Advanced time-to-event sample size calculation

nSurv() is used to calculate the sample size for a two-arm clinical trial with a time-to-event endpoint under the assumption of proportional hazards. The default method assumes a fixed enrollment duration and fixed trial duration; in this case the required sample size is achieved by increasing enrollment rates. nSurv() implements the Lachin and Foulkes (1986) method as default. Schoenfeld (1981), Freedman (1982), and Bernstein and Lagakos (1989) methods are also supported; see Details. gsSurv() combines nSurv() with gsDesign() to derive a group sequential design for a study with a time-to-event endpoint.

Usage

tEventsIA(x, timing = 0.25, tol = .Machine$double.eps^0.25)

nEventsIA(tIA = 5, x = NULL, target = 0, simple = TRUE)

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,
  method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)

# S3 method for class 'nSurv'
print(x, digits = 3, show_strata = TRUE, ...)

# 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",
  ...
)

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,
  method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)

# S3 method for class 'gsSurv'
print(x, digits = 3, show_gsDesign = FALSE, show_strata = TRUE, ...)

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.
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.
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.
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().
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. Note that rates corresponding the final time period are extended indefinitely.
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; otherwise enrollment after sum(R) is 0.
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. The final time period is extended indefinitely for each stratum.
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.
method: One of "LachinFoulkes" (default), "Schoenfeld", "Freedman", or "BernsteinLagakos". Note: "Schoenfeld" and "Freedman" methods only support superiority testing (hr0 = 1). "Freedman" does not support stratified populations.
digits: Number of digits past the decimal place to print (print.gsSurv.); also a pass through to generic xtable() from xtable.gsSurv().
show_strata: Logical; for print.gsSurv(), include strata summaries.
...: Other arguments that may be passed to generic functions underlying the methods here.
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")
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 (>= 1 and <= 80) controlling the number of numerical integration grid points. Default is 18, as recommended by Jennison and Turnbull (2000). Grid points are spread out in the tails for accurate probability calculations. Larger values provide more grid points and greater accuracy but slow down computation. Jennison and Turnbull (p. 350) note an accuracy of $10^{-6}$ with r = 16. This parameter is normally not changed by users.
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).
show_gsDesign: Logical; for print.gsSurv(), include gsDesign details.

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 an object of class gsDesign in a variable named gs; 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:

gs: A group sequential design (gsDesign) object.
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.
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.

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

The Lachin and Foulkes method uses both null and alternate hypothesis variances to derive sample size and is extended here to support non-inferiority, super-superiority, and stratified designs. As an alternative, the Kim and Tsiatis (1990) method can be used with fixed enrollment rates and either fixed enrollment duration or fixed minimum follow-up. The Schoenfeld approach uses the asymptotic distribution of the log-rank statistic under the assumption of proportional hazards and local alternatives (i.e., $\log(HR)$ is small). The Freedman approach uses the same asymptotic distribution and, like the Schoenfeld approach, uses just the null hypothesis variance to derive sample size. The Bernstein and Lagakos (1989) approach was derived to compute sample size for a stratified model with a common proportional hazards assumption across strata. Like the Lachin and Foulkes (1986) method, it uses both null and alternate hypothesis variances to compute sample size; however, the null hypothesis variance is computed differently. The Bernstein and Lagakos (1989) approach uses the alternate hypothesis failure rate assumptions for both the control and experimental groups, while the Lachin and Foulkes method uses null hypothesis rates that average the alternate hypothesis failure rates to get similar numbers of expected events under the null and alternate hypotheses. Since the Lachin and Foulkes method has fewer events under the null hypothesis (less statistical information), it calculates less power than the Bernstein and Lagakos method. 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.

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 gamma to obtain power
# T, minfup and R all specified, although R will be adjusted on output
# gamma as input will be multiplied in output to achieve desired power
# Default method is Lachin and Foulkes
x_nsurv <- nSurv(
  lambdaC = log(2) / 6, R = 10, hr = .5, eta = .001, gamma = 1,
  alpha = 0.02, beta = .15, T = 36, minfup = 12, method = "LachinFoulkes"
)
# Demonstrate print method
print(x_nsurv)
#> nSurv fixed-design summary (method=LachinFoulkes; target=Accrual rate)
#> HR=0.500 vs HR0=1.000 | alpha=0.020 (sided=1) | power=85.0%
#> N=95.7 subjects | D=78.1 events | T=36.0 study duration | accrual=24.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma 3.986     1
#>            Accrual rate duration(s)       R    24    10
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S
# Same assumptions for group sequential design
x_gs <- gsSurv(
  k = 4, sfl = gsDesign::sfPower, sflpar = .5, lambdaC = log(2) / 6, hr = .5,
  eta = .001, gamma = 1, T = 36, minfup = 12, method = "LachinFoulkes"
)
print(x_gs)
#> Group sequential design (method=LachinFoulkes; k=4 analyses; Two-sided asymmetric with non-binding futility)
#> N=140.7 subjects | D=114.8 events | T=36.0 study duration | accrual=24.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4.
#>   Futility bounds derived using a Kim-DeMets (power) spending function with rho = 0.
#> 
#> Analysis summary:
#> Method: LachinFoulkes 
#>     Analysis              Value Efficacy Futility
#>    IA 1: 25%                  Z   3.1554   0.2264
#>        N: 76        p (1-sided)   0.0008   0.4105
#>   Events: 29       ~HR at bound   0.3079   0.9190
#>    Month: 13   P(Cross) if HR=1   0.0008   0.5895
#>              P(Cross) if HR=0.5   0.0995   0.0500
#>    IA 2: 50%                  Z   2.8183   0.8619
#>       N: 116        p (1-sided)   0.0024   0.1944
#>   Events: 58       ~HR at bound   0.4752   0.7965
#>    Month: 20   P(Cross) if HR=1   0.0030   0.8366
#>              P(Cross) if HR=0.5   0.4388   0.0707
#>    IA 3: 75%                  Z   2.4390   1.4589
#>       N: 142        p (1-sided)   0.0074   0.0723
#>   Events: 87       ~HR at bound   0.5912   0.7302
#>    Month: 26   P(Cross) if HR=1   0.0085   0.9445
#>              P(Cross) if HR=0.5   0.7776   0.0866
#>        Final                  Z   2.0136   2.0136
#>       N: 142        p (1-sided)   0.0220   0.0220
#>  Events: 115       ~HR at bound   0.6867   0.6867
#>    Month: 36   P(Cross) if HR=1   0.0187   0.9813
#>              P(Cross) if HR=0.5   0.9000   0.1000
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma 5.863     1
#>            Accrual rate duration(s)       R    24    12
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S
# Demonstrate xtable method for gsSurv
print(xtable::xtable(x_gs,
  footnote = "This is a footnote; note that it can be wide.",
  caption = "Caption example for xtable output."
))
#> % latex table generated in R 4.5.2 by xtable 1.8-8 package
#> % Mon Feb 23 21:35:26 2026
#> \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: 76 & HR & 0.92 & 0.31 \\ 
#>   3 & Events: 29 & p (1-sided) & 0.4105 & 8e-04 \\ 
#>   4 & 12.8 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: 116 & HR & 0.8 & 0.48 \\ 
#>   8 & Events: 58 & p (1-sided) & 0.1944 & 0.0024 \\ 
#>   9 & 19.6 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: 142 & HR & 0.73 & 0.59 \\ 
#>   13 & Events: 87 & p (1-sided) & 0.0723 & 0.0074 \\ 
#>   14 & 25.7 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: 142 & HR & 0.69 & 0.69 \\ 
#>   18 & Events: 115 & 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 for xtable output.} 
#> \end{table}
# Demonstrate nEventsIA method
# find expected number of events at time 12 in the above trial
nEventsIA(x = x_gs, tIA = 10)
#> [1] 18.92635

# find time at which 1/4 of events are expected
tEventsIA(x = x_gs, timing = .25)
#> $T
#> [1] 12.76571
#> 
#> $eDC
#> [1] 17.79583
#> 
#> $eDE
#> [1] 10.90882
#> 
#> $eNC
#> [1] 37.42481
#> 
#> $eNE
#> [1] 37.42481
#> 

# Adjust accrual duration R to achieve desired power
# Trial duration T input as NULL and will be computed based on
# accrual duration R and minimum follow-up duration minfup
# Minimum follow-up duration minfup is specified
# We use the Schoenfeld method to compute accrual duration R
# Control median survival time is 6
nSurv(
  lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
  alpha = .025, beta = .1, minfup = 12, T = NULL, method = "Schoenfeld"
)
#> nSurv fixed-design summary (method=Schoenfeld; target=Accrual duration)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=1) | power=90.0%
#> N=109.5 subjects | D=86.2 events | T=30.2 study duration | accrual=18.2 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Key inputs (names preserved):
#>                                desc    item  value input
#>                     Accrual rate(s)   gamma      6     6
#>            Accrual rate duration(s)       R 18.243    12
#>              Control hazard rate(s) lambdaC  0.116 0.116
#>             Control dropout rate(s)     eta  0.001 0.001
#>        Experimental dropout rate(s)    etaE  0.001  etaE
#>  Event and dropout rate duration(s)       S   NULL     S
# Same assumptions for group sequential design
gsSurv(
  k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
  lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
  T = 36, minfup = 12, method = "Schoenfeld"
) |>
  print()
#> Group sequential design (method=Schoenfeld; k=4 analyses; Two-sided asymmetric with non-binding futility)
#> N=142.9 subjects | D=116.6 events | T=36.0 study duration | accrual=24.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4.
#>   Futility bounds derived using a Kim-DeMets (power) spending function with rho = 0.
#> 
#> Analysis summary:
#> Method: Schoenfeld 
#>     Analysis              Value Efficacy Futility
#>    IA 1: 25%                  Z   3.1554   0.2264
#>        N: 78        p (1-sided)   0.0008   0.4105
#>   Events: 30       ~HR at bound   0.3107   0.9196
#>    Month: 13   P(Cross) if HR=1   0.0008   0.5895
#>              P(Cross) if HR=0.5   0.0995   0.0500
#>    IA 2: 50%                  Z   2.8183   0.8619
#>       N: 118        p (1-sided)   0.0024   0.1944
#>   Events: 59       ~HR at bound   0.4780   0.7979
#>    Month: 20   P(Cross) if HR=1   0.0030   0.8366
#>              P(Cross) if HR=0.5   0.4388   0.0707
#>    IA 3: 75%                  Z   2.4390   1.4589
#>       N: 144        p (1-sided)   0.0074   0.0723
#>   Events: 88       ~HR at bound   0.5936   0.7320
#>    Month: 26   P(Cross) if HR=1   0.0085   0.9445
#>              P(Cross) if HR=0.5   0.7776   0.0866
#>        Final                  Z   2.0136   2.0136
#>       N: 144        p (1-sided)   0.0220   0.0220
#>  Events: 117       ~HR at bound   0.6887   0.6887
#>    Month: 36   P(Cross) if HR=1   0.0187   0.9813
#>              P(Cross) if HR=0.5   0.9000   0.1000
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma 5.955     6
#>            Accrual rate duration(s)       R    24    12
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S

# Vary minimum follow-up duration minfup to obtain power
# Accrual duration R rate gamma are fixed and will not change on output.
# Trial duration T and minimum follow-up minfup are input as NULL
# and will be computed on output.
# We will use the Freedman method to compute sample size
# Control median survival time is 6
# Often this method will fail as the accrual duration and rate provide too
# little or too much sample size.
nSurv(
  lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
  alpha = .025, beta = .1, minfup = NULL, T = NULL, method = "Freedman"
)
#> nSurv fixed-design summary (method=Freedman; target=Follow-up duration)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=1) | power=90.0%
#> N=150.0 subjects | D=86.8 events | T=25.3 study duration | accrual=25.0 Accrual duration | minfup=0.3 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma     6     6
#>            Accrual rate duration(s)       R    25    25
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S
# Same assumptions for group sequential design
gsSurv(
  k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
  lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
  T = 36, minfup = 12, method = "Freedman"
) |>
  print()
#> Group sequential design (method=Freedman; k=4 analyses; Two-sided asymmetric with non-binding futility)
#> N=154.5 subjects | D=126.1 events | T=36.0 study duration | accrual=24.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4.
#>   Futility bounds derived using a Kim-DeMets (power) spending function with rho = 0.
#> 
#> Analysis summary:
#> Method: Freedman 
#>     Analysis              Value Efficacy Futility
#>    IA 1: 25%                  Z   3.1554   0.2264
#>        N: 84        p (1-sided)   0.0008   0.4105
#>   Events: 32       ~HR at bound   0.3249   0.9225
#>    Month: 13   P(Cross) if HR=1   0.0008   0.5895
#>              P(Cross) if HR=0.5   0.0995   0.0500
#>    IA 2: 50%                  Z   2.8183   0.8619
#>       N: 128        p (1-sided)   0.0024   0.1944
#>   Events: 64       ~HR at bound   0.4916   0.8048
#>    Month: 20   P(Cross) if HR=1   0.0030   0.8366
#>              P(Cross) if HR=0.5   0.4388   0.0707
#>    IA 3: 75%                  Z   2.4390   1.4589
#>       N: 156        p (1-sided)   0.0074   0.0723
#>   Events: 95       ~HR at bound   0.6055   0.7407
#>    Month: 26   P(Cross) if HR=1   0.0085   0.9445
#>              P(Cross) if HR=0.5   0.7776   0.0866
#>        Final                  Z   2.0136   2.0136
#>       N: 156        p (1-sided)   0.0220   0.0220
#>  Events: 127       ~HR at bound   0.6986   0.6986
#>    Month: 36   P(Cross) if HR=1   0.0187   0.9813
#>              P(Cross) if HR=0.5   0.9000   0.1000
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma 6.437     6
#>            Accrual rate duration(s)       R    24    12
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S

# piecewise constant enrollment rates (vary accrual rate to achieve power)
# also piecewise constant failure rates
# will specify annualized enrollment and failure rates
nSurv(
  lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
  S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
  R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
  gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
  # multiplied by ratio to achieve desired power
  hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
  minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
  alpha = .025, beta = .1, method = "LachinFoulkes"
)
#> nSurv fixed-design summary (method=LachinFoulkes; target=Accrual rate)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=1) | power=90.0%
#> N=1088.8 subjects | D=91.1 events | T=5.0 study duration | accrual=2.0 Accrual duration | minfup=3.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Key inputs (names preserved):
#>                                desc    item                     value
#>                     Accrual rate(s)   gamma 108.876, 326.629, 653.258
#>            Accrual rate duration(s)       R           0.25, 0.25, 1.5
#>              Control hazard rate(s) lambdaC         0.051, 0.03, 0.02
#>             Control dropout rate(s)     eta          0.01, 0.01, 0.01
#>        Experimental dropout rate(s)    etaE          0.01, 0.01, 0.01
#>  Event and dropout rate duration(s)       S                      1, 1
#>              input
#>            1, 3, 6
#>    0.25, 0.25, 1.5
#>  0.051, 0.03, 0.02
#>               0.01
#>               etaE
#>               1, 1
# Same assumptions for group sequential design
gsSurv(
  k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
  lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
  S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
  R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
  gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
  # multiplied by ratio to achieve desired power
  hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
  minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
  alpha = .025, beta = .1, method = "LachinFoulkes"
) |>
  print()
#> Group sequential design (method=LachinFoulkes; k=4 analyses; Two-sided asymmetric with non-binding futility)
#> N=1451.2 subjects | D=121.4 events | T=5.0 study duration | accrual=2.0 Accrual duration | minfup=3.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4.
#>   Futility bounds derived using a Kim-DeMets (power) spending function with rho = 0.
#> 
#> Analysis summary:
#> Method: LachinFoulkes 
#>     Analysis              Value Efficacy Futility
#>    IA 1: 25%                  Z   3.1554   0.2264
#>      N: 1198        p (1-sided)   0.0008   0.4105
#>   Events: 31       ~HR at bound   0.3181   0.9211
#>     Month: 2   P(Cross) if HR=1   0.0008   0.5895
#>              P(Cross) if HR=0.5   0.0995   0.0500
#>    IA 2: 50%                  Z   2.8183   0.8619
#>      N: 1452        p (1-sided)   0.0024   0.1944
#>   Events: 61       ~HR at bound   0.4851   0.8015
#>     Month: 2   P(Cross) if HR=1   0.0030   0.8366
#>              P(Cross) if HR=0.5   0.4388   0.0707
#>    IA 3: 75%                  Z   2.4390   1.4589
#>      N: 1452        p (1-sided)   0.0074   0.0723
#>   Events: 92       ~HR at bound   0.5998   0.7366
#>     Month: 3   P(Cross) if HR=1   0.0085   0.9445
#>              P(Cross) if HR=0.5   0.7776   0.0866
#>        Final                  Z   2.0136   2.0136
#>      N: 1452        p (1-sided)   0.0220   0.0220
#>  Events: 122       ~HR at bound   0.6939   0.6939
#>     Month: 5   P(Cross) if HR=1   0.0187   0.9813
#>              P(Cross) if HR=0.5   0.9000   0.1000
#> 
#> Key inputs (names preserved):
#>                                desc    item                     value
#>                     Accrual rate(s)   gamma 145.125, 435.375, 870.749
#>            Accrual rate duration(s)       R           0.25, 0.25, 1.5
#>              Control hazard rate(s) lambdaC         0.051, 0.03, 0.02
#>             Control dropout rate(s)     eta          0.01, 0.01, 0.01
#>        Experimental dropout rate(s)    etaE          0.01, 0.01, 0.01
#>  Event and dropout rate duration(s)       S                      1, 1
#>              input
#>            1, 3, 6
#>    0.25, 0.25, 1.5
#>  0.051, 0.03, 0.02
#>               0.01
#>               etaE
#>               1, 1

# combine it all: 2 strata, 2 failure rate periods
# Note that method = "LachinFoulkes" may be preferred here
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,
  alpha = .025, beta = .1, method = "BernsteinLagakos"
)
#> nSurv fixed-design summary (method=BernsteinLagakos; target=Accrual rate)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=1) | power=90.0%
#> N=226.8 subjects | D=78.7 events | T=18.0 study duration | accrual=6.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Expected events by stratum (H1):
#>   control experimental  total
#> 1  26.517       16.431 42.949
#> 2  22.980       12.761 35.741
#> 
#> Key inputs (names preserved):
#>                                desc    item                           value
#>                     Accrual rate(s)   gamma  12.84, 25.68, 21.4 ... (len=4)
#>            Accrual rate duration(s)       R                            5, 1
#>              Control hazard rate(s) lambdaC 0.116, 0.058, 0.039 ... (len=4)
#>             Control dropout rate(s)     eta 0.017, 0.014, 0.015 ... (len=4)
#>        Experimental dropout rate(s)    etaE 0.017, 0.014, 0.015 ... (len=4)
#>  Event and dropout rate duration(s)       S                               3
#>                            input
#>              3, 6, 5 ... (len=4)
#>                            5, 10
#>  0.116, 0.058, 0.039 ... (len=4)
#>  0.017, 0.014, 0.015 ... (len=4)
#>                             etaE
#>                                3
# Same assumptions for group sequential design
gsSurv(
  k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
  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,
  alpha = .025, beta = .1, method = "BernsteinLagakos"
) |>
  print()
#> Group sequential design (method=BernsteinLagakos; k=4 analyses; Two-sided asymmetric with non-binding futility)
#> N=302.4 subjects | D=104.9 events | T=18.0 study duration | accrual=6.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4.
#>   Futility bounds derived using a Kim-DeMets (power) spending function with rho = 0.
#> 
#> Analysis summary:
#> Method: BernsteinLagakos 
#>     Analysis              Value Efficacy Futility
#>    IA 1: 25%                  Z   3.1554   0.2264
#>       N: 250        p (1-sided)   0.0008   0.4105
#>   Events: 27       ~HR at bound   0.2916   0.9154
#>     Month: 5   P(Cross) if HR=1   0.0008   0.5895
#>              P(Cross) if HR=0.5   0.0995   0.0500
#>    IA 2: 50%                  Z   2.8183   0.8619
#>       N: 304        p (1-sided)   0.0024   0.1944
#>   Events: 53       ~HR at bound   0.4592   0.7882
#>     Month: 8   P(Cross) if HR=1   0.0030   0.8366
#>              P(Cross) if HR=0.5   0.4388   0.0707
#>    IA 3: 75%                  Z   2.4390   1.4589
#>       N: 304        p (1-sided)   0.0074   0.0723
#>   Events: 79       ~HR at bound   0.5770   0.7197
#>    Month: 12   P(Cross) if HR=1   0.0085   0.9445
#>              P(Cross) if HR=0.5   0.7776   0.0866
#>        Final                  Z   2.0136   2.0136
#>       N: 304        p (1-sided)   0.0220   0.0220
#>  Events: 105       ~HR at bound   0.6749   0.6749
#>    Month: 18   P(Cross) if HR=1   0.0187   0.9813
#>              P(Cross) if HR=0.5   0.9000   0.1000
#> 
#> Key inputs (names preserved):
#>                                desc    item                             value
#>                     Accrual rate(s)   gamma 17.115, 28.525, 34.23 ... (len=4)
#>            Accrual rate duration(s)       R                              5, 1
#>              Control hazard rate(s) lambdaC   0.116, 0.039, 0.058 ... (len=4)
#>             Control dropout rate(s)     eta   0.017, 0.015, 0.014 ... (len=4)
#>        Experimental dropout rate(s)    etaE   0.017, 0.015, 0.014 ... (len=4)
#>  Event and dropout rate duration(s)       S                                 3
#>                            input
#>              3, 5, 6 ... (len=4)
#>                            5, 10
#>  0.116, 0.039, 0.058 ... (len=4)
#>  0.017, 0.015, 0.014 ... (len=4)
#>                             etaE
#>                                3
#> 
#> Expected events by stratum (H1) at final analysis:
#>    stratum control experimental  total
#>  Stratum 1  35.346       21.902 57.248
#>  Stratum 2  30.631       17.010 47.641

# Example to compute power for a fixed design.
# Trial duration T, minimum follow-up minfup and accrual duration R are all
# specified and will not change on output.
# beta=NULL will compute power and output will be the same as if beta were specified.
# This option is not available for group sequential designs.
nSurv(
  lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
  alpha = .025, beta = NULL, minfup = 12, T = 36, method = "LachinFoulkes"
) |>
  print()
#> nSurv fixed-design summary (method=LachinFoulkes; target=Power)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=1) | power=96.5%
#> N=144.0 subjects | D=117.5 events | T=36.0 study duration | accrual=24.0 Accrual duration | minfup=12.0 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma     6     6
#>            Accrual rate duration(s)       R    24    25
#>              Control hazard rate(s) lambdaC 0.116 0.116
#>             Control dropout rate(s)     eta 0.001 0.001
#>        Experimental dropout rate(s)    etaE 0.001  etaE
#>  Event and dropout rate duration(s)       S  NULL     S