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Fleming-Harrington weighted logrank tests

Source: R/fh_weight.R
fh_weight.Rd

With output from the function counting_process().

Usage

fh_weight(
  x = counting_process(cut_data_by_event(sim_pw_surv(n = 200), 150), arm =
    "experimental"),
  rho_gamma = data.frame(rho = c(0, 0, 1, 1), gamma = c(0, 1, 0, 1)),
  return_variance = FALSE,
  return_corr = FALSE
)

Arguments

x

A counting_process()-class data frame with a counting process dataset.

rho_gamma

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

return_variance

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

return_corr

A logical flag that, if TRUE, adds columns estimated correlation for weighted sum of observed minus expected; see details; Default: FALSE.

Value

A data frame with rho_gamma 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 (for example, to be used for covariances in a combination test), then this will be returned in the column Var.

Details

The input value x produced by counting_process() produces a counting process dataset grouped by stratum and sorted within stratum 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 counting_process().

  • \(N_{ije}\) - 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(dplyr)

# Example 1
# 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, 1)
fh_weight(x, rho_gamma = data.frame(rho = 0, gamma = 1))
#>   rho gamma         z
#> 1   0     1 -3.332801
fh_weight(x, rho_gamma = data.frame(rho = 0, gamma = 1), return_variance = TRUE)
#>   rho gamma         z      var
#> 1   0     1 -3.332801 2.477853

# Compute the corvariance between FH(0, 0), FH(0, 1) and FH(1, 0)
fh_weight(x, rho_gamma = data.frame(rho = c(0, 0, 1), gamma = c(0, 1, 0)))
#>   rho gamma         z
#> 1   0     0 -3.744963
#> 2   0     1 -3.332801
#> 3   1     0 -3.561863
fh_weight(x, rho_gamma = data.frame(rho = c(0, 0, 1), gamma = c(0, 1, 0)), return_variance = TRUE)
#>   rho gamma         z        v1       v2        v3
#> 1   0     0 -3.744963 24.195511 6.495887 17.699623
#> 2   0     1 -3.332801  6.495887 2.477853  4.018035
#> 3   1     0 -3.561863 17.699623 4.018035 13.681589
fh_weight(x, rho_gamma = data.frame(rho = c(0, 0, 1), gamma = c(0, 1, 0)), return_corr = TRUE)
#>   rho gamma         z        v1        v2        v3
#> 1   0     0 -3.744963 1.0000000 0.8389447 0.9728108
#> 2   0     1 -3.332801 0.8389447 1.0000000 0.6900931
#> 3   1     0 -3.561863 0.9728108 0.6900931 1.0000000

# Example 2
# Use default enrollment and event rates at cut of 100 events
set.seed(123)
x <- sim_pw_surv(n = 200) |>
  cut_data_by_event(100) |>
  counting_process(arm = "experimental") |>
  fh_weight(rho_gamma = data.frame(rho = c(0, 0), gamma = c(0, 1)), return_corr = TRUE)

# Compute p-value for MaxCombo
library(mvtnorm)
1 - pmvnorm(
  lower = rep(min(x$z), nrow(x)),
  corr = data.matrix(select(x, -c(rho, gamma, z))),
  algorithm = GenzBretz(maxpts = 50000, abseps = 0.00001)
)[1]
#> [1] 0.003336366

# Check that covariance is as expected
x <- sim_pw_surv(n = 200) |>
  cut_data_by_event(100) |>
  counting_process(arm = "experimental")

x |> fh_weight(
  rho_gamma = data.frame(
    rho = c(0, 0),
    gamma = c(0, 1)
  ),
  return_variance = TRUE
)
#>   rho gamma          z        v1       v2
#> 1   0     0 -0.2790263 24.554559 6.644167
#> 2   0     1 -1.1511132  6.644167 2.552420

# Off-diagonal element should be variance in following
x |> fh_weight(
  rho_gamma = data.frame(
    rho = 0,
    gamma = .5
  ),
  return_variance = TRUE
)
#>   rho gamma          z      var
#> 1   0   0.5 -0.9492403 6.644167

# Compare off diagonal result with fh_weight()
x |> fh_weight(rho_gamma = data.frame(rho = 0, gamma = .5))
#>   rho gamma          z
#> 1   0   0.5 -0.9492403