How to Create Summaries with rpact

Utilities

This 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.

Author

Friedrich Pahlke and Gernot Wassmer

Published

July 6, 2023

Global options

First, load the rpact package

library(rpact)
packageVersion("rpact")

[1] '3.4.0'

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.justify" = "left")
options("rpact.summary.justify" = "centre")
options("rpact.summary.justify" = "right")

Design summaries

kable(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, inflation factor 1.1247, ASN H1 0.6553, ASN H01 0.8792, ASN H0 0.7415.

Stage	1	2	3
Information rate	33.3%	66.7%	100%
Efficacy boundary (z-value scale)	2.394	2.294	2.200
Stage levels (one-sided)	0.0083	0.0109	0.0139
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
Futility probabilities under H1	<0.001	0.016

Design parameters and output of group sequential design

User defined parameters

Type of design: Kim & DeMets alpha spending
Information rates: 0.333, 0.667, 1.000
Type II error rate: 0.0500
Parameter for alpha spending function: 1
Type of beta spending: O’Brien & Fleming type beta spending

Default parameters

Maximum number of stages: 3
Stages: 1, 2, 3
Significance level: 0.0250
Two-sided power: FALSE
Binding futility: FALSE
Test: one-sided
Tolerance: 0.00000001

Output

Power: 0.4259, 0.8092, 0.9500
Futility bounds (non-binding): -0.993, 0.982
Cumulative alpha spending: 0.008333, 0.016667, 0.025000
Cumulative beta spending: 0.0006869, 0.0163747, 0.0500000
Critical values: 2.394, 2.294, 2.200
Stage levels (one-sided): 0.008333, 0.010902, 0.013906

kable(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
Stage levels (one-sided)	0.0250

Design parameters and output of group sequential design

User defined parameters

Maximum number of stages: 1

Default parameters

Type of design: O’Brien & Fleming
Significance level: 0.0250
Type II error rate: 0.2000
Two-sided power: FALSE
Binding futility: FALSE
Test: one-sided
Tolerance: 0.00000001

Output

Critical values: 1.960
Stage levels (one-sided): 0.0250

kable(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, inflation factor 1.017, ASN H1 0.8458, ASN H01 0.9876, ASN H0 1.0144.

Stage	1	2	3	4
Information rate	25%	50%	75%	100%
Efficacy boundary (z-value scale)	4.579	3.238	2.644	2.289
Stage levels (one-sided)	<0.0001	0.0006	0.0041	0.0110
Cumulative alpha spent	<0.0001	0.0012	0.0086	0.0250
Overall power	0.0012	0.1494	0.5227	0.8000

Design parameters and output of group sequential design

User defined parameters

Maximum number of stages: 4
Stages: 1, 2, 3, 4
Information rates: 0.250, 0.500, 0.750, 1.000
Test: two-sided

Default parameters

Type of design: O’Brien & Fleming
Significance level: 0.0250
Type II error rate: 0.2000
Two-sided power: FALSE
Tolerance: 0.00000001

Output

Cumulative alpha spending: 0.000004679, 0.001207215, 0.008644578, 0.024999990
Critical values: 4.579, 3.238, 2.644, 2.289
Stage levels (one-sided): 0.000002339, 0.000602619, 0.004102447, 0.011029355

kable(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, inflation factor 1.017, ASN H1 0.8458, ASN H01 0.9876, ASN H0 1.0144.

Stage	1	2	3	4
Information rate	25%	50%	75%	100%
Efficacy boundary (z-value scale)	4.579	3.238	2.644	2.289
Stage levels (one-sided)	0.000002339	0.000602619	0.004102447	0.011029355
Cumulative alpha spent	0.000004679	0.001207215	0.008644578	0.024999990
Overall power	0.001247	0.149399	0.522709	0.800000

Design parameters and output of group sequential design

User defined parameters

Maximum number of stages: 4
Stages: 1, 2, 3, 4
Information rates: 0.250, 0.500, 0.750, 1.000
Test: two-sided

Default parameters

Type of design: O’Brien & Fleming
Significance level: 0.0250
Type II error rate: 0.2000
Two-sided power: FALSE
Tolerance: 0.00000001

Output

Cumulative alpha spending: 0.000004679, 0.001207215, 0.008644578, 0.024999990
Critical values: 4.579, 3.238, 2.644, 2.289
Stage levels (one-sided): 0.000002339, 0.000602619, 0.004102447, 0.011029355

kable(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, inflation factor 1.0178, ASN H1 0.8529, ASN H01 0.9413, ASN H0 0.8457.

Stage	1	2	3
Information rate	33.3%	66.7%	100%
Efficacy boundary (z-value scale)	3.47109	2.45443	2.00404
Stage levels (one-sided)	0.000259	0.007055	0.022533
Futility boundary (z-value scale)	-Inf	0
Cumulative alpha spent	0.000259	0.007160	0.025000
Overall power	0.032939	0.442575	0.800000
Futility probabilities under H1	0	0.01051

Design parameters and output of group sequential design

User defined parameters

Information rates: 0.333, 0.667, 1.000
Futility bounds (non-binding): -Inf, 0.000

Derived from user defined parameters

Maximum number of stages: 3

Default parameters

Type of design: O’Brien & Fleming
Stages: 1, 2, 3
Significance level: 0.0250
Type II error rate: 0.2000
Two-sided power: FALSE
Binding futility: FALSE
Test: one-sided
Tolerance: 0.00000001

Output

Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Critical values: 3.471, 2.454, 2.004
Stage levels (one-sided): 0.0002592, 0.0070554, 0.0225331

Design plan summaries

Design plan summaries - means

kable(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
Lower efficacy boundary (t)	-0.364
Upper efficacy boundary (t)	0.364

Legend:

(t): treatment effect scale

Design plan parameters and output for means

Design parameters

Critical values: 2.241
Two-sided power: FALSE
Significance level: 0.0250
Type II error rate: 0.2000
Test: two-sided

User defined parameters

Alternatives: -0.5

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Number of subjects fixed: 154.6
Number of subjects fixed (1): 77.3
Number of subjects fixed (2): 77.3
Lower critical values (treatment effect scale): -0.364
Upper critical values (treatment effect scale): 0.364
Local two-sided significance levels: 0.0250

Legend

(i): values of treatment arm i

kable(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

Design plan parameters and output for means

Design parameters

Critical values: 1.96
Significance level: 0.0250
Test: one-sided

User defined parameters

Alternatives: -0.5, -0.3
Direction upper: FALSE
Maximum number of subjects: 100

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Effect: -0.5, -0.3
Overall reject: 0.6969, 0.3175
Number of subjects fixed: 100
Number of subjects fixed (1): 50
Number of subjects fixed (2): 50
Critical values (treatment effect scale): -0.397

Legend

(i): values of treatment arm i

kable(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	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

Design plan parameters and output for means

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Alternatives: 0.2, 0.4, 0.6, 0.8, 1.0
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Reject per stage [1]: 0.09667
Reject per stage [2]: 0.60637
Reject per stage [3]: 0.09696
Overall futility stop: 0.1968
Futility stop per stage [1]: 0.12094
Futility stop per stage [2]: 0.07581
Early stop: 0.8998
Maximum number of subjects: 1416.5, 356.8, 160.5, 91.9, 60.2
Maximum number of subjects (1): 708.3, 178.4, 80.3, 46, 30.1
Maximum number of subjects (2): 708.3, 178.4, 80.3, 46, 30.1
Number of subjects [1]: 472.2, 118.9, 53.5, 30.6, 20.1
Number of subjects [2]: 944.4, 237.8, 107, 61.3, 40.1
Number of subjects [3]: 1416.5, 356.8, 160.5, 91.9, 60.2
Expected number of subjects under H0: 552.7, 139.2, 62.6, 35.9, 23.5
Expected number of subjects under H0/H1: 772.7, 194.6, 87.6, 50.1, 32.8
Expected number of subjects under H1: 888.9, 223.9, 100.7, 57.7, 37.8
Critical values (treatment effect scale) [1]: 0.322, 0.655, 1.013, 1.413, 1.882
Critical values (treatment effect scale) [2]: 0.160, 0.321, 0.483, 0.646, 0.812
Critical values (treatment effect scale) [3]: 0.107, 0.213, 0.319, 0.424, 0.528
Futility bounds (treatment effect scale) [1]: 0.0921, 0.1842, 0.2761, 0.3678, 0.4592
Futility bounds (treatment effect scale) [2]: 0.1303, 0.2608, 0.3913, 0.5220, 0.6529
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

kable(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

Design plan parameters and output for means

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Test: one-sided

User defined parameters

Alternatives: 1
Direction upper: NA
Maximum number of subjects: 100

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Effect: 1
Overall reject: 0.9563
Reject per stage [1]: 0.26996
Reject per stage [2]: 0.65816
Reject per stage [3]: 0.02814
Overall futility stop: 0.04359
Futility stop per stage [1]: 0.03157
Futility stop per stage [2]: 0.01202
Early stop: 0.9717
Expected number of subjects: 57.6
Number of subjects [1]: 33.3
Number of subjects [2]: 66.7
Number of subjects [3]: 100
Critical values (treatment effect scale) [1]: 1.340
Critical values (treatment effect scale) [2]: 0.618
Critical values (treatment effect scale) [3]: 0.406
Futility bounds (treatment effect scale) [1]: 0.352
Futility bounds (treatment effect scale) [2]: 0.500
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

kable(summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 3))

Sample size calculation for a continuous 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)	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

Design plan parameters and output for means

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Alternatives: 0.2, 0.4, 0.6, 0.8, 1.0
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Reject per stage [1]: 0.09667
Reject per stage [2]: 0.60637
Reject per stage [3]: 0.09696
Overall futility stop: 0.1968
Futility stop per stage [1]: 0.12094
Futility stop per stage [2]: 0.07581
Early stop: 0.8998
Maximum number of subjects: 1416.5, 356.8, 160.5, 91.9, 60.2
Maximum number of subjects (1): 708.3, 178.4, 80.3, 46, 30.1
Maximum number of subjects (2): 708.3, 178.4, 80.3, 46, 30.1
Number of subjects [1]: 472.2, 118.9, 53.5, 30.6, 20.1
Number of subjects [2]: 944.4, 237.8, 107, 61.3, 40.1
Number of subjects [3]: 1416.5, 356.8, 160.5, 91.9, 60.2
Expected number of subjects under H0: 552.7, 139.2, 62.6, 35.9, 23.5
Expected number of subjects under H0/H1: 772.7, 194.6, 87.6, 50.1, 32.8
Expected number of subjects under H1: 888.9, 223.9, 100.7, 57.7, 37.8
Critical values (treatment effect scale) [1]: 0.322, 0.655, 1.013, 1.413, 1.882
Critical values (treatment effect scale) [2]: 0.160, 0.321, 0.483, 0.646, 0.812
Critical values (treatment effect scale) [3]: 0.107, 0.213, 0.319, 0.424, 0.528
Futility bounds (treatment effect scale) [1]: 0.0921, 0.1842, 0.2761, 0.3678, 0.4592
Futility bounds (treatment effect scale) [2]: 0.1303, 0.2608, 0.3913, 0.5220, 0.6529
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

kable(summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = 0))

Sample size calculation for a continuous 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)	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	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

Design plan parameters and output for means

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Alternatives: 0.2, 0.4, 0.6, 0.8, 1.0
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Reject per stage [1]: 0.09667
Reject per stage [2]: 0.60637
Reject per stage [3]: 0.09696
Overall futility stop: 0.1968
Futility stop per stage [1]: 0.12094
Futility stop per stage [2]: 0.07581
Early stop: 0.8998
Maximum number of subjects: 1416.5, 356.8, 160.5, 91.9, 60.2
Maximum number of subjects (1): 708.3, 178.4, 80.3, 46, 30.1
Maximum number of subjects (2): 708.3, 178.4, 80.3, 46, 30.1
Number of subjects [1]: 472.2, 118.9, 53.5, 30.6, 20.1
Number of subjects [2]: 944.4, 237.8, 107, 61.3, 40.1
Number of subjects [3]: 1416.5, 356.8, 160.5, 91.9, 60.2
Expected number of subjects under H0: 552.7, 139.2, 62.6, 35.9, 23.5
Expected number of subjects under H0/H1: 772.7, 194.6, 87.6, 50.1, 32.8
Expected number of subjects under H1: 888.9, 223.9, 100.7, 57.7, 37.8
Critical values (treatment effect scale) [1]: 0.322, 0.655, 1.013, 1.413, 1.882
Critical values (treatment effect scale) [2]: 0.160, 0.321, 0.483, 0.646, 0.812
Critical values (treatment effect scale) [3]: 0.107, 0.213, 0.319, 0.424, 0.528
Futility bounds (treatment effect scale) [1]: 0.0921, 0.1842, 0.2761, 0.3678, 0.4592
Futility bounds (treatment effect scale) [2]: 0.1303, 0.2608, 0.3913, 0.5220, 0.6529
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

kable(summary(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = -1))

Sample size calculation for a continuous endpoint

Stage	1	2	3
Information rate	33.3%	66.7%	100%
Efficacy boundary (z-value scale)	3.4710914446541	2.45443229863353	2.00403557995285
Futility boundary (z-value scale)	1	2
Overall power	0.0966650610605349	0.703040057014069	0.80000000002314
Expected number of subjects, alt. = 0.2	888.920027605035
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	57.6740643002558
Expected number of subjects, alt. = 1	37.767751680711
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	0.00705536161371012	0.0225331246048346
Efficacy boundary (t), alt. = 0.2	0.321710839190332	0.160038360077691	0.106587932791287
Efficacy boundary (t), alt. = 0.4	0.654823493383795	0.320689953155162	0.212954736915394
Efficacy boundary (t), alt. = 0.6	1.01268266453838	0.48256113296648	0.318855026247291
Efficacy boundary (t), alt. = 0.8	1.41302933799472	0.646240615813089	0.423996481981201
Efficacy boundary (t), alt. = 1	1.88164819392793	0.812280944461677	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)	0.217605032843877	0.682186663832684
Exit probability for efficacy (under H0)	0.000259173723496486	0.00623541950983217
Exit probability for efficacy (under H1)	0.0966650610605349	0.606374995953535
Exit probability for futility (under H0)	0.841344746068543	0.139987320252672
Exit probability for futility (under H1)	0.120939971783342	0.0758116678791494

Legend:

alt.: alternative
(t): treatment effect scale

Design plan parameters and output for means

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

Default parameters

Mean ratio: FALSE
Theta H0: 0
Normal approximation: FALSE
Alternatives: 0.2, 0.4, 0.6, 0.8, 1.0
Standard deviation: 1
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Reject per stage [1]: 0.09667
Reject per stage [2]: 0.60637
Reject per stage [3]: 0.09696
Overall futility stop: 0.1968
Futility stop per stage [1]: 0.12094
Futility stop per stage [2]: 0.07581
Early stop: 0.8998
Maximum number of subjects: 1416.5, 356.8, 160.5, 91.9, 60.2
Maximum number of subjects (1): 708.3, 178.4, 80.3, 46, 30.1
Maximum number of subjects (2): 708.3, 178.4, 80.3, 46, 30.1
Number of subjects [1]: 472.2, 118.9, 53.5, 30.6, 20.1
Number of subjects [2]: 944.4, 237.8, 107, 61.3, 40.1
Number of subjects [3]: 1416.5, 356.8, 160.5, 91.9, 60.2
Expected number of subjects under H0: 552.7, 139.2, 62.6, 35.9, 23.5
Expected number of subjects under H0/H1: 772.7, 194.6, 87.6, 50.1, 32.8
Expected number of subjects under H1: 888.9, 223.9, 100.7, 57.7, 37.8
Critical values (treatment effect scale) [1]: 0.322, 0.655, 1.013, 1.413, 1.882
Critical values (treatment effect scale) [2]: 0.160, 0.321, 0.483, 0.646, 0.812
Critical values (treatment effect scale) [3]: 0.107, 0.213, 0.319, 0.424, 0.528
Futility bounds (treatment effect scale) [1]: 0.0921, 0.1842, 0.2761, 0.3678, 0.4592
Futility bounds (treatment effect scale) [2]: 0.1303, 0.2608, 0.3913, 0.5220, 0.6529
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

Design plan summaries - rates

kable(summary(getSampleSizeRates(pi2 = 0.3)))

Sample size calculation for a binary endpoint

Fixed sample analysis, 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) as specified, control rate pi(2) = 0.3, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	1.960
Number of subjects, pi(1) = 0.4	711.9
Number of subjects, pi(1) = 0.5	186.0
Number of subjects, pi(1) = 0.6	83.9
One-sided local significance level	0.0250
Efficacy boundary (t), pi(1) = 0.4	0.069
Efficacy boundary (t), pi(1) = 0.5	0.139
Efficacy boundary (t), pi(1) = 0.6	0.210

Legend:

(t): treatment effect scale

Design plan parameters and output for rates

Design parameters

Critical values: 1.96
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

User defined parameters

Assumed control rate: 0.300

Default parameters

Risk ratio: FALSE
Theta H0: 0
Normal approximation: TRUE
Assumed treatment rate: 0.400, 0.500, 0.600
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Direction upper: TRUE, TRUE, TRUE
Number of subjects fixed: 711.9, 186, 83.9
Number of subjects fixed (1): 355.9, 93, 42
Number of subjects fixed (2): 355.9, 93, 42
Critical values (treatment effect scale): 0.0693, 0.1387, 0.2100

Legend

(i): values of treatment arm i

kable(summary(getSampleSizeRates(getDesignGroupSequential(futilityBounds = c(1, 2)))))

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) as specified, control rate pi(2) = 0.2, 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, pi(1) = 0.4	183.5
Expected number of subjects, pi(1) = 0.5	86.9
Expected number of subjects, pi(1) = 0.6	50.5
Number of subjects, pi(1) = 0.4	97.5	195.0	292.5
Number of subjects, pi(1) = 0.5	46.2	92.4	138.6
Number of subjects, pi(1) = 0.6	26.8	53.6	80.4
Cumulative alpha spent	0.0003	0.0072	0.0250
One-sided local significance level	0.0003	0.0071	0.0225
Efficacy boundary (t), pi(1) = 0.4	0.339	0.158	0.102
Efficacy boundary (t), pi(1) = 0.5	0.509	0.238	0.152
Efficacy boundary (t), pi(1) = 0.6	0.669	0.322	0.205
Futility boundary (t), pi(1) = 0.4	0.087	0.126
Futility boundary (t), pi(1) = 0.5	0.130	0.190
Futility boundary (t), pi(1) = 0.6	0.175	0.257
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:

(t): treatment effect scale

Design plan parameters and output for rates

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 3.471, 2.454, 2.004
Futility bounds (non-binding): 1.000, 2.000
Cumulative alpha spending: 0.0002592, 0.0071601, 0.0250000
Local one-sided significance levels: 0.0002592, 0.0070554, 0.0225331
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

Default parameters

Risk ratio: FALSE
Theta H0: 0
Normal approximation: TRUE
Assumed treatment rate: 0.400, 0.500, 0.600
Assumed control rate: 0.200
Treatment groups: 2
Planned allocation ratio: 1

Sample size and output

Direction upper: TRUE, TRUE, TRUE
Reject per stage [1]: 0.09667
Reject per stage [2]: 0.60637
Reject per stage [3]: 0.09696
Overall futility stop: 0.1968
Futility stop per stage [1]: 0.12094
Futility stop per stage [2]: 0.07581
Early stop: 0.8998
Maximum number of subjects: 292.5, 138.6, 80.4
Maximum number of subjects (1): 146.2, 69.3, 40.2
Maximum number of subjects (2): 146.2, 69.3, 40.2
Number of subjects [1]: 97.5, 46.2, 26.8
Number of subjects [2]: 195, 92.4, 53.6
Number of subjects [3]: 292.5, 138.6, 80.4
Expected number of subjects under H0: 114.1, 54.1, 31.4
Expected number of subjects under H0/H1: 159.5, 75.6, 43.9
Expected number of subjects under H1: 183.5, 86.9, 50.5
Critical values (treatment effect scale) [1]: 0.339, 0.509, 0.669
Critical values (treatment effect scale) [2]: 0.158, 0.238, 0.322
Critical values (treatment effect scale) [3]: 0.102, 0.152, 0.205
Futility bounds (treatment effect scale) [1]: 0.0869, 0.1299, 0.1748
Futility bounds (treatment effect scale) [2]: 0.1261, 0.1898, 0.2566
Futility bounds (one-sided p-value scale) [1]: 0.15866
Futility bounds (one-sided p-value scale) [2]: 0.02275

Legend

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

kable(summary(getSampleSizeRates(getDesignGroupSequential(kMax = 1, sided = 2),
    groups = 1, thetaH0 = 0.2, pi1 = c(0.4, 0.5)
)))

Sample size calculation for a binary endpoint

Fixed sample analysis, significance level 2.5% (two-sided). The sample size was calculated for a one-sample test for rates (normal approximation), H0: pi = 0.2, H1: treatment rate pi as specified, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	2.241
Number of subjects, pi(1) = 0.4	42.8
Number of subjects, pi(1) = 0.5	19.3
Two-sided local significance level	0.0250
Lower efficacy boundary (t), pi(1) = 0.4	0.063
Lower efficacy boundary (t), pi(1) = 0.5	-0.004
Upper efficacy boundary (t), pi(1) = 0.4	0.337
Upper efficacy boundary (t), pi(1) = 0.5	0.404

Legend:

(t): treatment effect scale

Design plan parameters and output for rates

Design parameters

Critical values: 2.241
Two-sided power: FALSE
Significance level: 0.0250
Type II error rate: 0.2000
Test: two-sided

User defined parameters

Theta H0: 0.2
Assumed treatment rate: 0.400, 0.500
Treatment groups: 1

Default parameters

Normal approximation: TRUE

Sample size and output

Direction upper: TRUE, TRUE
Number of subjects fixed: 42.8, 19.3
Lower critical values (treatment effect scale): 0.06300, -0.00417
Upper critical values (treatment effect scale): 0.337, 0.404
Local two-sided significance levels: 0.0250

Design plan summaries - survival

kable(summary(getSampleSizeSurvival(lambda2 = 0.3, hazardRatio = 1.2)))

Sample size calculation for a survival endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The sample size was calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: hazard ratio = 1.2, control lambda(2) = 0.3, accrual time = 12, accrual intensity = 81.6, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	1.960
Number of subjects	979.2
Number of events	944.5
Analysis time	18.0
Expected study duration	18.0
One-sided local significance level	0.0250
Efficacy boundary (t)	1.136

Legend:

(t): treatment effect scale

Design plan parameters and output for survival data

Design parameters

Critical values: 1.96
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

User defined parameters

lambda(2): 0.300
Hazard ratio: 1.200
Accrual time: 12.00

Default parameters

Theta H0: 1
Type of computation: Schoenfeld
Planned allocation ratio: 1
kappa: 1
Piecewise survival times: 0.00
Follow up time: 6.00
Drop-out rate (1): 0.000
Drop-out rate (2): 0.000
Drop-out time: 12.00

Sample size and output

Direction upper: TRUE
median(1): 1.9
median(2): 2.3
lambda(1): 0.360
Number of events: 944.5
Accrual intensity: 81.6
Number of events fixed: 944.5
Number of subjects fixed: 979.2
Number of subjects fixed (1): 489.6
Number of subjects fixed (2): 489.6
Analysis times: 18.00
Study duration: 18.00
Critical values (treatment effect scale): 1.136

Legend

(i): values of treatment arm i

kable(summary(getSampleSizeSurvival(median1 = c(3.1, 3.2), median2 = 2.3)))

Sample size calculation for a survival endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The sample size was calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: treatment median(1) as specified, control median(2) = 2.3, accrual time = 12, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	1.960
Number of subjects, median(1) = 3.1	377.9
Number of subjects, median(1) = 3.2	309.7
Number of events, median(1) = 3.1	352.4
Number of events, median(1) = 3.2	287.9
Analysis time	18.0
Expected study duration, median(1) = 3.1	18.0
Expected study duration, median(1) = 3.2	18.0
One-sided local significance level	0.0250
Efficacy boundary (t), median(1) = 3.1	0.812
Efficacy boundary (t), median(1) = 3.2	0.794

Legend:

(t): treatment effect scale

Design plan parameters and output for survival data

Design parameters

Critical values: 1.96
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

User defined parameters

median(1): 3.1, 3.2
median(2): 2.3
Accrual time: 12.00

Default parameters

Theta H0: 1
Type of computation: Schoenfeld
Planned allocation ratio: 1
kappa: 1
Follow up time: 6.00
Drop-out rate (1): 0.000
Drop-out rate (2): 0.000
Drop-out time: 12.00

Sample size and output

Direction upper: FALSE, FALSE
lambda(1): 0.224, 0.217
lambda(2): 0.301
Hazard ratio: 0.742, 0.719
Number of events: 352.4, 287.9
Accrual intensity: 31.5, 25.8
Number of events fixed: 352.4, 287.9
Number of subjects fixed: 377.9, 309.7
Number of subjects fixed (1): 188.9, 154.9
Number of subjects fixed (2): 188.9, 154.9
Analysis times: 18.00
Study duration: 18.00
Critical values (treatment effect scale): 0.812, 0.794

Legend

(i): values of treatment arm i

kable(summary(getSampleSizeSurvival(pi1 = 0.1, pi2 = 0.3)))

Sample size calculation for a survival endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The sample size was calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: treatment pi(1) = 0.1, control pi(2) = 0.3, event time = 12, accrual time = 12, accrual intensity = 8.9, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	1.960
Number of subjects	106.7
Number of events	21.1
Analysis time	18.0
Expected study duration	18.0
One-sided local significance level	0.0250
Efficacy boundary (t)	0.426

Legend:

(t): treatment effect scale

Design plan parameters and output for survival data

Design parameters

Critical values: 1.96
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

User defined parameters

Assumed treatment rate: 0.100
Assumed control rate: 0.300
Accrual time: 12.00

Default parameters

Theta H0: 1
Type of computation: Schoenfeld
Planned allocation ratio: 1
Event time: 12
kappa: 1
Follow up time: 6.00
Drop-out rate (1): 0.000
Drop-out rate (2): 0.000
Drop-out time: 12.00

Sample size and output

Direction upper: FALSE
median(1): 78.9
median(2): 23.3
lambda(1): 0.00878
lambda(2): 0.0297
Hazard ratio: 0.295
Number of events: 21.1
Accrual intensity: 8.9
Number of events fixed: 21.1
Number of subjects fixed: 106.7
Number of subjects fixed (1): 53.3
Number of subjects fixed (2): 53.3
Analysis times: 18.00
Study duration: 18.00
Critical values (treatment effect scale): 0.426

Legend

(i): values of treatment arm i

piecewiseSurvivalTime <- list(
    "0 - <6" = 0.025,
    "6 - <9" = 0.04,
    "9 - <15" = 0.015,
    "15 - <21" = 0.01,
    ">= 21" = 0.007
)
kable(summary(getSampleSizeSurvival(
    piecewiseSurvivalTime = piecewiseSurvivalTime,
    hazardRatio = 1.2
)))

Sample size calculation for a survival endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The sample size was calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: hazard ratio = 1.2, piecewise survival distribution, piecewise survival time = c(0, 6, 9, 15, 21), control lambda(2) = c(0.025, 0.04, 0.015, 0.01, 0.007), accrual time = 12, accrual intensity = 279.2, power 80%.

Stage	Fixed
Efficacy boundary (z-value scale)	1.960
Number of subjects	3350.9
Number of events	944.5
Analysis time	18.0
Expected study duration	18.0
One-sided local significance level	0.0250
Efficacy boundary (t)	1.136

Legend:

(t): treatment effect scale

Design plan parameters and output for survival data

Design parameters

Critical values: 1.96
Significance level: 0.0250
Type II error rate: 0.2000
Test: one-sided

User defined parameters

lambda(2): 0.025, 0.040, 0.015, 0.010, 0.007
Hazard ratio: 1.200
Accrual time: 12.00
Piecewise survival times: 0.00, 6.00, 9.00, 15.00, 21.00

Default parameters

Theta H0: 1
Type of computation: Schoenfeld
Planned allocation ratio: 1
kappa: 1
Follow up time: 6.00
Drop-out rate (1): 0.000
Drop-out rate (2): 0.000
Drop-out time: 12.00

Sample size and output

Direction upper: TRUE
lambda(1): 0.0300, 0.0480, 0.0180, 0.0120, 0.0084
Number of events: 944.5
Accrual intensity: 279.2
Number of events fixed: 944.5
Number of subjects fixed: 3350.9
Number of subjects fixed (1): 1675.5
Number of subjects fixed (2): 1675.5
Analysis times: 18.00
Study duration: 18.00
Critical values (treatment effect scale): 1.136

Legend

(i): values of treatment arm i

Simulation results summaries

Simulation results base

Simulation results base - means

design <- getDesignInverseNormal(
    kMax = 3, alpha = 0.025,
    futilityBounds = c(-0.5, 0), bindingFutility = FALSE,
    typeOfDesign = "WT", deltaWT = 0.25,
    informationRates = c(0.4, 0.7, 1)
)
kable(summary(getSimulationMeans(
    design = design, plannedSubjects = c(40, 70, 100),
    alternative = seq(0, 0.8, 0.2),
    stDev = 1.2,
    conditionalPower = 0.8,
    minNumberOfSubjectsPerStage = c(40, 20, 20),
    maxNumberOfSubjectsPerStage = c(40, 100, 100),
    thetaH1 = 0.6, stDevH1 = 1.5,
    maxNumberOfIterations = 1000,
    seed = 1234
)))

Simulation of a continuous endpoint

Sequential analysis with a maximum of 3 looks (inverse normal combination test design), overall significance level 2.5% (one-sided). The results were simulated for a two-sample t-test (normal approximation), H0: mu(1) - mu(2) = 0, power directed towards larger values, H1: effect as specified, standard deviation = 1.2, planned cumulative sample size = c(40, 70, 100), sample size reassessment: conditional power = 0.8, minimum subjects per stage = c(40, 20, 20), maximum subjects per stage = c(40, 100, 100), theta H1 = 0.6, standard deviation H1 = 1.5, simulation runs = 1000, seed = 1234.

Stage	1	2	3
Fixed weight	0.632	0.548	0.548
Efficacy boundary (z-value scale)	2.631	2.287	2.092
Stage levels (one-sided)	0.0043	0.0111	0.0182
Futility boundary (z-value scale)	-0.500	0.000
Overall power, alt. = 0	0.0040	0.0110	0.0270
Overall power, alt. = 0.2	0.0190	0.0960	0.2090
Overall power, alt. = 0.4	0.0390	0.3250	0.6260
Overall power, alt. = 0.6	0.1330	0.7100	0.9300
Overall power, alt. = 0.8	0.2830	0.9320	0.9930
Expected number of subjects, alt. = 0	151.5
Expected number of subjects, alt. = 0.2	186.1
Expected number of subjects, alt. = 0.4	179.3
Expected number of subjects, alt. = 0.6	139.1
Expected number of subjects, alt. = 0.8	108.2
Stagewise number of subjects, alt. = 0	40.0	99.3	98.1
Stagewise number of subjects, alt. = 0.2	40.0	98.6	95.0
Stagewise number of subjects, alt. = 0.4	40.0	96.8	86.6
Stagewise number of subjects, alt. = 0.6	40.0	93.5	75.7
Stagewise number of subjects, alt. = 0.8	40.0	88.9	73.3
Conditional power (achieved), alt. = 0		0.2206	0.3011
Conditional power (achieved), alt. = 0.2		0.3112	0.4507
Conditional power (achieved), alt. = 0.4		0.4009	0.5970
Conditional power (achieved), alt. = 0.6		0.5292	0.7151
Conditional power (achieved), alt. = 0.8		0.6066	0.7396
Exit probability for futility, alt. = 0	0.3200	0.2170
Exit probability for futility, alt. = 0.2	0.1500	0.0790
Exit probability for futility, alt. = 0.4	0.0610	0.0110
Exit probability for futility, alt. = 0.6	0.0230	0
Exit probability for futility, alt. = 0.8	0.0030	0
Exit probability for efficacy, alt. = 0	0.0040	0.0070	0.0160
Exit probability for efficacy, alt. = 0.2	0.0190	0.0770	0.1130
Exit probability for efficacy, alt. = 0.4	0.0390	0.2860	0.3010
Exit probability for efficacy, alt. = 0.6	0.1330	0.5770	0.2200
Exit probability for efficacy, alt. = 0.8	0.2830	0.6490	0.0610

Legend:

alt.: alternative

Simulation of means (inverse normal combination test design)

Design parameters

Information rates: 0.400, 0.700, 1.000
Critical values: 2.631, 2.287, 2.092
Futility bounds (non-binding): -0.500, 0.000
Cumulative alpha spending: 0.004262, 0.013396, 0.025000
Local one-sided significance levels: 0.004262, 0.011095, 0.018220
Significance level: 0.0250
Test: one-sided

User defined parameters

Seed: 1234
Conditional power: 0.8
Standard deviation: 1.2
Planned cumulative subjects: 40, 70, 100
Minimum number of subjects per stage: 40, 20, 20
Maximum number of subjects per stage: 40, 100, 100
Assumed effect under alternative: 0.6
Assumed standard deviation under alternative: 1.5
Alternatives: 0.0, 0.2, 0.4, 0.6, 0.8

Default parameters

Maximum number of iterations: 1000
Planned allocation ratio: 1
Mean ratio: FALSE
Theta H0: 0
Normal approximation: TRUE
Treatment groups: 2
Direction upper: TRUE

Results

Iterations [1]: 1000, 1000, 1000, 1000, 1000
Iterations [2]: 676, 831, 900, 844, 714
Iterations [3]: 452, 675, 603, 267, 65
Overall reject: 0.0270, 0.2090, 0.6260, 0.9300, 0.9930
Reject per stage [1]: 0.0040, 0.0190, 0.0390, 0.1330, 0.2830
Reject per stage [2]: 0.0070, 0.0770, 0.2860, 0.5770, 0.6490
Reject per stage [3]: 0.0160, 0.1130, 0.3010, 0.2200, 0.0610
Overall futility stop: 0.5370, 0.2290, 0.0720, 0.0230, 0.0030
Futility stop per stage [1]: 0.3200, 0.1500, 0.0610, 0.0230, 0.0030
Futility stop per stage [2]: 0.2170, 0.0790, 0.0110, 0.0000, 0.0000
Early stop: 0.5480, 0.3250, 0.3970, 0.7330, 0.9350
Expected number of subjects: 151.5, 186.1, 179.3, 139.1, 108.2
Sample sizes [1]: 40, 40, 40, 40, 40
Sample sizes [2]: 99.3, 98.6, 96.8, 93.5, 88.9
Sample sizes [3]: 98.1, 95, 86.6, 75.7, 73.3
Conditional power (achieved) [1]: NA, NA, NA, NA, NA
Conditional power (achieved) [2]: 0.2206, 0.3112, 0.4009, 0.5292, 0.6066
Conditional power (achieved) [3]: 0.3011, 0.4507, 0.5970, 0.7151, 0.7396

Legend

[k]: values at stage k

Simulation results base - rates

design <- getDesignFisher(
    kMax = 3, alpha = 0.025,
    alpha0Vec = c(0.5, 0.4), bindingFutility = FALSE,
    informationRates = c(0.4, 0.7, 1)
)
kable(summary(getSimulationRates(
    design = design, plannedSubjects = c(40, 70, 100),
    groups = 1,
    thetaH0 = 0.2,
    pi1 = seq(0.05, 0.2, 0.05),
    directionUpper = FALSE,
    maxNumberOfIterations = 1000,
    seed = 1234
)))

Simulation of a binary endpoint

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 2.5% (one-sided). The results were simulated for a one-sample test for rates (normal approximation), H0: pi = 0.2, power directed towards smaller values, H1: treatment rate pi as specified, planned cumulative sample size = c(40, 70, 100), simulation runs = 1000, seed = 1234.

Stage	1	2	3
Fixed weight	1	0.866	0.866
Efficacy boundary (p product scale)	0.013187	0.002705	0.000641
Stage levels (one-sided)	0.0132	0.0132	0.0132
Futility boundary (separate p-value scale)	0.500	0.400
Overall power, pi(1) = 0.05	0.6790	0.9040	0.9780
Overall power, pi(1) = 0.1	0.2150	0.3760	0.5540
Overall power, pi(1) = 0.15	0.0390	0.0670	0.1070
Overall power, pi(1) = 0.2	0.0090	0.0100	0.0110
Expected number of subjects, pi(1) = 0.05	52.4
Expected number of subjects, pi(1) = 0.1	78.0
Expected number of subjects, pi(1) = 0.15	76.9
Expected number of subjects, pi(1) = 0.2	58.0
Stagewise number of subjects, pi(1) = 0.05	40.0	30.0	30.0
Stagewise number of subjects, pi(1) = 0.1	40.0	30.0	30.0
Stagewise number of subjects, pi(1) = 0.15	40.0	30.0	30.0
Stagewise number of subjects, pi(1) = 0.2	40.0	30.0	30.0
Conditional power (achieved), pi(1) = 0.05		0.4489	0.5698
Conditional power (achieved), pi(1) = 0.1		0.2752	0.3434
Conditional power (achieved), pi(1) = 0.15		0.1405	0.1871
Conditional power (achieved), pi(1) = 0.2		0.0820	0.0932
Exit probability for futility, pi(1) = 0.05	0.0010	0.0020
Exit probability for futility, pi(1) = 0.1	0.0410	0.0600
Exit probability for futility, pi(1) = 0.15	0.2250	0.2150
Exit probability for futility, pi(1) = 0.2	0.5680	0.2440
Exit probability for efficacy, pi(1) = 0.05	0.6790	0.2250	0.0740
Exit probability for efficacy, pi(1) = 0.1	0.2150	0.1610	0.1780
Exit probability for efficacy, pi(1) = 0.15	0.0390	0.0280	0.0400
Exit probability for efficacy, pi(1) = 0.2	0.0090	0.0010	0.0010

Simulation of rates (Fisher’s combination test design)

Design parameters

Information rates: 0.400, 0.700, 1.000
Critical values: 0.013187, 0.002705, 0.000641
Alpha_0: 0.5000, 0.4000
Cumulative alpha spending: 0.01319, 0.01987, 0.02500
Local one-sided significance levels: 0.01319, 0.01319, 0.01319
Significance level: 0.0250
Test: one-sided

User defined parameters

Seed: 1234
Direction upper: FALSE
Planned cumulative subjects: 40, 70, 100
Theta H0: 0.2
Assumed treatment rate: 0.050, 0.100, 0.150, 0.200
Treatment groups: 1

Default parameters

Maximum number of iterations: 1000
Risk ratio: FALSE
Normal approximation: TRUE

Results

Effect: -0.15, -0.10, -0.05, 0.00
Iterations [1]: 1000, 1000, 1000, 1000
Iterations [2]: 320, 744, 736, 423
Iterations [3]: 93, 523, 493, 178
Overall reject: 0.9780, 0.5540, 0.1070, 0.0110
Reject per stage [1]: 0.6790, 0.2150, 0.0390, 0.0090
Reject per stage [2]: 0.2250, 0.1610, 0.0280, 0.0010
Reject per stage [3]: 0.0740, 0.1780, 0.0400, 0.0010
Overall futility stop: 0.0030, 0.1010, 0.4400, 0.8120
Futility stop per stage [1]: 0.0010, 0.0410, 0.2250, 0.5680
Futility stop per stage [2]: 0.0020, 0.0600, 0.2150, 0.2440
Early stop: 0.9070, 0.4770, 0.5070, 0.8220
Expected number of subjects: 52.4, 78, 76.9, 58
Sample sizes [1]: 40, 40, 40, 40
Sample sizes [2]: 30, 30, 30, 30
Sample sizes [3]: 30, 30, 30, 30
Conditional power (achieved) [1]: NA, NA, NA, NA
Conditional power (achieved) [2]: 0.4489, 0.2752, 0.1405, 0.0820
Conditional power (achieved) [3]: 0.5698, 0.3434, 0.1871, 0.0932

Legend

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

Simulation results base - survival

design <- getDesignInverseNormal(
    alpha = 0.05, kMax = 4, futilityBounds = c(0, 0, 0),
    sided = 1, typeOfDesign = "WT", deltaWT = 0.1
)
kable(summary(getSimulationSurvival(
    design = design,
    plannedEvents = c(40, 70, 100, 150),
    maxNumberOfSubjects = 600,
    thetaH0 = 1.2,
    pi1 = seq(0.1, 0.25, 0.05),
    pi2 = 0.2,
    allocation1 = 2,
    directionUpper = FALSE,
    conditionalPower = 0.8,
    minNumberOfEventsPerStage = c(40, 20, 20, 20),
    maxNumberOfEventsPerStage = c(40, 100, 100, 100),
    thetaH1 = 1,
    maxNumberOfIterations = 1000,
    seed = 1234
)))

Simulation of a survival endpoint

Sequential analysis with a maximum of 4 looks (inverse normal combination test design), overall significance level 5% (one-sided). The results were simulated for a two-sample logrank test, H0: hazard ratio = 1.2, power directed towards smaller values, H1: treatment pi(1) as specified, control pi(2) = 0.2, planned cumulative events = c(40, 70, 100, 150), maximum number of subjects = 600, planned allocation ratio = 2, event time = 12, accrual time = 12, accrual intensity = 50, sample size reassessment: conditional power = 0.8, minimum events per stage = c(40, 20, 20, 20), maximum events per stage = c(40, 100, 100, 100), thetaH1 = 0.833, simulation runs = 1000, seed = 1234.

Stage	1	2	3	4
Fixed weight	0.5	0.5	0.5	0.5
Efficacy boundary (z-value scale)	3.069	2.326	1.978	1.763
Stage levels (one-sided)	0.0011	0.0100	0.0240	0.0390
Futility boundary (z-value scale)	0	0	0
Overall power, pi(1) = 0.1	0.4490	0.9970	0.9980	0.9980
Overall power, pi(1) = 0.15	0.0720	0.6740	0.9060	0.9350
Overall power, pi(1) = 0.2	0.0110	0.1030	0.2520	0.3910
Overall power, pi(1) = 0.25	0.0010	0.0030	0.0110	0.0220
Expected number of subjects, pi(1) = 0.1	589.6
Expected number of subjects, pi(1) = 0.15	589.7
Expected number of subjects, pi(1) = 0.2	564.5
Expected number of subjects, pi(1) = 0.25	508.4
Number of subjects, pi(1) = 0.1	576.9	600.0	600.0
Number of subjects, pi(1) = 0.15	522.5	600.0	600.0	600.0
Number of subjects, pi(1) = 0.2	474.5	600.0	600.0	600.0
Number of subjects, pi(1) = 0.25	434.7	600.0	600.0	600.0
Expected number of events, pi(1) = 0.1	95.0
Expected number of events, pi(1) = 0.15	156.3
Expected number of events, pi(1) = 0.2	210.6
Expected number of events, pi(1) = 0.25	129.3
Cumulative number of events, pi(1) = 0.1	40.0	100.0	100.0
Cumulative number of events, pi(1) = 0.15	40.0	100.0	100.0	100.0
Cumulative number of events, pi(1) = 0.2	40.0	100.0	100.0	100.0
Cumulative number of events, pi(1) = 0.25	40.0	100.0	100.0	100.0
Analysis time, pi(1) = 0.1	11.8	28.5	52.1
Analysis time, pi(1) = 0.15	10.5	23.6	39.7	61.5
Analysis time, pi(1) = 0.2	9.5	20.4	33.5	51.1
Analysis time, pi(1) = 0.25	8.7	18.0	29.1	43.8
Expected study duration, pi(1) = 0.1	21.0
Expected study duration, pi(1) = 0.15	26.8
Expected study duration, pi(1) = 0.2	32.1
Expected study duration, pi(1) = 0.25	18.5
Conditional power (achieved), pi(1) = 0.1		0.4405	0.6022
Conditional power (achieved), pi(1) = 0.15		0.2556	0.4813	0.5406
Conditional power (achieved), pi(1) = 0.2		0.1305	0.2567	0.3413
Conditional power (achieved), pi(1) = 0.25		0.0794	0.1206	0.1487
Exit probability for futility, pi(1) = 0.1	0.0020	0	0
Exit probability for futility, pi(1) = 0.15	0.0610	0.0010	0
Exit probability for futility, pi(1) = 0.2	0.2720	0.0520	0.0080
Exit probability for futility, pi(1) = 0.25	0.5530	0.1760	0.0810
Exit probability for efficacy, pi(1) = 0.1	0.4490	0.5480	0.0010	0
Exit probability for efficacy, pi(1) = 0.15	0.0720	0.6020	0.2320	0.0290
Exit probability for efficacy, pi(1) = 0.2	0.0110	0.0920	0.1490	0.1390
Exit probability for efficacy, pi(1) = 0.25	0.0010	0.0020	0.0080	0.0110

Simulation of survival data (inverse normal combination test design)

Design parameters

Information rates: 0.250, 0.500, 0.750, 1.000
Critical values: 3.069, 2.326, 1.978, 1.763
Futility bounds (non-binding): 0.000, 0.000, 0.000
Cumulative alpha spending: 0.001074, 0.010527, 0.028206, 0.050000
Local one-sided significance levels: 0.001074, 0.010012, 0.023983, 0.038976
Significance level: 0.0500
Test: one-sided

User defined parameters

Seed: 1234
Planned allocation ratio: 2
Conditional power: 0.8
Direction upper: FALSE
Planned cumulative events: 40, 70, 100, 150
Minimum number of events per stage: 40, 20, 20, 20
Maximum number of events per stage: 40, 100, 100, 100
Assumed treatment rate: 0.100, 0.150, 0.200, 0.250
Maximum number of subjects: 600
Accrual time: 12.00
Theta H0: 1.2
Allocation 1: 2

Default parameters

Maximum number of iterations: 1000
Assumed control rate: 0.200
Drop-out rate (1): 0.000
Drop-out rate (2): 0.000
Drop-out time: 12.00
Event time: 12
Allocation 2: 1
kappa: 1

Results

Assumed effect under alternative: 0.8333
median(1): 78.9, 51.2, 37.3, 28.9
median(2): 37.3
Accrual intensity: 50.0
lambda(1): 0.00878, 0.01354, 0.01860, 0.02397
lambda(2): 0.0186
Hazard ratio: 0.472, 0.728, 1.000, 1.289
Analysis times [1]: 11.83, 10.47, 9.50, 8.70
Analysis times [2]: 28.46, 23.56, 20.36, 18.05
Analysis times [3]: 52.12, 39.69, 33.48, 29.11
Analysis times [4]: NA, 61.52, 51.13, 43.84
Expected study duration: 21.01, 26.76, 32.14, 18.48
Events not achieved [1]: 0.000, 0.000, 0.000, 0.000
Events not achieved [2]: 0.000, 0.000, 0.000, 0.000
Events not achieved [3]: 0.000, 0.000, 0.000, 0.000
Events not achieved [4]: 0.000, 0.000, 0.000, 0.000
Number of subjects [1]: 576.9, 522.5, 474.5, 434.7
Number of subjects [2]: 600, 600, 600, 600
Number of subjects [3]: 600, 600, 600, 600
Number of subjects [4]: NA, 600, 600, 600
Number of subjects (1) [1]: 384.6, 348.3, 316.4, 289.8
Number of subjects (1) [2]: 400, 400, 400, 400
Number of subjects (1) [3]: 400, 400, 400, 400
Number of subjects (1) [4]: NA, 400, 400, 400
Number of subjects (2) [1]: 192.3, 174.2, 158.2, 144.9
Number of subjects (2) [2]: 200, 200, 200, 200
Number of subjects (2) [3]: 200, 200, 200, 200
Number of subjects (2) [4]: NA, 200, 200, 200
Number of events per stage [1]: 40, 40, 40, 40
Number of events per stage [2]: 100, 100, 100, 100
Number of events per stage [3]: 100, 100, 100, 100
Number of events per stage [4]: NA, 100, 100, 100
Cumulative number of events [1]: 40, 40, 39.999, 40
Cumulative number of events [2]: 140, 140, 139.999, 140
Cumulative number of events [3]: 240, 240, 239.999, 240
Cumulative number of events [4]: 240, 340, 339.999, 340
Iterations [1]: 1000, 1000, 1000, 1000
Iterations [2]: 549, 867, 717, 446
Iterations [3]: 1, 264, 573, 268
Iterations [4]: 0, 32, 416, 179
Overall reject: 0.9980, 0.9350, 0.3910, 0.0220
Reject per stage [1]: 0.4490, 0.0720, 0.0110, 0.0010
Reject per stage [2]: 0.5480, 0.6020, 0.0920, 0.0020
Reject per stage [3]: 0.0010, 0.2320, 0.1490, 0.0080
Reject per stage [4]: 0.0000, 0.0290, 0.1390, 0.0110
Overall futility stop: 0.0020, 0.0620, 0.3320, 0.8100
Futility stop per stage [1]: 0.0020, 0.0610, 0.2720, 0.5530
Futility stop per stage [2]: 0.0000, 0.0010, 0.0520, 0.1760
Futility stop per stage [3]: 0.0000, 0.0000, 0.0080, 0.0810
Early stop: 1.0000, 0.9680, 0.5840, 0.8210
Expected number of subjects: 589.6, 589.7, 564.5, 508.4
Expected number of events: 95, 156.3, 210.6, 129.3
Conditional power (achieved) [1]: NA, NA, NA, NA
Conditional power (achieved) [2]: 0.4405, 0.2556, 0.1305, 0.0794
Conditional power (achieved) [3]: 0.6022, 0.4813, 0.2567, 0.1206
Conditional power (achieved) [4]: NA, 0.5406, 0.3413, 0.1487

Legend

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

Simulation results multi-arm

Simulation results multi-arm - means

options("rpact.summary.output.size" = "medium") # small, medium, large
design <- getDesignFisher(alpha = 0.05, kMax = 3)
kable(summary(getSimulationMultiArmMeans(
    design = design,
    plannedSubjects = c(40, 70, 100),
    activeArms = 3,
    typeOfShape = "sigmoidEmax",
    gED50 = 2,
    typeOfSelection = "rBest",
    rValue = 2,
    stDev = 1.2,
    maxNumberOfIterations = 100,
    seed = 1234
)))

Simulation of a continuous endpoint (multi-arm design)

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 5% (one-sided). The results were simulated for a multi-arm comparisons for means (3 treatments vs. control), H0: mu(i) - mu(control) = 0, power directed towards larger values, H1: effect as specified, standard deviation = 1.2, planned cumulative sample size = c(40, 70, 100), effect shape = sigmoid emax, slope = 1, ED50 = 2, intersection test = Dunnett, selection = r best, r = 2, effect measure based on effect estimate, success criterion: all, simulation runs = 100, seed = 1234.

Stage	1	2	3
Fixed weight	1	1	1
Efficacy boundary (p product scale)	0.0255136	0.0038966	0.0007481
Stage levels (one-sided)	0.0255	0.0255	0.0255
Reject at least one, mu_max = 0	0.0500
Reject at least one, mu_max = 0.2	0.1400
Reject at least one, mu_max = 0.4	0.2900
Reject at least one, mu_max = 0.6	0.4800
Reject at least one, mu_max = 0.8	0.7100
Reject at least one, mu_max = 1	0.9100
Success per stage, mu_max = 0	0	0.0200	0
Success per stage, mu_max = 0.2	0	0.0200	0.0100
Success per stage, mu_max = 0.4	0.0200	0.0400	0.0400
Success per stage, mu_max = 0.6	0.0400	0.1600	0.0500
Success per stage, mu_max = 0.8	0.0700	0.2400	0.1100
Success per stage, mu_max = 1	0.2000	0.4000	0.0900
Expected number of subjects, mu_max = 0	338.2
Expected number of subjects, mu_max = 0.2	338.2
Expected number of subjects, mu_max = 0.4	332.8
Expected number of subjects, mu_max = 0.6	318.4
Expected number of subjects, mu_max = 0.8	305.8
Expected number of subjects, mu_max = 1	268.0
Overall exit probability, mu_max = 0	0	0.0200
Overall exit probability, mu_max = 0.2	0	0.0200
Overall exit probability, mu_max = 0.4	0.0200	0.0400
Overall exit probability, mu_max = 0.6	0.0400	0.1600
Overall exit probability, mu_max = 0.8	0.0700	0.2400
Overall exit probability, mu_max = 1	0.2000	0.4000
Stagewise number of subjects, mu_max = 0
Treatment arm 1	40.0	19.2	19.3
Treatment arm 2	40.0	18.9	19.0
Treatment arm 3	40.0	21.9	21.7
Control arm	40.0	30.0	30.0
Stagewise number of subjects, mu_max = 0.2
Treatment arm 1	40.0	17.4	17.8
Treatment arm 2	40.0	20.1	19.9
Treatment arm 3	40.0	22.5	22.3
Control arm	40.0	30.0	30.0
Stagewise number of subjects, mu_max = 0.4
Treatment arm 1	40.0	17.1	17.6
Treatment arm 2	40.0	20.8	20.7
Treatment arm 3	40.0	22.0	21.7
Control arm	40.0	30.0	30.0
Stagewise number of subjects, mu_max = 0.6
Treatment arm 1	40.0	15.9	16.5
Treatment arm 2	40.0	18.8	19.1
Treatment arm 3	40.0	25.3	24.4
Control arm	40.0	30.0	30.0
Stagewise number of subjects, mu_max = 0.8
Treatment arm 1	40.0	12.6	12.2
Treatment arm 2	40.0	22.9	23.5
Treatment arm 3	40.0	24.5	24.3
Control arm	40.0	30.0	30.0
Stagewise number of subjects, mu_max = 1
Treatment arm 1	40.0	12.4	16.5
Treatment arm 2	40.0	20.2	18.0
Treatment arm 3	40.0	27.4	25.5
Control arm	40.0	30.0	30.0
Selected arms, mu_max = 0
Treatment arm 1	1.0000	0.6400	0.6300
Treatment arm 2	1.0000	0.6300	0.6200
Treatment arm 3	1.0000	0.7300	0.7100
Selected arms, mu_max = 0.2
Treatment arm 1	1.0000	0.5800	0.5800
Treatment arm 2	1.0000	0.6700	0.6500
Treatment arm 3	1.0000	0.7500	0.7300
Selected arms, mu_max = 0.4
Treatment arm 1	1.0000	0.5600	0.5500
Treatment arm 2	1.0000	0.6800	0.6500
Treatment arm 3	1.0000	0.7200	0.6800
Selected arms, mu_max = 0.6
Treatment arm 1	1.0000	0.5100	0.4400
Treatment arm 2	1.0000	0.6000	0.5100
Treatment arm 3	1.0000	0.8100	0.6500
Selected arms, mu_max = 0.8
Treatment arm 1	1.0000	0.3900	0.2800
Treatment arm 2	1.0000	0.7100	0.5400
Treatment arm 3	1.0000	0.7600	0.5600
Selected arms, mu_max = 1
Treatment arm 1	1.0000	0.3300	0.2200
Treatment arm 2	1.0000	0.5400	0.2400
Treatment arm 3	1.0000	0.7300	0.3400
Number of active arms, mu_max = 0	3.000	2.000	2.000
Number of active arms, mu_max = 0.2	3.000	2.000	2.000
Number of active arms, mu_max = 0.4	3.000	2.000	2.000
Number of active arms, mu_max = 0.6	3.000	2.000	2.000
Number of active arms, mu_max = 0.8	3.000	2.000	2.000
Number of active arms, mu_max = 1	3.000	2.000	2.000

Legend:

(i): treatment arm i

Simulation of multi-arm means (Fisher’s combination test design)

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 0.0255136, 0.0038966, 0.0007481
Alpha_0: 1.0000, 1.0000
Cumulative alpha spending: 0.02551, 0.03981, 0.05000
Local one-sided significance levels: 0.02551, 0.02551, 0.02551
Significance level: 0.0500
Test: one-sided

User defined parameters

Maximum number of iterations: 100
Seed: 1234
Standard deviation: 1.2
Planned cumulative subjects: 40, 70, 100
Type of shape: sigmoidEmax
ED50: 2
Type of selection: rBest
r value: 2

Default parameters

Planned allocation ratio: 1
Calculate subjects function: default
Active arms: 3
Effect matrix (1): 0.00000, 0.06667, 0.13333, 0.20000, 0.26667, 0.33333
Effect matrix (2): 0.00000, 0.10000, 0.20000, 0.30000, 0.40000, 0.50000
Effect matrix (3): 0.00000, 0.12000, 0.24000, 0.36000, 0.48000, 0.60000
mu_max: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0
Slope: 1
Intersection test: Dunnett
Adaptations: TRUE, TRUE
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
Threshold: -Inf

Results

Iterations [1]: 100, 100, 100, 100, 100, 100
Iterations [2]: 100, 100, 98, 96, 93, 80
Iterations [3]: 98, 98, 94, 80, 69, 40
Reject at least one: 0.0500, 0.1400, 0.2900, 0.4800, 0.7100, 0.9100
Rejected arms per stage (1) [1]: 0.0000, 0.0300, 0.0300, 0.1000, 0.1300, 0.2300
Rejected arms per stage (1) [2]: 0.0100, 0.0000, 0.0200, 0.0300, 0.0700, 0.0800
Rejected arms per stage (1) [3]: 0.0100, 0.0100, 0.0100, 0.0500, 0.0000, 0.0300
Rejected arms per stage (2) [1]: 0.0100, 0.0400, 0.0700, 0.1400, 0.2400, 0.3800
Rejected arms per stage (2) [2]: 0.0100, 0.0200, 0.0300, 0.0600, 0.0700, 0.1600
Rejected arms per stage (2) [3]: 0.0000, 0.0200, 0.0400, 0.0300, 0.1000, 0.0600
Rejected arms per stage (3) [1]: 0.0200, 0.0100, 0.1100, 0.1700, 0.2500, 0.5000
Rejected arms per stage (3) [2]: 0.0100, 0.0300, 0.0600, 0.1200, 0.2100, 0.2200
Rejected arms per stage (3) [3]: 0.0000, 0.0100, 0.0400, 0.0700, 0.1300, 0.1400
Overall futility stop: 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000
Futility stop per stage [1]: 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000
Futility stop per stage [2]: 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000
Early stop [1]: 0.0000, 0.0000, 0.0200, 0.0400, 0.0700, 0.2000
Early stop [2]: 0.0200, 0.0200, 0.0400, 0.1600, 0.2400, 0.4000
Success per stage [1]: 0.0000, 0.0000, 0.0200, 0.0400, 0.0700, 0.2000
Success per stage [2]: 0.0200, 0.0200, 0.0400, 0.1600, 0.2400, 0.4000
Success per stage [3]: 0.0000, 0.0100, 0.0400, 0.0500, 0.1100, 0.0900
Selected arms (1) [1]: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (1) [2]: 0.6400, 0.5800, 0.5600, 0.5100, 0.3900, 0.3300
Selected arms (1) [3]: 0.6300, 0.5800, 0.5500, 0.4400, 0.2800, 0.2200
Selected arms (2) [1]: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (2) [2]: 0.6300, 0.6700, 0.6800, 0.6000, 0.7100, 0.5400
Selected arms (2) [3]: 0.6200, 0.6500, 0.6500, 0.5100, 0.5400, 0.2400
Selected arms (3) [1]: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (3) [2]: 0.7300, 0.7500, 0.7200, 0.8100, 0.7600, 0.7300
Selected arms (3) [3]: 0.7100, 0.7300, 0.6800, 0.6500, 0.5600, 0.3400
Selected arms (4) [1]: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (4) [2]: 1.0000, 1.0000, 0.9800, 0.9600, 0.9300, 0.8000
Selected arms (4) [3]: 0.9800, 0.9800, 0.9400, 0.8000, 0.6900, 0.4000
Number of active arms [1]: 3.000, 3.000, 3.000, 3.000, 3.000, 3.000
Number of active arms [2]: 2.000, 2.000, 2.000, 2.000, 2.000, 2.000
Number of active arms [3]: 2.000, 2.000, 2.000, 2.000, 2.000, 2.000
Expected number of subjects: 338.2, 338.2, 332.8, 318.4, 305.8, 268
Sample sizes (1) [1]: 40, 40, 40, 40, 40, 40
Sample sizes (1) [2]: 19.2, 17.4, 17.1, 15.9, 12.6, 12.4
Sample sizes (1) [3]: 19.3, 17.8, 17.6, 16.5, 12.2, 16.5
Sample sizes (2) [1]: 40, 40, 40, 40, 40, 40
Sample sizes (2) [2]: 18.9, 20.1, 20.8, 18.8, 22.9, 20.2
Sample sizes (2) [3]: 19, 19.9, 20.7, 19.1, 23.5, 18
Sample sizes (3) [1]: 40, 40, 40, 40, 40, 40
Sample sizes (3) [2]: 21.9, 22.5, 22, 25.3, 24.5, 27.4
Sample sizes (3) [3]: 21.7, 22.3, 21.7, 24.4, 24.3, 25.5
Sample sizes (4) [1]: 40, 40, 40, 40, 40, 40
Sample sizes (4) [2]: 30, 30, 30, 30, 30, 30
Sample sizes (4) [3]: 30, 30, 30, 30, 30, 30
Conditional power (achieved) [1]: NA, NA, NA, NA, NA, NA
Conditional power (achieved) [2]: 0.0324, 0.0787, 0.1360, 0.2446, 0.3514, 0.4881
Conditional power (achieved) [3]: 0.0257, 0.1019, 0.1782, 0.2398, 0.4298, 0.4535

Legend

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

Simulation results multi-arm - rates

options("rpact.summary.output.size" = "medium") # small, medium, large
kable(summary(getSimulationMultiArmRates(
    design = design,
    plannedSubjects = c(40, 70, 100),
    activeArms = 3,
    typeOfShape = "userDefined",
    effectMatrix = matrix(c(
        0.1, 0.2, 0.3,
        0.2, 0.3, 0.4,
        0.2, 0.4, 0.4
    ), nrow = 3),
    typeOfSelection = "rBest",
    rValue = 2,
    directionUpper = FALSE,
    allocationRatioPlanned = 2,
    piControl = 0.4,
    conditionalPower = 0.8,
    minNumberOfSubjectsPerStage = c(40, 20, 20),
    maxNumberOfSubjectsPerStage = c(40, 100, 100),
    piH1 = 0.6, piControlH1 = 0.4,
    maxNumberOfIterations = 100,
    seed = 1234
)))

Warning: Argument unknown in getSimulationMultiArmRates(...): 'piH1' = 0.6 will
be ignored

Simulation of a binary endpoint (multi-arm design)

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 5% (one-sided). The results were simulated for a multi-arm comparisons for rates (3 treatments vs. control), H0: pi(i) - pi(control) = 0, power directed towards smaller values, H1: treatment rate pi_max as specified, control rate pi(control) = 0.4, planned cumulative sample size = c(40, 70, 100), planned allocation ratio = 2, effect shape = user defined, intersection test = Dunnett, selection = r best, r = 2, effect measure based on effect estimate, success criterion: all, sample size reassessment: conditional power = 0.8, minimum subjects per stage = c(40, 20, 20), maximum subjects per stage = c(40, 100, 100), pi(control)H1 = 0.4, simulation runs = 100, seed = 1234.

Stage	1	2	3
Fixed weight	1	1	1
Efficacy boundary (p product scale)	0.0255136	0.0038966	0.0007481
Stage levels (one-sided)	0.0255	0.0255	0.0255
Reject at least one [1]	1.0000
Reject at least one [2]	0.9400
Reject at least one [3]	0.4000
Success per stage [1]	0.1800	0.4800	0.1100
Success per stage [2]	0	0.1300	0.1100
Success per stage [3]	0.0100	0.0200	0.0200
Expected number of subjects [1]	259.9
Expected number of subjects [2]	484.6
Expected number of subjects [3]	622.7
Overall exit probability [1]	0.1800	0.4800
Overall exit probability [2]	0	0.1300
Overall exit probability [3]	0.0100	0.0200
Stagewise number of subjects [1]
Treatment arm 1	40.0	46.2	22.6
Treatment arm 2	40.0	27.0	14.4
Treatment arm 3	40.0	22.6	14.1
Control arm	20.0	23.9	12.7
Stagewise number of subjects [2]
Treatment arm 1	40.0	86.7	56.5
Treatment arm 2	40.0	70.9	43.1
Treatment arm 3	40.0	17.8	15.7
Control arm	20.0	43.8	28.8
Stagewise number of subjects [3]
Treatment arm 1	40.0	92.8	91.9
Treatment arm 2	40.0	44.4	44.5
Treatment arm 3	40.0	60.5	59.7
Control arm	20.0	49.5	49.0
Selected arms [1]
Treatment arm 1	1.0000	0.8000	0.3300
Treatment arm 2	1.0000	0.4300	0.1700
Treatment arm 3	1.0000	0.4100	0.1800
Selected arms [2]
Treatment arm 1	1.0000	0.9900	0.8600
Treatment arm 2	1.0000	0.8000	0.6800
Treatment arm 3	1.0000	0.2100	0.2000
Selected arms [3]
Treatment arm 1	1.0000	0.9300	0.9100
Treatment arm 2	1.0000	0.4400	0.4400
Treatment arm 3	1.0000	0.6100	0.5900
Number of active arms [1]	3.000	2.000	2.000
Number of active arms [2]	3.000	2.000	2.000
Number of active arms [3]	3.000	2.000	2.000

Legend:

(i): treatment arm i
[j]: effect matrix row j (situation to consider)

Simulation of multi-arm rates (Fisher’s combination test design)

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 0.0255136, 0.0038966, 0.0007481
Alpha_0: 1.0000, 1.0000
Cumulative alpha spending: 0.02551, 0.03981, 0.05000
Local one-sided significance levels: 0.02551, 0.02551, 0.02551
Significance level: 0.0500
Test: one-sided

User defined parameters

Maximum number of iterations: 100
Seed: 1234
Planned allocation ratio: 2
Conditional power: 0.8
Direction upper: FALSE
Planned cumulative subjects: 40, 70, 100
Minimum number of subjects per stage: 40, 20, 20
Maximum number of subjects per stage: 40, 100, 100
Effect matrix (1): 0.1, 0.2, 0.3
Effect matrix (2): 0.2, 0.3, 0.4
Effect matrix (3): 0.2, 0.4, 0.4
Type of shape: userDefined
Assumed control rate: 0.400
pi(control) under H1: 0.400
Type of selection: rBest
r value: 2

Derived from user defined parameters

pi_max: 0.200, 0.400, 0.400

Default parameters

Calculate subjects function: default
Active arms: 3
pi(treatment) under H1: NA
Slope: 1
Intersection test: Dunnett
Adaptations: TRUE, TRUE
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
Threshold: -Inf

Results

Iterations [1]: 100, 100, 100
Iterations [2]: 82, 100, 99
Iterations [3]: 34, 87, 97
Reject at least one: 1.0000, 0.9400, 0.4000
Rejected arms per stage (1) [1]: 0.6400, 0.2500, 0.0800
Rejected arms per stage (1) [2]: 0.3200, 0.5300, 0.1100
Rejected arms per stage (1) [3]: 0.0200, 0.1600, 0.2100
Rejected arms per stage (2) [1]: 0.2700, 0.0300, 0.0100
Rejected arms per stage (2) [2]: 0.1700, 0.0900, 0.0000
Rejected arms per stage (2) [3]: 0.0500, 0.1000, 0.0000
Rejected arms per stage (3) [1]: 0.2400, 0.0100, 0.0300
Rejected arms per stage (3) [2]: 0.1700, 0.0000, 0.0000
Rejected arms per stage (3) [3]: 0.0700, 0.0100, 0.0200
Overall futility stop: 0.0000, 0.0000, 0.0000
Futility stop per stage [1]: 0.0000, 0.0000, 0.0000
Futility stop per stage [2]: 0.0000, 0.0000, 0.0000
Early stop [1]: 0.1800, 0.0000, 0.0100
Early stop [2]: 0.4800, 0.1300, 0.0200
Success per stage [1]: 0.1800, 0.0000, 0.0100
Success per stage [2]: 0.4800, 0.1300, 0.0200
Success per stage [3]: 0.1100, 0.1100, 0.0200
Selected arms (1) [1]: 1.0000, 1.0000, 1.0000
Selected arms (1) [2]: 0.8000, 0.9900, 0.9300
Selected arms (1) [3]: 0.3300, 0.8600, 0.9100
Selected arms (2) [1]: 1.0000, 1.0000, 1.0000
Selected arms (2) [2]: 0.4300, 0.8000, 0.4400
Selected arms (2) [3]: 0.1700, 0.6800, 0.4400
Selected arms (3) [1]: 1.0000, 1.0000, 1.0000
Selected arms (3) [2]: 0.4100, 0.2100, 0.6100
Selected arms (3) [3]: 0.1800, 0.2000, 0.5900
Selected arms (4) [1]: 1.0000, 1.0000, 1.0000
Selected arms (4) [2]: 0.8200, 1.0000, 0.9900
Selected arms (4) [3]: 0.3400, 0.8700, 0.9700
Number of active arms [1]: 3.000, 3.000, 3.000
Number of active arms [2]: 2.000, 2.000, 2.000
Number of active arms [3]: 2.000, 2.000, 2.000
Expected number of subjects: 259.9, 484.6, 622.7
Sample sizes (1) [1]: 40, 40, 40
Sample sizes (1) [2]: 46.2, 86.7, 92.8
Sample sizes (1) [3]: 22.6, 56.5, 91.9
Sample sizes (2) [1]: 40, 40, 40
Sample sizes (2) [2]: 27, 70.9, 44.4
Sample sizes (2) [3]: 14.4, 43.1, 44.5
Sample sizes (3) [1]: 40, 40, 40
Sample sizes (3) [2]: 22.6, 17.8, 60.5
Sample sizes (3) [3]: 14.1, 15.7, 59.7
Sample sizes (4) [1]: 20, 20, 20
Sample sizes (4) [2]: 23.9, 43.8, 49.5
Sample sizes (4) [3]: 12.7, 28.8, 49
Conditional power (achieved) [1]: NA, NA, NA
Conditional power (achieved) [2]: 0.7129, 0.1913, 0.0452
Conditional power (achieved) [3]: 0.9512, 0.7013, 0.1104

Legend

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

Simulation results multi-arm - survival

options("rpact.summary.output.size" = "medium") # small, medium, large
kable(summary(getSimulationMultiArmSurvival(
    seed = 1234,
    getDesignInverseNormal(informationRates = c(0.2, 0.6, 1)),
    typeOfShape = "linear", activeArms = 4,
    plannedEvents = c(10, 30, 50), omegaMaxVector = seq(0.3, 0.6, 0.1),
    adaptations = rep(TRUE, 2), directionUpper = FALSE,
    minNumberOfEventsPerStage = c(10, 4, 4), maxNumberOfEventsPerStage = c(10, 100, 100),
    maxNumberOfIterations = 10,
    calcEventsFunction = function(..., stage, minNumberOfEventsPerStage) {
        return(ifelse(stage == 3, 33, minNumberOfEventsPerStage[stage]))
    }
)))

Simulation of a survival endpoint (multi-arm design)

Sequential analysis with a maximum of 3 looks (inverse normal combination test design), overall significance level 2.5% (one-sided). The results were simulated for a multi-arm logrank test (4 treatments vs. control), H0: hazard ratio(i) = 1, power directed towards smaller values, H1: omega_max as specified, planned cumulative events = c(10, 30, 50), effect shape = linear, intersection test = Dunnett, selection = best, effect measure based on effect estimate, success criterion: all, sample size reassessment: user defined ‘calcEventsFunction’, minimum events per stage = c(10, 4, 4), maximum events per stage = c(10, 100, 100), simulation runs = 10, seed = 1234.

Stage	1	2	3
Fixed weight	0.447	0.632	0.632
Efficacy boundary (z-value scale)	4.455	2.572	1.992
Stage levels (one-sided)	<0.0001	0.0051	0.0232
Reject at least one, omega_max = 0.3	0.3000
Reject at least one, omega_max = 0.4	0.4000
Reject at least one, omega_max = 0.5	0.7000
Reject at least one, omega_max = 0.6	0.3000
Success per stage, omega_max = 0.3	0	0.1000	0.2000
Success per stage, omega_max = 0.4	0	0	0.4000
Success per stage, omega_max = 0.5	0	0.2000	0.5000
Success per stage, omega_max = 0.6	0	0.1000	0.2000
Expected number of events, omega_max = 0.3	43.7
Expected number of events, omega_max = 0.4	47.0
Expected number of events, omega_max = 0.5	40.4
Expected number of events, omega_max = 0.6	43.7
Overall exit probability, omega_max = 0.3	0	0.1000
Overall exit probability, omega_max = 0.4	0	0
Overall exit probability, omega_max = 0.5	0	0.2000
Overall exit probability, omega_max = 0.6	0	0.1000
Cumulative number of events, omega_max = 0.3
Treatment arm 1 vs. control	5.6	8.5	32.5
Treatment arm 2 vs. control	5.1	8.1	33.4
Treatment arm 3 vs. control	4.5	7.8	33.7
Treatment arm 4 vs. control	4.0	7.0	31.9
Cumulative number of events, omega_max = 0.4
Treatment arm 1 vs. control	5.3	8.0	30.2
Treatment arm 2 vs. control	4.9	7.7	31.3
Treatment arm 3 vs. control	4.4	7.7	34.6
Treatment arm 4 vs. control	4.0	7.3	34.2
Cumulative number of events, omega_max = 0.5
Treatment arm 1 vs. control	5.0	7.7	30.0
Treatment arm 2 vs. control	4.7	7.3	29.5
Treatment arm 3 vs. control	4.3	7.4	32.6
Treatment arm 4 vs. control	4.0	7.0	31.6
Cumulative number of events, omega_max = 0.6
Treatment arm 1 vs. control	4.8	7.4	28.5
Treatment arm 2 vs. control	4.5	7.0	28.0
Treatment arm 3 vs. control	4.2	7.4	34.3
Treatment arm 4 vs. control	4.0	6.6	28.7
Selected arms, omega_max = 0.3
Treatment arm 1 vs. control	1.0000	0.1000	0.1000
Treatment arm 2 vs. control	1.0000	0.2000	0.2000
Treatment arm 3 vs. control	1.0000	0.4000	0.3000
Treatment arm 4 vs. control	1.0000	0.3000	0.3000
Selected arms, omega_max = 0.4
Treatment arm 1 vs. control	1.0000	0	0
Treatment arm 2 vs. control	1.0000	0.1000	0.1000
Treatment arm 3 vs. control	1.0000	0.4000	0.4000
Treatment arm 4 vs. control	1.0000	0.5000	0.5000
Selected arms, omega_max = 0.5
Treatment arm 1 vs. control	1.0000	0.1000	0.1000
Treatment arm 2 vs. control	1.0000	0.1000	0.1000
Treatment arm 3 vs. control	1.0000	0.4000	0.3000
Treatment arm 4 vs. control	1.0000	0.4000	0.3000
Selected arms, omega_max = 0.6
Treatment arm 1 vs. control	1.0000	0.2000	0.1000
Treatment arm 2 vs. control	1.0000	0.1000	0.1000
Treatment arm 3 vs. control	1.0000	0.5000	0.5000
Treatment arm 4 vs. control	1.0000	0.2000	0.2000
Number of active arms, omega_max = 0.3	4.000	1.000	1.000
Number of active arms, omega_max = 0.4	4.000	1.000	1.000
Number of active arms, omega_max = 0.5	4.000	1.000	1.000
Number of active arms, omega_max = 0.6	4.000	1.000	1.000

Legend:

(i): results of treatment arm i vs. control arm

Simulation of multi-arm survival data (inverse normal combination test design)

Design parameters

Information rates: 0.200, 0.600, 1.000
Critical values: 4.455, 2.572, 1.992
Futility bounds (non-binding): -Inf, -Inf
Cumulative alpha spending: 0.000004202, 0.005059438, 0.024999990
Local one-sided significance levels: 0.000004202, 0.005057577, 0.023176744
Significance level: 0.0250
Test: one-sided

User defined parameters

Maximum number of iterations: 10
Seed: 1234
Direction upper: FALSE
Planned cumulative events: 10, 30, 50
Minimum number of events per stage: 10, 4, 4
Maximum number of events per stage: 10, 100, 100
Calculate events function: user defined
Active arms: 4
omega_max: 0.300, 0.400, 0.500, 0.600

Default parameters

Planned allocation ratio: 1
Effect matrix (1): 0.825, 0.850, 0.875, 0.900
Effect matrix (2): 0.650, 0.700, 0.750, 0.800
Effect matrix (3): 0.475, 0.550, 0.625, 0.700
Effect matrix (4): 0.300, 0.400, 0.500, 0.600
Type of shape: linear
Slope: 1
Intersection test: Dunnett
Adaptations: TRUE, TRUE
Type of selection: best
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
r value: NA
Threshold: -Inf

Results

Number of events per stage (1) [1]: 5.6, 5.3, 5, 4.8
Number of events per stage (1) [2]: 8.5, 8, 7.7, 7.4
Number of events per stage (1) [3]: 32.5, 30.2, 30, 28.5
Number of events per stage (2) [1]: 5.1, 4.9, 4.7, 4.5
Number of events per stage (2) [2]: 8.1, 7.7, 7.3, 7
Number of events per stage (2) [3]: 33.4, 31.3, 29.5, 28
Number of events per stage (3) [1]: 4.5, 4.4, 4.3, 4.2
Number of events per stage (3) [2]: 7.8, 7.7, 7.4, 7.4
Number of events per stage (3) [3]: 33.7, 34.6, 32.6, 34.3
Number of events per stage (4) [1]: 4, 4, 4, 4
Number of events per stage (4) [2]: 7, 7.3, 7, 6.6
Number of events per stage (4) [3]: 31.9, 34.2, 31.6, 28.7
Iterations [1]: 10, 10, 10, 10
Iterations [2]: 10, 10, 10, 10
Iterations [3]: 9, 10, 8, 9
Reject at least one: 0.3000, 0.4000, 0.7000, 0.3000
Rejected arms per stage (1) [1]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (1) [2]: 0.0000, 0.0000, 0.0000, 0.1000
Rejected arms per stage (1) [3]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (2) [1]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (2) [2]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (2) [3]: 0.0000, 0.1000, 0.0000, 0.0000
Rejected arms per stage (3) [1]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (3) [2]: 0.1000, 0.0000, 0.1000, 0.0000
Rejected arms per stage (3) [3]: 0.1000, 0.0000, 0.2000, 0.1000
Rejected arms per stage (4) [1]: 0.0000, 0.0000, 0.0000, 0.0000
Rejected arms per stage (4) [2]: 0.0000, 0.0000, 0.1000, 0.0000
Rejected arms per stage (4) [3]: 0.1000, 0.3000, 0.3000, 0.1000
Overall futility stop: 0.0000, 0.0000, 0.0000, 0.0000
Futility stop per stage [1]: 0.0000, 0.0000, 0.0000, 0.0000
Futility stop per stage [2]: 0.0000, 0.0000, 0.0000, 0.0000
Early stop [1]: 0.0000, 0.0000, 0.0000, 0.0000
Early stop [2]: 0.1000, 0.0000, 0.2000, 0.1000
Success per stage [1]: 0.0000, 0.0000, 0.0000, 0.0000
Success per stage [2]: 0.1000, 0.0000, 0.2000, 0.1000
Success per stage [3]: 0.2000, 0.4000, 0.5000, 0.2000
Selected arms (1) [1]: 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (1) [2]: 0.1000, 0.0000, 0.1000, 0.2000
Selected arms (1) [3]: 0.1000, 0.0000, 0.1000, 0.1000
Selected arms (2) [1]: 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (2) [2]: 0.2000, 0.1000, 0.1000, 0.1000
Selected arms (2) [3]: 0.2000, 0.1000, 0.1000, 0.1000
Selected arms (3) [1]: 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (3) [2]: 0.4000, 0.4000, 0.4000, 0.5000
Selected arms (3) [3]: 0.3000, 0.4000, 0.3000, 0.5000
Selected arms (4) [1]: 1.0000, 1.0000, 1.0000, 1.0000
Selected arms (4) [2]: 0.3000, 0.5000, 0.4000, 0.2000
Selected arms (4) [3]: 0.3000, 0.5000, 0.3000, 0.2000
Number of active arms [1]: 4.000, 4.000, 4.000, 4.000
Number of active arms [2]: 1.000, 1.000, 1.000, 1.000
Number of active arms [3]: 1.000, 1.000, 1.000, 1.000
Expected number of events: 43.7, 47, 40.4, 43.7
Single number of events {1} [1]: 2.5, 2.4, 2.3, 2.2
Single number of events {1} [2]: 0.2, 0, 0.2, 0.4
Single number of events {1} [3]: 1.7, 0, 1.9, 1.7
Single number of events {2} [1]: 2, 2, 2, 2
Single number of events {2} [2]: 0.3, 0.2, 0.2, 0.2
Single number of events {2} [3]: 2.9, 1.4, 1.8, 1.6
Single number of events {3} [1]: 1.5, 1.6, 1.7, 1.8
Single number of events {3} [2]: 0.5, 0.6, 0.6, 0.8
Single number of events {3} [3]: 3.5, 4.7, 4.8, 7.5
Single number of events {4} [1]: 0.9, 1.1, 1.3, 1.5
Single number of events {4} [2]: 0.3, 0.6, 0.5, 0.3
Single number of events {4} [3]: 2.5, 4.7, 4.1, 2.8
Single number of events {control} [1]: 3.1, 2.9, 2.7, 2.5
Single number of events {control} [2]: 2.7, 2.7, 2.5, 2.3
Single number of events {control} [3]: 22.4, 22.2, 20.4, 19.3
Conditional power (achieved) [1]: NA, NA, NA, NA
Conditional power (achieved) [2]: 0.1323, 0.3350, 0.3248, 0.1917
Conditional power (achieved) [3]: 0.2868, 0.6077, 0.6094, 0.3748

Legend

(i): values of treatment arm i compared to control
{j}: values of treatment arm j
[k]: values at stage k

Simulation results enrichment

Simulation results enrichment - means

options("rpact.summary.output.size" = "medium") # small, medium, large

design <- getDesignFisher(alpha = 0.05, kMax = 3)

subGroups <- c("S", "R")
prevalences <- c(0.1, 0.9)
alternative <- c(0.4, 0.5)
effectList <- list(
    subGroups = subGroups, prevalences = prevalences,
    stDevs = 1,
    effects = matrix(alternative, byrow = TRUE, ncol = 2)
)

kable(summary(getSimulationEnrichmentMeans(
    design = design,
    plannedSubjects = c(40, 70, 100),
    effectList = effectList,
    typeOfSelection = "rBest",
    rValue = 2,
    maxNumberOfIterations = 100,
    seed = 1234
)))

Warning: Simulation of enrichment designs is experimental and hence not fully
validated (see www.rpact.com/experimental)

Simulation of a continuous endpoint (enrichment design)

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 5% (one-sided). The results were simulated for a population enrichment comparisons for means (treatment vs. control, 2 populations), H0: mu(treatment) - mu(control) = 0, power directed towards larger values, H1: effects = c(0.4, 0.5), subgroups = c(S, R), prevalences = c(0.1, 0.9), standard deviation = 1, planned cumulative sample size = c(40, 70, 100), intersection test = Simes, selection = r best, r = 2, effect measure based on effect estimate, success criterion: all, simulation runs = 100, seed = 1234.

Stage	1	2	3
Fixed weight	1	1	1
Efficacy boundary (p product scale)	0.0255136	0.0038966	0.0007481
Stage levels (one-sided)	0.0255	0.0255	0.0255
Reject at least one	0.5600
Success per stage	0.0300	0.0400	0.0400
Expected number of subjects	97.0
Overall exit probability	0.0300	0.0400
Stagewise number of subjects
Subset S	4.0	3.0	3.0
Remaining population R	36.0	27.0	27.0
Selected populations
Subset S	1.0000	0.9700	0.9300
Full population F	1.0000	0.9700	0.9300
Number of populations	2.000	2.000	2.000

Simulation of enrichment means (Fisher’s combination test design)

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 0.0255136, 0.0038966, 0.0007481
Alpha_0: 1.0000, 1.0000
Cumulative alpha spending: 0.02551, 0.03981, 0.05000
Local one-sided significance levels: 0.02551, 0.02551, 0.02551
Significance level: 0.0500
Test: one-sided

User defined parameters

Maximum number of iterations: 100
Seed: 1234
Planned cumulative subjects: 40, 70, 100
Effect list [Sub-groups]: S, R
Effect list [Prevalences]: 0.1, 0.9
Effect list [Standard deviations]: 1, 1
Effect list [Effects]: 0.4, 0.5
Type of selection: rBest
r value: 2

Derived from user defined parameters

Populations: 2

Default parameters

Planned allocation ratio: 1
Calculate subjects function: default
Intersection test: Simes
Stratified analysis: TRUE
Adaptations: TRUE, TRUE
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
Threshold: -Inf

Results

Iterations [1]: 100
Iterations [2]: 97
Iterations [3]: 93
Reject at least one: 0.5600
Rejected populations per stage (S) [1]: 0.04
Rejected populations per stage (S) [2]: 0.05
Rejected populations per stage (S) [3]: 0.02
Rejected populations per stage (F) [1]: 0.28
Rejected populations per stage (F) [2]: 0.17
Rejected populations per stage (F) [3]: 0.11
Overall futility stop: 0.0000
Futility stop per stage [1]: 0.0000
Futility stop per stage [2]: 0.0000
Early stop [1]: 0.0300
Early stop [2]: 0.0400
Success per stage [1]: 0.0300
Success per stage [2]: 0.0400
Success per stage [3]: 0.0400
Selected populations (S) [1]: 1
Selected populations (S) [2]: 0.97
Selected populations (S) [3]: 0.93
Selected populations (F) [1]: 1
Selected populations (F) [2]: 0.97
Selected populations (F) [3]: 0.93
Number of populations [1]: 2
Number of populations [2]: 2
Number of populations [3]: 2
Expected number of subjects: 97
Sample sizes (S) [1]: 4
Sample sizes (S) [2]: 3
Sample sizes (S) [3]: 3
Sample sizes (R) [1]: 36
Sample sizes (R) [2]: 27
Sample sizes (R) [3]: 27
Conditional power (achieved) [1]: NA
Conditional power (achieved) [2]: 0.3092
Conditional power (achieved) [3]: 0.3745

Legend

[k]: values at stage k
S[i]: population i
F: full population
R: remaining population

Simulation results enrichment - rates

options("rpact.summary.output.size" = "large") # small, medium, large

design <- getDesignFisher(alpha = 0.05, kMax = 3)

subGroups <- c("S", "R")
prevalences <- c(0.1, 0.9)
pi2 <- c(0.3, 0.4)
piTreatments <- c(0.4, 0.5)
effectList <- list(
    subGroups = subGroups, prevalences = prevalences,
    piControl = pi2, piTreatments = matrix(piTreatments, byrow = TRUE, ncol = 2)
)

kable(summary(getSimulationEnrichmentRates(
    design = design,
    plannedSubjects = c(40, 70, 100),
    effectList = effectList,
    typeOfSelection = "rBest",
    rValue = 2,
    maxNumberOfIterations = 100,
    seed = 1234
)))

Warning: Simulation of enrichment designs is experimental and hence not fully
validated (see www.rpact.com/experimental)

Simulation of a binary endpoint (enrichment design)

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 5% (one-sided). The results were simulated for a population enrichment comparisons for rates (treatment vs. control, 2 populations), H0: pi(treatment) - pi(control) = 0, power directed towards larger values, H1: assumed treatment rate pi(treatment) = c(0.4, 0.5), subgroups = c(S, R), prevalences = c(0.1, 0.9), control rates pi(control) = c(0.3, 0.4), planned cumulative sample size = c(40, 70, 100), intersection test = Simes, selection = r best, r = 2, effect measure based on effect estimate, success criterion: all, simulation runs = 100, seed = 1234.

Stage	1	2	3
Fixed weight	1	1	1
Efficacy boundary (p product scale)	0.0255136	0.0038966	0.0007481
Stage levels (one-sided)	0.0255	0.0255	0.0255
Reject at least one	0.1800
Rejected populations per stage
Subset S	0.0200	0.0500	0.0400
Full population F	0.0500	0.0400	0.0700
Success per stage	0.0200	0.0300	0.0400
Expected number of subjects	97.9
Overall exit probability	0.0200	0.0300
Stagewise number of subjects
Subset S	4.0	3.0	3.0
Remaining population R	36.0	27.0	27.0
Selected populations
Subset S	1.0000	0.9800	0.9500
Full population F	1.0000	0.9800	0.9500
Number of populations	2.000	2.000	2.000
Conditional power (achieved)		0.0689	0.0731

Simulation of enrichment rates (Fisher’s combination test design)

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 0.0255136, 0.0038966, 0.0007481
Alpha_0: 1.0000, 1.0000
Cumulative alpha spending: 0.02551, 0.03981, 0.05000
Local one-sided significance levels: 0.02551, 0.02551, 0.02551
Significance level: 0.0500
Test: one-sided

User defined parameters

Maximum number of iterations: 100
Seed: 1234
Planned cumulative subjects: 40, 70, 100
Effect list [Sub-groups]: S, R
Effect list [Prevalences]: 0.1, 0.9
Effect list [Assumed control rates]: 0.3, 0.4
Effect list [Assumed treatment rates]: 0.4, 0.5
Type of selection: rBest
r value: 2

Derived from user defined parameters

Populations: 2

Default parameters

Planned allocation ratio: 1
Direction upper: TRUE
Calculate subjects function: default
Intersection test: Simes
Stratified analysis: TRUE
Adaptations: TRUE, TRUE
pi(treatment) under H1: NA
pi(control) under H1: NA
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
Threshold: -Inf

Results

Iterations [1]: 100
Iterations [2]: 98
Iterations [3]: 95
Reject at least one: 0.1800
Rejected populations per stage (S) [1]: 0.02
Rejected populations per stage (S) [2]: 0.05
Rejected populations per stage (S) [3]: 0.04
Rejected populations per stage (F) [1]: 0.05
Rejected populations per stage (F) [2]: 0.04
Rejected populations per stage (F) [3]: 0.07
Overall futility stop: 0.0000
Futility stop per stage [1]: 0.0000
Futility stop per stage [2]: 0.0000
Early stop [1]: 0.0200
Early stop [2]: 0.0300
Success per stage [1]: 0.0200
Success per stage [2]: 0.0300
Success per stage [3]: 0.0400
Selected populations (S) [1]: 1
Selected populations (S) [2]: 0.98
Selected populations (S) [3]: 0.95
Selected populations (F) [1]: 1
Selected populations (F) [2]: 0.98
Selected populations (F) [3]: 0.95
Number of populations [1]: 2
Number of populations [2]: 2
Number of populations [3]: 2
Expected number of subjects: 97.9
Sample sizes (S) [1]: 4
Sample sizes (S) [2]: 3
Sample sizes (S) [3]: 3
Sample sizes (R) [1]: 36
Sample sizes (R) [2]: 27
Sample sizes (R) [3]: 27
Conditional power (achieved) [1]: NA
Conditional power (achieved) [2]: 0.0689
Conditional power (achieved) [3]: 0.0731

Legend

[k]: values at stage k
S[i]: population i
F: full population
R: remaining population

Simulation results enrichment - survival

options("rpact.summary.output.size" = "medium") # small, medium, large

design <- getDesignFisher(alpha = 0.05, kMax = 3)

subGroups <- c("S1", "S2", "S12", "R")
prevalences <- c(0.1, 0.3, 0.4, 0.2)
hazardRatios <- c(0.4, 0.5, 0.6, 0.7, 0.6, 0.6, 0.6, 0.8)
effectList <- list(
    subGroups = subGroups, prevalences = prevalences,
    hazardRatios = matrix(hazardRatios, byrow = TRUE, ncol = 4)
)

kable(summary(getSimulationEnrichmentSurvival(
    design = design,
    plannedEvents = c(40, 70, 100),
    effectList = effectList,
    conditionalPower = 0.8,
    minNumberOfEventsPerStage = c(40, 20, 20),
    maxNumberOfEventsPerStage = c(40, 100, 100),
    maxNumberOfIterations = 100,
    seed = 1234
)))

Warning: Simulation of enrichment designs is experimental and hence not fully
validated (see www.rpact.com/experimental)

Simulation of a survival endpoint (enrichment design)

Sequential analysis with a maximum of 3 looks (Fisher’s combination test design), overall significance level 5% (one-sided). The results were simulated for a population enrichment logrank test (treatment vs. control, 3 populations), H0: hazard ratio = 1, power directed towards larger values, H1: hazard ratios as specified, subgroups = c(S1, S2, S12, R), prevalences = c(0.1, 0.3, 0.4, 0.2), planned cumulative events = c(40, 70, 100), intersection test = Simes, selection = best, effect measure based on effect estimate, success criterion: all, sample size reassessment: conditional power = 0.8, minimum events per stage = c(40, 20, 20), maximum events per stage = c(40, 100, 100), simulation runs = 100, seed = 1234.

Stage	1	2	3
Fixed weight	1	1	1
Efficacy boundary (p product scale)	0.0255136	0.0038966	0.0007481
Stage levels (one-sided)	0.0255	0.0255	0.0255
Reject at least one, hazard ratios = c(0.4, 0.5, 0.6, 0.7)	0
Reject at least one, hazard ratios = c(0.6, 0.6, 0.6, 0.8)	0
Success per stage, hazard ratios = c(0.4, 0.5, 0.6, 0.7)	0	0	0
Success per stage, hazard ratios = c(0.6, 0.6, 0.6, 0.8)	0	0	0
Expected number of events, hazard ratios = c(0.4, 0.5, 0.6, 0.7)	240.0
Expected number of events, hazard ratios = c(0.6, 0.6, 0.6, 0.8)	238.6
Overall exit probability, hazard ratios = c(0.4, 0.5, 0.6, 0.7)	0	0
Overall exit probability, hazard ratios = c(0.6, 0.6, 0.6, 0.8)	0	0
Single number of events, hazard ratios = c(0.4, 0.5, 0.6, 0.7)
Subset S1 only	3.6	8.1	8.1
Subset S2 only	11.5	25.3	25.3
Subset S12	16.3	59.0	59.0
Remaining population R	8.7	7.6	7.6
Single number of events, hazard ratios = c(0.6, 0.6, 0.6, 0.8)
Subset S1 only	3.9	12.1	12.4
Subset S2 only	11.7	18.7	18.7
Subset S12	15.6	60.6	61.7
Remaining population R	8.8	7.2	7.2
Selected populations, hazard ratios = c(0.4, 0.5, 0.6, 0.7)
Subset S1	1.0000	0.2800	0.2800
Subset S2	1.0000	0.3700	0.3700
Full population F	1.0000	0.3500	0.3500
Selected populations, hazard ratios = c(0.6, 0.6, 0.6, 0.8)
Subset S1	1.0000	0.4600	0.4600
Subset S2	1.0000	0.2100	0.2100
Full population F	1.0000	0.3300	0.3300
Number of populations, hazard ratios = c(0.4, 0.5, 0.6, 0.7)	3.000	1.000	1.000
Number of populations, hazard ratios = c(0.6, 0.6, 0.6, 0.8)	3.000	1.000	1.000

Simulation of enrichment survival data (Fisher’s combination test design)

Design parameters

Information rates: 0.333, 0.667, 1.000
Critical values: 0.0255136, 0.0038966, 0.0007481
Alpha_0: 1.0000, 1.0000
Cumulative alpha spending: 0.02551, 0.03981, 0.05000
Local one-sided significance levels: 0.02551, 0.02551, 0.02551
Significance level: 0.0500
Test: one-sided

User defined parameters

Maximum number of iterations: 100
Seed: 1234
Conditional power: 0.8
Planned cumulative events: 40, 70, 100
Minimum number of events per stage: 40, 20, 20
Maximum number of events per stage: 40, 100, 100
Effect list [Sub-groups]: S1, S2, S12, R
Effect list [Prevalences]: 0.1, 0.3, 0.4, 0.2
Effect list [Assumed control rates]:
Effect list [Hazard ratios] (1): 0.4, 0.5, 0.6, 0.7
Effect list [Hazard ratios] (2): 0.6, 0.6, 0.6, 0.8

Derived from user defined parameters

Populations: 3

Default parameters

Planned allocation ratio: 1
Direction upper: TRUE
Calculate events function: default
Intersection test: Simes
Stratified analysis: TRUE
Adaptations: TRUE, TRUE
Type of selection: best
Effect measure: effectEstimate
Success criterion: all
Epsilon value: NA
r value: NA
Threshold: -Inf

Results

Iterations [1]: 100, 100
Iterations [2]: 100, 100
Iterations [3]: 100, 100
Reject at least one: 0.0000, 0.0000
Rejected populations per stage (S1) [1]: 0, 0
Rejected populations per stage (S1) [2]: 0, 0
Rejected populations per stage (S1) [3]: 0, 0
Rejected populations per stage (S2) [1]: 0, 0
Rejected populations per stage (S2) [2]: 0, 0
Rejected populations per stage (S2) [3]: 0, 0
Rejected populations per stage (F) [1]: 0, 0
Rejected populations per stage (F) [2]: 0, 0
Rejected populations per stage (F) [3]: 0, 0
Overall futility stop: 0.0000, 0.0000
Futility stop per stage [1]: 0.0000, 0.0000
Futility stop per stage [2]: 0.0000, 0.0000
Early stop [1]: 0.0000, 0.0000
Early stop [2]: 0.0000, 0.0000
Success per stage [1]: 0.0000, 0.0000
Success per stage [2]: 0.0000, 0.0000
Success per stage [3]: 0.0000, 0.0000
Selected populations (S1) [1]: 1, 1
Selected populations (S1) [2]: 0.28, 0.46
Selected populations (S1) [3]: 0.28, 0.46
Selected populations (S2) [1]: 1, 1
Selected populations (S2) [2]: 0.37, 0.21
Selected populations (S2) [3]: 0.37, 0.21
Selected populations (F) [1]: 1, 1
Selected populations (F) [2]: 0.35, 0.33
Selected populations (F) [3]: 0.35, 0.33
Number of populations [1]: 3, 3
Number of populations [2]: 1, 1
Number of populations [3]: 1, 1
Expected number of events: 240, 238.6
Single number of events (S1 only) [1]: 3.6, 3.9
Single number of events (S1 only) [2]: 8.1, 12.1
Single number of events (S1 only) [3]: 8.1, 12.4
Single number of events (S2 only) [1]: 11.5, 11.7
Single number of events (S2 only) [2]: 25.3, 18.7
Single number of events (S2 only) [3]: 25.3, 18.7
Single number of events (S12) [1]: 16.3, 15.6
Single number of events (S12) [2]: 59, 60.6
Single number of events (S12) [3]: 59, 61.7
Single number of events (R) [1]: 8.7, 8.8
Single number of events (R) [2]: 7.6, 7.2
Single number of events (R) [3]: 7.6, 7.2
Conditional power (achieved) [1]: NA, NA
Conditional power (achieved) [2]: 0.0006, 0.0114
Conditional power (achieved) [3]: 0, 0.0001

Legend

(i): results of situation i
[k]: values at stage k
S[i]: population i
F: full population
R: remaining population

Analysis results summaries

Create three different designs

design1 <- getDesignInverseNormal(
    kMax = 4, alpha = 0.02,
    futilityBounds = c(-0.5, 0, 0.5), bindingFutility = FALSE,
    typeOfDesign = "asKD", gammaA = 1.2,
    informationRates = c(0.15, 0.4, 0.7, 1)
)

design3 <- getDesignConditionalDunnett(
    alpha = 0.02,
    informationAtInterim = 0.4, secondStageConditioning = TRUE
)

Analysis results base

Analysis results base - means

simpleDataExampleMeans1 <- getDataset(
    n = c(120, 130, 130),
    means = c(0.45, 0.51, 0.45) * 100,
    stDevs = c(1.3, 1.4, 1.2) * 100
)

kable(summary(getAnalysisResults(
    design = design1, dataInput = simpleDataExampleMeans1,
    nPlanned = 130, thetaH0 = 30, thetaH1 = 60, assumedStDev = 100
)))

Calculation of final confidence interval performed for kMax = 4 (for kMax > 2, it is theoretically shown that it is valid only if no sample size change was performed)

Analysis results for a continuous endpoint

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a one-sample t-test (one-sided, alpha = 0.02). H0: mu = 30 against H1: mu > 30. The conditional power calculation with planned sample size is based on assumed effect = 60 and assumed standard deviation = 100.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size	45.000	48.120	47.053
Cumulative standard deviation	130.000	135.055	129.950
Stage-wise test statistic	1.264	1.710	1.425
Stage-wise p-value	0.1044	0.0448	0.0783
Inverse normal combination	1.257	2.112	2.524
Test action	continue	continue	reject and stop
Conditional rejection probability	0.0433	0.1706	0.3851
Planned sample size				130
Conditional power				0.9991
96% repeated confidence interval	10.269 - 79.731	25.952 - 70.719	30.900 - 63.004
Repeated p-value	>0.5	0.0662	0.0141
Final p-value			0.0108
Final confidence interval			31.784 - 60.146
Median unbiased estimate			46.183

Analysis results (means of one group, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Theta H0: 30
Planned sample size: NA, NA, NA, 130
Assumed effect under alternative: 60
Assumed standard deviation: 100

Default parameters

Normal approximation: FALSE
Direction upper: TRUE

Stage results

Cumulative effect sizes: 45.00, 48.12, 47.05, NA
Cumulative standard deviations: 130.0, 135.1, 130.0, NA
Stage-wise test statistics: 1.264, 1.710, 1.425, NA
Stage-wise p-values: 0.10435, 0.04481, 0.07825, NA
Combination test statistics: 1.257, 2.112, 2.524, NA

Analysis results

Actions: continue, continue, reject and stop, NA
Conditional rejection probability: 0.04331, 0.17064, 0.38514, NA
Conditional power: NA, NA, NA, 0.9991
Repeated confidence intervals (lower): 10.27, 25.95, 30.90, NA
Repeated confidence intervals (upper): 79.73, 70.72, 63.00, NA
Repeated p-values: >0.5, 0.06620, 0.01409, NA
Final stage: 3
Final p-value: NA, NA, 0.01083, NA
Final CIs (lower): NA, NA, 31.78, NA
Final CIs (upper): NA, NA, 60.15, NA
Median unbiased estimate: NA, NA, 46.18, NA

simpleDataExampleMeans2 <- getDataset(
    n1 = c(23, 13, 22, 13),
    n2 = c(22, 11, 22, 11),
    means1 = c(2.7, 2.5, 4.5, 2.5) * 100,
    means2 = c(1, 1.1, 1.3, 1) * 100,
    stds1 = c(1.3, 2.4, 2.2, 1.3) * 100,
    stds2 = c(1.2, 2.2, 2.1, 1.3) * 100
)

kable(summary(getAnalysisResults(
    design = design1, dataInput = simpleDataExampleMeans2,
    equalVariances = TRUE, directionUpper = TRUE
)))

Analysis results for a continuous endpoint

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample t-test (one-sided, alpha = 0.02), equal variances option. H0: mu(1) - mu(2) = 0 against H1: mu(1) - mu(2) > 0.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size	170.000	159.444	219.793	206.784
Cumulative (pooled) standard deviation	125.216	166.324	196.455	187.224
Stage-wise test statistic	4.553	1.479	4.935	2.817
Stage-wise p-value	<0.0001	0.0767	<0.0001	0.0050
Inverse normal combination	4.090	3.633	5.599	6.094
Test action	reject and stop	reject and stop	reject and stop	reject
Conditional rejection probability	0.5469	0.7525	1.0000
96% repeated confidence interval	56.780 - 283.220	50.708 - 267.898	129.271 - 304.606	127.414 - 266.016
Repeated p-value	0.0002	0.0006	<0.0001	<0.0001
Final p-value	<0.0001
Final confidence interval	93.310 - 246.690
Median unbiased estimate	170.000

Analysis results (means of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

Default parameters

Normal approximation: FALSE
Direction upper: TRUE
Theta H0: 0
Equal variances: TRUE

Stage results

Cumulative effect sizes: 170.0, 159.4, 219.8, 206.8
Cumulative (pooled) standard deviations: 125.2, 166.3, 196.5, 187.2
Stage-wise test statistics: 4.553, 1.479, 4.935, 2.817
Stage-wise p-values: 0.00002158, 0.07671383, <0.00001, 0.00502569
Combination test statistics: 4.090, 3.633, 5.599, 6.094

Analysis results

Assumed standard deviation: 187.2
Actions: reject and stop, reject and stop, reject and stop, reject
Conditional rejection probability: 0.5469, 0.7525, 1.0000, NA
Conditional power: NA, NA, NA, NA
Repeated confidence intervals (lower): 56.78, 50.71, 129.27, 127.41
Repeated confidence intervals (upper): 283.2, 267.9, 304.6, 266.0
Repeated p-values: 0.000209854, 0.000587509, 0.000001954, 0.000001954
Final stage: 1
Final p-value: <0.0001, NA, NA, NA
Final CIs (lower): 93.31, NA, NA, NA
Final CIs (upper): 246.7, NA, NA, NA
Median unbiased estimate: 170, NA, NA, NA

Analysis results base - rates

simpleDataExampleRates1 <- getDataset(
    n = c(8, 10, 9, 11),
    events = c(4, 5, 5, 6)
)

kable(summary(getAnalysisResults(
    design = design1, dataInput = simpleDataExampleRates1,
    stage = 3, thetaH0 = 0.75, normalApproximation = TRUE, directionUpper = FALSE,
    nPlanned = 10
)))

Calculation of final confidence interval performed for kMax = 4 (for kMax > 2, it is theoretically shown that it is valid only if no sample size change was performed)

Analysis results for a binary endpoint

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a one-sample test for rates (one-sided, alpha = 0.02), normal approximation test. H0: pi = 0.75 against H1: pi < 0.75. The conditional power calculation with planned sample size is based on overall treatment rate = 0.52.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative treatment rate	0.500	0.500	0.519
Stage-wise test statistic	-1.633	-1.826	-1.347
Stage-wise p-value	0.0512	0.0339	0.0890
Inverse normal combination	1.633	2.443	2.729
Test action	continue	continue	reject and stop
Conditional rejection probability	0.0669	0.2722	0.5084
Planned sample size				10
Conditional power				0.9310
96% repeated confidence interval	0.144 - 0.856	0.240 - 0.760	0.307 - 0.727
Repeated p-value	0.4992	0.0285	0.0079
Final p-value			0.0087
Final confidence interval			0.300 - 0.714
Median unbiased estimate			0.502

Analysis results (rates of one group, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Direction upper: FALSE
Theta H0: 0.75
Planned sample size: NA, NA, NA, 10

Default parameters

Normal approximation: TRUE

Stage results

Cumulative treatment rate: 0.5, 0.5, 0.519, NA
Stage-wise test statistics: -1.633, -1.826, -1.347, NA
Stage-wise p-values: 0.05124, 0.03394, 0.08897, NA
Combination test statistics: 1.633, 2.443, 2.729, NA

Analysis results

Actions: continue, continue, reject and stop, NA
Conditional rejection probability: 0.06689, 0.27218, 0.50844, NA
Conditional power: NA, NA, NA, 0.931
Repeated confidence intervals (lower): 0.1439, 0.2403, 0.3074, NA
Repeated confidence intervals (upper): 0.8561, 0.7597, 0.7274, NA
Repeated p-values: 0.499183, 0.028536, 0.007935, NA
Final stage: 3
Final p-value: NA, NA, 0.008716, NA
Final CIs (lower): NA, NA, 0.2997, NA
Final CIs (upper): NA, NA, 0.7143, NA
Median unbiased estimate: NA, NA, 0.5024, NA

simpleDataExampleRates2 <- getDataset(
    n1 = c(17, 23, 22),
    n2 = c(18, 20, 19),
    events1 = c(11, 12, 17),
    events2 = c(5, 10, 7)
)

kable(summary(getAnalysisResults(design1, simpleDataExampleRates2,
    thetaH0 = 0, stage = 2, directionUpper = TRUE,
    normalApproximation = FALSE, pi1 = 0.9, pi2 = 0.3, nPlanned = c(20, 20)
)))

Repeated confidence intervals will be calculated under the normal approximation

Analysis results for a binary endpoint

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample test for rates (one-sided, alpha = 0.02), exact test of Fisher. H0: pi(1) - pi(2) = 0 against H1: pi(1) - pi(2) > 0. The conditional power calculation with planned sample size is based on assumed treatment rate = 0.9 and assumed control rate = 0.3.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size	0.369	0.180
Cumulative treatment rate	0.647	0.575
Cumulative control rate	0.278	0.395
Stage-wise test statistic
Stage-wise p-value	0.0313	0.5648
Inverse normal combination	1.862	1.011
Test action	continue	continue
Conditional rejection probability	0.0863	0.0213
Planned sample size			20	20
Conditional power			0.7257	0.9884
96% repeated confidence interval	-0.111 - 0.711	-0.125 - 0.429
Repeated p-value	0.3048	>0.5

Analysis results (rates of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Normal approximation: FALSE
Assumed treatment rate: 0.900
Assumed control rate: 0.300
Planned sample size: NA, NA, 20, 20

Default parameters

Direction upper: TRUE
Theta H0: 0
Planned allocation ratio: 1

Stage results

Cumulative effect sizes: 0.3693, 0.1803, NA, NA
Cumulative treatment rate: 0.647, 0.575, NA, NA
Cumulative control rate: 0.278, 0.395, NA, NA
Stage-wise test statistics: NA, NA, NA, NA
Stage-wise p-values: 0.03129, 0.56476, NA, NA
Combination test statistics: 1.862, 1.011, NA, NA

Analysis results

Actions: continue, continue, NA, NA
Conditional rejection probability: 0.08632, 0.02125, NA, NA
Conditional power: NA, NA, 0.7257, 0.9884
Repeated confidence intervals (lower): -0.1113, -0.1246, NA, NA
Repeated confidence intervals (upper): 0.7108, 0.4290, NA, NA
Repeated p-values: 0.3048, >0.5, NA, NA
Final stage: NA
Final p-value: NA, NA, NA, NA
Final CIs (lower): NA, NA, NA, NA
Final CIs (upper): NA, NA, NA, NA
Median unbiased estimate: NA, NA, NA, NA

Legend

(i): values of treatment arm i

Analysis results base - survival

simpleDataExampleSurvival <- getDataset(
    overallEvents = c(8, 15, 29),
    overallAllocationRatios = c(1, 1, 1),
    overallLogRanks = c(1.52, 1.38, 2.9)
)

kable(simpleDataExampleSurvival$getNumberOfGroups())

x
2

kable(summary(getAnalysisResults(design1, simpleDataExampleSurvival, directionUpper = TRUE)))

Calculation of final confidence interval performed for kMax = 4 (for kMax > 2, it is theoretically shown that it is valid only if no sample size change was performed)

Analysis results for a survival endpoint

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample logrank test (one-sided, alpha = 0.02). H0: hazard ratio = 1 against H1: hazard ratio > 1.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size	2.929	2.039	2.936
Stage-wise test statistic	1.520	0.395	2.745
Stage-wise p-value	0.0643	0.3464	0.0030
Inverse normal combination	1.520	1.243	2.737
Test action	continue	continue	reject and stop
Conditional rejection probability	0.0588	0.0350	0.5134
96% repeated confidence interval	0.385 - 22.290	0.499 - 7.358	1.138 - 6.825
Repeated p-value	>0.5	0.3740	0.0077
Final p-value			0.0087
Final confidence interval			1.160 - 5.900
Median unbiased estimate			2.686

Analysis results (survival of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

Default parameters

Normal approximation: TRUE
Direction upper: TRUE
Theta H0: 1

Stage results

Cumulative effect sizes: 2.929, 2.039, 2.936, NA
Stage-wise test statistics: 1.520, 0.395, 2.745, NA
Stage-wise p-values: 0.064255, 0.346361, 0.003022, NA
Combination test statistics: 1.520, 1.243, 2.737, NA

Analysis results

Actions: continue, continue, reject and stop, NA
Conditional rejection probability: 0.05881, 0.03502, 0.51339, NA
Conditional power: NA, NA, NA, NA
Repeated confidence intervals (lower): 0.385, 0.499, 1.138, NA
Repeated confidence intervals (upper): 22.290, 7.358, 6.825, NA
Repeated p-values: >0.5, 0.37400, 0.00775, NA
Final stage: 3
Final p-value: NA, NA, 0.008657, NA
Final CIs (lower): NA, NA, 1.16, NA
Final CIs (upper): NA, NA, 5.9, NA
Median unbiased estimate: NA, NA, 2.686, NA

Analysis results multi-arm

Analysis results multi-arm - means

dataExampleMeans <- getDataset(
    n1 = c(13, 25),
    n2 = c(15, NA),
    n3 = c(14, 27),
    n4 = c(12, 29),
    means1 = c(242, 222),
    means2 = c(188, NA),
    means3 = c(267, 277),
    means4 = c(92, 122),
    stDevs1 = c(244, 221),
    stDevs2 = c(212, NA),
    stDevs3 = c(256, 232),
    stDevs4 = c(215, 227)
)

kable(summary(getAnalysisResults(
    design = design3, dataInput = dataExampleMeans, stage = 2, thetaH0 = 120,
    directionUpper = TRUE, normalApproximation = TRUE,
    assumedStDevs = c(24, 25, 23)
)))

Warning: 'assumedStDevs' (24, 25, 23) will be ignored because 'nPlanned' is not
defined

Multi-arm analysis results for a continuous endpoint (3 active arms vs. control)

Sequential analysis with 2 looks (conditional Dunnett test design). The results were calculated using a multi-arm t-test (one-sided, alpha = 0.02), normal approximation test, overall pooled variances option, conditional second stage p-values. H0: mu(1) = 120 against H1: mu(1) > 120.

Stage	1	2
Fixed information at interim	0.4
Cumulative effect size (1)	150.000	115.623
Cumulative effect size (2)	96.000
Cumulative effect size (3)	175.000	160.366
Cumulative (pooled) standard deviation	232.555	228.359
Stage-wise test statistic (1)	0.322	-0.323
Stage-wise test statistic (2)	-0.266
Stage-wise test statistic (3)	0.601	0.577
Stage-wise p-value (1)	0.3736	0.6267
Stage-wise p-value (2)	0.6051
Stage-wise p-value (3)	0.2739	0.2820
Conditional error rate (1, 2, 3)	0.0062
Second stage p-value (1, 2, 3)		0.3810
Test action: reject (1)	FALSE	FALSE
Test action: reject (2)	FALSE	FALSE
Test action: reject (3)	FALSE	FALSE
96% overall confidence interval (1)		-7.244 - 225.388
96% overall confidence interval (2)
96% overall confidence interval (3)		39.389 - 267.536
Overall p-value (1)		>0.5
Overall p-value (2)
Overall p-value (3)		0.3768

Legend:

(i): results of treatment arm i vs. control arm
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Multi-arm analysis results (means of 4 groups, conditional Dunnett design)

Design parameters

Stages: 1, 2
Significance level: 0.0200
Information at interim: 0.400
Conditional second stage p-values: TRUE
Test: one-sided

User defined parameters

Normal approximation: TRUE
Theta H0: 120
Assumed standard deviations (1): 24
Assumed standard deviations (2): 25
Assumed standard deviations (3): 23

Default parameters

Direction upper: TRUE
Intersection test: Dunnett
Variance option: overallPooled

Stage results

Cumulative effect sizes (1): 150.0, 115.6
Cumulative effect sizes (2): 96.0, NA
Cumulative effect sizes (3): 175.0, 160.4
Cumulative (pooled) standard deviations (1): 230.6, 223.6
Cumulative (pooled) standard deviations (2): 213.3, NA
Cumulative (pooled) standard deviations (3): 238.1, 229.4
Stage-wise test statistics (1): 0.322, -0.323
Stage-wise test statistics (2): -0.266, NA
Stage-wise test statistics (3): 0.601, 0.577
Separate p-values (1): 0.3736, 0.6267
Separate p-values (2): 0.6051, NA
Separate p-values (3): 0.2739, 0.2820

Conditional error rate

Treatments 1, 2, 3 vs. control: 0.006162, NA
Treatments 1, 2 vs. control: 0.004067, NA
Treatments 1, 3 vs. control: 0.009580, NA
Treatments 2, 3 vs. control: 0.007271, NA
Treatment 1 vs. control: 0.008464, NA
Treatment 2 vs. control: 0.002059, NA
Treatment 3 vs. control: 0.015366, NA

Second stage p-values

Treatments 1, 2, 3 vs. control: NA, 0.3810
Treatments 1, 2 vs. control: NA, 0.6267
Treatments 1, 3 vs. control: NA, 0.3810
Treatments 2, 3 vs. control: NA, 0.2820
Treatment 1 vs. control: NA, 0.6267
Treatment 2 vs. control: NA, NA
Treatment 3 vs. control: NA, 0.2820

Test actions

Rejected (1): FALSE, FALSE
Rejected (2): FALSE, FALSE
Rejected (3): FALSE, FALSE

Further analysis results

Conditional rejection probability (1): NA, 0.004067
Conditional rejection probability (2): NA, 0.002059
Conditional rejection probability (3): NA, 0.006162
Conditional power (1): NA, NA
Conditional power (2): NA, NA
Conditional power (3): NA, NA
Overall confidence intervals (lower) (1): NA, -7.244
Overall confidence intervals (lower) (2): NA, NA
Overall confidence intervals (lower) (3): NA, 39.389
Overall confidence intervals (upper) (1): NA, 225.4
Overall confidence intervals (upper) (2): NA, NA
Overall confidence intervals (upper) (3): NA, 267.5
Overall p-values (1): NA, >0.5
Overall p-values (2): NA, NA
Overall p-values (3): NA, 0.3768

Legend

(i): results of treatment arm i vs. control group 4

Analysis results multi-arm - rates

dataExampleRates <- getDataset(
    n1 = c(23, 25),
    n2 = c(25, NA),
    n3 = c(24, 27),
    n4 = c(22, 29),
    events1 = c(15, 12),
    events2 = c(19, NA),
    events3 = c(18, 22),
    events4 = c(12, 13)
)


kable(summary(getAnalysisResults(
    design = design1, dataInput = dataExampleRates,
    intersectionTest = "Bonferroni", nPlanned = c(20, 20),
    directionUpper = TRUE, piTreatments = c(0.4, 0.6, 0.5)
)))

Multi-arm analysis results for a binary endpoint (3 active arms vs. control)

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a multi-arm test for rates (one-sided, alpha = 0.02), Bonferroni intersection test, normal approximation test. H0: pi(i) - pi(control) = 0 against H1: pi(i) - pi(control) > 0. The conditional power calculation with planned sample size is based on assumed treatment rate: pi(1) = 0.4, pi(2) = 0.6, pi(3) = 0.5 and overall control rate = 0.49.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size (1)	0.107	0.072
Cumulative effect size (2)	0.215
Cumulative effect size (3)	0.205	0.294
Cumulative treatment rate (1)	0.652	0.562
Cumulative treatment rate (2)	0.760
Cumulative treatment rate (3)	0.750	0.784
Cumulative control rate	0.545	0.490
Stage-wise test statistic (1)	0.730	0.233
Stage-wise test statistic (2)	1.549
Stage-wise test statistic (3)	1.455	2.831
Stage-wise p-value (1)	0.2325	0.4078
Stage-wise p-value (2)	0.0607
Stage-wise p-value (3)	0.0728	0.0023
Adjusted stage-wise p-value (1, 2, 3)	0.1821	0.0046
Overall adjusted test statistic (1, 2, 3)	0.907	2.612
Test action: reject (1)	FALSE	FALSE
Test action: reject (2)	FALSE	FALSE
Test action: reject (3)	FALSE	TRUE
Conditional rejection probability (1)	0.0227	0.0088
Conditional rejection probability (2)	0.0283
Conditional rejection probability (3)	0.0283	0.3346
Planned sample size			20	20
Conditional power (1)			0.0004	0.0016
Conditional power (2)
Conditional power (3)			0.2769	0.3544
96% repeated confidence interval (1)	-0.336 - 0.513	-0.236 - 0.356
96% repeated confidence interval (2)	-0.222 - 0.588
96% repeated confidence interval (3)	-0.237 - 0.583	-0.003 - 0.549
Repeated p-value (1)	>0.5	>0.5
Repeated p-value (2)	>0.5
Repeated p-value (3)	>0.5	0.0179

Legend:

(i): results of treatment arm i vs. control arm
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Multi-arm analysis results (rates of 4 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Planned sample size: NA, NA, 20, 20
Assumed treatment rates (1): 0.400
Assumed treatment rates (2): 0.600
Assumed treatment rates (3): 0.500
Intersection test: Bonferroni

Default parameters

Normal approximation: TRUE
Direction upper: TRUE
Theta H0: 0
Planned allocation ratio: 1

Stage results

Cumulative effect sizes (1): 0.1067, 0.0723, NA, NA
Cumulative effect sizes (2): 0.2145, NA, NA, NA
Cumulative effect sizes (3): 0.2045, 0.2941, NA, NA
Cumulative treatment rate (1): 0.652, 0.562, NA, NA
Cumulative treatment rate (2): 0.760, NA, NA, NA
Cumulative treatment rate (3): 0.750, 0.784, NA, NA
Cumulative control rate: 0.545, 0.490, NA, NA
Stage-wise test statistics (1): 0.730, 0.233, NA, NA
Stage-wise test statistics (2): 1.549, NA, NA, NA
Stage-wise test statistics (3): 1.455, 2.831, NA, NA
Separate p-values (1): 0.23255, 0.40783, NA, NA
Separate p-values (2): 0.06071, NA, NA, NA
Separate p-values (3): 0.07283, 0.00232, NA, NA

Adjusted stage-wise p-values

Treatments 1, 2, 3 vs. control: 0.182127, 0.004639, NA, NA
Treatments 1, 2 vs. control: 0.121418, 0.407833, NA, NA
Treatments 1, 3 vs. control: 0.145667, 0.004639, NA, NA
Treatments 2, 3 vs. control: 0.121418, 0.002320, NA, NA
Treatment 1 vs. control: 0.232549, 0.407833, NA, NA
Treatment 2 vs. control: 0.060709, NA, NA, NA
Treatment 3 vs. control: 0.072834, 0.002320, NA, NA

Overall adjusted test statistics

Treatments 1, 2, 3 vs. control: 0.9073, 2.6124, NA, NA
Treatments 1, 2 vs. control: 1.1679, 0.8995, NA, NA
Treatments 1, 3 vs. control: 1.0552, 2.7029, NA, NA
Treatments 2, 3 vs. control: 1.1679, 2.9534, NA, NA
Treatment 1 vs. control: 0.7305, 0.6316, NA, NA
Treatment 2 vs. control: 1.5488, NA, NA, NA
Treatment 3 vs. control: 1.4550, 3.1292, NA, NA

Test actions

Rejected (1): FALSE, FALSE, NA, NA
Rejected (2): FALSE, FALSE, NA, NA
Rejected (3): FALSE, TRUE, NA, NA

Further analysis results

Conditional rejection probability (1): 0.022713, 0.008799, NA, NA
Conditional rejection probability (2): 0.028283, NA, NA, NA
Conditional rejection probability (3): 0.028283, 0.334554, NA, NA
Conditional power (1): NA, NA, 0.0004, 0.0016
Conditional power (2): NA, NA, NA, NA
Conditional power (3): NA, NA, 0.2769, 0.3544
Repeated confidence intervals (lower) (1): -0.336171, -0.236083, NA, NA
Repeated confidence intervals (lower) (2): -0.221567, NA, NA, NA
Repeated confidence intervals (lower) (3): -0.237142, -0.003183, NA, NA
Repeated confidence intervals (upper) (1): 0.5132, 0.3556, NA, NA
Repeated confidence intervals (upper) (2): 0.5884, NA, NA, NA
Repeated confidence intervals (upper) (3): 0.5829, 0.5490, NA, NA
Repeated p-values (1): >0.5, >0.5, NA, NA
Repeated p-values (2): >0.5, NA, NA, NA
Repeated p-values (3): >0.5, 0.01786, NA, NA

Legend

(i): results of treatment arm i vs. control group 4

Analysis results multi-arm - survival

dataExampleSurvival <- getDataset(
    events1 = c(25, 32),
    events2 = c(18, NA),
    events3 = c(22, 36),
    logRanks1 = c(2.2, 1.8),
    logRanks2 = c(1.99, NA),
    logRanks3 = c(2.32, 2.11)
)

kable(summary(getAnalysisResults(
    design = design3, dataInput = dataExampleSurvival,
    intersectionTest = "Dunnett", directionUpper = TRUE, thetaH0 = 2
)))

Multi-arm analysis results for a survival endpoint (3 active arms vs. control)

Sequential analysis with 2 looks (conditional Dunnett test design). The results were calculated using a multi-arm logrank test (one-sided, alpha = 0.02), conditional second stage p-values. H0: hazard ratio = 2 against H1: hazard ratio > 2.

Stage	1	2
Fixed information at interim	0.4
Cumulative effect size (1)	2.411	2.103
Cumulative effect size (2)	2.555
Cumulative effect size (3)	2.689	2.252
Stage-wise test statistic (1)	0.467	-0.161
Stage-wise test statistic (2)	0.520
Stage-wise test statistic (3)	0.694	0.031
Stage-wise p-value (1)	0.3202	0.5638
Stage-wise p-value (2)	0.3017
Stage-wise p-value (3)	0.2437	0.4878
Conditional error rate (1, 2, 3)	0.0103
Second stage p-value (1, 2, 3)		0.6188
Test action: reject (1)	FALSE	FALSE
Test action: reject (2)	FALSE	FALSE
Test action: reject (3)	FALSE	FALSE
96% overall confidence interval (1)		1.122 - 3.911
96% overall confidence interval (2)
96% overall confidence interval (3)		1.217 - 4.197
Overall p-value (1)		0.4913
Overall p-value (2)
Overall p-value (3)		0.4913

Legend:

(i): results of treatment arm i vs. control arm
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Multi-arm analysis results (survival of 4 groups, conditional Dunnett design)

Design parameters

Stages: 1, 2
Significance level: 0.0200
Information at interim: 0.400
Conditional second stage p-values: TRUE
Test: one-sided

User defined parameters

Theta H0: 2

Default parameters

Direction upper: TRUE
Intersection test: Dunnett

Stage results

Cumulative effect sizes (1): 2.411, 2.103
Cumulative effect sizes (2): 2.555, NA
Cumulative effect sizes (3): 2.689, 2.252
Stage-wise test statistics (1): 0.4671, -0.1605
Stage-wise test statistics (2): 0.5196, NA
Stage-wise test statistics (3): 0.6944, 0.0306
Separate p-values (1): 0.3202, 0.5638
Separate p-values (2): 0.3017, NA
Separate p-values (3): 0.2437, 0.4878

Conditional error rate

Treatments 1, 2, 3 vs. control: 0.010257, NA
Treatments 1, 2 vs. control: 0.009761, NA
Treatments 1, 3 vs. control: 0.012282, NA
Treatments 2, 3 vs. control: 0.012854, NA
Treatment 1 vs. control: 0.011605, NA
Treatment 2 vs. control: 0.012970, NA
Treatment 3 vs. control: 0.018563, NA

Second stage p-values

Treatments 1, 2, 3 vs. control: NA, 0.6188
Treatments 1, 2 vs. control: NA, 0.5638
Treatments 1, 3 vs. control: NA, 0.6188
Treatments 2, 3 vs. control: NA, 0.4878
Treatment 1 vs. control: NA, 0.5638
Treatment 2 vs. control: NA, NA
Treatment 3 vs. control: NA, 0.4878

Test actions

Rejected (1): FALSE, FALSE
Rejected (2): FALSE, FALSE
Rejected (3): FALSE, FALSE

Further analysis results

Conditional rejection probability (1): NA, 0.009761
Conditional rejection probability (2): NA, 0.009761
Conditional rejection probability (3): NA, 0.010257
Conditional power (1): NA, NA
Conditional power (2): NA, NA
Conditional power (3): NA, NA
Overall confidence intervals (lower) (1): NA, 1.122
Overall confidence intervals (lower) (2): NA, NA
Overall confidence intervals (lower) (3): NA, 1.217
Overall confidence intervals (upper) (1): NA, 3.911
Overall confidence intervals (upper) (2): NA, NA
Overall confidence intervals (upper) (3): NA, 4.197
Overall p-values (1): NA, 0.4913
Overall p-values (2): NA, NA
Overall p-values (3): NA, 0.4913

Legend

(i): results of treatment arm i vs. control group 4

Analysis results enrichment

Analysis results enrichment - means

dataS1 <- getDataset(
    means1 = c(13.2, 12.8),
    means2 = c(11.1, 10.8),
    stDev1 = c(3.4, 3.3),
    stDev2 = c(2.9, 3.5),
    n1 = c(21, 22),
    n2 = c(19, 21)
)
dataNotS1 <- getDataset(
    means1 = c(11.8, NA),
    means2 = c(11.5, NA),
    stDev1 = c(3.6, NA),
    stDev2 = c(2.7, NA),
    n1 = c(15, NA),
    n2 = c(13, NA)
)
dataExampleMeans <- getDataset(S1 = dataS1, R = dataNotS1)

kable(summary(getAnalysisResults(
    design = design1, dataInput = dataExampleMeans, varianceOption = "pooledFromFull",
    intersectionTest = "SpiessensDebois", nPlanned = c(20, 20), directionUpper = TRUE, assumedStDevs = 5
)))

Enrichment analysis results for a continuous endpoint (2 populations)

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample t-test (one-sided, alpha = 0.02), Spiessens and Debois intersection test, pooled from full population variances option, stratified analysis. H0: mu(treatment) - mu(control) = 0 against H1: mu(treatment) - mu(control) > 0. The conditional power calculation with planned sample size is based on overall effect: thetaH1(S1) = 2.05, thetaH1(F) = NA and assumed standard deviation = 5.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size S1	2.100	2.053
Cumulative effect size F	1.354
Cumulative (pooled) standard deviation S1	3.173	3.256
Cumulative (pooled) standard deviation F	3.186
Stage-wise test statistic S1	2.079	1.929
Stage-wise test statistic F	1.755
Stage-wise p-value S1	0.0222	0.0304
Stage-wise p-value F	0.0421
Adjusted stage-wise p-value (1, 2)	0.0334	0.0304
Overall adjusted test statistic (1, 2)	1.833	2.605
Test action: reject S1	FALSE	TRUE
Test action: reject F	FALSE	FALSE
Conditional rejection probability S1	0.0835	0.3318
Conditional rejection probability F	0.0743
Planned sample size			20	20
Conditional power S1			0.6080	0.7684
Conditional power F
96% repeated confidence interval S1	-1.101 - 5.301	0.025 - 4.066
96% repeated confidence interval F	-1.096 - 3.816
Repeated p-value S1	0.3255	0.0182
Repeated p-value F	0.4097

Legend:

F: full population
S[i]: population i
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Enrichment analysis results (means of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Planned sample size: NA, NA, 20, 20
Assumed standard deviations: 5
Intersection test: SpiessensDebois
Variance option: pooledFromFull

Default parameters

Normal approximation: FALSE
Direction upper: TRUE
Theta H0: 0
Planned allocation ratio: 1
Stratified analysis: TRUE

Stage results

Cumulative effect sizes S1: 2.100, 2.053, NA, NA
Cumulative effect sizes F: 1.354, NA, NA, NA
Cumulative (pooled) standard deviations S1: 3.173, 3.256, NA, NA
Cumulative (pooled) standard deviations F: 3.186, NA, NA, NA
Stage-wise test statistics S1: 2.079, 1.929, NA, NA
Stage-wise test statistics F: 1.755, NA, NA, NA
Separate p-values S1: 0.02220, 0.03036, NA, NA
Separate p-values F: 0.04205, NA, NA, NA

Adjusted stage-wise p-values

Adjusted stage-wise p-values S1, F: 0.03341, 0.03036, NA, NA
Adjusted stage-wise p-values S1: 0.02220, 0.03036, NA, NA
Adjusted stage-wise p-values F: 0.04205, NA, NA, NA

Overall adjusted test statistics

Overall adjusted test statistics S1, F: 1.833, 2.605, NA, NA
Overall adjusted test statistics S1: 2.010, 2.714, NA, NA
Overall adjusted test statistics F: 1.727, NA, NA, NA

Test actions

Rejected S1: FALSE, TRUE, NA, NA
Rejected F: FALSE, FALSE, NA, NA

Further analysis results

Conditional rejection probability S1: 0.08350, 0.33178, NA, NA
Conditional rejection probability F: 0.07427, NA, NA, NA
Conditional power S1: NA, NA, 0.6080, 0.7684
Conditional power F: NA, NA, NA, NA
Repeated confidence intervals (lower) S1: -1.1011, 0.0251, NA, NA
Repeated confidence intervals (lower) F: -1.0965, NA, NA, NA
Repeated confidence intervals (upper) S1: 5.301, 4.066, NA, NA
Repeated confidence intervals (upper) F: 3.816, NA, NA, NA
Repeated p-values S1: 0.32551, 0.01823, NA, NA
Repeated p-values F: 0.40971, NA, NA, NA

Legend

S[i]: population i
F: full population

Analysis results enrichment - rates

S1 <- getDataset(
    events2 = c(16, 19),
    sampleSizes2 = c(33, 34),
    events1 = c(26, 29),
    sampleSizes1 = c(35, 32)
)
S2 <- getDataset(
    events2 = c(12, 15),
    sampleSizes2 = c(36, 31),
    events1 = c(22, 24),
    sampleSizes1 = c(31, 39)
)
F <- getDataset(
    events2 = c(65, 54),
    sampleSizes2 = c(83, 84),
    events1 = c(66, 59),
    sampleSizes1 = c(85, 82)
)

dataExampleRates <- getDataSet(S1 = S1, S2 = S2, F = F)

kable(summary(getAnalysisResults(
    design = design1, dataInput = dataExampleRates, stratifiedAnalysis = FALSE,
    intersectionTest = "Simes", nPlanned = c(20, 20),
    piControls = c(0.6, 0.2, 0.3),
    directionUpper = TRUE
)))

Enrichment analysis results for a binary endpoint (3 populations)

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample test for rates (one-sided, alpha = 0.02), Simes intersection test, normal approximation test, non-stratified analysis. H0: pi(treatment) - pi(control) = 0 against H1: pi(treatment) - pi(control) > 0. The conditional power calculation with planned sample size is based on overall treatment rate: pi(S1) = 0.82, pi(S2) = 0.66, pi(F) = 0.75 and assumed control rate: pi(S1) = 0.6, pi(S2) = 0.2, pi(F) = 0.3.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size S1	0.258	0.299
Cumulative effect size S2	0.376	0.254
Cumulative effect size F	-0.007	0.036
Cumulative treatment rate S1	0.743	0.821
Cumulative treatment rate S2	0.710	0.657
Cumulative treatment rate F	0.776	0.749
Cumulative control rate	0.485	0.333	0.783
Stage-wise test statistic S1	2.188	3.167
Stage-wise test statistic S2	3.072	1.100
Stage-wise test statistic F	-0.104	1.059
Stage-wise p-value S1	0.0143	0.0008
Stage-wise p-value S2	0.0011	0.1356
Stage-wise p-value F	0.5415	0.1448
Adjusted stage-wise p-value (1, 2, 3)	0.0032	0.0023
Overall adjusted test statistic (1, 2, 3)	2.728	3.910
Test action: reject S1	FALSE	TRUE
Test action: reject S2	FALSE	TRUE
Test action: reject F	FALSE	FALSE
Conditional rejection probability S1	0.0900	0.7054
Conditional rejection probability S2	0.2063	0.3252
Conditional rejection probability F	0.0076	0.0123
Planned sample size			20	20
Conditional power S1			0.9368	0.9735
Conditional power S2			0.9478	0.9948
Conditional power F			0.2417	0.7339
96% repeated confidence interval S1	-0.118 - 0.573	0.052 - 0.533
96% repeated confidence interval S2	-0.016 - 0.667	-0.046 - 0.491
96% repeated confidence interval F	-0.214 - 0.202	-0.120 - 0.197
Repeated p-value S1	0.2793	0.0010
Repeated p-value S2	0.0310	0.0191
Repeated p-value F	>0.5	>0.5

Legend:

F: full population
S[i]: population i
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Enrichment analysis results (rates of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Planned sample size: NA, NA, 20, 20
Assumed control rates S1: 0.600
Assumed control rates S2: 0.200
Assumed control rates F: 0.300
Stratified analysis: FALSE

Default parameters

Normal approximation: TRUE
Direction upper: TRUE
Theta H0: 0
Planned allocation ratio: 1
Intersection test: Simes

Stage results

Cumulative effect sizes S1: 0.258009, 0.298507, NA, NA
Cumulative effect sizes S2: 0.376344, 0.254158, NA, NA
Cumulative effect sizes F: -0.006662, 0.035928, NA, NA
Cumulative treatment rate S1: 0.743, 0.821, NA, NA
Cumulative treatment rate S2: 0.710, 0.657, NA, NA
Cumulative treatment rate F: 0.776, 0.749, NA, NA
Cumulative control rate S1: 0.485, 0.522, NA, NA
Cumulative control rate S2: 0.333, 0.403, NA, NA
Cumulative control rate F: 0.783, 0.713, NA, NA
Stage-wise test statistics S1: 2.188, 3.167, NA, NA
Stage-wise test statistics S2: 3.072, 1.100, NA, NA
Stage-wise test statistics F: -0.104, 1.059, NA, NA
Separate p-values S1: 0.0143309, 0.0007692, NA, NA
Separate p-values S2: 0.0010622, 0.1355959, NA, NA
Separate p-values F: 0.5414851, 0.1447620, NA, NA

Adjusted stage-wise p-values

Adjusted stage-wise p-values S1, S2, F: 0.0031867, 0.0023077, NA, NA
Adjusted stage-wise p-values S1, S2: 0.0021244, 0.0015385, NA, NA
Adjusted stage-wise p-values S1, F: 0.0286619, 0.0015385, NA, NA
Adjusted stage-wise p-values S2, F: 0.0021244, 0.1447620, NA, NA
Adjusted stage-wise p-values S1: 0.0143309, 0.0007692, NA, NA
Adjusted stage-wise p-values S2: 0.0010622, 0.1355959, NA, NA
Adjusted stage-wise p-values F: 0.5414851, 0.1447620, NA, NA

Overall adjusted test statistics

Overall adjusted test statistics S1, S2, F: 2.7279, 3.9100, NA, NA
Overall adjusted test statistics S1, S2: 2.8591, 4.0909, NA, NA
Overall adjusted test statistics S1, F: 1.9008, 3.5041, NA, NA
Overall adjusted test statistics S2, F: 2.8591, 2.5882, NA, NA
Overall adjusted test statistics S1: 2.1881, 3.8439, NA, NA
Overall adjusted test statistics S2: 3.0723, 2.7512, NA, NA
Overall adjusted test statistics F: -0.1042, 0.7736, NA, NA

Test actions

Rejected S1: FALSE, TRUE, NA, NA
Rejected S2: FALSE, TRUE, NA, NA
Rejected F: FALSE, FALSE, NA, NA

Further analysis results

Conditional rejection probability S1: 0.089957, 0.705416, NA, NA
Conditional rejection probability S2: 0.206347, 0.325240, NA, NA
Conditional rejection probability F: 0.007631, 0.012332, NA, NA
Conditional power S1: NA, NA, 0.9368, 0.9735
Conditional power S2: NA, NA, 0.9478, 0.9948
Conditional power F: NA, NA, 0.2417, 0.7339
Repeated confidence intervals (lower) S1: -0.11793, 0.05233, NA, NA
Repeated confidence intervals (lower) S2: -0.01576, -0.04578, NA, NA
Repeated confidence intervals (lower) F: -0.21370, -0.11998, NA, NA
Repeated confidence intervals (upper) S1: 0.5728, 0.5326, NA, NA
Repeated confidence intervals (upper) S2: 0.6671, 0.4905, NA, NA
Repeated confidence intervals (upper) F: 0.2019, 0.1969, NA, NA
Repeated p-values S1: 0.2792515, 0.0009566, NA, NA
Repeated p-values S2: 0.0310478, 0.0191307, NA, NA
Repeated p-values F: >0.5, >0.5, NA, NA

Legend

S[i]: population i
F: full population

Analysis results enrichment - survival

S <- getDataset(
    events = c(16, 19),
    logRanks = c(1.5, 1.3)
)

R <- getDataset(
    events = c(16, 29),
    logRanks = c(1.5, 1.3)
)
dataExampleSurvival <- getDataset(S1 = S, F = R)

kable(summary(getAnalysisResults(
    design = design1, dataInput = dataExampleSurvival,
    intersectionTest = "Simes", nPlanned = c(20, 20), directionUpper = TRUE
)))

Test statistics from full (and sub-populations) need to be stratified log-rank tests

Enrichment analysis results for a survival endpoint (2 populations)

Sequential analysis with 4 looks (inverse normal combination test design). The results were calculated using a two-sample logrank test (one-sided, alpha = 0.02), Simes intersection test, stratified analysis. H0: hazard ratio = 1 against H1: hazard ratio > 1. The conditional power calculation with planned sample size is based on overall effect: thetaH1(S1) = 1.95, thetaH1(F) = 1.78.

Stage	1	2	3	4
Fixed weight	0.387	0.5	0.548	0.548
Efficacy boundary (z-value scale)	2.870	2.572	2.391	2.272
Futility boundary (z-value scale)	-0.500	0.000	0.500
Cumulative alpha spent	0.0021	0.0067	0.0130	0.0200
Stage level	0.0021	0.0050	0.0084	0.0116
Cumulative effect size S1	2.117	1.948
Cumulative effect size F	2.117	1.782
Stage-wise test statistic S1	1.500	1.300
Stage-wise test statistic F	1.500	1.300
Stage-wise p-value S1	0.0668	0.0968
Stage-wise p-value F	0.0668	0.0968
Adjusted stage-wise p-value (1, 2)	0.0668	0.0968
Overall adjusted test statistic (1, 2)	1.500	1.946
Test action: reject S1	FALSE	FALSE
Test action: reject F	FALSE	FALSE
Conditional rejection probability S1	0.0574	0.1312
Conditional rejection probability F	0.0574	0.1312
Planned sample size			20	20
Conditional power S1			0.5342	0.8076
Conditional power F			0.4551	0.7248
96% repeated confidence interval S1	0.453 - 9.887	0.703 - 5.334
96% repeated confidence interval F	0.453 - 9.887	0.734 - 4.381
Repeated p-value S1	>0.5	0.0970
Repeated p-value F	>0.5	0.0970

Legend:

F: full population
S[i]: population i
(i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Enrichment analysis results (survival of 2 groups, inverse normal combination test design)

Design parameters

Fixed weights: 0.387, 0.500, 0.548, 0.548
Critical values: 2.870, 2.572, 2.391, 2.272
Futility bounds (non-binding): -0.500, 0.000, 0.500
Cumulative alpha spending: 0.002053, 0.006660, 0.013036, 0.020000
Local one-sided significance levels: 0.002053, 0.005050, 0.008402, 0.011555
Significance level: 0.0200
Test: one-sided

User defined parameters

Theta H0: 1
Planned sample size: NA, NA, 20, 20

Default parameters

Direction upper: TRUE
Planned allocation ratio: 1
Intersection test: Simes
Stratified analysis: TRUE

Stage results

Cumulative effect sizes S1: 2.117, 1.948, NA, NA
Cumulative effect sizes F: 2.117, 1.782, NA, NA
Stage-wise test statistics S1: 1.500, 1.300, NA, NA
Stage-wise test statistics F: 1.500, 1.300, NA, NA
Separate p-values S1: 0.06681, 0.09680, NA, NA
Separate p-values F: 0.06681, 0.09680, NA, NA

Adjusted stage-wise p-values

Adjusted stage-wise p-values S1, F: 0.06681, 0.09680, NA, NA
Adjusted stage-wise p-values S1: 0.06681, 0.09680, NA, NA
Adjusted stage-wise p-values F: 0.06681, 0.09680, NA, NA

Overall adjusted test statistics

Overall adjusted test statistics S1, F: 1.500, 1.946, NA, NA
Overall adjusted test statistics S1: 1.500, 1.946, NA, NA
Overall adjusted test statistics F: 1.500, 1.946, NA, NA

Test actions

Rejected S1: FALSE, FALSE, NA, NA
Rejected F: FALSE, FALSE, NA, NA

Further analysis results

Conditional rejection probability S1: 0.05736, 0.13117, NA, NA
Conditional rejection probability F: 0.05736, 0.13117, NA, NA
Conditional power S1: NA, NA, 0.5342, 0.8076
Conditional power F: NA, NA, 0.4551, 0.7248
Repeated confidence intervals (lower) S1: 0.4533, 0.7029, NA, NA
Repeated confidence intervals (lower) F: 0.4533, 0.7336, NA, NA
Repeated confidence intervals (upper) S1: 9.887, 5.334, NA, NA
Repeated confidence intervals (upper) F: 9.887, 4.381, NA, NA
Repeated p-values S1: >0.5, 0.09696, NA, NA
Repeated p-values F: >0.5, 0.09696, NA, NA

Legend

S[i]: population i
F: full population

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/.

Reuse

https://creativecommons.org/licenses/by-sa/4.0/