Package 'gsDesign2'

Average hazard ratio under non-proportional hazards

Description

Provides a geometric average hazard ratio under various non-proportional hazards assumptions for either single or multiple strata studies. The piecewise exponential distribution allows a simple method to specify a distribution and enrollment pattern where the enrollment, failure and dropout rates changes over time.

Usage

ahr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  total_duration = 30,
  ratio = 1
)

Arguments

enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().
fail_rate	A fail_rate data frame with or without stratum created by define_fail_rate().
total_duration	Total follow-up from start of enrollment to data cutoff; this can be a single value or a vector of positive numbers.
ratio	Ratio of experimental to control randomization.

Value

A data frame with time (from total_duration), ahr (average hazard ratio), n (sample size), event (expected number of events), info (information under given scenarios), and info0 (information under related null hypothesis) for each value of total_duration input.

Specification

The contents of this section are shown in PDF user manual only.

Examples

# Example 1: default
ahr()

# Example 2: default with multiple analysis times (varying total_duration)
ahr(total_duration = c(15, 30))

# Example 3: stratified population
enroll_rate <- define_enroll_rate(
  stratum = c(rep("Low", 2), rep("High", 3)),
  duration = c(2, 10, 4, 4, 8),
  rate = c(5, 10, 0, 3, 6)
)
fail_rate <- define_fail_rate(
  stratum = c(rep("Low", 2), rep("High", 2)),
  duration = c(1, Inf, 1, Inf),
  fail_rate = c(.1, .2, .3, .4),
  dropout_rate = .001,
  hr = c(.9, .75, .8, .6)
)
ahr(enroll_rate = enroll_rate, fail_rate = fail_rate, total_duration = c(15, 30))

---

Blinded estimation of average hazard ratio

Description

Based on blinded data and assumed hazard ratios in different intervals, compute a blinded estimate of average hazard ratio (AHR) and corresponding estimate of statistical information. This function is intended for use in computing futility bounds based on spending assuming the input hazard ratio (hr) values for intervals specified here.

Usage

ahr_blinded(
  surv = survival::Surv(time = simtrial::ex1_delayed_effect$month, event =
    simtrial::ex1_delayed_effect$evntd),
  intervals = c(3, Inf),
  hr = c(1, 0.6),
  ratio = 1
)

Arguments

surv	Input survival object (see survival::Surv()); note that only 0 = censored, 1 = event for survival::Surv().
intervals	Vector containing positive values indicating interval lengths where the exponential rates are assumed. Note that a final infinite interval is added if any events occur after the final interval specified.
hr	Vector of hazard ratios assumed for each interval.
ratio	Ratio of experimental to control randomization.

Value

A tibble with one row containing

• ahr - Blinded average hazard ratio based on assumed period-specific hazard ratios input in fail_rate and observed events in the corresponding intervals.

• event - Total observed number of events.

• info0 - Information under related null hypothesis.

• theta - Natural parameter for group sequential design representing expected incremental drift at all analyses.

Specification

The contents of this section are shown in PDF user manual only.

Examples

ahr_blinded(
  surv = survival::Surv(
    time = simtrial::ex2_delayed_effect$month,
    event = simtrial::ex2_delayed_effect$evntd
  ),
  intervals = c(4, 100),
  hr = c(1, .55),
  ratio = 1
)

---

Convert summary table of a fixed or group sequential design object to a gt object

Description

Convert summary table of a fixed or group sequential design object to a gt object

Usage

as_gt(x, ...)

## S3 method for class 'fixed_design'
as_gt(x, title = NULL, footnote = NULL, ...)

## S3 method for class 'gs_design'
as_gt(
  x,
  title = NULL,
  subtitle = NULL,
  colname_spanner = "Cumulative boundary crossing probability",
  colname_spannersub = c("Alternate hypothesis", "Null hypothesis"),
  footnote = NULL,
  display_bound = c("Efficacy", "Futility"),
  display_columns = NULL,
  display_inf_bound = FALSE,
  ...
)

Arguments

x	A summary object of a fixed or group sequential design.
...	Additional arguments (not used).
title	A string to specify the title of the gt table.
footnote	A list containing content, location, and attr. content is a vector of string to specify the footnote text; location is a vector of string to specify the locations to put the superscript of the footnote index; attr is a vector of string to specify the attributes of the footnotes, for example, c("colname", "title", "subtitle", "analysis", "spanner"); users can use the functions in the gt package to customize the table.
subtitle	A string to specify the subtitle of the gt table.
colname_spanner	A string to specify the spanner of the gt table.
colname_spannersub	A vector of strings to specify the spanner details of the gt table.
display_bound	A vector of strings specifying the label of the bounds. The default is c("Efficacy", "Futility").
display_columns	A vector of strings specifying the variables to be displayed in the summary table.
display_inf_bound	Logical, whether to display the +/-inf bound.

Value

A gt_tbl object.

Examples

library(dplyr)

# Enrollment rate
enroll_rate <- define_enroll_rate(
  duration = 18,
  rate = 20
)

# Failure rates
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 12,
  dropout_rate = .001,
  hr = c(1, .6)
)

# Study duration in months
study_duration <- 36

# Experimental / Control randomization ratio
ratio <- 1

# 1-sided Type I error
alpha <- 0.025

# Type II error (1 - power)
beta <- 0.1

# Example 1 ----
fixed_design_ahr(
  alpha = alpha, power = 1 - beta,
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  study_duration = study_duration, ratio = ratio
) %>%
  summary() %>%
  as_gt()

# Example 2 ----
fixed_design_fh(
  alpha = alpha, power = 1 - beta,
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  study_duration = study_duration, ratio = ratio
) %>%
  summary() %>%
  as_gt()


library(dplyr)
# Example 1 ----
# The default output

gs_design_ahr() %>%
  summary() %>%
  as_gt()

gs_power_ahr() %>%
  summary() %>%
  as_gt()

gs_design_wlr() %>%
  summary() %>%
  as_gt()

gs_power_wlr() %>%
  summary() %>%
  as_gt()

gs_power_combo() %>%
  summary() %>%
  as_gt()

gs_design_rd() %>%
  summary() %>%
  as_gt()

gs_power_rd() %>%
  summary() %>%
  as_gt()

# Example 2 ----
# Usage of title = ..., subtitle = ...
# to edit the title/subtitle
gs_power_wlr() %>%
  summary() %>%
  as_gt(
    title = "Bound Summary",
    subtitle = "from gs_power_wlr"
  )

# Example 3 ----
# Usage of colname_spanner = ..., colname_spannersub = ...
# to edit the spanner and its sub-spanner
gs_power_wlr() %>%
  summary() %>%
  as_gt(
    colname_spanner = "Cumulative probability to cross boundaries",
    colname_spannersub = c("under H1", "under H0")
  )

# Example 4 ----
# Usage of footnote = ...
# to edit the footnote
gs_power_wlr() %>%
  summary() %>%
  as_gt(
    footnote = list(
      content = c(
        "approximate weighted hazard ratio to cross bound.",
        "wAHR is the weighted AHR.",
        "the crossing probability.",
        "this table is generated by gs_power_wlr."
      ),
      location = c("~wHR at bound", NA, NA, NA),
      attr = c("colname", "analysis", "spanner", "title")
    )
  )

# Example 5 ----
# Usage of display_bound = ...
# to either show efficacy bound or futility bound, or both(default)
gs_power_wlr() %>%
  summary() %>%
  as_gt(display_bound = "Efficacy")

# Example 6 ----
# Usage of display_columns = ...
# to select the columns to display in the summary table
gs_power_wlr() %>%
  summary() %>%
  as_gt(display_columns = c("Analysis", "Bound", "Nominal p", "Z", "Probability"))

---

Write summary table of a fixed or group sequential design object to an RTF file

Description

Write summary table of a fixed or group sequential design object to an RTF file

Usage

as_rtf(x, ...)

## S3 method for class 'fixed_design'
as_rtf(
  x,
  title = NULL,
  footnote = NULL,
  col_rel_width = NULL,
  orientation = c("portrait", "landscape"),
  text_font_size = 9,
  file,
  ...
)

## S3 method for class 'gs_design'
as_rtf(
  x,
  title = NULL,
  subtitle = NULL,
  colname_spanner = "Cumulative boundary crossing probability",
  colname_spannersub = c("Alternate hypothesis", "Null hypothesis"),
  footnote = NULL,
  display_bound = c("Efficacy", "Futility"),
  display_columns = NULL,
  display_inf_bound = TRUE,
  col_rel_width = NULL,
  orientation = c("portrait", "landscape"),
  text_font_size = 9,
  file,
  ...
)

Arguments

x	A summary object of a fixed or group sequential design.
...	Additional arguments (not used).
title	A string to specify the title of the RTF table.
footnote	A list containing content, location, and attr. content is a vector of string to specify the footnote text; location is a vector of string to specify the locations to put the superscript of the footnote index; attr is a vector of string to specify the attributes of the footnotes, for example, c("colname", "title", "subtitle", "analysis", "spanner"); users can use the functions in the gt package to customize the table.
col_rel_width	Column relative width in a vector e.g. c(2,1,1) refers to 2:1:1. Default is NULL for equal column width.
orientation	Orientation in 'portrait' or 'landscape'.
text_font_size	Text font size. To vary text font size by column, use numeric vector with length of vector equal to number of columns displayed e.g. c(9,20,40).
file	File path for the output.
subtitle	A string to specify the subtitle of the RTF table.
colname_spanner	A string to specify the spanner of the RTF table.
colname_spannersub	A vector of strings to specify the spanner details of the RTF table.
display_bound	A vector of strings specifying the label of the bounds. The default is c("Efficacy", "Futility").
display_columns	A vector of strings specifying the variables to be displayed in the summary table.
display_inf_bound	Logical, whether to display the +/-inf bound.

Value

as_rtf() returns the input x invisibly.

Examples

library(dplyr)

# Enrollment rate
enroll_rate <- define_enroll_rate(
  duration = 18,
  rate = 20
)

# Failure rates
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 12,
  dropout_rate = .001,
  hr = c(1, .6)
)

# Study duration in months
study_duration <- 36

# Experimental / Control randomization ratio
ratio <- 1

# 1-sided Type I error
alpha <- 0.025

# Type II error (1 - power)
beta <- 0.1

# AHR ----
# under fixed power
x <- fixed_design_ahr(
  alpha = alpha, power = 1 - beta,
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  study_duration = study_duration, ratio = ratio
) %>% summary()
x %>% as_rtf(file = tempfile(fileext = ".rtf"))
x %>% as_rtf(title = "Fixed design", file = tempfile(fileext = ".rtf"))
x %>% as_rtf(
  footnote = "Power computed with average hazard ratio method given the sample size",
  file = tempfile(fileext = ".rtf")
)
x %>% as_rtf(text_font_size = 10, file = tempfile(fileext = ".rtf"))

# FH ----
# under fixed power
fixed_design_fh(
  alpha = alpha, power = 1 - beta,
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  study_duration = study_duration, ratio = ratio
) %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))
#'

# the default output
library(dplyr)

gs_design_ahr() %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

gs_power_ahr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

gs_design_wlr() %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))


gs_power_combo() %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

gs_design_rd() %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

gs_power_rd() %>%
  summary() %>%
  as_rtf(file = tempfile(fileext = ".rtf"))

# usage of title = ..., subtitle = ...
# to edit the title/subtitle
gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(
    title = "Bound Summary",
    subtitle = "from gs_power_wlr",
    file = tempfile(fileext = ".rtf")
  )

# usage of colname_spanner = ..., colname_spannersub = ...
# to edit the spanner and its sub-spanner
gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(
    colname_spanner = "Cumulative probability to cross boundaries",
    colname_spannersub = c("under H1", "under H0"),
    file = tempfile(fileext = ".rtf")
  )

# usage of footnote = ...
# to edit the footnote
gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(
    footnote = list(
      content = c(
        "approximate weighted hazard ratio to cross bound.",
        "wAHR is the weighted AHR.",
        "the crossing probability.",
        "this table is generated by gs_power_wlr."
      ),
      location = c("~wHR at bound", NA, NA, NA),
      attr = c("colname", "analysis", "spanner", "title")
    ),
    file = tempfile(fileext = ".rtf")
  )

# usage of display_bound = ...
# to either show efficacy bound or futility bound, or both(default)
gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(
    display_bound = "Efficacy",
    file = tempfile(fileext = ".rtf")
  )

# usage of display_columns = ...
# to select the columns to display in the summary table
gs_power_wlr(lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1)) %>%
  summary() %>%
  as_rtf(
    display_columns = c("Analysis", "Bound", "Nominal p", "Z", "Probability"),
    file = tempfile(fileext = ".rtf")
  )

---

Define enrollment rate

Description

Define the enrollment rate of subjects for a study as following a piecewise exponential distribution.

Usage

define_enroll_rate(duration, rate, stratum = "All")

Arguments

duration	A numeric vector of ordered piecewise study duration interval.
rate	A numeric vector of enrollment rate in each duration.
stratum	A character vector of stratum name.

Details

The duration are ordered piecewise for a duration equal to $t_i - t_{i-1}$, where $0 = t_0 < t_i < \cdots < t_M = \infty$. The enrollment rates are defined in each duration with the same length.

For a study with multiple strata, different duration and rates can be specified in each stratum.

Value

An enroll_rate data frame.

Examples

# Define enroll rate without stratum
define_enroll_rate(
  duration = c(2, 2, 10),
  rate = c(3, 6, 9)
)

# Define enroll rate with stratum
define_enroll_rate(
  duration = rep(c(2, 2, 2, 18), 3),
  rate = c((1:4) / 3, (1:4) / 2, (1:4) / 6),
  stratum = c(array("High", 4), array("Moderate", 4), array("Low", 4))
)

---

Define failure rate

Description

Define subject failure rate for a study with two treatment groups. Also supports stratified designs that have different failure rates in each stratum.

Usage

define_fail_rate(duration, fail_rate, dropout_rate, hr = 1, stratum = "All")

Arguments

duration	A numeric vector of ordered piecewise study duration interval.
fail_rate	A numeric vector of failure rate in each duration in the control group.
dropout_rate	A numeric vector of dropout rate in each duration.
hr	A numeric vector of hazard ratio between treatment and control group.
stratum	A character vector of stratum name.

Details

Define the failure and dropout rate of subjects for a study as following a piecewise exponential distribution. The duration are ordered piecewise for a duration equal to $t_i - t_{i-1}$, where $0 = t_0 < t_i < \cdots < t_M = \infty$. The failure rate, dropout rate, and hazard ratio in a study duration can be specified.

For a study with multiple strata, different duration, failure rates, dropout rates, and hazard ratios can be specified in each stratum.

Value

A fail_rate data frame.

Examples

# Define enroll rate
define_fail_rate(
  duration = c(3, 100),
  fail_rate = log(2) / c(9, 18),
  hr = c(.9, .6),
  dropout_rate = .001
)

# Define enroll rate with stratum
define_fail_rate(
  stratum = c(rep("Low", 2), rep("High", 2)),
  duration = 1,
  fail_rate = c(.1, .2, .3, .4),
  dropout_rate = .001,
  hr = c(.9, .75, .8, .6)
)

---

Piecewise constant expected accrual

Description

Computes the expected cumulative enrollment (accrual) given a set of piecewise constant enrollment rates and times.

Usage

expected_accrual(
  time = 0:24,
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

Arguments

time	Times at which enrollment is to be computed.
enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().

Value

A vector with expected cumulative enrollment for the specified times.

Specification

The contents of this section are shown in PDF user manual only.

Examples

library(tibble)

# Example 1: default
expected_accrual()

# Example 2: unstratified design
expected_accrual(
  time = c(5, 10, 20),
  enroll_rate = define_enroll_rate(
    duration = c(3, 3, 18),
    rate = c(5, 10, 20)
  )
)

expected_accrual(
  time = c(5, 10, 20),
  enroll_rate = define_enroll_rate(
    duration = c(3, 3, 18),
    rate = c(5, 10, 20),
  )
)

# Example 3: stratified design
expected_accrual(
  time = c(24, 30, 40),
  enroll_rate = define_enroll_rate(
    stratum = c("subgroup", "complement"),
    duration = c(33, 33),
    rate = c(30, 30)
  )
)

# Example 4: expected accrual over time
# Scenario 4.1: for the enrollment in the first 3 months,
# it is exactly 3 * 5 = 15.
expected_accrual(
  time = 3,
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

# Scenario 4.2: for the enrollment in the first 6 months,
# it is exactly 3 * 5 + 3 * 10 = 45.
expected_accrual(
  time = 6,
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

# Scenario 4.3: for the enrollment in the first 24 months,
# it is exactly 3 * 5 + 3 * 10 + 18 * 20 = 405.
expected_accrual(
  time = 24,
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

# Scenario 4.4: for the enrollment after 24 months,
# it is the same as that from the 24 months, since the enrollment is stopped.
expected_accrual(
  time = 25,
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

# Instead of compute the enrolled subjects one time point by one time point,
# we can also compute it once.
expected_accrual(
  time = c(3, 6, 24, 25),
  enroll_rate = define_enroll_rate(duration = c(3, 3, 18), rate = c(5, 10, 20))
)

---

Expected events observed under piecewise exponential model

Description

Computes expected events over time and by strata under the assumption of piecewise constant enrollment rates and piecewise exponential failure and censoring rates. The piecewise exponential distribution allows a simple method to specify a distribution and enrollment pattern where the enrollment, failure and dropout rates changes over time. While the main purpose may be to generate a trial that can be analyzed at a single point in time or using group sequential methods, the routine can also be used to simulate an adaptive trial design. The intent is to enable sample size calculations under non-proportional hazards assumptions for stratified populations.

Usage

expected_event(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18),
    dropout_rate = 0.001),
  total_duration = 25,
  simple = TRUE
)

Arguments

enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().
fail_rate	A fail_rate data frame with or without stratum created by define_fail_rate().
total_duration	Total follow-up from start of enrollment to data cutoff.
simple	If default (TRUE), return numeric expected number of events, otherwise a data frame as described below.

Details

More periods will generally be supplied in output than those that are input. The intent is to enable expected event calculations in a tidy format to maximize flexibility for a variety of purposes.

Value

The default when simple = TRUE is to return the total expected number of events as a real number. Otherwise, when simple = FALSE, a data frame is returned with the following variables for each period specified in fail_rate:

• t: start of period.

• fail_rate: failure rate during the period.

• event: expected events during the period.

The records in the returned data frame correspond to the input data frame fail_rate.

Specification

Examples

library(gsDesign2)

# Default arguments, simple output (total event count only)
expected_event()

# Event count by time period
expected_event(simple = FALSE)

# Early cutoff
expected_event(total_duration = .5)

# Single time period example
expected_event(
  enroll_rate = define_enroll_rate(duration = 10, rate = 10),
  fail_rate = define_fail_rate(duration = 100, fail_rate = log(2) / 6, dropout_rate = .01),
  total_duration = 22,
  simple = FALSE
)

# Single time period example, multiple enrollment periods
expected_event(
  enroll_rate = define_enroll_rate(duration = c(5, 5), rate = c(10, 20)),
  fail_rate = define_fail_rate(duration = 100, fail_rate = log(2) / 6, dropout_rate = .01),
  total_duration = 22, simple = FALSE
)

---

Predict time at which a targeted event count is achieved

Description

expected_time() is made to match input format with ahr() and to solve for the time at which the expected accumulated events is equal to an input target. Enrollment and failure rate distributions are specified as follows. The piecewise exponential distribution allows a simple method to specify a distribution and enrollment pattern where the enrollment, failure and dropout rates changes over time.

Usage

expected_time(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9) * 5),
  fail_rate = define_fail_rate(stratum = "All", duration = c(3, 100), fail_rate =
    log(2)/c(9, 18), hr = c(0.9, 0.6), dropout_rate = rep(0.001, 2)),
  target_event = 150,
  ratio = 1,
  interval = c(0.01, 100)
)

Arguments

enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().
fail_rate	A fail_rate data frame with or without stratum created by define_fail_rate().
target_event	The targeted number of events to be achieved.
ratio	Experimental:Control randomization ratio.
interval	An interval that is presumed to include the time at which expected event count is equal to target_event.

Value

A data frame with Time (computed to match events in target_event), AHR (average hazard ratio), Events (target_event input), info (information under given scenarios), and info0 (information under related null hypothesis) for each value of total_duration input.

Specification

Examples

# Example 1 ----
# default

expected_time()


# Example 2 ----
# check that result matches a finding using AHR()
# Start by deriving an expected event count
enroll_rate <- define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9) * 5)
fail_rate <- define_fail_rate(
  duration = c(3, 100),
  fail_rate = log(2) / c(9, 18),
  hr = c(.9, .6),
  dropout_rate = .001
)
total_duration <- 20
xx <- ahr(enroll_rate, fail_rate, total_duration)
xx

# Next we check that the function confirms the timing of the final analysis.

expected_time(enroll_rate, fail_rate,
  target_event = xx$event, interval = c(.5, 1.5) * xx$time
)


# Example 3 ----
# In this example, we verify `expected_time()` by `ahr()`.

x <- ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  ratio = 1, total_duration = 20
)

cat("The number of events by 20 months is ", x$event, ".\n")

y <- expected_time(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  ratio = 1, target_event = x$event
)

cat("The time to get ", x$event, " is ", y$time, "months.\n")

---

Fixed design under non-proportional hazards

Description

Computes fixed design sample size (given power) or power (given sample size) by:

• fixed_design_ahr() - Average hazard ratio method.

• fixed_design_fh() - Weighted logrank test with Fleming-Harrington weights (Farrington and Manning, 1990).

• fixed_design_mb() - Weighted logrank test with Magirr-Burman weights.

• fixed_design_lf() - Lachin-Foulkes method (Lachin and Foulkes, 1986).

• fixed_design_maxcombo() - MaxCombo method.

• fixed_design_rmst() - RMST method.

• fixed_design_milestone() - Milestone method.

Additionally, fixed_design_rd() provides fixed design for binary endpoint with treatment effect measuring in risk difference.

Usage

fixed_design_ahr(
  enroll_rate,
  fail_rate,
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  event = NULL
)

fixed_design_fh(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  enroll_rate,
  fail_rate,
  rho = 0,
  gamma = 0
)

fixed_design_lf(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  enroll_rate,
  fail_rate
)

fixed_design_maxcombo(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  enroll_rate,
  fail_rate,
  rho = c(0, 0, 1),
  gamma = c(0, 1, 0),
  tau = rep(-1, 3)
)

fixed_design_mb(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  enroll_rate,
  fail_rate,
  tau = 6,
  w_max = Inf
)

fixed_design_milestone(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  enroll_rate,
  fail_rate,
  study_duration = 36,
  tau = NULL
)

fixed_design_rd(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  p_c,
  p_e,
  rd0 = 0,
  n = NULL
)

fixed_design_rmst(
  alpha = 0.025,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  enroll_rate,
  fail_rate,
  tau = NULL
)

Arguments

enroll_rate	Enrollment rates.
fail_rate	Failure and dropout rates.
alpha	One-sided Type I error (strictly between 0 and 1).
power	Power (NULL to compute power or strictly between 0 and 1 - alpha otherwise).
ratio	Experimental:Control randomization ratio.
study_duration	Study duration.
event	Targeted event at each analysis.
rho	A vector of numbers paring with gamma and tau for MaxCombo test.
gamma	A vector of numbers paring with rho and tau for MaxCombo test.
tau	Test parameter in RMST.
w_max	Test parameter of Magirr-Burman method.
p_c	A numerical value of the control arm rate.
p_e	A numerical value of the experimental arm rate.
rd0	Risk difference under null hypothesis, default is 0.
n	Sample size. If NULL with power input, the sample size will be computed to achieve the targeted power

Value

A list of design characteristic summary.

Examples

# AHR method ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_ahr(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_ahr(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36
)
x %>% summary()

# WLR test with FH weights ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_fh(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  rho = 1, gamma = 1
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_fh(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  rho = 1, gamma = 1
)
x %>% summary()

# LF method ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_lf(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_lf(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36
)
x %>% summary()

# MaxCombo test ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_maxcombo(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  rho = c(0, 0.5), gamma = c(0, 0), tau = c(-1, -1)
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_maxcombo(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  rho = c(0, 0.5), gamma = c(0, 0), tau = c(-1, -1)
)
x %>% summary()

# WLR test with MB weights ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_mb(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 4,
  w_max = 2
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_mb(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = c(4, 100),
    fail_rate = log(2) / 12,
    hr = c(1, .6),
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 4,
  w_max = 2
)
x %>% summary()

# Milestone method ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_milestone(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 18
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_milestone(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 18
)
x %>% summary()

# Binary endpoint with risk differences ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_rd(
  alpha = 0.025, power = 0.9, p_c = .15, p_e = .1,
  rd0 = 0, ratio = 1
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_rd(
  alpha = 0.025, power = NULL, p_c = .15, p_e = .1,
  rd0 = 0, n = 2000, ratio = 1
)
x %>% summary()

# RMST method ----
library(dplyr)

# Example 1: given power and compute sample size
x <- fixed_design_rmst(
  alpha = .025, power = .9,
  enroll_rate = define_enroll_rate(duration = 18, rate = 1),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 18
)
x %>% summary()

# Example 2: given sample size and compute power
x <- fixed_design_rmst(
  alpha = .025,
  enroll_rate = define_enroll_rate(duration = 18, rate = 20),
  fail_rate = define_fail_rate(
    duration = 100,
    fail_rate = log(2) / 12,
    hr = .7,
    dropout_rate = .001
  ),
  study_duration = 36,
  tau = 18
)
x %>% summary()

---

Default boundary generation

Description

gs_b() is the simplest version of a function to be used with the upper and lower arguments in gs_power_npe() and gs_design_npe() or the upper_bound and lower_bound arguments in gs_prob_combo() and pmvnorm_combo(). It simply returns the vector of Z-values in the input vector par or, if k is specified, par[k] is returned. Note that if bounds need to change with changing information at analyses, gs_b() should not be used. For instance, for spending function bounds use gs_spending_bound().

Usage

gs_b(par = NULL, k = NULL, ...)

Arguments

par	For gs_b(), this is just Z-values for the boundaries; can include infinite values.
k	Is NULL (default), return par, else return par[k].
...	Further arguments passed to or from other methods.

Value

Returns the vector input par if k is NULL, otherwise, par[k].

Specification

The contents of this section are shown in PDF user manual only.

Examples

# Simple: enter a vector of length 3 for bound
gs_b(par = 4:2)

# 2nd element of par
gs_b(par = 4:2, k = 2)

# Generate an efficacy bound using a spending function
# Use Lan-DeMets spending approximation of O'Brien-Fleming bound
# as 50%, 75% and 100% of final spending
# Information fraction
IF <- c(.5, .75, 1)
gs_b(par = gsDesign::gsDesign(
  alpha = .025, k = length(IF),
  test.type = 1, sfu = gsDesign::sfLDOF,
  timing = IF
)$upper$bound)

---

Create npsurvSS arm object

Description

Create npsurvSS arm object

Usage

gs_create_arm(enroll_rate, fail_rate, ratio, total_time = 1e+06)

Arguments

enroll_rate	Enrollment rates from define_enroll_rate().
fail_rate	Failure and dropout rates from define_fail_rate().
ratio	Experimental:Control randomization ratio.
total_time	Total analysis time.

Value

A list of the two arms.

Specification

The contents of this section are shown in PDF user manual only.

Examples

enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 10),
  rate = c(3, 6, 9)
)

fail_rate <- define_fail_rate(
  duration = c(3, 100),
  fail_rate = log(2) / c(9, 18),
  hr = c(.9, .6),
  dropout_rate = .001
)

gs_create_arm(enroll_rate, fail_rate, ratio = 1)

---

Group sequential design using average hazard ratio under non-proportional hazards

Description

Group sequential design using average hazard ratio under non-proportional hazards

Usage

gs_design_ahr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  alpha = 0.025,
  beta = 0.1,
  info_frac = NULL,
  analysis_time = 36,
  ratio = 1,
  binding = FALSE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
  h1_spending = TRUE,
  test_upper = TRUE,
  test_lower = TRUE,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  r = 18,
  tol = 1e-06,
  interval = c(0.01, 1000)
)

Arguments

enroll_rate	Enrollment rates.
fail_rate	Failure and dropout rates.
alpha	One-sided Type I error.
beta	Type II error.
info_frac	Targeted information fraction at each analysis.
analysis_time	Minimum time of analysis.
ratio	Experimental:Control randomization ratio (not yet implemented).
binding	Indicator of whether futility bound is binding; default of FALSE is recommended.
upper	Function to compute upper bound.
upar	Parameters passed to upper.
lower	Function to compute lower bound.
lpar	Parameters passed to lower.
h1_spending	Indicator that lower bound to be set by spending under alternate hypothesis (input fail_rate) if spending is used for lower bound.
test_upper	Indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound.
test_lower	Indicator of which analyses should include an lower bound; single value of TRUE (default) indicates all analyses; single value FALSE indicated no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.
info_scale	Information scale for calculation. Options are: "h0_h1_info" (default): variance under both null and alternative hypotheses is used. "h0_info": variance under null hypothesis is used. "h1_info": variance under alternative hypothesis is used.
r	Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally, r will not be changed by the user.
tol	Tolerance parameter for boundary convergence (on Z-scale).
interval	An interval that is presumed to include the time at which expected event count is equal to targeted event.

Details

To be added.

Value

A list with input parameters, enrollment rate, analysis, and bound.

Specification

The contents of this section are shown in PDF user manual only.

Examples

library(gsDesign)
library(gsDesign2)
library(dplyr)

# Example 1 ----
# call with defaults
gs_design_ahr()

# Example 2 ----
# Single analysis
gs_design_ahr(analysis_time = 40)

# Example 3 ----
# Multiple analysis_time
gs_design_ahr(analysis_time = c(12, 24, 36))

# Example 4 ----
# Specified information fraction

gs_design_ahr(info_frac = c(.25, .75, 1), analysis_time = 36)


# Example 5 ----
# multiple analysis times & info_frac
# driven by times
gs_design_ahr(info_frac = c(.25, .75, 1), analysis_time = c(12, 25, 36))
# driven by info_frac

gs_design_ahr(info_frac = c(1 / 3, .8, 1), analysis_time = c(12, 25, 36))


# Example 6 ----
# 2-sided symmetric design with O'Brien-Fleming spending

gs_design_ahr(
  analysis_time = c(12, 24, 36),
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  h1_spending = FALSE
)

# 2-sided asymmetric design with O'Brien-Fleming upper spending
# Pocock lower spending under H1 (NPH)

gs_design_ahr(
  analysis_time = c(12, 24, 36),
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDPocock, total_spend = 0.1, param = NULL, timing = NULL),
  h1_spending = TRUE
)


# Example 7 ----

gs_design_ahr(
  alpha = 0.0125,
  analysis_time = c(12, 24, 36),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.0125, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = rep(-Inf, 3)
)

gs_design_ahr(
  alpha = 0.0125,
  analysis_time = c(12, 24, 36),
  upper = gs_b,
  upar = gsDesign::gsDesign(
    k = 3, test.type = 1, n.I = c(.25, .75, 1),
    sfu = sfLDOF, sfupar = NULL, alpha = 0.0125
  )$upper$bound,
  lower = gs_b,
  lpar = rep(-Inf, 3)
)

---

Group sequential design using MaxCombo test under non-proportional hazards

Description

Group sequential design using MaxCombo test under non-proportional hazards

Usage

gs_design_combo(
  enroll_rate = define_enroll_rate(duration = 12, rate = 500/12),
  fail_rate = define_fail_rate(duration = c(4, 100), fail_rate = log(2)/15, hr = c(1,
    0.6), dropout_rate = 0.001),
  fh_test = rbind(data.frame(rho = 0, gamma = 0, tau = -1, test = 1, analysis = 1:3,
    analysis_time = c(12, 24, 36)), data.frame(rho = c(0, 0.5), gamma = 0.5, tau = -1,
    test = 2:3, analysis = 3, analysis_time = 36)),
  ratio = 1,
  alpha = 0.025,
  beta = 0.2,
  binding = FALSE,
  upper = gs_b,
  upar = c(3, 2, 1),
  lower = gs_b,
  lpar = c(-1, 0, 1),
  algorithm = mvtnorm::GenzBretz(maxpts = 1e+05, abseps = 1e-05),
  n_upper_bound = 1000,
  ...
)

Arguments

enroll_rate	Enrollment rates.
fail_rate	Failure and dropout rates.
fh_test	A data frame to summarize the test in each analysis. See examples for its data structure.
ratio	Experimental:Control randomization ratio (not yet implemented).
alpha	One-sided Type I error.
beta	Type II error.
binding	Indicator of whether futility bound is binding; default of FALSE is recommended.
upper	Function to compute upper bound.
upar	Parameters passed to upper.
lower	Function to compute lower bound.
lpar	Parameters passed to lower.
algorithm	an object of class GenzBretz, Miwa or TVPACK specifying both the algorithm to be used as well as the associated hyper parameters.
n_upper_bound	A numeric value of upper limit of sample size.
...	Additional parameters passed to mvtnorm::pmvnorm.

Value

A list with input parameters, enrollment rate, analysis, and bound.

Examples

# The example is slow to run
library(dplyr)
library(mvtnorm)
library(gsDesign)

enroll_rate <- define_enroll_rate(
  duration = 12,
  rate = 500 / 12
)

fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 15, # median survival 15 month
  hr = c(1, .6),
  dropout_rate = 0.001
)

fh_test <- rbind(
  data.frame(
    rho = 0, gamma = 0, tau = -1,
    test = 1, analysis = 1:3, analysis_time = c(12, 24, 36)
  ),
  data.frame(
    rho = c(0, 0.5), gamma = 0.5, tau = -1,
    test = 2:3, analysis = 3, analysis_time = 36
  )
)

x <- gsSurv(
  k = 3,
  test.type = 4,
  alpha = 0.025,
  beta = 0.2,
  astar = 0,
  timing = 1,
  sfu = sfLDOF,
  sfupar = 0,
  sfl = sfLDOF,
  sflpar = 0,
  lambdaC = 0.1,
  hr = 0.6,
  hr0 = 1,
  eta = 0.01,
  gamma = 10,
  R = 12,
  S = NULL,
  T = 36,
  minfup = 24,
  ratio = 1
)

# Example 1 ----
# User-defined boundary

gs_design_combo(
  enroll_rate,
  fail_rate,
  fh_test,
  alpha = 0.025, beta = 0.2,
  ratio = 1,
  binding = FALSE,
  upar = x$upper$bound,
  lpar = x$lower$bound
)

# Example 2 ----

# Boundary derived by spending function
gs_design_combo(
  enroll_rate,
  fail_rate,
  fh_test,
  alpha = 0.025,
  beta = 0.2,
  ratio = 1,
  binding = FALSE,
  upper = gs_spending_combo,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025), # alpha spending
  lower = gs_spending_combo,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2), # beta spending
)

---

Group sequential design computation with non-constant effect and information

Description

Derives group sequential design size, bounds and boundary crossing probabilities based on proportionate information and effect size at analyses. It allows a non-constant treatment effect over time, but also can be applied for the usual homogeneous effect size designs. It requires treatment effect and proportionate statistical information at each analysis as well as a method of deriving bounds, such as spending. The routine enables two things not available in the gsDesign package:

1. non-constant effect, 2) more flexibility in boundary selection. For many applications, the non-proportional-hazards design function gs_design_nph() will be used; it calls this function. Initial bound types supported are 1) spending bounds,

2. fixed bounds, and 3) Haybittle-Peto-like bounds. The requirement is to have a boundary update method that can each bound without knowledge of future bounds. As an example, bounds based on conditional power that require knowledge of all future bounds are not supported by this routine; a more limited conditional power method will be demonstrated. Boundary family designs Wang-Tsiatis designs including the original (non-spending-function-based) O'Brien-Fleming and Pocock designs are not supported by gs_power_npe().

Usage

gs_design_npe(
  theta = 0.1,
  theta0 = NULL,
  theta1 = NULL,
  info = 1,
  info0 = NULL,
  info1 = NULL,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  alpha = 0.025,
  beta = 0.1,
  upper = gs_b,
  upar = qnorm(0.975),
  lower = gs_b,
  lpar = -Inf,
  test_upper = TRUE,
  test_lower = TRUE,
  binding = FALSE,
  r = 18,
  tol = 1e-06
)

Arguments

theta	Natural parameter for group sequential design representing expected incremental drift at all analyses; used for power calculation.
theta0	Natural parameter used for upper bound spending; if NULL, this will be set to 0.
theta1	Natural parameter used for lower bound spending; if NULL, this will be set to theta which yields the usual beta-spending. If set to 0, spending is 2-sided under null hypothesis.
info	Proportionate statistical information at all analyses for input theta.
info0	Proportionate statistical information under null hypothesis, if different than alternative; impacts null hypothesis bound calculation.
info1	Proportionate statistical information under alternate hypothesis; impacts null hypothesis bound calculation.
info_scale	Information scale for calculation. Options are: "h0_h1_info" (default): variance under both null and alternative hypotheses is used. "h0_info": variance under null hypothesis is used. "h1_info": variance under alternative hypothesis is used.
alpha	One-sided Type I error.
beta	Type II error.
upper	Function to compute upper bound.
upar	Parameters passed to the function provided in upper.
lower	Function to compare lower bound.
lpar	Parameters passed to the function provided in lower.
test_upper	Indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound.
test_lower	Indicator of which analyses should include an lower bound; single value of TRUE (default) indicates all analyses; single value FALSE indicates no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.
binding	Indicator of whether futility bound is binding; default of FALSE is recommended.
r	Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally r will not be changed by the user.
tol	Tolerance parameter for boundary convergence (on Z-scale).

Details

The inputs info and info0 should be vectors of the same length with increasing positive numbers. The design returned will change these by some constant scale factor to ensure the design has power 1 - beta. The bound specifications in upper, lower, upar, lpar will be used to ensure Type I error and other boundary properties are as specified.

Value

A tibble with columns analysis, bound, z, probability, theta, info, info0.

Specification

The contents of this section are shown in PDF user manual only.

Author(s)

Keaven Anderson [email protected]

Examples

library(dplyr)
library(gsDesign)

# Example 1 ----
# Single analysis
# Lachin book p 71 difference of proportions example
pc <- .28 # Control response rate
pe <- .40 # Experimental response rate
p0 <- (pc + pe) / 2 # Ave response rate under H0

# Information per increment of 1 in sample size
info0 <- 1 / (p0 * (1 - p0) * 4)
info <- 1 / (pc * (1 - pc) * 2 + pe * (1 - pe) * 2)

# Result should round up to next even number = 652
# Divide information needed under H1 by information per patient added
gs_design_npe(theta = pe - pc, info = info, info0 = info0)


# Example 2 ----
# Fixed bound
x <- gs_design_npe(
  alpha = 0.0125,
  theta = c(.1, .2, .3),
  info = (1:3) * 80,
  info0 = (1:3) * 80,
  upper = gs_b,
  upar = gsDesign::gsDesign(k = 3, sfu = gsDesign::sfLDOF, alpha = 0.0125)$upper$bound,
  lower = gs_b,
  lpar = c(-1, 0, 0)
)
x

# Same upper bound; this represents non-binding Type I error and will total 0.025
gs_power_npe(
  theta = rep(0, 3),
  info = (x %>% filter(bound == "upper"))$info,
  upper = gs_b,
  upar = (x %>% filter(bound == "upper"))$z,
  lower = gs_b,
  lpar = rep(-Inf, 3)
)

# Example 3 ----
# Spending bound examples
# Design with futility only at analysis 1; efficacy only at analyses 2, 3
# Spending bound for efficacy; fixed bound for futility
# NOTE: test_upper and test_lower DO NOT WORK with gs_b; must explicitly make bounds infinite
# test_upper and test_lower DO WORK with gs_spending_bound
gs_design_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  info0 = (1:3) * 40,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = c(-1, -Inf, -Inf),
  test_upper = c(FALSE, TRUE, TRUE)
)

# one can try `info_scale = "h1_info"` or `info_scale = "h0_info"` here
gs_design_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  info0 = (1:3) * 30,
  info_scale = "h1_info",
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = c(-1, -Inf, -Inf),
  test_upper = c(FALSE, TRUE, TRUE)
)

# Example 4 ----
# Spending function bounds
# 2-sided asymmetric bounds
# Lower spending based on non-zero effect
gs_design_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  info0 = (1:3) * 30,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfHSD, total_spend = 0.1, param = -1, timing = NULL)
)

# Example 5 ----
# Two-sided symmetric spend, O'Brien-Fleming spending
# Typically, 2-sided bounds are binding
xx <- gs_design_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
)
xx

# Re-use these bounds under alternate hypothesis
# Always use binding = TRUE for power calculations
gs_power_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  binding = TRUE,
  upper = gs_b,
  lower = gs_b,
  upar = (xx %>% filter(bound == "upper"))$z,
  lpar = -(xx %>% filter(bound == "upper"))$z
)

---

Group sequential design of binary outcome measuring in risk difference

Description

Group sequential design of binary outcome measuring in risk difference

Usage

gs_design_rd(
  p_c = tibble::tibble(stratum = "All", rate = 0.2),
  p_e = tibble::tibble(stratum = "All", rate = 0.15),
  info_frac = 1:3/3,
  rd0 = 0,
  alpha = 0.025,
  beta = 0.1,
  ratio = 1,
  stratum_prev = NULL,
  weight = c("unstratified", "ss", "invar"),
  upper = gs_b,
  lower = gs_b,
  upar = gsDesign(k = 3, test.type = 1, sfu = sfLDOF, sfupar = NULL)$upper$bound,
  lpar = c(qnorm(0.1), rep(-Inf, 2)),
  test_upper = TRUE,
  test_lower = TRUE,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  binding = FALSE,
  r = 18,
  tol = 1e-06,
  h1_spending = TRUE
)

Arguments

p_c	Rate at the control group.
p_e	Rate at the experimental group.
info_frac	Statistical information fraction.
rd0	Treatment effect under super-superiority designs, the default is 0.
alpha	One-sided Type I error.
beta	Type II error.
ratio	Experimental:Control randomization ratio (not yet implemented).
stratum_prev	Randomization ratio of different stratum. If it is unstratified design then NULL. Otherwise it is a tibble containing two columns (stratum and prevalence).
weight	The weighting scheme for stratified population.
upper	Function to compute upper bound.
lower	Function to compute lower bound.
upar	Parameters passed to upper.
lpar	Parameters passed to lower.
test_upper	Indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound.
test_lower	Indicator of which analyses should include an lower bound; single value of TRUE (default) indicates all analyses; single value of FALSE indicates no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.
info_scale	Information scale for calculation. Options are: "h0_h1_info" (default): variance under both null and alternative hypotheses is used. "h0_info": variance under null hypothesis is used. "h1_info": variance under alternative hypothesis is used.
binding	Indicator of whether futility bound is binding; default of FALSE is recommended.
r	Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally, r will not be changed by the user.
tol	Tolerance parameter for boundary convergence (on Z-scale).
h1_spending	Indicator that lower bound to be set by spending under alternate hypothesis (input fail_rate) if spending is used for lower bound.

Details

To be added.

Value

A list with input parameters, analysis, and bound.

Examples

library(gsDesign)

# Example 1 ----
# unstratified group sequential design
x <- gs_design_rd(
  p_c = tibble::tibble(stratum = "All", rate = .2),
  p_e = tibble::tibble(stratum = "All", rate = .15),
  info_frac = c(0.7, 1),
  rd0 = 0,
  alpha = .025,
  beta = .1,
  ratio = 1,
  stratum_prev = NULL,
  weight = "unstratified",
  upper = gs_b,
  lower = gs_b,
  upar = gsDesign(k = 2, test.type = 1, sfu = sfLDOF, sfupar = NULL)$upper$bound,
  lpar = c(qnorm(.1), rep(-Inf, 2))
)

y <- gs_power_rd(
  p_c = tibble::tibble(stratum = "All", rate = .2),
  p_e = tibble::tibble(stratum = "All", rate = .15),
  n = tibble::tibble(stratum = "All", n = x$analysis$n, analysis = 1:2),
  rd0 = 0,
  ratio = 1,
  weight = "unstratified",
  upper = gs_b,
  lower = gs_b,
  upar = gsDesign(k = 2, test.type = 1, sfu = sfLDOF, sfupar = NULL)$upper$bound,
  lpar = c(qnorm(.1), rep(-Inf, 2))
)

# The above 2 design share the same power with the same sample size and treatment effect
x$bound$probability[x$bound$bound == "upper" & x$bound$analysis == 2]
y$bound$probability[y$bound$bound == "upper" & y$bound$analysis == 2]

# Example 2 ----
# stratified group sequential design
gs_design_rd(
  p_c = tibble::tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    rate = c(.2, .25)
  ),
  p_e = tibble::tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    rate = c(.15, .22)
  ),
  info_frac = c(0.7, 1),
  rd0 = 0,
  alpha = .025,
  beta = .1,
  ratio = 1,
  stratum_prev = tibble::tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    prevalence = c(.4, .6)
  ),
  weight = "ss",
  upper = gs_spending_bound, lower = gs_b,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lpar = rep(-Inf, 2)
)

---

Group sequential design using weighted log-rank test under non-proportional hazards

Description

Group sequential design using weighted log-rank test under non-proportional hazards

Usage

gs_design_wlr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = tibble(stratum = "All", duration = c(3, 100), fail_rate = log(2)/c(9, 18),
    hr = c(0.9, 0.6), dropout_rate = rep(0.001, 2)),
  weight = wlr_weight_fh,
  approx = "asymptotic",
  alpha = 0.025,
  beta = 0.1,
  ratio = 1,
  info_frac = NULL,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  analysis_time = 36,
  binding = FALSE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
  test_upper = TRUE,
  test_lower = TRUE,
  h1_spending = TRUE,
  r = 18,
  tol = 1e-06,
  interval = c(0.01, 1000)
)

Arguments

enroll_rate	Enrollment rates.
fail_rate	Failure and dropout rates.
weight	Weight of weighted log rank test: "1" = unweighted. "n" = Gehan-Breslow. "sqrtN" = Tarone-Ware. "FH_p[a]_q[b]" = Fleming-Harrington with p=a and q=b.
approx	Approximate estimation method for Z statistics. "event_driven" = only work under proportional hazard model with log rank test. "asymptotic".
alpha	One-sided Type I error.
beta	Type II error.
ratio	Experimental:Control randomization ratio (not yet implemented).
info_frac	Targeted information fraction at each analysis.
info_scale	Information scale for calculation. Options are: "h0_h1_info" (default): variance under both null and alternative hypotheses is used. "h0_info": variance under null hypothesis is used. "h1_info": variance under alternative hypothesis is used.
analysis_time	Minimum time of analysis.
binding	Indicator of whether futility bound is binding; default of FALSE is recommended.
upper	Function to compute upper bound.
upar	Parameters passed to upper.
lower	Function to compute lower bound.
lpar	Parameters passed to lower.
test_upper	Indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound.
test_lower	Indicator of which analyses should include an lower bound; single value of TRUE (default) indicates all analyses; single value FALSE indicated no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.
h1_spending	Indicator that lower bound to be set by spending under alternate hypothesis (input fail_rate) if spending is used for lower bound.
r	Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally, r will not be changed by the user.
tol	Tolerance parameter for boundary convergence (on Z-scale).
interval	An interval that is presumed to include the time at which expected event count is equal to targeted event.

Value

A list with input parameters, enrollment rate, analysis, and bound.

Specification

The contents of this section are shown in PDF user manual only.

Examples

library(dplyr)
library(mvtnorm)
library(gsDesign)
library(gsDesign2)

# set enrollment rates
enroll_rate <- define_enroll_rate(duration = 12, rate = 1)

# set failure rates
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 15, # median survival 15 month
  hr = c(1, .6),
  dropout_rate = 0.001
)

# Example 1 ----
# Information fraction driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = function(x, arm0, arm1) {
    wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0.5)
  },
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = 36,
  info_frac = c(0.6, 1)
)

# Example 2 ----
# Calendar time driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = function(x, arm0, arm1) {
    wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0.5)
  },
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = c(24, 36),
  info_frac = NULL
)

# Example 3 ----
# Both calendar time and information fraction driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = function(x, arm0, arm1) {
    wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0.5)
  },
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = c(24, 36),
  info_frac = c(0.6, 1)
)

---

Information and effect size based on AHR approximation

Description

Based on piecewise enrollment rate, failure rate, and dropout rates computes approximate information and effect size using an average hazard ratio model.

Usage

gs_info_ahr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  ratio = 1,
  event = NULL,
  analysis_time = NULL,
  interval = c(0.01, 1000)
)

Arguments

enroll_rate	Enrollment rates from define_enroll_rate().
fail_rate	Failure and dropout rates from define_fail_rate().
ratio	Experimental:Control randomization ratio.
event	Targeted minimum events at each analysis.
analysis_time	Targeted minimum study duration at each analysis.
interval	An interval that is presumed to include the time at which expected event count is equal to targeted event.

Details

The ahr() function computes statistical information at targeted event times. The expected_time() function is used to get events and average HR at targeted analysis_time.

Value

A data frame with columns analysis, time, ahr, event, theta, info, info0. The columns info and info0 contain statistical information under H1, H0, respectively. For analysis k, time[k] is the maximum of analysis_time[k] and the expected time required to accrue the targeted event[k]. ahr is the expected average hazard ratio at each analysis.

Specification

The contents of this section are shown in PDF user manual only.

Examples

library(gsDesign)
library(gsDesign2)

# Example 1 ----

# Only put in targeted events
gs_info_ahr(event = c(30, 40, 50))


# Example 2 ----
# Only put in targeted analysis times
gs_info_ahr(analysis_time = c(18, 27, 36))

# Example 3 ----

# Some analysis times after time at which targeted event accrue
# Check that both Time >= input analysis_time and event >= input event
gs_info_ahr(event = c(30, 40, 50), analysis_time = c(16, 19, 26))
gs_info_ahr(event = c(30, 40, 50), analysis_time = c(14, 20, 24))

---

Information and effect size for MaxCombo test

Description

Information and effect size for MaxCombo test

Usage

gs_info_combo(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  ratio = 1,
  event = NULL,
  analysis_time = NULL,
  rho,
  gamma,
  tau = rep(-1, length(rho)),
  approx = "asymptotic"
)

Arguments

enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().
fail_rate	A fail_rate data frame with or without stratum created by define_fail_rate().
ratio	Experimental:Control randomization ratio (not yet implemented).
event	Targeted events at each analysis.
analysis_time	Minimum time of analysis.
rho	Weighting parameters.
gamma	Weighting parameters.
tau	Weighting parameters.
approx	Approximation method.

Value

A tibble with columns as test index, analysis index, analysis time, sample size, number of events, ahr, delta, sigma2, theta, and statistical information.

Examples

gs_info_combo(rho = c(0, 0.5), gamma = c(0.5, 0), analysis_time = c(12, 24))

---

Information and effect size under risk difference

Description

Information and effect size under risk difference

Usage

gs_info_rd(
  p_c = tibble::tibble(stratum = "All", rate = 0.2),
  p_e = tibble::tibble(stratum = "All", rate = 0.15),
  n = tibble::tibble(stratum = "All", n = c(100, 200, 300), analysis = 1:3),
  rd0 = 0,
  ratio = 1,
  weight = c("unstratified", "ss", "invar")
)

Arguments

p_c	Rate at the control group.
p_e	Rate at the experimental group.
n	Sample size.
rd0	The risk difference under H0.
ratio	Experimental:Control randomization ratio.
weight	Weighting method, can be "unstratified", "ss", or "invar".

Value

A tibble with columns as analysis index, sample size, risk difference, risk difference under null hypothesis, theta1 (standardized treatment effect under alternative hypothesis), theta0 (standardized treatment effect under null hypothesis), and statistical information.

Examples

# Example 1 ----
# unstratified case with H0: rd0 = 0
gs_info_rd(
  p_c = tibble::tibble(stratum = "All", rate = .15),
  p_e = tibble::tibble(stratum = "All", rate = .1),
  n = tibble::tibble(stratum = "All", n = c(100, 200, 300), analysis = 1:3),
  rd0 = 0,
  ratio = 1
)

# Example 2 ----
# unstratified case with H0: rd0 != 0
gs_info_rd(
  p_c = tibble::tibble(stratum = "All", rate = .2),
  p_e = tibble::tibble(stratum = "All", rate = .15),
  n = tibble::tibble(stratum = "All", n = c(100, 200, 300), analysis = 1:3),
  rd0 = 0.005,
  ratio = 1
)

# Example 3 ----
# stratified case under sample size weighting and H0: rd0 = 0
gs_info_rd(
  p_c = tibble::tibble(stratum = c("S1", "S2", "S3"), rate = c(.15, .2, .25)),
  p_e = tibble::tibble(stratum = c("S1", "S2", "S3"), rate = c(.1, .16, .19)),
  n = tibble::tibble(
    stratum = rep(c("S1", "S2", "S3"), each = 3),
    analysis = rep(1:3, 3),
    n = c(50, 100, 200, 40, 80, 160, 60, 120, 240)
  ),
  rd0 = 0,
  ratio = 1,
  weight = "ss"
)

# Example 4 ----
# stratified case under inverse variance weighting and H0: rd0 = 0
gs_info_rd(
  p_c = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.15, .2, .25)
  ),
  p_e = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.1, .16, .19)
  ),
  n = tibble::tibble(
    stratum = rep(c("S1", "S2", "S3"), each = 3),
    analysis = rep(1:3, 3),
    n = c(50, 100, 200, 40, 80, 160, 60, 120, 240)
  ),
  rd0 = 0,
  ratio = 1,
  weight = "invar"
)

# Example 5 ----
# stratified case under sample size weighting and H0: rd0 != 0
gs_info_rd(
  p_c = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.15, .2, .25)
  ),
  p_e = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.1, .16, .19)
  ),
  n = tibble::tibble(
    stratum = rep(c("S1", "S2", "S3"), each = 3),
    analysis = rep(1:3, 3),
    n = c(50, 100, 200, 40, 80, 160, 60, 120, 240)
  ),
  rd0 = 0.02,
  ratio = 1,
  weight = "ss"
)

# Example 6 ----
# stratified case under inverse variance weighting and H0: rd0 != 0
gs_info_rd(
  p_c = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.15, .2, .25)
  ),
  p_e = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.1, .16, .19)
  ),
  n = tibble::tibble(
    stratum = rep(c("S1", "S2", "S3"), each = 3),
    analysis = rep(1:3, 3),
    n = c(50, 100, 200, 40, 80, 160, 60, 120, 240)
  ),
  rd0 = 0.02,
  ratio = 1,
  weight = "invar"
)

# Example 7 ----
# stratified case under inverse variance weighting and H0: rd0 != 0 and
# rd0 difference for different statum
gs_info_rd(
  p_c = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.15, .2, .25)
  ),
  p_e = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rate = c(.1, .16, .19)
  ),
  n = tibble::tibble(
    stratum = rep(c("S1", "S2", "S3"), each = 3),
    analysis = rep(1:3, 3),
    n = c(50, 100, 200, 40, 80, 160, 60, 120, 240)
  ),
  rd0 = tibble::tibble(
    stratum = c("S1", "S2", "S3"),
    rd0 = c(0.01, 0.02, 0.03)
  ),
  ratio = 1,
  weight = "invar"
)

---

Information and effect size for weighted log-rank test

Description

Based on piecewise enrollment rate, failure rate, and dropout rates computes approximate information and effect size using an average hazard ratio model.

Usage

gs_info_wlr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  ratio = 1,
  event = NULL,
  analysis_time = NULL,
  weight = wlr_weight_fh,
  approx = "asymptotic",
  interval = c(0.01, 1000)
)

Arguments

enroll_rate	An enroll_rate data frame with or without stratum created by define_enroll_rate().
fail_rate	Failure and dropout rates.
ratio	Experimental:Control randomization ratio.
event	Targeted minimum events at each analysis.
analysis_time	Targeted minimum study duration at each analysis.
weight	Weight of weighted log rank test: "1" = unweighted. "n" = Gehan-Breslow. "sqrtN" = Tarone-Ware. "FH_p[a]_q[b]" = Fleming-Harrington with p=a and q=b.
approx	Approximate estimation method for Z statistics. "event_driven" = only work under proportional hazard model with log rank test. "asymptotic".
interval	An interval that is presumed to include the time at which expected event count is equal to targeted event.

Details

The ahr() function computes statistical information at targeted event times. The expected_time() function is used to get events and average HR at targeted analysis_time.

Value

A tibble with columns Analysis, Time, N, Events, AHR, delta, sigma2, theta, info, info0. info and info0 contain statistical information under H1, H0, respectively. For analysis k, Time[k] is the maximum of analysis_time[k] and the expected time required to accrue the targeted event[k]. AHR is the expected average hazard ratio at each analysis.

Examples

library(gsDesign2)

# Set enrollment rates
enroll_rate <- define_enroll_rate(duration = 12, rate = 500 / 12)

# Set failure rates
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 15, # median survival 15 month
  hr = c(1, .6),
  dropout_rate = 0.001
)

# Set the targeted number of events and analysis time
event <- c(30, 40, 50)
analysis_time <- c(10, 24, 30)

gs_info_wlr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  event = event, analysis_time = analysis_time
)

---