Number of Subjects Having an Event by Calendar Time

Obtains the number of subjects having an event by given calendar times for each treatment group.

Usage

nevent(
  time = NA_real_,
  allocationRatioPlanned = 1,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  accrualDuration = NA_real_,
  maxFollowupTime = NA_real_
)

Arguments

time: A vector of calendar times at which to calculate the number of patients having an event.
allocationRatioPlanned: Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
accrualTime: A vector that specifies the starting time of piecewise Poisson enrollment time intervals. Must start with 0, e.g., c(0, 3) breaks the time axis into 2 accrual intervals: $[0, 3)$ and $[3, \infty)$.
accrualIntensity: A vector of accrual intensities. One for each accrual time interval.
piecewiseSurvivalTime: A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: $[0, 6)$ and $[6, \infty)$. Defaults to 0 for exponential distribution.
lambda1: A vector of hazard rates for the event for the active treatment group. One for each analysis time interval.
lambda2: A vector of hazard rates for the event for the control group. One for each analysis time interval.
gamma1: The hazard rate for exponential dropout, or a vector of hazard rates for piecewise exponential dropout for the active treatment group.
gamma2: The hazard rate for exponential dropout, or a vector of hazard rates for piecewise exponential dropout for the control group.
accrualDuration: Duration of the enrollment period.
maxFollowupTime: Follow-up time for the first enrolled subject. For fixed follow-up, maxFollowupTime = minFollowupTime. For variable follow-up, maxFollowupTime = accrualDuration + minFollowupTime.

Value

A matrix of the number of patients having an event at the specified calendar times (row) for each treatment group (column).

Details

For a given treatment group $g$ and calendar time $\tau$, the number of patients having an event by calendar time $\tau$ is calculated as $I_1 + I_2$, where $$I_1 = \phi_g A(\tau - T_{\rm{fmax}}) P_g(T_{\rm{fmax}}),$$ and $$I_2 = \phi_g \int_{\tau - T_{\rm{fmax}}}^{\tau} a(u) P_g(\tau - u) du,$$ where $\phi_g$ is the probability of randomization to treatment group $g$, $A(\tau - T_{\rm{fmax}})$ is the number of patients enrolled by calendar time $\tau - T_{\rm{fmax}}$, $P_g(T_{\rm{fmax}})$ is the probability of having an event by the maximum follow-up time $T_{\rm{fmax}}$ for a patient in treatment group $g$ after enrollment, $a(u)$ is the accrual intensity at calendar time $u$, and $P_g(\tau - u)$ is the probability of having an event by calendar time $\tau$ for a patient in treatment group $g$ enrolled at calendar time $u$.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Piecewise accrual, piecewise exponential survivals, and 5% dropout by
# the end of 1 year.
nevent(time = c(9, 24), allocationRatioPlanned = 1,
       accrualTime = c(0, 3), accrualIntensity = c(10, 20),
       piecewiseSurvivalTime = c(0, 6),
       lambda1 = c(0.0533, 0.0309), lambda2 = c(0.0533, 0.0533),
       gamma1 = -log(1-0.05)/12, gamma2 = -log(1-0.05)/12,
       accrualDuration = 12, maxFollowupTime = 30)
#>          [,1]     [,2]
#> [1,] 13.10990 13.43671
#> [2,] 49.61969 60.81305