# Designing Group Sequential Trials with a Binary Endpoint with rpact

Planning
Rates
This document provides examples for designing trials with binary endpoints using rpact.
Author

Marcel Wolbers, Gernot Wassmer, and Friedrich Pahlke

Published

July 6, 2023

# Sample size calculation for a superiority trial with two groups without interim analyses

The sample size for a trial with binary endpoints can be calculated using the function getSampleSizeRates(). This function is fully documented in the help page (?getSampleSizeRates). Hence, we only provide some examples below.

library(rpact)
packageVersion("rpact") 
[1] '3.4.0'

To get the direction of the effects correctly, note that in rpact the index “2” in an argument name always refers to the control group, “1” to the intervention group, and treatment effects compare treatment versus control. Specifically, for binary endpoints, the probabilities of an event in the control group and intervention group, respectively, are given by arguments pi2 and pi1. The default treatment effect is the absolute risk difference pi1 - pi2 but the relative risk scale pi1/pi2 is also supported if the argument riskRatio is set to TRUE.

# Example of a standard trial:
# - probability 25% in control (pi2 = 0.25) vs. 40% (pi1 = 0.4) in intervention
# - one-sided test (sided = 1)
# - Type I error 0.025 (alpha = 0.025) and power 80% (beta = 0.2)
sampleSizeResult <- getSampleSizeRates(
pi2 = 0.25, pi1 = 0.4,
sided = 1, alpha = 0.025, beta = 0.2
)
kable(sampleSizeResult)

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 treatment rate: 0.400
• Assumed control rate: 0.250

Default parameters

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

Sample size and output

• Direction upper: TRUE
• Number of subjects fixed: 303.7
• Number of subjects fixed (1): 151.9
• Number of subjects fixed (2): 151.9
• Critical values (treatment effect scale): 0.103

Legend

• (i): values of treatment arm i

As per the output above, the required total sample size is 304 and the critical value corresponds to a minimal detectable difference (on the absolute risk difference scale, the default) of approximately 0.103. This calculation assumes that pi2 = 0.25 is the observed rate in treatment group 2.

A useful summary is provided with the generic summary() function:

kable(summary(sampleSizeResult))

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) = 0.4, control rate pi(2) = 0.25, power 80%.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Number of subjects 303.7
One-sided local significance level 0.0250
Efficacy boundary (t) 0.103

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 treatment rate: 0.400
• Assumed control rate: 0.250

Default parameters

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

Sample size and output

• Direction upper: TRUE
• Number of subjects fixed: 303.7
• Number of subjects fixed (1): 151.9
• Number of subjects fixed (2): 151.9
• Critical values (treatment effect scale): 0.103

Legend

• (i): values of treatment arm i

You can change the randomization allocation between the treatment groups using allocationRatioPlanned:

# Example: Extension of standard trial
# - 2(intervention):1(control) randomization (allocationRatioPlanned = 2)
kable(summary(getSampleSizeRates(
pi2 = 0.25, pi1 = 0.4,
sided = 1, alpha = 0.025, beta = 0.2,
allocationRatioPlanned = 2
)))

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) = 0.4, control rate pi(2) = 0.25, planned allocation ratio = 2, power 80%.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Number of subjects 346.3
One-sided local significance level 0.0250
Efficacy boundary (t) 0.104

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 treatment rate: 0.400
• Assumed control rate: 0.250
• Planned allocation ratio: 2

Default parameters

• Risk ratio: FALSE
• Theta H0: 0
• Normal approximation: TRUE
• Treatment groups: 2

Sample size and output

• Direction upper: TRUE
• Number of subjects fixed: 346.3
• Number of subjects fixed (1): 230.9
• Number of subjects fixed (2): 115.4
• Critical values (treatment effect scale): 0.104

Legend

• (i): values of treatment arm i

allocationRatioPlanned = 0 can be defined in order to obtain the optimum allocation ratio minimizing the overall sample size (the optimum ample size is only slightly smaller than sample size with equal allocation; practically, this has no effect):

# Example: Extension of standard trial
# optimum randomization ratio
kable(summary(getSampleSizeRates(
pi2 = 0.25, pi1 = 0.4,
sided = 1, alpha = 0.025, beta = 0.2,
allocationRatioPlanned = 0
)))

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) = 0.4, control rate pi(2) = 0.25, optimum planned allocation ratio = 0.953, power 80%.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Number of subjects 303.6
One-sided local significance level 0.0250
Efficacy boundary (t) 0.103

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 treatment rate: 0.400
• Assumed control rate: 0.250
• Optimum allocation ratio: TRUE

Default parameters

• Risk ratio: FALSE
• Theta H0: 0
• Normal approximation: TRUE
• Treatment groups: 2

Sample size and output

• Planned allocation ratio: 0.953
• Direction upper: TRUE
• Number of subjects fixed: 303.6
• Number of subjects fixed (1): 148.1
• Number of subjects fixed (2): 155.5
• Critical values (treatment effect scale): 0.103

Legend

• (i): values of treatment arm i

Power at given sample size can be calculated using the function getPowerRates(). This function has the same arguments as getSampleSizeRates() except that the maximum total sample size needs to be defined (maxNumberOfSubjects) and the Type II error beta is no longer needed. For one-sided tests, the direction of the test is also required. The default directionUpper = TRUE indicates that for the alternative the probability in the intervention group pi1 is larger than the probability in the control group pi2 (directionUpper = FALSE is the other direction):

# Example: Calculate power for a simple trial with total sample size 304
# as in the example above in case of pi2 = 0.25 (control) and
# pi1 = 0.37 (intervention)
powerResult <- getPowerRates(
pi2 = 0.25, pi1 = 0.37,
maxNumberOfSubjects = 304, sided = 1, alpha = 0.025
)
kable(powerResult)

Design plan parameters and output for rates

Design parameters

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

User defined parameters

• Assumed treatment rate: 0.370
• Assumed control rate: 0.250
• Direction upper: NA
• Maximum number of subjects: 304

Default parameters

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

Sample size and output

• Effect: 0.12
• Number of subjects fixed: 304
• Number of subjects fixed (1): 152
• Number of subjects fixed (2): 152
• Overall reject: 0.6196
• Critical values (treatment effect scale): 0.103

Legend

• (i): values of treatment arm i

The calculated power is provided in the output as “Overall reject” and is 0.620 for the example.

The summary() command produces the output

kable(summary(powerResult))

Power calculation for a binary endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The results were calculated for a two-sample test for rates (normal approximation), H0: pi(1) - pi(2) = 0, power directed towards larger values, H1: treatment rate pi(1) = 0.37, control rate pi(2) = 0.25, number of subjects = 304.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Power 0.6196
Number of subjects 304.0
One-sided local significance level 0.0250
Efficacy boundary (t) 0.103

Legend:

• (t): treatment effect scale

Design plan parameters and output for rates

Design parameters

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

User defined parameters

• Assumed treatment rate: 0.370
• Assumed control rate: 0.250
• Direction upper: NA
• Maximum number of subjects: 304

Default parameters

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

Sample size and output

• Effect: 0.12
• Number of subjects fixed: 304
• Number of subjects fixed (1): 152
• Number of subjects fixed (2): 152
• Overall reject: 0.6196
• Critical values (treatment effect scale): 0.103

Legend

• (i): values of treatment arm i

The getPowerRates() (as well as getSampleSizeRates()) functions can also be called with a vector argument for the probability pi1 in the intervention group. This is illustrated below via a plot of power depending on this probability. For examples of all available plots, see the R Markdown document How to create admirable plots with rpact.

# Example: Calculate power for simple design (with sample size 304 as above)
# for probabilities in intervention ranging from 0.3 to 0.5
powerResult <- getPowerRates(
pi2 = 0.25, pi1 = seq(0.3, 0.5, by = 0.01),
maxNumberOfSubjects = 304, sided = 1, alpha = 0.025
)

# one of several possible plots, this one plotting true effect size vs power
plot(powerResult, type = 7)

# Sample size calculation for a non-inferiority trial with two groups without interim analyses

Sample size calculation proceeds in the same fashion as for superiority trials except that the role of the null and the alternative hypothesis are reversed. I.e., in this case, the non-inferiority margin $$\Delta$$ corresponds to the treatment effect under the null hypothesis (thetaH0) which one aims to reject. Testing in non-inferiority trials is always one-sided.

# Example: Sample size for a non-inferiority trial
# Assume pi(control) = pi(intervention) = 0.2
# Test H0: pi1 - pi2 = 0.1 (risk increase in intervention >= Delta = 0.1)
# vs. H1: pi1 - pi2 < 0.1
sampleSizeNoninf <- getSampleSizeRates(
pi2 = 0.2, pi1 = 0.2,
thetaH0 = 0.1, sided = 1, alpha = 0.025, beta = 0.2
)
kable(sampleSizeNoninf)

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

• Theta H0: 0.1
• Assumed treatment rate: 0.200

Default parameters

• Risk ratio: FALSE
• Normal approximation: TRUE
• Assumed control rate: 0.200
• Treatment groups: 2
• Planned allocation ratio: 1

Sample size and output

• Direction upper: FALSE
• Number of subjects fixed: 508.4
• Number of subjects fixed (1): 254.2
• Number of subjects fixed (2): 254.2
• Critical values (treatment effect scale): 0.0285

Legend

• (i): values of treatment arm i
kable(summary(sampleSizeNoninf))

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) = 0.1, H1: treatment rate pi(1) = 0.2, control rate pi(2) = 0.2, power 80%.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Number of subjects 508.4
One-sided local significance level 0.0250
Efficacy boundary (t) 0.028

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

• Theta H0: 0.1
• Assumed treatment rate: 0.200

Default parameters

• Risk ratio: FALSE
• Normal approximation: TRUE
• Assumed control rate: 0.200
• Treatment groups: 2
• Planned allocation ratio: 1

Sample size and output

• Direction upper: FALSE
• Number of subjects fixed: 508.4
• Number of subjects fixed (1): 254.2
• Number of subjects fixed (2): 254.2
• Critical values (treatment effect scale): 0.0285

Legend

• (i): values of treatment arm i

# Sample size calculation for a single arm trial without interim analyses

The function getSampleSizeRates() allows to set the number of groups (which is 2 by default) to 1 for the design of single arm trials. The probability under the null hypothesis can be specified with the argument thetaH0and the specific alternative hypothesis which is used for the sample size calculation with the argument pi1. The sample size calculation can be based either on a normal approximation (normalApproximation = TRUE, the default) or on exact binomial probabilities (normalApproximation = FALSE).

# Example: Sample size for a single arm trial which tests
# H0: pi = 0.1 vs. H1: pi = 0.25
# (use conservative exact binomial calculation)
samplesSizeResults <- getSampleSizeRates(
groups = 1, thetaH0 = 0.1, pi1 = 0.25,
normalApproximation = FALSE, sided = 1, alpha = 0.025, beta = 0.2
)

kable(summary(samplesSizeResults))

Sample size calculation for a binary endpoint

Fixed sample analysis, significance level 2.5% (one-sided). The sample size was calculated for a one-sample test for rates (exact test), H0: pi = 0.1, H1: treatment rate pi = 0.25, power 80%.

Stage Fixed
Efficacy boundary (z-value scale) 1.960
Number of subjects 53.0
One-sided local significance level 0.0250
Efficacy boundary (t) 0.181

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

• Theta H0: 0.1
• Normal approximation: FALSE
• Assumed treatment rate: 0.250
• Treatment groups: 1

Sample size and output

• Direction upper: TRUE
• Number of subjects fixed: 53
• Critical values (treatment effect scale): 0.181

# Sample size calculation for group sequential designs

Sample size calculation for a group sequential trial is performed in two steps:

1. Define the (abstract) group sequential design using the function getDesignGroupSequential(). For details regarding this step, see the vignette Defining group sequential boundaries with rpact.
2. Calculate sample size for the binary endpoint by feeding the abstract design into the function getSampleSizeRates(). Note that the power 1 - beta needs to be defined in the design function, and not in getSampleSizeRates().

In general, rpact supports both one-sided and two-sided group sequential designs. However, if futility boundaries are specified, only one-sided tests are permitted.

R code for a simple example is provided below:

# Example: Group-sequential design with  O'Brien & Fleming type alpha-spending and
# one interim at 60% information
design <- getDesignGroupSequential(
sided = 1, alpha = 0.025, beta = 0.2,
informationRates = c(0.6, 1), typeOfDesign = "asOF"
)

# Sample size calculation assuming event probabilities are 25% in control
# (pi2 = 0.25) vs 40% (pi1 = 0.4) in intervention
sampleSizeResultGS <- getSampleSizeRates(design, pi2 = 0.25, pi1 = 0.4)
# Standard rpact output (sample size object only, not design object)
kable(sampleSizeResultGS)

Design plan parameters and output for rates

Design parameters

• Information rates: 0.600, 1.000
• Critical values: 2.669, 1.981
• Futility bounds (non-binding): -Inf
• Cumulative alpha spending: 0.003808, 0.025000
• Local one-sided significance levels: 0.003808, 0.023798
• Significance level: 0.0250
• Type II error rate: 0.2000
• Test: one-sided

User defined parameters

• Assumed treatment rate: 0.400
• Assumed control rate: 0.250

Default parameters

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

Sample size and output

• Direction upper: TRUE
• Reject per stage [1]: 0.3123
• Reject per stage [2]: 0.4877
• Early stop: 0.3123
• Maximum number of subjects: 306.3
• Maximum number of subjects (1): 153.2
• Maximum number of subjects (2): 153.2
• Number of subjects [1]: 183.8
• Number of subjects [2]: 306.3
• Expected number of subjects under H0: 305.9
• Expected number of subjects under H0/H1: 299.3
• Expected number of subjects under H1: 268.1
• Critical values (treatment effect scale) [1]: 0.187
• Critical values (treatment effect scale) [2]: 0.104

Legend

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

The summary() command produces the output

kable(summary(sampleSizeResultGS))

Sample size calculation for a binary endpoint

Sequential analysis with a maximum of 2 looks (group sequential design), overall significance level 2.5% (one-sided). The sample size was calculated for a two-sample test for rates (normal approximation), H0: pi(1) - pi(2) = 0, H1: treatment rate pi(1) = 0.4, control rate pi(2) = 0.25, power 80%.

Stage 1 2
Information rate 60% 100%
Efficacy boundary (z-value scale) 2.669 1.981
Overall power 0.3123 0.8000
Expected number of subjects 268.1
Number of subjects 183.8 306.3
Cumulative alpha spent 0.0038 0.0250
One-sided local significance level 0.0038 0.0238
Efficacy boundary (t) 0.187 0.104
Exit probability for efficacy (under H0) 0.0038
Exit probability for efficacy (under H1) 0.3123

Legend:

• (t): treatment effect scale

Design plan parameters and output for rates

Design parameters

• Information rates: 0.600, 1.000
• Critical values: 2.669, 1.981
• Futility bounds (non-binding): -Inf
• Cumulative alpha spending: 0.003808, 0.025000
• Local one-sided significance levels: 0.003808, 0.023798
• Significance level: 0.0250
• Type II error rate: 0.2000
• Test: one-sided

User defined parameters

• Assumed treatment rate: 0.400
• Assumed control rate: 0.250

Default parameters

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

Sample size and output

• Direction upper: TRUE
• Reject per stage [1]: 0.3123
• Reject per stage [2]: 0.4877
• Early stop: 0.3123
• Maximum number of subjects: 306.3
• Maximum number of subjects (1): 153.2
• Maximum number of subjects (2): 153.2
• Number of subjects [1]: 183.8
• Number of subjects [2]: 306.3
• Expected number of subjects under H0: 305.9
• Expected number of subjects under H0/H1: 299.3
• Expected number of subjects under H1: 268.1
• Critical values (treatment effect scale) [1]: 0.187
• Critical values (treatment effect scale) [2]: 0.104

Legend

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

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/.