#' @param test.type
#'
#' @param alpha Type I error rate. Default is 0.025 since 1-sided testing is default.
#' @param sided 1 for 1-sided testing, 2 for 2-sided testing.
#' @param beta Type II error rate. Default is 0.10 (90% power); NULL if power is to be computed based on other input values.
#' @param astar Normally not specified. If \code{test.type=5} or \code{6}, \code{astar} specifies the total probability of crossing a lower bound at all analyses combined. This will be changed to \code{1−alpha} when default value of \code{0} is used. Since this is the expected usage, normally \code{astar} is not specified by the user.
#' @param 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 (\code{test.type=1, 2}), \code{sfu}. The default value is \code{sfHSD} which is a Hwang-Shih-DeCani spending function.
#' @param 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 (\code{sfLDOF}, \code{sfLDPocock}) or bound types (\code{“OF”, “Pocock”}) that do not require parameters.
#' @param sfl Specifies the spending function for lower boundary crossing probabilities when asymmetric, two-sided testing is performed (\code{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 \code{sfHSD} which is a Hwang-Shih-DeCani spending function. The parameter \code{sfl} is ignored for one-sided testing (\code{test.type=1}) or symmetric 2-sided testing (\code{test.type=2}).
#' @param 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.
#' @param calendarTime Vector of increasing positive numbers with calendar times of analyses. Time 0 is start of randomization.
#' @param spending Select between calendar-based spending and information-based spending.
#' @param 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.
#' @param hr hazard ratio (experimental/control) under the alternate hypothesis (scalar).
#' @param hr0 hazard ratio (experimental/control) under the null hypothesis (scalar).
#' @param 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.
#' @param etaE matrix dropout hazard rates for the experimental group specified in like form as \code{eta}; if \code{NULL}, this is set equal to \code{eta}.
#' @param 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.
#' @param 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.
#' @param 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.
#' @param minfup A non-negative scalar less than the maximum value in \code{calendarTime}. Enrollment will be cut off at the difference between the maximum value in \code{calendarTime} and \code{minfup}.
#' @param ratio randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata.
#' @param 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.
#'
#' @export
gsSurvCalendar <- function(
test.type = 4, alpha = 0.025, sided = 1, beta = 0.1, astar = 0,
sfu = gsDesign::sfHSD, sfupar = -4,
sfl = gsDesign::sfHSD, sflpar = -2,
calendarTime = c(12, 24, 36),
spending = c("information", "calendar"),
lambdaC = log(2) / 6, hr = .6, hr0 = 1, eta = 0, etaE = NULL,
gamma = 1, R = 12, S = NULL, minfup = 18, ratio = 1,
r = 18 # , tol = .Machine$double.eps^0.25
) {
x <- nSurv(
lambdaC = lambdaC, hr = hr, hr0 = hr0, eta = eta, etaE = etaE,
gamma = gamma, R = R, S = S, T = max(calendarTime),
minfup = minfup, ratio = ratio,
alpha = alpha, beta = beta, sided = sided # , tol = tol
)
# Get interim expected event counts and sample size based on
# input gamma, eta, lambdaC, R, S, minfup
eDC <- NULL
eDE <- NULL
eNC <- NULL
eNE <- NULL
T <- NULL
k <- length(calendarTime)
for (i in 1:k) {
xx <- nEventsIA(tIA = calendarTime[i], x = x, simple = FALSE)
eDC <- rbind(eDC, xx$eDC)
eDE <- rbind(eDE, xx$eDE)
eNC <- rbind(eNC, xx$eNC)
eNE <- rbind(eNE, xx$eNE)
}
timing <- rowSums(eDC) + rowSums(eNC)
timing <- timing / max(timing)
# if calendar spending, set usTime, lsTime
if (spending[1] == "calendar") {
lsTime <- calendarTime / max(calendarTime)
} else {
lsTime <- NULL
}
usTime <- lsTime
# Now inflate events to get targeted power
y <- gsDesign::gsDesign(
k = k, test.type = test.type, alpha = alpha / sided,
beta = beta, astar = astar, n.fix = x$d, timing = timing,
sfu = sfu, sfupar = sfupar, sfl = sfl, sflpar = sflpar, # tol = tol,
delta1 = log(hr), delta0 = log(hr0),
usTime = usTime, lsTime = lsTime
)
y$hr <- hr
y$hr0 <- hr0
y$R <- x$R
y$S <- x$S
y$minfup <- x$minfup
# Inflate fixed design enrollment to get targeted events
inflate <- max(y$n.I) / x$d
y$gamma <- x$gamma * inflate
y$eDC <- inflate * eDC
y$eDE <- inflate * eDE
y$eNC <- inflate * eNC
y$eNE <- inflate * eNE
y$ratio <- ratio
y$lambdaC <- x$lambdaC
y$etaC <- x$etaC
y$etaE <- x$etaE
y$variable <- x$variable
# y$tol <- tol
y$T <- calendarTime
class(y) <- c("gsSurv", "gsDesign")
nameR <- nameperiod(cumsum(y$R))
stratnames <- paste("Stratum", 1:ncol(y$lambdaC))
if (is.null(y$S)) {
nameS <- "0-Inf"
} else {
nameS <- nameperiod(cumsum(c(y$S, Inf)))
}
rownames(y$lambdaC) <- nameS
colnames(y$lambdaC) <- stratnames
rownames(y$etaC) <- nameS
colnames(y$etaC) <- stratnames
rownames(y$etaE) <- nameS
colnames(y$etaE) <- stratnames
rownames(y$gamma) <- nameR
colnames(y$gamma) <- stratnames
return(y)
}4 Calendar-based design
We create the function gsSurvCalendar() as an extension to gsSurv() to set interim analysis timing at time of design to calendar times. We demonstrate how this can be used with either information-based or calendar-based spending. The coding is quite simple in that input enrollment rates, failure rates and dropout rates are used to compute expected events over time. Calendar timing of all analyses is specified by the user.
Now for gsSurvCalendar() which is actually a bit simpler and more flexible than gsSurv(). We swap in the argument calendarTime for the timing and T arguments above in gsSurv(); here the timing (information fraction) will be computed, while in gsSurv() the calendar times are computed. The parameter hrbeta is used when \(\beta\)-spending should be based on a different hazard ratio than specified in hr (only used when test.type is 3 or 4).
Example: default arguments
xxx <- gsSurvCalendar()
gsDesign::gsBoundSummary(xxx) Analysis Value Efficacy Futility
IA 1: 50% Z 2.7508 0.4687
N: 132 p (1-sided) 0.0030 0.3197
Events: 88 ~HR at bound 0.5551 0.9046
Month: 12 P(Cross) if HR=1 0.0030 0.6803
P(Cross) if HR=0.6 0.3621 0.0268
IA 2: 92% Z 2.1282 1.8018
N: 198 p (1-sided) 0.0167 0.0358
Events: 161 ~HR at bound 0.7148 0.7526
Month: 24 P(Cross) if HR=1 0.0175 0.9654
P(Cross) if HR=0.6 0.8661 0.0827
Final Z 2.0453 2.0453
N: 198 p (1-sided) 0.0204 0.0204
Events: 175 ~HR at bound 0.7339 0.7339
Month: 36 P(Cross) if HR=1 0.0222 0.9778
P(Cross) if HR=0.6 0.9000 0.1000We check this with a call to gsSurv() that should give an identical design.
yyy <- gsSurv(
k = 3, test.type = 4, alpha = 0.025, sided = 1, beta = 0.1, astar = 0,
sfu = gsDesign::sfHSD, sfupar = -4,
sfl = gsDesign::sfHSD, sflpar = -2,
timing = xxx$timing,
T = max(xxx$T),
lambdaC = log(2) / 6, hr = .6, hr0 = 1, eta = 0, etaE = NULL,
gamma = 1, R = 12, S = NULL, minfup = 18, ratio = 1,
r = 18, tol = .Machine$double.eps^0.25
)
gsDesign::gsBoundSummary(yyy) Analysis Value Efficacy Futility
IA 1: 50% Z 2.7508 0.4686
N: 182 p (1-sided) 0.0030 0.3197
Events: 88 ~HR at bound 0.5550 0.9046
Month: 17 P(Cross) if HR=1 0.0030 0.6803
P(Cross) if HR=0.6 0.3621 0.0268
IA 2: 92% Z 2.1282 1.8017
N: 198 p (1-sided) 0.0167 0.0358
Events: 161 ~HR at bound 0.7148 0.7526
Month: 30 P(Cross) if HR=1 0.0175 0.9653
P(Cross) if HR=0.6 0.8661 0.0827
Final Z 2.0453 2.0453
N: 198 p (1-sided) 0.0204 0.0204
Events: 175 ~HR at bound 0.7339 0.7339
Month: 36 P(Cross) if HR=1 0.0222 0.9778
P(Cross) if HR=0.6 0.9000 0.1000Example: default arguments, but information fraction
Note that calendar fraction spending below gives much more conservative interim bounds than information fraction spending above. This behavior is due to event accumulation slowing down as the trial proceeds; this is not always the case.
gsDesign::gsBoundSummary(gsSurvCalendar(spending = "calendar")) Analysis Value Efficacy Futility
IA 1: 50% Z 3.0107 0.1378
N: 122 p (1-sided) 0.0013 0.4452
Events: 82 ~HR at bound 0.5126 0.9699
Month: 12 P(Cross) if HR=1 0.0013 0.5548
P(Cross) if HR=0.6 0.2425 0.0148
IA 2: 92% Z 2.5422 1.3488
N: 184 p (1-sided) 0.0055 0.0887
Events: 150 ~HR at bound 0.6597 0.8020
Month: 24 P(Cross) if HR=1 0.0062 0.9156
P(Cross) if HR=0.6 0.7278 0.0437
Final Z 1.9682 1.9682
N: 184 p (1-sided) 0.0245 0.0245
Events: 163 ~HR at bound 0.7344 0.7344
Month: 36 P(Cross) if HR=1 0.0244 0.9756
P(Cross) if HR=0.6 0.9000 0.1000Example: computing power rather than sample size
Here we set beta = NULL so that gsSurvCalendar() will compuute power rather than sample size. This needs additional testing. Right now it does not work as gsDesign() will not compute power with the given call. This can possibly be fixed by using the maxn.IPlan argument when in the gsSurvCalendar() code.
gsDesign::gsBoundSummary(gsSurvCalendar(beta = NULL))Example: stratified design
Stratified designs still need fixing of weights for strata. However, gsSurvCalendar() will generate an answer with inappropriate weights until that time.
gsSurvCalendar(
alpha = .05, beta = .2, hr = 2 / 3, ratio = 1,
R = 2, S = NULL, calendarTime = c(2, 4),
minfup = 2,
lambdaC = matrix(c(1, .8, .5), nrow = 1),
eta = matrix(0, nrow = 1, ncol = 3),
etaE = matrix(0, nrow = 1, ncol = 3),
gamma = matrix(c(.4, .4, .2), nrow = 1)
)Time to event group sequential design with HR= 0.6666667
Equal randomization: ratio=1
Asymmetric two-sided group sequential design with
80 % power and 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++
1 128 1.13 0.8703 0.122 2.02 0.0214 0.0214
2 160 1.69 0.9544 0.078 1.69 0.0456 0.0286
Total 0.2000 0.0500
+ lower bound beta spending (under H1):
Hwang-Shih-DeCani spending function with gamma = -2.
++ 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 Total E{N}
0.0000 0.0214 0.0248 0.0462 130.6
0.2034 0.6056 0.1944 0.8000 136.0
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 Total
0.0000 0.8703 0.0835 0.9538
0.2034 0.1220 0.0780 0.2000
T n Events HR futility HR efficacy
IA 1 2 191.3497 127.0654 0.819 0.698
Final 4 191.3497 159.9666 0.766 0.766
Accrual rates:
Stratum 1 Stratum 2 Stratum 3
0-2 38.27 38.27 19.13
Control event rates (H1):
Stratum 1 Stratum 2 Stratum 3
0-Inf 1 0.8 0.5
Censoring rates:
Stratum 1 Stratum 2 Stratum 3
0-Inf 0 0 0