Free examples and use-cases:   rpact vignettes
rpact: Confirmatory Adaptive Clinical Trial Design and Analysis

Summary

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.

1 Global options

First, load the rpact package

library(rpact)
packageVersion("rpact") 
## [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")

2 Design summaries

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
summary(getDesignGroupSequential(kMax = 1))
## 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
summary(getDesignGroupSequential(kMax = 4, sided = 2))
## 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
summary(getDesignGroupSequential(kMax = 4, sided = 2), digits = 0)
## 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
summary(getDesignGroupSequential(kMax = 1, sided = 2))
## 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
summary(getDesignGroupSequential(futilityBounds = c(-6, 0)))
## 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
summary(getDesignGroupSequential(futilityBounds = c(-6, 0)), digits = 5)
## 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
summary(getDesignInverseNormal(kMax = 1))
## 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
summary(getDesignInverseNormal(futilityBounds = c(0, 1)))
## 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
summary(getDesignInverseNormal(kMax = 1))
## 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
summary(getDesignInverseNormal(kMax = 4, sided = 2))
## 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
summary(getDesignInverseNormal(kMax = 4, sided = 2), digits = 0)
## 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
summary(getDesignInverseNormal(kMax = 1, sided = 2))
## 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
summary(getDesignFisher())
## 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
summary(getDesignFisher(alpha0Vec = c(0.1, 0.2)))
## 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
summary(getDesignFisher(kMax = 1))
## 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
summary(getDesignFisher(kMax = 4), digits = 5)
## 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
summary(getDesignFisher(kMax = 4), digits = 0)
## 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

3 Design plan summaries

3.1 Design plan summaries - means

summary(getSampleSizeMeans(sided = 2, alternative = -0.5))
## 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
summary(getSampleSizeMeans(sided = 2), alternative = -0.5) # warning expected
## 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
summary(getSampleSizeMeans(thetaH0 = 0, alternative = 0.5, sided = 1, stDev = 2.5))
## 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
summary(getSampleSizeMeans(thetaH0 = 0, alternative = 0.5, sided = 1, stDev = 1, groups = 1))
## 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
summary(getSampleSizeMeans(thetaH0 = 0, sided = 2, stDev = 1, groups = 1))
## 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
summary(getSampleSizeMeans(thetaH0 = 0, alternative = 1.2, sided = 2, stDev = 5))
## 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
summary(getSampleSizeMeans(thetaH0 = 0, alternative = 1.2, sided = 2, stDev = 5, groups = 1))
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))))
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 0)
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(kMax = 4, sided = 2)))
## 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
summary(getPowerMeans(getDesignGroupSequential(kMax = 4, sided = 2), 
    maxNumberOfSubjects = 100))
## 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
summary(getSampleSizeMeans(getDesignInverseNormal(futilityBounds = c(1, 2))))
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 4)
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 3)
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 2)
## 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
summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 0)
## 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(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = -1)
## 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