`nSurv()`

is used to calculate the sample size for a clinical trial
with a time-to-event endpoint and an assumption of proportional hazards.
This set of routines is new with version 2.7 and will continue to be
modified and refined to improve input error checking and output format with
subsequent versions. It allows both the Lachin and Foulkes (1986) method
(fixed trial duration) as well as the Kim and Tsiatis(1990) method (fixed
enrollment rates and either fixed enrollment duration or fixed minimum
follow-up). Piecewise exponential survival is supported as well as piecewise
constant enrollment and dropout rates. The methods are for a 2-arm trial
with treatment groups referred to as experimental and control. A stratified
population is allowed as in Lachin and Foulkes (1986); this method has been
extended to derive non-inferiority as well as superiority trials.
Stratification also allows power calculation for meta-analyses.
`gsSurv()`

combines `nSurv()`

with `gsDesign()`

to derive a
group sequential design for a study with a time-to-event endpoint.

## Usage

```
# S3 method for nSurv
print(x, digits = 4, ...)
nSurv(
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
alpha = 0.025,
beta = 0.1,
sided = 1,
tol = .Machine$double.eps^0.25
)
tEventsIA(x, timing = 0.25, tol = .Machine$double.eps^0.25)
nEventsIA(tIA = 5, x = NULL, target = 0, simple = TRUE)
gsSurv(
k = 3,
test.type = 4,
alpha = 0.025,
sided = 1,
beta = 0.1,
astar = 0,
timing = 1,
sfu = sfHSD,
sfupar = -4,
sfl = sfHSD,
sflpar = -2,
r = 18,
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
tol = .Machine$double.eps^0.25,
usTime = NULL,
lsTime = NULL
)
# S3 method for gsSurv
print(x, digits = 2, ...)
# S3 method for gsSurv
xtable(
x,
caption = NULL,
label = NULL,
align = NULL,
digits = NULL,
display = NULL,
auto = FALSE,
footnote = NULL,
fnwid = "9cm",
timename = "months",
...
)
```

## Arguments

- x
An object of class

`nSurv`

or`gsSurv`

.`print.nSurv()`

is used for an object of class`nSurv`

which will generally be output from`nSurv()`

. For`print.gsSurv()`

is used for an object of class`gsSurv`

which will generally be output from`gsSurv()`

.`nEventsIA`

and`tEventsIA`

operate on both the`nSurv`

and`gsSurv`

class.- digits
Number of digits past the decimal place to print (

`print.gsSurv.`

); also a pass through to generic`xtable()`

from`xtable.gsSurv()`

.- ...
other arguments that may be passed to generic functions underlying the methods here.

- lambdaC
scalar, vector or matrix of event hazard rates for the control group; rows represent time periods while columns represent strata; a vector implies a single stratum.

- hr
hazard ratio (experimental/control) under the alternate hypothesis (scalar).

- hr0
hazard ratio (experimental/control) under the null hypothesis (scalar).

- eta
scalar, vector or matrix of dropout hazard rates for the control group; rows represent time periods while columns represent strata; if entered as a scalar, rate is constant across strata and time periods; if entered as a vector, rates are constant across strata.

- etaE
matrix dropout hazard rates for the experimental group specified in like form as

`eta`

; if NULL, this is set equal to`eta`

.- gamma
a scalar, vector or matrix of rates of entry by time period (rows) and strata (columns); if entered as a scalar, rate is constant across strata and time periods; if entered as a vector, rates are constant across strata.

- R
a scalar or vector of durations of time periods for recruitment rates specified in rows of

`gamma`

. Length is the same as number of rows in`gamma`

. Note that when variable enrollment duration is specified (input`T=NULL`

), the final enrollment period is extended as long as needed.- S
a scalar or vector of durations of piecewise constant event rates specified in rows of

`lambda`

,`eta`

and`etaE`

; this is NULL if there is a single event rate per stratum (exponential failure) or length of the number of rows in`lambda`

minus 1, otherwise.- T
study duration; if

`T`

is input as`NULL`

, this will be computed on output; see details.- minfup
follow-up of last patient enrolled; if

`minfup`

is input as`NULL`

, this will be computed on output; see details.- ratio
randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata.

- alpha
type I error rate. Default is 0.025 since 1-sided testing is default.

- beta
type II error rate. Default is 0.10 (90% power); NULL if power is to be computed based on other input values.

- sided
1 for 1-sided testing, 2 for 2-sided testing.

- tol
for cases when

`T`

or`minfup`

values are derived through root finding (`T`

or`minfup`

input as`NULL`

),`tol`

provides the level of error input to the`uniroot()`

root-finding function. The default is the same as for`uniroot`

.- timing
Sets relative timing of interim analyses in

`gsSurv`

. Default of 1 produces equally spaced analyses. Otherwise, this is a vector of length`k`

or`k-1`

. The values should satisfy`0 < timing[1] < timing[2] < ... < timing[k-1] < timing[k]=1`

. For`tEventsIA`

, this is a scalar strictly between 0 and 1 that indicates the targeted proportion of final planned events available at an interim analysis.- tIA
Timing of an interim analysis; should be between 0 and

`y$T`

.- target
The targeted proportion of events at an interim analysis. This is used for root-finding will be 0 for normal use.

- simple
See output specification for

`nEventsIA()`

.- k
Number of analyses planned, including interim and final.

- test.type
`1=`

one-sided`2=`

two-sided symmetric`3=`

two-sided, asymmetric, beta-spending with binding lower bound`4=`

two-sided, asymmetric, beta-spending with non-binding lower bound`5=`

two-sided, asymmetric, lower bound spending under the null hypothesis with binding lower bound`6=`

two-sided, asymmetric, lower bound spending under the null hypothesis with non-binding lower bound.

See details, examples and manual.- astar
Normally not specified. If

`test.type=5`

or`6`

,`astar`

specifies the total probability of crossing a lower bound at all analyses combined. This will be changed to \(1 - \)`alpha`

when default value of 0 is used. Since this is the expected usage, normally`astar`

is not specified by the user.- sfu
A spending function or a character string indicating a boundary type (that is, “WT” for Wang-Tsiatis bounds, “OF” for O'Brien-Fleming bounds and “Pocock” for Pocock bounds). For one-sided and symmetric two-sided testing is used to completely specify spending (

`test.type=1, 2`

),`sfu`

. The default value is`sfHSD`

which is a Hwang-Shih-DeCani spending function. See details,`vignette("SpendingFunctionOverview")`

, manual and examples.- sfupar
Real value, default is \(-4\) which is an O'Brien-Fleming-like conservative bound when used with the default Hwang-Shih-DeCani spending function. This is a real-vector for many spending functions. The parameter

`sfupar`

specifies any parameters needed for the spending function specified by`sfu`

; this will be ignored for spending functions (`sfLDOF`

,`sfLDPocock`

) or bound types (“OF”, “Pocock”) that do not require parameters.- sfl
Specifies the spending function for lower boundary crossing probabilities when asymmetric, two-sided testing is performed (

`test.type = 3`

,`4`

,`5`

, or`6`

). Unlike the upper bound, only spending functions are used to specify the lower bound. The default value is`sfHSD`

which is a Hwang-Shih-DeCani spending function. The parameter`sfl`

is ignored for one-sided testing (`test.type=1`

) or symmetric 2-sided testing (`test.type=2`

). See details, spending functions, manual and examples.- sflpar
Real value, default is \(-2\), which, with the default Hwang-Shih-DeCani spending function, specifies a less conservative spending rate than the default for the upper 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.- usTime
Default is NULL in which case upper bound spending time is determined by

`timing`

. Otherwise, this should be a vector of length`k`

with the spending time at each analysis (see Details in help for`gsDesign`

).- lsTime
Default is NULL in which case lower bound spending time is determined by

`timing`

. Otherwise, this should be a vector of length`k`

with the spending time at each analysis (see Details in help for`gsDesign`

).- caption
passed through to generic

`xtable()`

.- label
passed through to generic

`xtable()`

.- align
passed through to generic

`xtable()`

.- display
passed through to generic

`xtable()`

.- auto
passed through to generic

`xtable()`

.- footnote
footnote for xtable output; may be useful for describing some of the design parameters.

- fnwid
a text string controlling the width of footnote text at the bottom of the xtable output.

- timename
character string with plural of time units (e.g., "months")

## Value

`nSurv()`

returns an object of type `nSurv`

with the
following components:

- alpha
As input.

- sided
As input.

- beta
Type II error; if missing, this is computed.

- power
Power corresponding to input

`beta`

or computed if output`beta`

is computed.- lambdaC
As input.

- etaC
As input.

- etaE
As input.

- gamma
As input unless none of the following are

`NULL`

:`T`

,`minfup`

,`beta`

; otherwise, this is a constant times the input value required to power the trial given the other input variables.- ratio
As input.

- R
As input unless

`T`

was`NULL`

on input.- S
As input.

- T
As input.

- minfup
As input.

- hr
As input.

- hr0
As input.

- n
Total expected sample size corresponding to output accrual rates and durations.

- d
Total expected number of events under the alternate hypothesis.

- tol
As input, except when not used in computations in which case this is returned as

`NULL`

. This and the remaining output below are not printed by the`print()`

extension for the`nSurv`

class.- eDC
A vector of expected number of events by stratum in the control group under the alternate hypothesis.

- eDE
A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.

- eDC0
A vector of expected number of events by stratum in the control group under the null hypothesis.

- eDE0
A vector of expected number of events by stratum in the experimental group under the null hypothesis.

- eNC
A vector of the expected accrual in each stratum in the control group.

- eNE
A vector of the expected accrual in each stratum in the experimental group.

- variable
A text string equal to "Accrual rate" if a design was derived by varying the accrual rate, "Accrual duration" if a design was derived by varying the accrual duration, "Follow-up duration" if a design was derived by varying follow-up duration, or "Power" if accrual rates and duration as well as follow-up duration was specified and

`beta=NULL`

was input.

`gsSurv()`

returns much of the above plus variables in the class
`gsDesign`

; see `gsDesign`

for general documentation on what is returned in `gs`

. The value of
`gs$n.I`

represents the number of endpoints required at each analysis
to adequately power the trial. Other items returned by `gsSurv()`

are:

- lambdaC
As input.

- etaC
As input.

- etaE
As input.

- gamma
As input unless none of the following are

`NULL`

:`T`

,`minfup`

,`beta`

; otherwise, this is a constant times the input value required to power the trial given the other input variables.- ratio
As input.

- R
As input unless

`T`

was`NULL`

on input.- S
As input.

- T
As input.

- minfup
As input.

- hr
As input.

- hr0
As input.

- eNC
Total expected sample size corresponding to output accrual rates and durations.

- eNE
Total expected sample size corresponding to output accrual rates and durations.

- eDC
Total expected number of events under the alternate hypothesis.

- eDE
Total expected number of events under the alternate hypothesis.

- tol
As input, except when not used in computations in which case this is returned as

`NULL`

. This and the remaining output below are not printed by the`print()`

extension for the`nSurv`

class.- eDC
A vector of expected number of events by stratum in the control group under the alternate hypothesis.

- eDE
A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.

- eNC
A vector of the expected accrual in each stratum in the control group.

- eNE
A vector of the expected accrual in each stratum in the experimental group.

- variable
A text string equal to "Accrual rate" if a design was derived by varying the accrual rate, "Accrual duration" if a design was derived by varying the accrual duration, "Follow-up duration" if a design was derived by varying follow-up duration, or "Power" if accrual rates and duration as well as follow-up duration was specified and

`beta=NULL`

was input.

`nEventsIA()`

returns the expected proportion of the final planned
events observed at the input analysis time minus `target`

when
`simple=TRUE`

. When `simple=FALSE`

, `nEventsIA`

returns a
list with following components:

- T
The input value

`tIA`

.- eDC
The expected number of events in the control group at time the output time

`T`

.- eDE
The expected number of events in the experimental group at the output time

`T`

.- eNC
The expected enrollment in the control group at the output time

`T`

.- eNE
The expected enrollment in the experimental group at the output time

`T`

.

`tEventsIA()`

returns the same structure as `nEventsIA(..., simple=TRUE)`

when

## Details

`print()`

, `xtable()`

and `summary()`

methods are provided to
operate on the returned value from `gsSurv()`

, an object of class
`gsSurv`

. `print()`

is also extended to `nSurv`

objects. The
functions `gsBoundSummary`

(data frame for tabular output),
`xprint`

(application of `xtable`

for tabular output) and
`summary.gsSurv`

(textual summary of `gsDesign`

or `gsSurv`

object) may be preferred summary functions; see example in vignettes. See
also gsBoundSummary for output
of tabular summaries of bounds for designs produced by `gsSurv()`

.

Both `nEventsIA`

and `tEventsIA`

require a group sequential design
for a time-to-event endpoint of class `gsSurv`

as input.
`nEventsIA`

calculates the expected number of events under the
alternate hypothesis at a given interim time. `tEventsIA`

calculates
the time that the expected number of events under the alternate hypothesis
is a given proportion of the total events planned for the final analysis.

`nSurv()`

produces an object of class `nSurv`

with the number of
subjects and events for a set of pre-specified trial parameters, such as
accrual duration and follow-up period. The underlying power calculation is
based on Lachin and Foulkes (1986) method for proportional hazards assuming
a fixed underlying hazard ratio between 2 treatment groups. The method has
been extended here to enable designs to test non-inferiority. Piecewise
constant enrollment and failure rates are assumed and a stratified
population is allowed. See also `nSurvival`

for other Lachin and
Foulkes (1986) methods assuming a constant hazard difference or exponential
enrollment rate.

When study duration (`T`

) and follow-up duration (`minfup`

) are
fixed, `nSurv`

applies exactly the Lachin and Foulkes (1986) method of
computing sample size under the proportional hazards assumption when For
this computation, enrollment rates are altered proportionately to those
input in `gamma`

to achieve the power of interest.

Given the specified enrollment rate(s) input in `gamma`

, `nSurv`

may also be used to derive enrollment duration required for a trial to have
defined power if `T`

is input as `NULL`

; in this case, both
`R`

(enrollment duration for each specified enrollment rate) and
`T`

(study duration) will be computed on output.

Alternatively and also using the fixed enrollment rate(s) in `gamma`

,
if minimum follow-up `minfup`

is specified as `NULL`

, then the
enrollment duration(s) specified in `R`

are considered fixed and
`minfup`

and `T`

are computed to derive the desired power. This
method will fail if the specified enrollment rates and durations either
over-powers the trial with no additional follow-up or underpowers the trial
with infinite follow-up. This method produces a corresponding error message
in such cases.

The input to `gsSurv`

is a combination of the input to `nSurv()`

and `gsDesign()`

.

`nEventsIA()`

is provided to compute the expected number of events at a
given point in time given enrollment, event and censoring rates. The routine
is used with a root finding routine to approximate the approximate timing of
an interim analysis. It is also used to extend enrollment or follow-up of a
fixed design to obtain a sufficient number of events to power a group
sequential design.

## References

Kim KM and Tsiatis AA (1990), Study duration for clinical trials
with survival response and early stopping rule. *Biometrics*, 46, 81-92

Lachin JM and Foulkes MA (1986), Evaluation of Sample Size and Power for
Analyses of Survival with Allowance for Nonuniform Patient Entry, Losses to
Follow-Up, Noncompliance, and Stratification. *Biometrics*, 42,
507-519.

Schoenfeld D (1981), The Asymptotic Properties of Nonparametric Tests for
Comparing Survival Distributions. *Biometrika*, 68, 316-319.

## Author

Keaven Anderson keaven_anderson@merck.com

## Examples

```
# vary accrual rate to obtain power
nSurv(lambdaC = log(2) / 6, hr = .5, eta = log(2) / 40, gamma = 1, T = 36, minfup = 12)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=36
#> Accrual duration: 24
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 86.3258
#> Expected sample size (total): 119.8184
#> Accrual rates:
#> Stratum 1
#> 0-24 4.9924
#> Control event rates (H1):
#> Stratum 1
#> 0-Inf 0.1155
#> Censoring rates:
#> Stratum 1
#> 0-Inf 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# vary accrual duration to obtain power
nSurv(lambdaC = log(2) / 6, hr = .5, eta = log(2) / 40, gamma = 6, minfup = 12)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=18
#> Accrual duration: 6
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 86.8217
#> Expected sample size (total): 137.2033
#> Accrual rates:
#> Stratum 1
#> 0-6 22.8672
#> Control event rates (H1):
#> Stratum 1
#> 0-Inf 0.1155
#> Censoring rates:
#> Stratum 1
#> 0-Inf 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# vary follow-up duration to obtain power
nSurv(lambdaC = log(2) / 6, hr = .5, eta = log(2) / 40, gamma = 6, R = 25)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=18
#> Accrual duration: 12
#> Min. end-of-study follow-up: minfup=6
#> Expected events (total, H1): 87.2677
#> Expected sample size (total): 155.8581
#> Accrual rates:
#> Stratum 1
#> 0-12 12.9882
#> Control event rates (H1):
#> Stratum 1
#> 0-Inf 0.1155
#> Censoring rates:
#> Stratum 1
#> 0-Inf 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# piecewise constant enrollment rates (vary accrual duration)
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = log(2) / 40, gamma = c(1, 3, 6),
R = c(3, 6, 9), T = NULL, minfup = 12
)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual duration
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=37.7809
#> Accrual duration: 25.7809
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 86.3632
#> Expected sample size (total): 121.6855
#> Accrual rates:
#> Stratum 1
#> 0-3 1
#> 3-9 3
#> 9-25.78 6
#> Control event rates (H1):
#> Stratum 1
#> 0-Inf 0.1155
#> Censoring rates:
#> Stratum 1
#> 0-Inf 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# stratified population (vary accrual duration)
nSurv(
lambdaC = matrix(log(2) / c(6, 12), ncol = 2), hr = .5, eta = log(2) / 40,
gamma = matrix(c(2, 4), ncol = 2), minfup = 12
)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=18
#> Accrual duration: 6
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 87.6736
#> Expected sample size (total): 179.9025
#> Accrual rates:
#> Stratum 1 Stratum 2
#> 0-6 9.9946 19.9892
#> Control event rates (H1):
#> Stratum 1 Stratum 2
#> 0-Inf 0.1155 0.0578
#> Censoring rates:
#> Stratum 1 Stratum 2
#> 0-Inf 0.0173 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# piecewise exponential failure rates (vary accrual duration)
nSurv(lambdaC = log(2) / c(6, 12), hr = .5, eta = log(2) / 40, S = 3, gamma = 6, minfup = 12)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=18
#> Accrual duration: 6
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 87.9869
#> Expected sample size (total): 183.8174
#> Accrual rates:
#> Stratum 1
#> 0-6 30.6362
#> Control event rates (H1):
#> Stratum 1
#> 0-3 0.1155
#> 3-Inf 0.0578
#> Censoring rates:
#> Stratum 1
#> 0-3 0.0173
#> 3-Inf 0.0173
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# combine it all: 2 strata, 2 failure rate periods
nSurv(
lambdaC = matrix(log(2) / c(6, 12, 18, 24), ncol = 2), hr = .5,
eta = matrix(log(2) / c(40, 50, 45, 55), ncol = 2), S = 3,
gamma = matrix(c(3, 6, 5, 7), ncol = 2), R = c(5, 10), minfup = 12
)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.5/1
#> Study duration: T=18
#> Accrual duration: 6
#> Min. end-of-study follow-up: minfup=12
#> Expected events (total, H1): 88.7326
#> Expected sample size (total): 255.7917
#> Accrual rates:
#> Stratum 1 Stratum 2
#> 0-5 14.4788 24.1313
#> 5-6 28.9576 33.7838
#> Control event rates (H1):
#> Stratum 1 Stratum 2
#> 0-3 0.1155 0.0385
#> 3-Inf 0.0578 0.0289
#> Censoring rates:
#> Stratum 1 Stratum 2
#> 0-3 0.0173 0.0154
#> 3-Inf 0.0139 0.0126
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# example where only 1 month of follow-up is desired
# set failure rate to 0 after 1 month using lambdaC and S
nSurv(lambdaC = c(.4, 0), hr = 2 / 3, S = 1, minfup = 1)
#> Fixed design, two-arm trial with time-to-event
#> outcome (Lachin and Foulkes, 1986).
#> Solving for: Accrual rate
#> Hazard ratio H1/H0=0.6667/1
#> Study duration: T=18
#> Accrual duration: 17
#> Min. end-of-study follow-up: minfup=1
#> Expected events (total, H1): 257.7578
#> Expected sample size (total): 914.4375
#> Accrual rates:
#> Stratum 1
#> 0-17 53.7904
#> Control event rates (H1):
#> Stratum 1
#> 0-1 0.4
#> 1-Inf 0.0
#> Censoring rates:
#> Stratum 1
#> 0-1 0
#> 1-Inf 0
#> Power: 100*(1-beta)=90%
#> Type I error (1-sided): 100*alpha=2.5%
#> Equal randomization: ratio=1
# group sequential design (vary accrual rate to obtain power)
x <- gsSurv(
k = 4, sfl = sfPower, sflpar = .5, lambdaC = log(2) / 6, hr = .5,
eta = log(2) / 40, gamma = 1, T = 36, minfup = 12
)
x
#> Time to event group sequential design with HR= 0.5
#> Equal randomization: ratio=1
#> Asymmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Upper bound spending computations assume
#> trial continues if lower bound is crossed.
#>
#> ----Lower bounds---- ----Upper bounds-----
#> Analysis N Z Nominal p Spend+ Z Nominal p Spend++
#> 1 29 0.23 0.5895 0.0500 3.16 0.0008 0.0008
#> 2 58 0.86 0.8056 0.0207 2.82 0.0024 0.0022
#> 3 87 1.46 0.9277 0.0159 2.44 0.0074 0.0059
#> 4 116 2.01 0.9780 0.0134 2.01 0.0220 0.0161
#> Total 0.1000 0.0250
#> + lower bound beta spending (under H1):
#> Kim-DeMets (power) spending function with rho = 0.5.
#> ++ alpha spending:
#> Hwang-Shih-DeCani spending function with gamma = -4.
#>
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#>
#> Upper boundary (power or Type I Error)
#> Analysis
#> Theta 1 2 3 4 Total E{N}
#> 0.0000 0.0008 0.0022 0.0055 0.0102 0.0187 46.5
#> 0.3489 0.0995 0.3393 0.3388 0.1224 0.9000 71.2
#>
#> Lower boundary (futility or Type II Error)
#> Analysis
#> Theta 1 2 3 4 Total
#> 0.0000 0.5895 0.2470 0.1079 0.0369 0.9813
#> 0.3489 0.0500 0.0207 0.0159 0.0134 0.1000
#> T n Events HR futility HR efficacy
#> IA 1 12.24228 81.46723 28.76662 0.919 0.308
#> IA 2 18.97078 126.24254 57.53324 0.797 0.476
#> IA 3 25.02728 159.70989 86.29986 0.730 0.591
#> Final 36.00000 159.70989 115.06648 0.687 0.687
#> Accrual rates:
#> Stratum 1
#> 0-24 6.65
#> Control event rates (H1):
#> Stratum 1
#> 0-Inf 0.12
#> Censoring rates:
#> Stratum 1
#> 0-Inf 0.02
print(xtable::xtable(x,
footnote = "This is a footnote; note that it can be wide.",
caption = "Caption example."
))
#> % latex table generated in R 4.3.2 by xtable 1.8-4 package
#> % Wed Feb 21 21:33:51 2024
#> \begin{table}[ht]
#> \centering
#> \begin{tabular}{rllll}
#> \hline
#> & Analysis & Value & Futility & Efficacy \\
#> \hline
#> 1 & IA 1: 25$\backslash$\% & Z-value & 0.23 & 3.16 \\
#> 2 & N: 82 & HR & 0.92 & 0.31 \\
#> 3 & Events: 29 & p (1-sided) & 0.4105 & 8e-04 \\
#> 4 & 12.2 months & P$\backslash$\{Cross$\backslash$\} if HR=1 & 0.5895 & 8e-04 \\
#> 5 & & P$\backslash$\{Cross$\backslash$\} if HR=0.5 & 0.05 & 0.0995 \\
#> 6 & $\backslash$hline IA 2: 50$\backslash$\% & Z-value & 0.86 & 2.82 \\
#> 7 & N: 128 & HR & 0.8 & 0.48 \\
#> 8 & Events: 58 & p (1-sided) & 0.1944 & 0.0024 \\
#> 9 & 19 months & P$\backslash$\{Cross$\backslash$\} if HR=1 & 0.8366 & 0.003 \\
#> 10 & & P$\backslash$\{Cross$\backslash$\} if HR=0.5 & 0.0707 & 0.4388 \\
#> 11 & $\backslash$hline IA 3: 75$\backslash$\% & Z-value & 1.46 & 2.44 \\
#> 12 & N: 160 & HR & 0.73 & 0.59 \\
#> 13 & Events: 87 & p (1-sided) & 0.0723 & 0.0074 \\
#> 14 & 25 months & P$\backslash$\{Cross$\backslash$\} if HR=1 & 0.9445 & 0.0085 \\
#> 15 & & P$\backslash$\{Cross$\backslash$\} if HR=0.5 & 0.0866 & 0.7776 \\
#> 16 & $\backslash$hline Final analysis & Z-value & 2.01 & 2.01 \\
#> 17 & N: 160 & HR & 0.69 & 0.69 \\
#> 18 & Events: 116 & p (1-sided) & 0.022 & 0.022 \\
#> 19 & 36 months & P$\backslash$\{Cross$\backslash$\} if HR=1 & 0.9813 & 0.0187 \\
#> 20 & & P$\backslash$\{Cross$\backslash$\} if HR=0.5 & 0.1 & 0.9 $\backslash$$\backslash$ $\backslash$hline $\backslash$multicolumn\{4\}\{p\{ 9cm \}\}\{$\backslash$footnotesize This is a footnote; note that it can be wide. \} \\
#> \hline
#> \end{tabular}
#> \caption{Caption example.}
#> \end{table}
# find expected number of events at time 12 in the above trial
nEventsIA(x = x, tIA = 10)
#> [1] 20.51876
# find time at which 1/4 of events are expected
tEventsIA(x = x, timing = .25)
#> $T
#> [1] 12.24228
#>
#> $eDC
#> [1] 17.92465
#>
#> $eDE
#> [1] 10.84196
#>
#> $eNC
#> [1] 40.73361
#>
#> $eNE
#> [1] 40.73361
#>
```