Weighted logrank test

Usage

wlr(data, weight, return_variance = FALSE)

Arguments

data: Dataset that has been cut, generated by sim_pw_surv().
weight: Weighting functions, such as fh(), mb(), and early_zero().
return_variance: A logical flag that, if TRUE, adds columns estimated variance for weighted sum of observed minus expected; see details; Default: FALSE.

Value

A list containing the test method (method), parameters of this test method (parameter), point estimate of the treatment effect (estimate), standardized error of the treatment effect (se), Z-score (z), p-values (p_value).

Details

$z$ - 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

x <- sim_pw_surv(n = 200) |> cut_data_by_event(100)

# Example 1: WLR test with FH wights
x |> wlr(weight = fh(rho = 0, gamma = 1))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=1)"
#> 
#> $estimate
#> [1] -5.581453
#> 
#> $se
#> [1] 1.668562
#> 
#> $z
#> [1] -3.345067
#> 
x |> wlr(weight = fh(rho = 0, gamma = 1), return_variance = TRUE)
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=1)"
#> 
#> $estimate
#> [1] -5.581453
#> 
#> $se
#> [1] 1.668562
#> 
#> $z
#> [1] -3.345067
#> 

# Example 2: WLR test with MB wights
x |> wlr(weight = mb(delay = 4, w_max = 2))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "MB(delay = 4, max_weight = 2)"
#> 
#> $estimate
#> [1] -18.61212
#> 
#> $se
#> [1] 5.949729
#> 
#> $z
#> [1] -3.12823
#> 

# Example 3: WLR test with early zero wights
x |> wlr(weight = early_zero(early_period = 4))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "Xu 2017 with first 4 months of 0 weights"
#> 
#> $estimate
#> [1] -12.85986
#> 
#> $se
#> [1] 3.811585
#> 
#> $z
#> [1] -3.373888
#>