Expected number of events for a time-to-event study

eEvents() is used to calculate the expected number of events for a population with a time-to-event endpoint. It is based on calculations demonstrated in Lachin and Foulkes (1986) and is fundamental in computations for the sample size method they propose. Piecewise exponential survival and dropout rates are supported as well as piecewise uniform enrollment. A stratified population is allowed. Output is the expected number of events observed given a trial duration and the above rate parameters.

eEvents() produces an object of class eEvents 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.

print.eEvents() formats the output for an object of class eEvents and returns the input value.

Usage

eEvents(
  lambda = 1,
  eta = 0,
  gamma = 1,
  R = 1,
  S = NULL,
  T = 2,
  Tfinal = NULL,
  minfup = 0,
  digits = 4
)

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

Arguments

lambda: scalar, vector or matrix of event hazard rates; rows represent time periods while columns represent strata; a vector implies a single stratum.
eta: scalar, vector or matrix of dropout hazard rates; 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.
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 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: time of analysis; if Tfinal=NULL, this is also the study duration.
Tfinal: Study duration; if NULL, this will be replaced with T on output.
minfup: time from end of planned enrollment (sum(R) from output value of R) until Tfinal.
digits: which controls number of digits for printing.
x: an object of class eEvents returned from eEvents().
...: Other arguments that may be passed to the generic print function.

Value

eEvents() and print.eEvents() return an object of class eEvents which contains the following items:

lambda: as input; converted to a matrix on output.
eta: as input; converted to a matrix on output.
gamma: as input.
R: as input.
S: as input.
T: as input.
Tfinal: planned duration of study.
minfup: as input.
d: expected number of events.
n: expected sample size.
digits: as input.

References

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.

Bernstein D and Lagakos S (1978), Sample size and power determination for stratified clinical trials. Journal of Statistical Computation and Simulation, 8:65-73.

Author

Keaven Anderson keaven_anderson@merck.com

Examples


# 3 enrollment periods, 3 piecewise exponential failure rates
str(eEvents(
  lambda = c(.05, .02, .01), eta = .01, gamma = c(5, 10, 20),
  R = c(2, 1, 2), S = c(1, 1), T = 20
))
#> List of 11
#>  $ lambda: num [1:3, 1] 0.05 0.02 0.01
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:3] "0-1" "1-2" "2-20"
#>   .. ..$ : chr "Stratum 1"
#>  $ eta   : num [1:3, 1] 0.01 0.01 0.01
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:3] "0-1" "1-2" "2-20"
#>   .. ..$ : chr "Stratum 1"
#>  $ gamma : num [1:3, 1] 5 10 20
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:3] "0-2" "2-3" "3-5"
#>   .. ..$ : chr "Stratum 1"
#>  $ R     : num [1:3] 2 1 2
#>  $ S     : num [1:2] 1 1
#>  $ T     : num 20
#>  $ Tfinal: num 20
#>  $ minfup: num 0
#>  $ d     : num 11
#>  $ n     : num 60
#>  $ digits: num 4
#>  - attr(*, "class")= chr "eEvents"

# control group for example from Bernstein and Lagakos (1978)
lamC <- c(1, .8, .5)
n <- eEvents(
  lambda = matrix(c(lamC, lamC * 2 / 3), ncol = 6), eta = 0,
  gamma = matrix(.5, ncol = 6), R = 2, T = 4
)