Package {bbssr}

bbssr: Blinded Sample Size Re-Estimation for Binary Endpoints

Description

Tools for blinded sample size re-estimation (BSSR) in two-arm clinical trials with binary endpoints, together with the exact power and sample size calculations the re-estimation relies on. Five exact tests are supported, each available with a one-sided or a two-sided alternative, and the exact unconditional tests can be combined with the Berger-Boos procedure.

Main functions

BinaryRR

Rejection region of an exact test

BinaryPower

Exact power at a given sample size

BinarySampleSize

Sample size attaining a target power

BinaryPowerBSSR

Operating characteristics of a BSSR design

BinaryBSSR

Sample size re-estimation from observed interim data

Author(s)

Maintainer: Gosuke Homma my.name.is.gosuke@gmail.com

Authors:

• Gosuke Homma my.name.is.gosuke@gmail.com

See Also

Useful links:

• https://github.com/gosukehommaEX/bbssr

• Report bugs at https://github.com/gosukehommaEX/bbssr/issues

Sample Size Re-estimation from Observed Blinded Interim Data

Description

Re-estimates the sample size of an ongoing two-arm trial with a binary endpoint from the blinded data available at an interim analysis, and reports how many patients still have to be enrolled in each group during the second stage. Only the total number of patients and the total number of responders are required, so the treatment allocation remains concealed.

Usage

BinaryBSSR(
  n1,
  n2,
  S,
  Delta.A,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  N1 = NULL,
  N2 = NULL,
  alternative = c("greater", "two.sided"),
  tsmethod = c("minlike", "central"),
  n.grid = 100,
  bb.gamma = 0
)

Arguments

Details

The blinded estimate of the pooled response probability is hat.p = S / (n1 + n2). Group-specific proportions are recovered as hat.p1 = hat.p + Delta.A / (1 + r) and hat.p2 = hat.p - r Delta.A / (1 + r), truncated to the unit interval, and the sample size is then re-estimated by BinarySampleSize.

Under the unrestricted rule the final size of group 2 is the larger of the re-estimated size and what has already been observed. Under the restricted rule it is raised to the planned size first, so the trial can only grow. The final size of group 1 is then ceiling(r N2.final), so an imbalance already present at the interim is corrected by the remaining enrolment instead of being carried forward. Neither second-stage size is ever negative.

While BinaryPowerBSSR evaluates the operating characteristics of a BSSR design at the planning stage, this function is applied once, to the data of a trial that is under way.

Value

An object of class bbssr_bssr, a data frame with one row containing:

n1

Interim sample size of group 1

n2

Interim sample size of group 2

n

Total interim sample size

S

Total number of interim responders

hat.p

Blinded estimate of the pooled response probability

hat.p1

Recovered response probability of group 1

hat.p2

Recovered response probability of group 2

N1.re

Re-estimated total sample size of group 1

N2.re

Re-estimated total sample size of group 2

N.re

Re-estimated total sample size

n1.stage2

Number of additional patients to enrol in group 1

n2.stage2

Number of additional patients to enrol in group 2

n.stage2

Total number of additional patients to enrol

N1.final

Final sample size of group 1

N2.final

Final sample size of group 2

N.final

Final total sample size

Power

Exact power at the final sample size under the recovered proportions

Author(s)

Gosuke Homma (my.name.is.gosuke@gmail.com)

See Also

BinaryPowerBSSR, BinarySampleSize

Examples

# Interim data: 20 patients per group, 11 responders in total
BinaryBSSR(n1 = 20, n2 = 20, S = 11, Delta.A = 0.3, r = 1,
           alpha = 0.025, tar.power = 0.8, Test = 'Chisq')


# Restricted rule with a planned sample size of 40 per group
BinaryBSSR(n1 = 20, n2 = 20, S = 11, Delta.A = 0.3, r = 1,
           alpha = 0.025, tar.power = 0.8, Test = 'Boschloo',
           restricted = TRUE, N1 = 40, N2 = 40)

Power Calculation for Two-Arm Trials with Binary Endpoints

Description

Calculates power for two-arm trials with binary endpoints using exact statistical tests. Five tests are supported, each of which can be applied with a one-sided or a two-sided alternative, and vectors of response probabilities are accepted.

Usage

BinaryPower(
  p1,
  p2,
  N1,
  N2,
  alpha,
  Test,
  alternative = c("greater", "two.sided"),
  tsmethod = c("minlike", "central"),
  n.grid = 100,
  bb.gamma = 0
)

Arguments

Details

The power is obtained by summing the joint probability mass function of the two independent binomial counts over the rejection region returned by BinaryRR. The summation covers the whole rejection region rather than a row-wise tail, so it remains valid for two-sided tests, whose rejection regions are not contiguous within a row of the outcome grid.

Value

An object of class bbssr_power, a data frame with one row per element of p1 containing:

p1

True probability of responders for group 1

p2

True probability of responders for group 2

N1

Sample size for group 1

N2

Sample size for group 2

alpha

Level of significance

Test

Name of the statistical test

alternative

Direction of the alternative hypothesis

Power

Exact power

Author(s)

Gosuke Homma (my.name.is.gosuke@gmail.com)

Examples

# Power of the one-sided chi-squared test
BinaryPower(p1 = 0.5, p2 = 0.2, N1 = 5, N2 = 5, alpha = 0.025, Test = 'Chisq')


# Power over a range of response probabilities for the two-sided Boschloo test
pw <- BinaryPower(p1 = c(0.5, 0.6, 0.7, 0.8), p2 = rep(0.2, 4),
                  N1 = 20, N2 = 20, alpha = 0.05, Test = 'Boschloo',
                  alternative = 'two.sided')
print(pw)
plot(pw)

Power of a Blinded Sample Size Re-estimation Design for Binary Endpoints

Description

Calculates the power of a two-arm trial with a binary endpoint when blinded sample size re-estimation (BSSR) is implemented, together with the power of the corresponding fixed-sample design. Five exact tests are supported, each of which can be applied with a one-sided or a two-sided alternative, under either a restricted or an unrestricted design rule.

Usage

BinaryPowerBSSR(
  p,
  Delta.A,
  Delta.T,
  N1,
  N2,
  omega,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  alternative = c("greater", "two.sided"),
  tsmethod = c("minlike", "central"),
  n.grid = 100,
  bb.gamma = 0
)

Arguments

Details

Both the interim and the final sample size of group 1 are obtained from the size of group 2 by a single application of ceiling(r ...), so the allocation ratio is preserved as closely as whole numbers allow and is exact whenever r is a whole number. The size of the second stage follows as the difference between the final and the interim size, which keeps the two stages adding up to the final size. The argument N1 enters only through the fixed-sample comparator, and a warning is issued when it is not ceiling(r N2).

At the interim analysis the pooled number of responders is observed without unblinding. The pooled proportion is combined with the assumed treatment effect Delta.A to recover group-specific proportions, from which the sample size is re-estimated. The power is then averaged over the distribution of the interim outcome.

Setting Delta.T to 0 makes the two groups identical, so power.BSSR and power.TRAD become rejection probabilities under the null hypothesis. They are evaluated only at the values supplied through p, since the function does not search over the unit interval on its own. The largest type I error rate of the design is therefore obtained by passing a grid such as p = seq(0.02, 0.98, by = 0.02) and taking the maximum of the power.BSSR column. The rejection region and the re-estimated sample size are shared across the elements of p, so a fine grid costs much less than the same number of separate calls.

The weighted approach available in earlier versions of the package has been removed.

Value

An object of class bbssr_powerbssr, a data frame with one row per element of p containing:

p1

True probability of responders for group 1

p2

True probability of responders for group 2

p

True pooled probability of responders from both groups

power.BSSR

Power of the BSSR design

power.TRAD

Power of the fixed-sample design

E.N

Expected total sample size of the BSSR design

The interim sample sizes are stored as the attributes n1.interim and n2.interim.

Author(s)

Gosuke Homma (my.name.is.gosuke@gmail.com)

Examples

# Small BSSR calculation with the chi-squared test
BinaryPowerBSSR(
  p = 0.45,
  Delta.A = 0.3, Delta.T = 0.3,
  N1 = 5, N2 = 5, omega = 0.5, r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq'
)


res <- BinaryPowerBSSR(
  p = seq(0.19, 0.37, by = 0.03),
  Delta.A = 0.36, Delta.T = 0.36,
  N1 = 24, N2 = 24, omega = 0.5, r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Z-pool'
)
print(res)
plot(res)

Rejection Region for Two-Arm Trials with Binary Endpoints

Description

Provides a rejection region (RR) for two-arm trials with binary endpoints using various exact statistical tests. Five tests are supported, each of which can be applied with a one-sided or a two-sided alternative.

Usage

BinaryRR(
  N1,
  N2,
  alpha,
  Test,
  alternative = c("greater", "two.sided"),
  tsmethod = c("minlike", "central"),
  n.grid = 100,
  bb.gamma = 0
)

Arguments

Details

The function supports the following five tests:

• The Pearson chi-squared test (Chisq)

• The Fisher exact test (Fisher)

• The Fisher mid-p test (Fisher-midP)

• The Z-pooled exact unconditional test (Z-pool)

• The Boschloo exact unconditional test (Boschloo)

For the two-sided versions of the conditional tests, 'minlike' sums the null probabilities of all tables that are no more likely than the observed table, which is the convention of stats::fisher.test, whereas 'central' doubles the smaller of the two one-sided tail probabilities. The two-sided versions of 'Chisq' and 'Z-pool' order the outcomes by the absolute value of the Z statistic.

The unconditional tests maximize the null tail probability of an ordering statistic over the common response probability, which is a nuisance parameter. Outcomes sharing the same value of the ordering statistic receive the same p-value.

Value

An object of class bbssr_rr, which is a logical matrix of dimension (N1 + 1) by (N2 + 1) whose entry [i + 1, j + 1] is TRUE when the null hypothesis is rejected at i responders in group 1 and j responders in group 2. The design settings are stored as attributes

Author(s)

Gosuke Homma (my.name.is.gosuke@gmail.com)

Examples

# Simple example with small sample sizes
RR <- BinaryRR(N1 = 5, N2 = 5, alpha = 0.025, Test = 'Chisq')
print(RR)


# Two-sided Boschloo test with the Berger-Boos procedure
RR <- BinaryRR(N1 = 20, N2 = 10, alpha = 0.05, Test = 'Boschloo',
               alternative = 'two.sided', bb.gamma = 0.0001)
print(RR)
plot(RR)

Sample Size Calculation for Two-Arm Trials with Binary Endpoints

Description

Calculates the required sample size for two-arm trials with binary endpoints using exact statistical tests. Five tests are supported, each of which can be applied with a one-sided or a two-sided alternative.

Usage

BinarySampleSize(
  p1,
  p2,
  r,
  alpha,
  tar.power,
  Test,
  alternative = c("greater", "two.sided"),
  tsmethod = c("minlike", "central"),
  n.grid = 100,
  bb.gamma = 0
)

Arguments

Details

The calculation uses a three-step approach:

1. Calculate an initial sample size from the normal approximation to the chi-squared test

2. Evaluate the exact power at the initial sample size

3. Move the sample size up or down one unit at a time until the smallest sample size attaining the target power is found

The normal approximation of the first step uses alpha for a one-sided alternative and alpha / 2 for a two-sided alternative. Only the starting value of the search is affected, so the returned sample size is exact in either case.

Value

An object of class bbssr_samplesize, a data frame with one row containing:

p1

True probability of responders for group 1

p2

True probability of responders for group 2

r

Allocation ratio to group 1

alpha

Level of significance

tar.power

Target power

Test

Name of the statistical test

alternative

Direction of the alternative hypothesis

Power

Exact power at the selected sample size

N1

Required sample size of group 1

N2

Required sample size of group 2

N

Total required sample size

Author(s)

Gosuke Homma (my.name.is.gosuke@gmail.com)

Examples

# One-sided chi-squared test
BinarySampleSize(p1 = 0.4, p2 = 0.2, r = 1, alpha = 0.025,
                 tar.power = 0.8, Test = 'Chisq')


# Two-sided Fisher exact test
BinarySampleSize(p1 = 0.5, p2 = 0.2, r = 2, alpha = 0.05,
                 tar.power = 0.9, Test = 'Fisher', alternative = 'two.sided')

Plot Exact Power against the Response Probability of Group 1

Description

Plot Exact Power against the Response Probability of Group 1

Usage

## S3 method for class 'bbssr_power'
plot(
  x,
  main = NULL,
  sub = NULL,
  xlab = NULL,
  ylab = NULL,
  ylim = c(0, 1),
  show.points = TRUE,
  colours = NULL,
  base_size = 11,
  ...
)

Arguments

Details

The vertical range is imposed with coord_cartesian, so points outside ylim are hidden rather than removed from the data.

Value

A ggplot object

Examples

pw <- BinaryPower(p1 = seq(0.3, 0.8, by = 0.1), p2 = rep(0.2, 6),
                  N1 = 20, N2 = 20, alpha = 0.025, Test = 'Chisq')
plot(pw)

# Enlarge the type, rescale the vertical axis, rename the axes and drop the markers
plot(pw, base_size = 14, ylim = c(0.2, 1), sub = NA, show.points = FALSE,
     xlab = 'Response probability, experimental group', ylab = 'Exact power')

Plot the Power of a BSSR Design against the Fixed-Sample Design

Description

Plot the Power of a BSSR Design against the Fixed-Sample Design

Usage

## S3 method for class 'bbssr_powerbssr'
plot(
  x,
  main = NULL,
  sub = NULL,
  xlab = NULL,
  ylab = NULL,
  ylim = NULL,
  ref.line = NULL,
  legend.title = NULL,
  legend.labels = NULL,
  show.points = TRUE,
  colours = NULL,
  base_size = 11,
  ...
)

Arguments

Details

Setting Delta.T to 0 in BinaryPowerBSSR turns the two power columns into rejection probabilities under the null hypothesis. The default reference line at the target power is then out of place, and the level of significance passed through ref.line together with a rescaled ylim gives a readable plot of the type I error rate.

The vertical range is imposed with coord_cartesian, so points outside ylim are hidden rather than removed from the data.

Value

A ggplot object

Examples

res <- BinaryPowerBSSR(
  p = seq(0.19, 0.37, by = 0.03),
  Delta.A = 0.36, Delta.T = 0.36,
  N1 = 24, N2 = 24, omega = 0.5, r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq'
)
plot(res)

# Type I error rate, with the reference line moved to the level of significance
tie <- BinaryPowerBSSR(
  p = seq(0.19, 0.37, by = 0.03),
  Delta.A = 0.36, Delta.T = 0,
  N1 = 24, N2 = 24, omega = 0.5, r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq'
)
plot(tie, ref.line = 0.025, ylim = c(0.015, 0.030), base_size = 14,
     show.points = FALSE,
     main = 'Type I error rate of the BSSR design', ylab = 'Type I error rate',
     legend.title = 'Design', legend.labels = c('BSSR', 'Fixed'))

Plot a Rejection Region

Description

Displays the outcome grid of a two-arm trial with a binary endpoint and shades the outcomes for which the null hypothesis is rejected.

Usage

## S3 method for class 'bbssr_rr'
plot(
  x,
  main = NULL,
  sub = NULL,
  xlab = NULL,
  ylab = NULL,
  legend.title = NULL,
  legend.labels = NULL,
  colours = NULL,
  base_size = 11,
  ...
)

Arguments

Details

Both axes count responders, so the breaks are restricted to integers. The vertical axis runs downwards, which places the outcome with no responders in the top left corner and matches the layout of the map printed by print.bbssr_rr.

Value

A ggplot object

Examples

RR <- BinaryRR(N1 = 10, N2 = 10, alpha = 0.025, Test = 'Chisq')
plot(RR)

# Enlarge the type and relabel the legend
plot(RR, base_size = 14, legend.title = 'Decision',
     legend.labels = c('do not reject', 'reject'),
     colours = c('white', 'grey30'))

Plot the Exact Power Curve around a Sample Size Solution

Description

Recomputes the exact power over a range of sample sizes of group 2 and marks the selected sample size and the target power.

Usage

## S3 method for class 'bbssr_samplesize'
plot(
  x,
  N2.range = NULL,
  main = NULL,
  sub = NULL,
  xlab = NULL,
  ylab = NULL,
  ylim = NULL,
  ref.line = NULL,
  ref.line.N2 = NULL,
  show.points = TRUE,
  colours = NULL,
  base_size = 11,
  ...
)

Arguments

Details

The power is recomputed at every point of N2.range, so a wide range combined with one of the unconditional tests can take a long time to evaluate.

The vertical range is imposed with coord_cartesian, so points outside ylim are hidden rather than removed from the data.

Value

A ggplot object

Examples

ss <- BinarySampleSize(p1 = 0.4, p2 = 0.2, r = 1, alpha = 0.025,
                       tar.power = 0.8, Test = 'Chisq')
plot(ss)

# Enlarge the type, rename the axes and draw neither reference line
plot(ss, base_size = 14, ref.line = NA, ref.line.N2 = NA,
     xlab = 'Sample size of the control group', ylab = 'Exact power')

Print a Sample Size Re-estimation from Interim Data

Description

Print a Sample Size Re-estimation from Interim Data

Usage

## S3 method for class 'bbssr_bssr'
print(x, digits = 4, ...)

Arguments

Value

The object x, returned invisibly

Examples

res <- BinaryBSSR(n1 = 20, n2 = 20, S = 11, Delta.A = 0.3, r = 1,
                  alpha = 0.025, tar.power = 0.8, Test = 'Chisq')
print(res)

Print Exact Power Results

Description

Print Exact Power Results

Usage

## S3 method for class 'bbssr_power'
print(x, digits = 4, ...)

Arguments

Value

The object x, returned invisibly

Examples

pw <- BinaryPower(p1 = 0.5, p2 = 0.2, N1 = 5, N2 = 5, alpha = 0.025, Test = 'Chisq')
print(pw)

Print the Operating Characteristics of a BSSR Design

Description

Print the Operating Characteristics of a BSSR Design

Usage

## S3 method for class 'bbssr_powerbssr'
print(x, digits = 4, ...)

Arguments

Value

The object x, returned invisibly

Examples

res <- BinaryPowerBSSR(
  p = 0.45,
  Delta.A = 0.3, Delta.T = 0.3,
  N1 = 5, N2 = 5, omega = 0.5, r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq'
)
print(res)

Print a Rejection Region

Description

Print a Rejection Region

Usage

## S3 method for class 'bbssr_rr'
print(x, show.map = NULL, ...)

Arguments

Value

The object x, returned invisibly

Examples

RR <- BinaryRR(N1 = 5, N2 = 5, alpha = 0.025, Test = 'Chisq')
print(RR)

Print a Sample Size Calculation

Description

Print a Sample Size Calculation

Usage

## S3 method for class 'bbssr_samplesize'
print(x, digits = 4, ...)

Arguments

Value

The object x, returned invisibly

Examples

ss <- BinarySampleSize(p1 = 0.4, p2 = 0.2, r = 1, alpha = 0.025,
                       tar.power = 0.8, Test = 'Chisq')
print(ss)