stopping and futility stopping probabilities for a range of alternative values. This can be done by specifying the lower and the upper bound for \(\pi_1\) in `getPowerRates()`

and use the generic `plot()`

command with `type = 6`

:

```
<- getPowerRates(dGS,
power maxNumberOfSubjects = 280,
pi1 = c(0.3, 0.6), pi2 = 0.6, directionUpper = FALSE
)plot(power, type = 6)
```

# Sample size reassessment for testing rates

Suppose that, using an adaptive design, the sample size from the above example can be increased *in the last interim* up to a 4-fold of the originally planned sample size for the last stage. Conditional power 90% *based on the observed effect sizes (failure rates)* should be used to increase the sample size. We want to use the inverse normal method to allow for the sample size increase and compare the test characteristics with the group sequential design from the above example.

## Assess power

To assess the test characteristics of this adaptive design we first define the inverse normal design and then perform two simulations, one without and one with SSR:

```
<- getDesignInverseNormal(
dIN informationRates = c(0.5, 0.75, 1),
alpha = 0.025, beta = 0.1, futilityBounds = c(0, 0.5)
)
<- getSimulationRates(dIN,
sim1 plannedSubjects = c(140, 210, 280),
pi1 = seq(0.4, 0.5, 0.01), pi2 = 0.6, directionUpper = FALSE,
maxNumberOfIterations = 1000, conditionalPower = 0.9,
minNumberOfSubjectsPerStage = c(140, 70, 70),
maxNumberOfSubjectsPerStage = c(140, 70, 70), seed = 1234
)
<- getSimulationRates(dIN,
sim2 plannedSubjects = c(140, 210, 280),
pi1 = seq(0.4, 0.5, 0.01), pi2 = 0.6, directionUpper = FALSE,
maxNumberOfIterations = 1000, conditionalPower = 0.9,
minNumberOfSubjectsPerStage = c(NA, 70, 70),
maxNumberOfSubjectsPerStage = c(NA, 70, 4 * 70), seed = 1234
)
```

Note that the sample sizes will be calculated under the assumption that the *conditional power for the subsequent stage* is 90%. If the resulting sample size is larger, the upper bound (4*70 = 280) is used.

Note also that `sim1`

can also be *calculated* using `getPowerRates()`

or can also *easier be simulated* without specifying `conditionalPower`

, `minNumberOfSubjectsPerStage`

, and `maxNumberOfSubjectsPerStage`

(which obviously is redundant for `sim1`

) but this way ensures that the calculated objects `sim1`

and `sim2`

*contain exactly the same parameters* and therefore can easier be combined (see below).

We can look at the power and the expected sample size of the two procedures and assess the power gain of using the adaptive design which comes along with an increased expected sample size:

`$pi1 sim1`

` [1] 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50`

`round(sim1$overallReject, 3)`

` [1] 0.921 0.890 0.853 0.810 0.752 0.721 0.675 0.582 0.526 0.469 0.405`

`round(sim2$overallReject, 3)`

` [1] 0.976 0.971 0.940 0.912 0.882 0.869 0.800 0.718 0.695 0.601 0.475`

`round(sim1$expectedNumberOfSubjects, 1)`

` [1] 202.1 209.3 216.4 219.5 222.0 230.9 231.1 238.3 234.2 237.9 236.5`

`round(sim2$expectedNumberOfSubjects, 1)`

` [1] 240.7 251.6 270.6 278.8 286.6 305.8 323.8 330.9 336.4 336.8 349.3`

## Illustrate power difference

We now want to graphically illustrate the gain in power when using the adaptive sample size recalculation. We use ggplot2 (see ggplot2.tidyverse.org) for doing this. First, a dataset `df`

combining `sim1`

and `sim2`

is defined with the additional variable SSR. Defining `mytheme`

and using the following ggplot2 commands, the difference in power and ASN of the two strategies is illustrated. It shows that at least for (absolute) effect difference > 0.15 an overall power of more than around 85% can be achieved with the proposed sample size recalculation strategy.

```
library(ggplot2)
<- as.data.frame(sim1, niceColumnNamesEnabled = FALSE)
dataSim1 <- as.data.frame(sim2, niceColumnNamesEnabled = FALSE)
dataSim2
$SSR <- rep("no SSR", nrow(dataSim1))
dataSim1$SSR <- rep("SSR", nrow(dataSim2))
dataSim2<- rbind(dataSim1, dataSim2)
df
<- theme(
myTheme axis.title.x = element_text(size = 12), axis.text.x = element_text(size = 12),
axis.title.y = element_text(size = 12), axis.text.y = element_text(size = 12),
plot.title = element_text(size = 14, hjust = 0.5),
plot.subtitle = element_text(size = 12, hjust = 0.5)
)
<- ggplot(
p data = df,
aes(x = effect, y = overallReject, group = SSR, color = SSR)
+
) geom_line(size = 1.1) +
geom_line(aes(
x = effect, y = expectedNumberOfSubjects / 400,
group = SSR, color = SSR
size = 1.1, linetype = "dashed") +
), scale_y_continuous("Power",
sec.axis = sec_axis(~ . * 400, name = "ASN"),
limits = c(0.2, 1)
+
) xlab("effect") +
ggtitle("Power and ASN", "Power solid, ASN dashed") +
geom_hline(size = 0.5, yintercept = 0.8, linetype = "dotted") +
geom_hline(size = 0.5, yintercept = 0.9, linetype = "dotted") +
geom_vline(size = 0.5, xintercept = c(-0.2, -0.15), linetype = "dashed") +
theme_classic() +
myTheme
plot(p)
```

For saving the graph, use

`ggplot2::ggsave(filename = "c:/yourdirectory/comparison.png",`

`plot = ggplot2::last_plot(), device = NULL, path = NULL,`

`scale = 1.2, width = 20, height = 12, units = "cm", dpi = 600,`

`limitsize = TRUE)`

For another example of using ggplot2 in rpact see also the vignette Supplementing and enhancing rpact’s graphical capabilities with ggplot2.

## Histogram of sample sizes

Finally, we create a histogram for the attained sample size of the study *when using the adaptive sample size recalculation*.

With the `getData()`

command the simulation results are obtained and `str(simdata)`

provides information of the struture of this data:

```
<- getData(sim2)
simData str(simData)
```

```
'data.frame': 24579 obs. of 19 variables:
$ iterationNumber : num 1 2 2 2 3 3 4 4 4 5 ...
$ stageNumber : num 1 1 2 3 1 2 1 2 3 1 ...
$ pi1 : num 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 ...
$ pi2 : num 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 ...
$ numberOfSubjects : num 140 140 70 147 140 70 140 70 91 140 ...
$ numberOfCumulatedSubjects: num 140 140 210 357 140 210 140 210 301 140 ...
$ rejectPerStage : num 1 0 0 1 0 1 0 0 1 1 ...
$ futilityPerStage : num 0 0 0 0 0 0 0 0 0 0 ...
$ testStatistic : num 3.05 2.03 2.07 4.07 2.03 ...
$ testStatisticsPerStage : num 3.054 2.028 0.718 4.547 2.029 ...
$ overallRate1 : num 0.329 0.414 0.438 0.369 0.429 ...
$ overallRate2 : num 0.586 0.586 0.581 0.607 0.6 ...
$ stagewiseRates1 : num 0.329 0.414 0.486 0.27 0.429 ...
$ stagewiseRates2 : num 0.586 0.586 0.571 0.644 0.6 ...
$ sampleSizesPerStage1 : num 70 70 35 74 70 35 70 35 46 70 ...
$ sampleSizesPerStage2 : num 70 70 35 73 70 35 70 35 45 70 ...
$ trialStop : logi TRUE FALSE FALSE TRUE FALSE TRUE ...
$ conditionalPowerAchieved : num NA NA 0.602 0.9 NA ...
$ pValue : num 0.00112984 0.02126124 0.23628281 0.00000272 0.02121903 ...
```

Depending on \(\pi_1\) (in this example, for \(\pi_1 = 0.5\)), you can create the histogram of the simulated total sample size as follows:

```
<- simData[simData$pi1 == 0.5, ]
simDataPart
<-
overallSampleSizes sapply(1:1000, function(i) {
sum(simDataPart[simDataPart$iterationNumber == i, ]$numberOfSubjects)
})
hist(overallSampleSizes, main = "Histogram", xlab = "Achieved sample size")
```

How often the maximum and other sample sizes are reached over the stages can be obtained as follows:

```
<- cut(simDataPart$numberOfSubjects, c(69, 70, 139, 140, 210, 279, 280),
subjectsRange labels = c(
"(69,70]", "(70,139]", "(139,140]",
"(140,210]", "(210,279]", "(279,280]"
)
)
kable(round(prop.table(table(simDataPart$stageNumber, subjectsRange), margin = 1) * 100, 1))
```

(69,70] | (70,139] | (139,140] | (140,210] | (210,279] | (279,280] |
---|---|---|---|---|---|

0 | 0.0 | 100.0 | 0.0 | 0.0 | 0.0 |

100 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

0 | 9.1 | 0.3 | 7.9 | 7.1 | 75.5 |

For this simulation, the originally planned sample size (70) was never selected for the third stage and in most of the cases the maximum of sample size (280) was used.

# References

Gernot Wassmer and Werner Brannath,

*Group Sequential and Confirmatory Adaptive Designs in Clinical Trials*, Springer 2016, ISBN 978-3319325606R-Studio,

*Data Visualization with ggplot2 - Cheat Sheet*, version 2.1, 2016, https://www.rstudio.com/wp-content/uploads/2016/11/ggplot2-cheatsheet-2.1.pdf

System: rpact 3.4.0, R version 4.2.2 (2022-10-31 ucrt), platform: x86_64-w64-mingw32

To cite R in publications use:

R Core Team (2022). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

To cite package ‘rpact’ in publications use:

Wassmer G, Pahlke F (2023). *rpact: Confirmatory Adaptive Clinical Trial Design and Analysis*. https://www.rpact.org, https://www.rpact.com, https://github.com/rpact-com/rpact, https://rpact-com.github.io/rpact/.