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Fleming-Harrington Weighted Logrank Tests

Source: R/tenFH.R
tenFH.Rd

With output from the function counting_process

Usage

tenFH(
  x = sim_pw_surv(n = 200) %>% cut_data_by_event(150) %>% counting_process(arm =
    "Experimental"),
  rg = tibble(rho = c(0, 0, 1, 1), gamma = c(0, 1, 0, 1)),
  returnVariance = FALSE
)

Arguments

x

a counting_process-class tibble with a counting process dataset

rg

a tibble with variables rho and gamma, both greater than equal to zero, to specify one Fleming-Harrington weighted logrank test per row; Default: tibble(rho = c(0, 0, 1, 1), gamma = c(0, 1, 0, 1))

returnVariance

a logical flag that, if true, adds columns estimated variance for weighted sum of observed minus expected; see details; Default: FALSE

Value

a tibble with rg as input and the FH test statistic for the data in x

(Z, a directional square root of the usual weighted logrank test); if variance calculations are specified (e.g., to be used for covariances in a combination test), the this will be returned in the column Var

Details

The input value x produced by counting_process() produces a counting process dataset grouped by strata and sorted within strata by increasing times where events occur.

  • \(Z\) - standardized normal Fleming-Harrington weighted logrank test

  • \(i\) - stratum index

  • \(d_i\) - number of distinct times at which events occurred in stratum \(i\)

  • \(t_{ij}\) - ordered times at which events in stratum \(i\), \(j=1,2,\ldots d_i\) were observed; for each observation, \(t_{ij}\) represents the time post study entry

  • \(O_{ij.}\) - total number of events in stratum \(i\) that occurred at time \(t_{ij}\)

  • \(O_{ije}\) - total number of events in stratum \(i\) in the experimental treatment group that occurred at time \(t_{ij}\)

  • \(N_{ij.}\) - total number of study subjects in stratum \(i\) who were followed for at least duration

  • \(E_{ije}\) - expected observations in experimental treatment group given random selection of \(O_{ij.}\) from those in stratum \(i\) at risk at time \(t_{ij}\)

  • \(V_{ije}\) - hypergeometric variance for \(E_{ije}\) as produced in Var from the counting_process() routine

  • \(N_{ije}\) - total number of study subjects in stratum \(i\) in the experimental treatment group who were followed for at least duration \(t_{ij}\)

  • \(E_{ije}\) - expected observations in experimental group in stratum \(i\) at time \(t_{ij}\) conditioning on the overall number of events and at risk populations at that time and sampling at risk observations without replacement: $$E_{ije} = O_{ij.} N_{ije}/N_{ij.}$$

  • \(S_{ij}\) - Kaplan-Meier estimate of survival in combined treatment groups immediately prior to time \(t_{ij}\)

  • \(\rho, \gamma\) - real parameters for Fleming-Harrington test

  • \(X_i\) - Numerator for signed logrank test in stratum \(i\) $$X_i = \sum_{j=1}^{d_{i}} S_{ij}^\rho(1-S_{ij}^\gamma)(O_{ije}-E_{ije})$$

  • \(V_{ij}\) - variance used in denominator for Fleming-Harrington weighted logrank tests $$V_i = \sum_{j=1}^{d_{i}} (S_{ij}^\rho(1-S_{ij}^\gamma))^2V_{ij})$$

    The stratified Fleming-Harrington weighted logrank test is then computed as: $$Z = \sum_i X_i/\sqrt{\sum_i V_i}$$

Examples

library(tidyr)
# Use default enrollment and event rates at cut at 100 events
x <- sim_pw_surv(n = 200) %>%
  cut_data_by_event(100) %>%
  counting_process(arm ="Experimental")
# compute logrank (FH(0,0)) and FH(0,1)
tenFH(x, rg = tibble(rho = c(0, 0), gamma = c(0, 1)))
#> # A tibble: 2 × 3
#>     rho gamma     Z
#>   <dbl> <dbl> <dbl>
#> 1     0     0 -1.36
#> 2     0     1 -2.08