This R Markdown document provides many different examples that illustrate the usage of the *R generic function* `summary`

with rpact. This is a technical vignette and is to be considered mainly as a comprehensive overview of the possible summaries in rpact.

**First, load the rpact package**

`## [1] '3.2.1'`

The following options can be set globally:

`rpact.summary.output.size`

: one of c(“small”, “medium”, “large”); defines how many details will be included into the summary; default is “large”, i.e., all available details are displayed.

`rpact.summary.justify`

: one of c(“right”, “left”, “centre”); shall the values be right-justified (the default), left-justified or centered.

`rpact.summary.intervalFormat`

: defines how intervals will be displayed in the summary, default is “[%s; %s]”.

`rpact.summary.digits`

: defines how many digits are to be used for numeric values (default is 3).

`rpact.summary.digits.probs`

: defines how many digits are to be used for numeric values (default is one more than value of rpact.summary.digits, i.e., 4).

`rpact.summary.trim.zeroes`

: if TRUE (default) zeroes will always displayed as “0”, e.g. “0.000” will become “0”.

**Examples**

```
options("rpact.summary.output.size" = "small") # small, medium, large
options("rpact.summary.output.size" = "medium") # small, medium, large
options("rpact.summary.output.size" = "large") # small, medium, large
options("rpact.summary.intervalFormat" = "[%s; %s]")
options("rpact.summary.intervalFormat" = "%s - %s")
options("rpact.summary.enforceIntervalView" = TRUE)
options("rpact.summary.justify" = "left")
options("rpact.summary.justify" = "centre")
options("rpact.summary.justify" = "right")
```

```
summary(getDesignGroupSequential(beta = 0.05, typeOfDesign = "asKD", gammaA = 1,
typeBetaSpending = "bsOF"))
```

```
## Sequential analysis with a maximum of 3 looks (group sequential design)
##
## Kim & DeMets alpha spending design (gammaA = 1) and
## O'Brien & Fleming type beta spending, non-binding futility, one-sided overall
## significance level 2.5%, power 95%, undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 2.394 2.294 2.200
## Futility boundary (z-value scale) -0.993 0.982
## Cumulative alpha spent 0.0083 0.0167 0.0250
## Cumulative beta spent 0.0007 0.0164 0.0500
## Overall power 0.4259 0.8092 0.9500
```

```
## Fixed sample analysis
##
## O'Brien & Fleming design, one-sided significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
```

```
## Sequential analysis with a maximum of 4 looks (group sequential design)
##
## O'Brien & Fleming design, two-sided overall significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Cumulative alpha spent <0.0001 0.0012 0.0086 0.0250
## Overall power 0.0012 0.1494 0.5227 0.8000
```

```
## Sequential analysis with a maximum of 4 looks (group sequential design)
##
## O'Brien & Fleming design, two-sided overall significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Cumulative alpha spent 0.000004679 0.001207215 0.008644578 0.024999990
## Overall power 0.001247 0.149399 0.522709 0.800000
```

```
## Fixed sample analysis
##
## O'Brien & Fleming design, two-sided significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
```

```
## Sequential analysis with a maximum of 3 looks (group sequential design)
##
## O'Brien & Fleming design, non-binding futility, one-sided overall
## significance level 2.5%, power 80%, undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) -6.000 0.000
## Cumulative alpha spent 0.0003 0.0072 0.0250
## Overall power 0.0329 0.4426 0.8000
```

```
## Sequential analysis with a maximum of 3 looks (group sequential design)
##
## O'Brien & Fleming design, non-binding futility, one-sided overall
## significance level 2.5%, power 80%, undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.47109 2.45443 2.00404
## Futility boundary (z-value scale) -6.00000 0.00000
## Cumulative alpha spent 0.000259 0.007160 0.025000
## Overall power 0.032939 0.442575 0.800000
```

```
## Fixed sample analysis
##
## O'Brien & Fleming design, one-sided significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
```

```
## Sequential analysis with a maximum of 3 looks
## (inverse normal combination test design)
##
## O'Brien & Fleming design, non-binding futility, one-sided overall
## significance level 2.5%, power 80%, undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 0 1.000
## Cumulative alpha spent 0.0003 0.0072 0.0250
## Overall power 0.0377 0.4763 0.8000
```

```
## Fixed sample analysis
##
## O'Brien & Fleming design, one-sided significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
```

```
## Sequential analysis with a maximum of 4 looks
## (inverse normal combination test design)
##
## O'Brien & Fleming design, two-sided overall significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Cumulative alpha spent <0.0001 0.0012 0.0086 0.0250
## Overall power 0.0012 0.1494 0.5227 0.8000
```

```
## Sequential analysis with a maximum of 4 looks
## (inverse normal combination test design)
##
## O'Brien & Fleming design, two-sided overall significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Cumulative alpha spent 0.000004679 0.001207215 0.008644578 0.024999990
## Overall power 0.001247 0.149399 0.522709 0.800000
```

```
## Fixed sample analysis
##
## O'Brien & Fleming design, two-sided significance level 2.5%, power 80%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
```

```
## Sequential analysis with a maximum of 3 looks (Fisher's combination test design)
##
## Fisher's combination test design, one-sided overall significance level 2.5%,
## undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (p product scale) 0.0123085 0.0016636 0.0002911
## Cumulative alpha spent 0.0123 0.0196 0.0250
```

```
## Sequential analysis with a maximum of 3 looks (Fisher's combination test design)
##
## Fisher's combination test design, binding futility, one-sided overall
## significance level 2.5%, undefined endpoint.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (p product scale) 0.0193942 0.0028231 0.0005226
## Futility boundary (separate p-value scale) 0.100 0.200
## Cumulative alpha spent 0.0194 0.0240 0.0250
```

```
## Fixed sample analysis
##
## Fisher's combination test design, one-sided significance level 2.5%,
## undefined endpoint.
##
## Stage Fixed
## Efficacy boundary (p product scale) 0.025
```

```
## Sequential analysis with a maximum of 4 looks (Fisher's combination test design)
##
## Fisher's combination test design, one-sided overall significance level 2.5%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (p product scale) 0.01040479 0.00137037 0.00023506 0.00004581
## Cumulative alpha spent 0.010405 0.016661 0.021286 0.025000
```

```
## Sequential analysis with a maximum of 4 looks (Fisher's combination test design)
##
## Fisher's combination test design, one-sided overall significance level 2.5%,
## undefined endpoint.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (p product scale) 0.01040479 0.00137037 0.00023506 0.00004581
## Cumulative alpha spent 0.01040 0.01666 0.02129 0.02500
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect = -0.5, standard deviation = 1, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects 154.6
## Two-sided local significance level 0.0250
## Efficacy boundary (t) -0.364 - 0.364
##
## Legend:
## (t): treatment effect scale
```

`## Warning: Argument unknown in summary(...): 'alternative' = -0.5 will be ignored`

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects, alt. = 0.2 953.0
## Number of subjects, alt. = 0.4 240.2
## Number of subjects, alt. = 0.6 108.2
## Number of subjects, alt. = 0.8 62.0
## Number of subjects, alt. = 1 40.6
## Two-sided local significance level 0.0250
## Efficacy boundary (t), alt. = 0.2 -0.145 - 0.145
## Efficacy boundary (t), alt. = 0.4 -0.291 - 0.291
## Efficacy boundary (t), alt. = 0.6 -0.437 - 0.437
## Efficacy boundary (t), alt. = 0.8 -0.584 - 0.584
## Efficacy boundary (t), alt. = 1 -0.732 - 0.732
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
summary(getPowerMeans(sided = 1, alternative = c(-0.5,-0.3),
maxNumberOfSubjects = 100, directionUpper = FALSE))
```

```
## Power calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (one-sided).
## The results were calculated for a two-sample t-test,
## H0: mu(1) - mu(2) = 0, power directed towards smaller values,
## H1: effect as specified, standard deviation = 1, number of subjects = 100.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
## Power, alt. = -0.5 0.6969
## Power, alt. = -0.3 0.3175
## Number of subjects 100.0
## One-sided local significance level 0.0250
## Efficacy boundary (t) -0.397
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (one-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect = 0.5, standard deviation = 2.5, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
## Number of subjects 786.8
## One-sided local significance level 0.0250
## Efficacy boundary (t) 0.350
##
## Legend:
## (t): treatment effect scale
```

```
summary(getPowerMeans(thetaH0 = 0, alternative = 0.5, sided = 1, stDev = 2.5,
maxNumberOfSubjects = 100, directionUpper = FALSE))
```

```
## Power calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (one-sided).
## The results were calculated for a two-sample t-test,
## H0: mu(1) - mu(2) = 0, power directed towards smaller values, H1: effect = 0.5,
## standard deviation = 2.5, number of subjects = 100.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
## Power 0.0016
## Number of subjects 100.0
## One-sided local significance level 0.0250
## Efficacy boundary (t) -0.992
##
## Legend:
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (one-sided).
## The sample size was calculated for a one-sample t-test, H0: mu = 0,
## H1: effect = 0.5, standard deviation = 1, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
## Number of subjects 33.4
## One-sided local significance level 0.0250
## Efficacy boundary (t) 0.352
##
## Legend:
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a one-sample t-test, H0: mu = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects, alt. = 0.2 240.1
## Number of subjects, alt. = 0.4 61.9
## Number of subjects, alt. = 0.6 29.0
## Number of subjects, alt. = 0.8 17.5
## Number of subjects, alt. = 1 12.2
## Two-sided local significance level 0.0250
## Efficacy boundary (t), alt. = 0.2 -0.146 - 0.146
## Efficacy boundary (t), alt. = 0.4 -0.292 - 0.292
## Efficacy boundary (t), alt. = 0.6 -0.440 - 0.440
## Efficacy boundary (t), alt. = 0.8 -0.590 - 0.590
## Efficacy boundary (t), alt. = 1 -0.742 - 0.742
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect = 1.2, standard deviation = 5, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects 662.6
## Two-sided local significance level 0.0250
## Efficacy boundary (t) -0.873 - 0.873
##
## Legend:
## (t): treatment effect scale
```

```
summary(getSampleSizeMeans(thetaH0 = 0, alternative = 1.2, sided = 2, stDev = 5,
allocationRatioPlanned = 0))
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect = 1.2, standard deviation = 5, optimum planned allocation ratio = 1,
## power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects 662.6
## Two-sided local significance level 0.0250
## Efficacy boundary (t) -0.873 - 0.873
##
## Legend:
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The sample size was calculated for a one-sample t-test, H0: mu = 0,
## H1: effect = 1.2, standard deviation = 5, power 80%.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Number of subjects 167.5
## Two-sided local significance level 0.0250
## Efficacy boundary (t) -0.874 - 0.874
##
## Legend:
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.0967 0.7030 0.8000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.0003 0.0072 0.0250
## One-sided local significance level 0.0003 0.0071 0.0225
## Efficacy boundary (t), alt. = 0.2 0.322 0.160 0.107
## Efficacy boundary (t), alt. = 0.4 0.655 0.321 0.213
## Efficacy boundary (t), alt. = 0.6 1.013 0.483 0.319
## Efficacy boundary (t), alt. = 0.8 1.413 0.646 0.424
## Efficacy boundary (t), alt. = 1 1.882 0.812 0.528
## Futility boundary (t), alt. = 0.2 0.092 0.130
## Futility boundary (t), alt. = 0.4 0.184 0.261
## Futility boundary (t), alt. = 0.6 0.276 0.391
## Futility boundary (t), alt. = 0.8 0.368 0.522
## Futility boundary (t), alt. = 1 0.459 0.653
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.2176 0.6822
## Exit probability for efficacy (under H0) 0.0003 0.0062
## Exit probability for efficacy (under H1) 0.0967 0.6064
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.1209 0.0758
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.09667 0.70304 0.80000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.0002592 0.0071601 0.0250000
## One-sided local significance level 0.0002592 0.0070554 0.0225331
## Efficacy boundary (t), alt. = 0.2 0.322 0.160 0.107
## Efficacy boundary (t), alt. = 0.4 0.655 0.321 0.213
## Efficacy boundary (t), alt. = 0.6 1.013 0.483 0.319
## Efficacy boundary (t), alt. = 0.8 1.413 0.646 0.424
## Efficacy boundary (t), alt. = 1 1.882 0.812 0.528
## Futility boundary (t), alt. = 0.2 0.0921 0.1303
## Futility boundary (t), alt. = 0.4 0.184 0.261
## Futility boundary (t), alt. = 0.6 0.276 0.391
## Futility boundary (t), alt. = 0.8 0.368 0.522
## Futility boundary (t), alt. = 1 0.459 0.653
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.2176 0.6822
## Exit probability for efficacy (under H0) 0.0002592 0.0062354
## Exit probability for efficacy (under H1) 0.09667 0.60637
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.12094 0.07581
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
summary(getPowerMeans(getDesignGroupSequential(futilityBounds = c(1, 2)),
maxNumberOfSubjects = 100, alternative = 1))
```

```
## Power calculation for a continuous endpoint
##
## Sequential analysis with a maximum of 3 looks (group sequential design), overall
## significance level 2.5% (one-sided).
## The results were calculated for a two-sample t-test,
## H0: mu(1) - mu(2) = 0, power directed towards larger values, H1: effect = 1,
## standard deviation = 1, maximum number of subjects = 100.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.2700 0.9281 0.9563
## Expected number of subjects 57.6
## Number of subjects 33.3 66.7 100.0
## Cumulative alpha spent 0.0003 0.0072 0.0250
## One-sided local significance level 0.0003 0.0071 0.0225
## Efficacy boundary (t) 1.340 0.618 0.406
## Futility boundary (t) 0.352 0.500
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.3015 0.6702
## Exit probability for efficacy (under H0) 0.0003 0.0062
## Exit probability for efficacy (under H1) 0.2700 0.6582
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.0316 0.0120
##
## Legend:
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Sequential analysis with a maximum of 4 looks (group sequential design), overall
## significance level 2.5% (two-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Overall power 0.0012 0.1494 0.5227 0.8000
## Expected number of subjects, alt. = 0.2 806.0
## Expected number of subjects, alt. = 0.4 203.1
## Expected number of subjects, alt. = 0.6 91.5
## Expected number of subjects, alt. = 0.8 52.4
## Expected number of subjects, alt. = 1 34.4
## Number of subjects, alt. = 0.2 242.3 484.6 726.9 969.2
## Number of subjects, alt. = 0.4 61.1 122.1 183.2 244.2
## Number of subjects, alt. = 0.6 27.5 55.0 82.5 110.0
## Number of subjects, alt. = 0.8 15.8 31.5 47.3 63.0
## Number of subjects, alt. = 1 10.3 20.7 31.0 41.3
## Cumulative alpha spent <0.0001 0.0012 0.0086 0.0250
## Two-sided local significance level <0.0001 0.0012 0.0082 0.0221
## Efficacy boundary (t), alt. = 0.2 -0.602 - 0.602 -0.296 - 0.296 -0.197 - 0.197 -0.147 - 0.147
## Efficacy boundary (t), alt. = 0.4 -1.290 - 1.290 -0.600 - 0.600 -0.395 - 0.395 -0.295 - 0.295
## Efficacy boundary (t), alt. = 0.6 -2.204 - 2.204 -0.923 - 0.923 -0.597 - 0.597 -0.443 - 0.443
## Efficacy boundary (t), alt. = 0.8 -3.652 - 3.652 -1.276 - 1.276 -0.804 - 0.804 -0.592 - 0.592
## Efficacy boundary (t), alt. = 1 -6.468 - 6.468 -1.678 - 1.678 -1.019 - 1.019 -0.742 - 0.742
## Exit probability for efficacy (under H0) <0.0001 0.0012 0.0074
## Exit probability for efficacy (under H1) 0.0012 0.1482 0.3733
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Power calculation for a continuous endpoint
##
## Sequential analysis with a maximum of 4 looks (group sequential design), overall
## significance level 2.5% (two-sided).
## The results were calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, maximum number of subjects = 100.
##
## Stage 1 2 3 4
## Information rate 25% 50% 75% 100%
## Efficacy boundary (z-value scale) 4.579 3.238 2.644 2.289
## Overall power, alt. = 0 <0.0001 0.0012 0.0086 0.0250
## Overall power, alt. = 0.2 <0.0001 0.0056 0.0382 0.1033
## Overall power, alt. = 0.4 0.0002 0.0328 0.1779 0.3870
## Overall power, alt. = 0.6 0.0010 0.1264 0.4718 0.7564
## Overall power, alt. = 0.8 0.0046 0.3280 0.7831 0.9533
## Overall power, alt. = 1 0.0174 0.5996 0.9491 0.9961
## Expected number of subjects, alt. = 0 99.8
## Expected number of subjects, alt. = 0.2 98.9
## Expected number of subjects, alt. = 0.4 94.7
## Expected number of subjects, alt. = 0.6 85.0
## Expected number of subjects, alt. = 0.8 72.1
## Expected number of subjects, alt. = 1 60.8
## Number of subjects 25.0 50.0 75.0 100.0
## Cumulative alpha spent <0.0001 0.0012 0.0086 0.0250
## Two-sided local significance level <0.0001 0.0012 0.0082 0.0221
## Efficacy boundary (t) -2.376 - 2.376 -0.974 - 0.974 -0.628 - 0.628 -0.465 - 0.465
## Exit probability for efficacy (under H0) <0.0001 0.0012 0.0074
## Exit probability for efficacy (under H1), alt. = 0 <0.0001 0.0012 0.0074
## Exit probability for efficacy (under H1), alt. = 0.2 <0.0001 0.0056 0.0326
## Exit probability for efficacy (under H1), alt. = 0.4 0.0002 0.0326 0.1451
## Exit probability for efficacy (under H1), alt. = 0.6 0.0010 0.1255 0.3454
## Exit probability for efficacy (under H1), alt. = 0.8 0.0046 0.3234 0.4551
## Exit probability for efficacy (under H1), alt. = 1 0.0174 0.5822 0.3495
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
summary(getPowerMeans(getDesignGroupSequential(kMax = 1, sided = 2),
maxNumberOfSubjects = 100, directionUpper = TRUE))
```

```
## Power calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (two-sided).
## The results were calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, number of subjects = 100.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 2.241
## Power, alt. = 0 0.0250
## Power, alt. = 0.2 0.1055
## Power, alt. = 0.4 0.3947
## Power, alt. = 0.6 0.7642
## Power, alt. = 0.8 0.9561
## Power, alt. = 1 0.9965
## Number of subjects 100.0
## Two-sided local significance level 0.0250
## Efficacy boundary (t) -0.455 - 0.455
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
summary(getPowerMeans(getDesignGroupSequential(kMax = 1, sided = 1),
maxNumberOfSubjects = 100, directionUpper = FALSE))
```

```
## Power calculation for a continuous endpoint
##
## Fixed sample analysis, significance level 2.5% (one-sided).
## The results were calculated for a two-sample t-test,
## H0: mu(1) - mu(2) = 0, power directed towards smaller values,
## H1: effect as specified, standard deviation = 1, number of subjects = 100.
##
## Stage Fixed
## Efficacy boundary (z-value scale) 1.960
## Power, alt. = 0 0.0250
## Power, alt. = 0.2 0.0016
## Power, alt. = 0.4 <0.0001
## Power, alt. = 0.6 <0.0001
## Power, alt. = 0.8 <0.0001
## Power, alt. = 1 0
## Number of subjects 100.0
## One-sided local significance level 0.0250
## Efficacy boundary (t) -0.397
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous endpoint
##
## Sequential analysis with a maximum of 3 looks
## (inverse normal combination test design), overall significance level 2.5%
## (one-sided).
## The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.0967 0.7030 0.8000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.0003 0.0072 0.0250
## One-sided local significance level 0.0003 0.0071 0.0225
## Efficacy boundary (t), alt. = 0.2 0.322 0.160 0.107
## Efficacy boundary (t), alt. = 0.4 0.655 0.321 0.213
## Efficacy boundary (t), alt. = 0.6 1.013 0.483 0.319
## Efficacy boundary (t), alt. = 0.8 1.413 0.646 0.424
## Efficacy boundary (t), alt. = 1 1.882 0.812 0.528
## Futility boundary (t), alt. = 0.2 0.092 0.130
## Futility boundary (t), alt. = 0.4 0.184 0.261
## Futility boundary (t), alt. = 0.6 0.276 0.391
## Futility boundary (t), alt. = 0.8 0.368 0.522
## Futility boundary (t), alt. = 1 0.459 0.653
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.2176 0.6822
## Exit probability for efficacy (under H0) 0.0003 0.0062
## Exit probability for efficacy (under H1) 0.0967 0.6064
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.1209 0.0758
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.4711 2.4544 2.0040
## Futility boundary (z-value scale) 1.0000 2.0000
## Overall power 0.09667 0.70304 0.80000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.00026 0.00716 0.02500
## One-sided local significance level 0.00026 0.00706 0.02253
## Efficacy boundary (t), alt. = 0.2 0.3217 0.1600 0.1066
## Efficacy boundary (t), alt. = 0.4 0.6548 0.3207 0.2130
## Efficacy boundary (t), alt. = 0.6 1.0127 0.4826 0.3189
## Efficacy boundary (t), alt. = 0.8 1.4130 0.6462 0.4240
## Efficacy boundary (t), alt. = 1 1.8816 0.8123 0.5280
## Futility boundary (t), alt. = 0.2 0.0921 0.1303
## Futility boundary (t), alt. = 0.4 0.1842 0.2608
## Futility boundary (t), alt. = 0.6 0.2761 0.3913
## Futility boundary (t), alt. = 0.8 0.3678 0.5220
## Futility boundary (t), alt. = 1 0.4592 0.6529
## Overall exit probability (under H0) 0.84160 0.14622
## Overall exit probability (under H1) 0.21761 0.68219
## Exit probability for efficacy (under H0) 0.00026 0.00624
## Exit probability for efficacy (under H1) 0.09667 0.60637
## Exit probability for futility (under H0) 0.84134 0.13999
## Exit probability for futility (under H1) 0.12094 0.07581
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.0967 0.7030 0.8000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.0003 0.0072 0.0250
## One-sided local significance level 0.0003 0.0071 0.0225
## Efficacy boundary (t), alt. = 0.2 0.322 0.160 0.107
## Efficacy boundary (t), alt. = 0.4 0.655 0.321 0.213
## Efficacy boundary (t), alt. = 0.6 1.013 0.483 0.319
## Efficacy boundary (t), alt. = 0.8 1.413 0.646 0.424
## Efficacy boundary (t), alt. = 1 1.882 0.812 0.528
## Futility boundary (t), alt. = 0.2 0.092 0.130
## Futility boundary (t), alt. = 0.4 0.184 0.261
## Futility boundary (t), alt. = 0.6 0.276 0.391
## Futility boundary (t), alt. = 0.8 0.368 0.522
## Futility boundary (t), alt. = 1 0.459 0.653
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.2176 0.6822
## Exit probability for efficacy (under H0) 0.0003 0.0062
## Exit probability for efficacy (under H1) 0.0967 0.6064
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.1209 0.0758
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.47 2.45 2.00
## Futility boundary (z-value scale) 1.00 2.00
## Overall power 0.097 0.703 0.800
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent <0.001 0.007 0.025
## One-sided local significance level <0.001 0.007 0.023
## Efficacy boundary (t), alt. = 0.2 0.32 0.16 0.11
## Efficacy boundary (t), alt. = 0.4 0.65 0.32 0.21
## Efficacy boundary (t), alt. = 0.6 1.01 0.48 0.32
## Efficacy boundary (t), alt. = 0.8 1.41 0.65 0.42
## Efficacy boundary (t), alt. = 1 1.88 0.81 0.53
## Futility boundary (t), alt. = 0.2 0.09 0.13
## Futility boundary (t), alt. = 0.4 0.18 0.26
## Futility boundary (t), alt. = 0.6 0.28 0.39
## Futility boundary (t), alt. = 0.8 0.37 0.52
## Futility boundary (t), alt. = 1 0.46 0.65
## Overall exit probability (under H0) 0.842 0.146
## Overall exit probability (under H1) 0.218 0.682
## Exit probability for efficacy (under H0) <0.001 0.006
## Exit probability for efficacy (under H1) 0.097 0.606
## Exit probability for futility (under H0) 0.841 0.140
## Exit probability for futility (under H1) 0.121 0.076
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.471 2.454 2.004
## Futility boundary (z-value scale) 1.000 2.000
## Overall power 0.09667 0.70304 0.80000
## Expected number of subjects, alt. = 0.2 888.9
## Expected number of subjects, alt. = 0.4 223.9
## Expected number of subjects, alt. = 0.6 100.7
## Expected number of subjects, alt. = 0.8 57.7
## Expected number of subjects, alt. = 1 37.8
## Number of subjects, alt. = 0.2 472.2 944.4 1416.5
## Number of subjects, alt. = 0.4 118.9 237.8 356.8
## Number of subjects, alt. = 0.6 53.5 107.0 160.5
## Number of subjects, alt. = 0.8 30.6 61.3 91.9
## Number of subjects, alt. = 1 20.1 40.1 60.2
## Cumulative alpha spent 0.0002592 0.0071601 0.0250000
## One-sided local significance level 0.0002592 0.0070554 0.0225331
## Efficacy boundary (t), alt. = 0.2 0.322 0.160 0.107
## Efficacy boundary (t), alt. = 0.4 0.655 0.321 0.213
## Efficacy boundary (t), alt. = 0.6 1.013 0.483 0.319
## Efficacy boundary (t), alt. = 0.8 1.413 0.646 0.424
## Efficacy boundary (t), alt. = 1 1.882 0.812 0.528
## Futility boundary (t), alt. = 0.2 0.0921 0.1303
## Futility boundary (t), alt. = 0.4 0.184 0.261
## Futility boundary (t), alt. = 0.6 0.276 0.391
## Futility boundary (t), alt. = 0.8 0.368 0.522
## Futility boundary (t), alt. = 1 0.459 0.653
## Overall exit probability (under H0) 0.8416 0.1462
## Overall exit probability (under H1) 0.2176 0.6822
## Exit probability for efficacy (under H0) 0.0002592 0.0062354
## Exit probability for efficacy (under H1) 0.09667 0.60637
## Exit probability for futility (under H0) 0.8413 0.1400
## Exit probability for futility (under H1) 0.12094 0.07581
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```

```
## Sample size calculation for a continuous 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 t-test, H0: mu(1) - mu(2) = 0,
## H1: effect as specified, standard deviation = 1, power 80%.
##
## Stage 1 2 3
## Information rate 33.3% 66.7% 100%
## Efficacy boundary (z-value scale) 3.4710914446541 2.45443229863352 2.00403557995285
## Futility boundary (z-value scale) 1 2
## Overall power 0.0966650610605351 0.70304005701407 0.80000000002314
## Expected number of subjects, alt. = 0.2 888.920027605034
## Expected number of subjects, alt. = 0.4 223.87696109837
## Expected number of subjects, alt. = 0.6 100.743836089913
## Expected number of subjects, alt. = 0.8 57.6740643002557
## Expected number of subjects, alt. = 1 37.7677516807109
## Number of subjects, alt. = 0.2 472.175971190466 944.351942380932 1416.5279135714
## Number of subjects, alt. = 0.4 118.918820873684 237.837641747368 356.756462621052
## Number of subjects, alt. = 0.6 53.5130463596028 107.026092719206 160.539139078808
## Number of subjects, alt. = 0.8 30.6352725529709 61.2705451059418 91.9058176589127
## Number of subjects, alt. = 1 20.0614501594328 40.1229003188657 60.1843504782985
## Cumulative alpha spent 0.000259173723496486 0.00716005940148245 0.02499999
## One-sided local significance level 0.000259173723496486 0.00705536161371023 0.0225331246048346
## Efficacy boundary (t), alt. = 0.2 0.321710839190332 0.16003836007769 0.106587932791287
## Efficacy boundary (t), alt. = 0.4 0.654823493383795 0.320689953155161 0.212954736915394
## Efficacy boundary (t), alt. = 0.6 1.01268266453838 0.482561132966479 0.318855026247291
## Efficacy boundary (t), alt. = 0.8 1.41302933799472 0.646240615813088 0.423996481981201
## Efficacy boundary (t), alt. = 1 1.88164819392793 0.812280944461675 0.528020838325742
## Futility boundary (t), alt. = 0.2 0.09213829 0.13033753
## Futility boundary (t), alt. = 0.4 0.18418994 0.26075187
## Futility boundary (t), alt. = 0.6 0.27608063 0.39130341
## Futility boundary (t), alt. = 0.8 0.36776309 0.52201903
## Futility boundary (t), alt. = 1 0.45923643 0.65287366
## Overall exit probability (under H0) 0.841603919792039 0.146222739762505
## Overall exit probability (under H1) 0.217605032843878 0.682186663832684
## Exit probability for efficacy (under H0) 0.000259173723496486 0.00623541950983228
## Exit probability for efficacy (under H1) 0.0966650610605351 0.606374995953535
## Exit probability for futility (under H0) 0.841344746068543 0.139987320252672
## Exit probability for futility (under H1) 0.120939971783343 0.0758116678791493
##
## Legend:
## alt.: alternative
## (t): treatment effect scale
```