library(rpact)
packageVersion("rpact")
How to Create Summaries with rpact
Global options
First, load the rpact package
[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
Sequential analysis with a maximum of 3 looks (group sequential design), overall significance level 2.5% (one-sided). The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0, H1: effect as specified, standard deviation = 1, power 80%.
Stage | 1 | 2 | 3 |
---|---|---|---|
Information rate | 33.3% | 66.7% | 100% |
Efficacy boundary (z-value scale) | 3.471 | 2.454 | 2.004 |
Futility boundary (z-value scale) | 1.000 | 2.000 | |
Overall power | 0.0967 | 0.7030 | 0.8000 |
Expected number of subjects, alt. = 0.2 | 888.9 | ||
Expected number of subjects, alt. = 0.4 | 223.9 | ||
Expected number of subjects, alt. = 0.6 | 100.7 | ||
Expected number of subjects, alt. = 0.8 | 57.7 | ||
Expected number of subjects, alt. = 1 | 37.8 | ||
Number of subjects, alt. = 0.2 | 472.2 | 944.4 | 1416.5 |
Number of subjects, alt. = 0.4 | 118.9 | 237.8 | 356.8 |
Number of subjects, alt. = 0.6 | 53.5 | 107.0 | 160.5 |
Number of subjects, alt. = 0.8 | 30.6 | 61.3 | 91.9 |
Number of subjects, alt. = 1 | 20.1 | 40.1 | 60.2 |
Cumulative alpha spent | 0.0003 | 0.0072 | 0.0250 |
One-sided local significance level | 0.0003 | 0.0071 | 0.0225 |
Efficacy boundary (t), alt. = 0.2 | 0.322 | 0.160 | 0.107 |
Efficacy boundary (t), alt. = 0.4 | 0.655 | 0.321 | 0.213 |
Efficacy boundary (t), alt. = 0.6 | 1.013 | 0.483 | 0.319 |
Efficacy boundary (t), alt. = 0.8 | 1.413 | 0.646 | 0.424 |
Efficacy boundary (t), alt. = 1 | 1.882 | 0.812 | 0.528 |
Futility boundary (t), alt. = 0.2 | 0.092 | 0.130 | |
Futility boundary (t), alt. = 0.4 | 0.184 | 0.261 | |
Futility boundary (t), alt. = 0.6 | 0.276 | 0.391 | |
Futility boundary (t), alt. = 0.8 | 0.368 | 0.522 | |
Futility boundary (t), alt. = 1 | 0.459 | 0.653 | |
Overall exit probability (under H0) | 0.8416 | 0.1462 | |
Overall exit probability (under H1) | 0.2176 | 0.6822 | |
Exit probability for efficacy (under H0) | 0.0003 | 0.0062 | |
Exit probability for efficacy (under H1) | 0.0967 | 0.6064 | |
Exit probability for futility (under H0) | 0.8413 | 0.1400 | |
Exit probability for futility (under H1) | 0.1209 | 0.0758 |
Legend:
- alt.: alternative
- (t): treatment effect scale
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
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(getSampleSizeMeans(getDesignGroupSequential(futilityBounds = c(1, 2))), digits = -1))
Sample size calculation for a continuous endpoint
Sequential analysis with a maximum of 3 looks (group sequential design), overall significance level 2.5% (one-sided). The sample size was calculated for a two-sample t-test, H0: mu(1) - mu(2) = 0, H1: effect as specified, standard deviation = 1, power 80%.
Stage | 1 | 2 | 3 |
---|---|---|---|
Information rate | 33.3% | 66.7% | 100% |
Efficacy boundary (z-value scale) | 3.4710914446541 | 2.45443229863353 | 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
<- list(
piecewiseSurvivalTime "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
<- getDesignInverseNormal(
design 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
<- getDesignFisher(
design 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
<- getDesignInverseNormal(
design 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
<- getDesignFisher(alpha = 0.05, kMax = 3)
design 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
<- getDesignFisher(alpha = 0.05, kMax = 3)
design
<- c("S", "R")
subGroups <- c(0.1, 0.9)
prevalences <- c(0.4, 0.5)
alternative <- list(
effectList 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
<- getDesignFisher(alpha = 0.05, kMax = 3)
design
<- c("S", "R")
subGroups <- c(0.1, 0.9)
prevalences <- c(0.3, 0.4)
pi2 <- c(0.4, 0.5)
piTreatments <- list(
effectList 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
<- getDesignFisher(alpha = 0.05, kMax = 3)
design
<- c("S1", "S2", "S12", "R")
subGroups <- c(0.1, 0.3, 0.4, 0.2)
prevalences <- c(0.4, 0.5, 0.6, 0.7, 0.6, 0.6, 0.6, 0.8)
hazardRatios <- list(
effectList 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
<- getDesignInverseNormal(
design1 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)
)
<- getDesignConditionalDunnett(
design3 alpha = 0.02,
informationAtInterim = 0.4, secondStageConditioning = TRUE
)
Analysis results base
Analysis results base - means
<- getDataset(
simpleDataExampleMeans1 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
<- getDataset(
simpleDataExampleMeans2 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
<- getDataset(
simpleDataExampleRates1 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
<- getDataset(
simpleDataExampleRates2 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
<- getDataset(
simpleDataExampleSurvival 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
<- getDataset(
dataExampleMeans 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
<- getDataset(
dataExampleRates 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
<- getDataset(
dataExampleSurvival 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
<- getDataset(
dataS1 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)
)<- getDataset(
dataNotS1 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)
)<- getDataset(S1 = dataS1, R = dataNotS1)
dataExampleMeans
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
<- getDataset(
S1 events2 = c(16, 19),
sampleSizes2 = c(33, 34),
events1 = c(26, 29),
sampleSizes1 = c(35, 32)
)<- getDataset(
S2 events2 = c(12, 15),
sampleSizes2 = c(36, 31),
events1 = c(22, 24),
sampleSizes1 = c(31, 39)
)<- getDataset(
F events2 = c(65, 54),
sampleSizes2 = c(83, 84),
events1 = c(66, 59),
sampleSizes1 = c(85, 82)
)
<- getDataSet(S1 = S1, S2 = S2, F = F)
dataExampleRates
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
<- getDataset(
S events = c(16, 19),
logRanks = c(1.5, 1.3)
)
<- getDataset(
R events = c(16, 29),
logRanks = c(1.5, 1.3)
)<- getDataset(S1 = S, F = R)
dataExampleSurvival
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/.