Advanced time-to-event sample size calculation

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 class '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 class 'gsSurv'
print(x, digits = 2, ...)

# S3 method for class '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. Note that sfupar can be specified as a positive scalar for sfLDOF for a generalized O'Brien-Fleming spending function.
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 4.4.2 by xtable 1.8-4 package
#> % Fri Nov 15 23:55:11 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
#>