Weighted logrank test

Usage

wlr(data, weight, return_variance = FALSE, ratio = NULL)

# S3 method for class 'tte_data'
wlr(data, weight, return_variance = FALSE, ratio = NULL)

# S3 method for class 'counting_process'
wlr(data, weight, return_variance = FALSE, ratio = NULL)

Arguments

data

Dataset (generated by sim_pw_surv()) that has been cut by counting_process(), cut_data_by_date(), or cut_data_by_event().

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.

ratio

randomization ratio (experimental:control).

If the data is generated by simtrial, such as
- data = sim_pw_surv(...) |> cut_data_by_date(...)
- data = sim_pw_surv(...) |> cut_data_by_event(...)
- data = sim_pw_surv(...) |> cut_data_by_date(...) |> counting_process(...)
- data = sim_pw_surv(...) |> cut_data_by_event(...) |> counting_process(...) there is no need to input the ratio, as simtrial gets the ratio via the block arguments in sim_pw_surv().
If the data is a custom dataset (see Example 2) below,
- Users are suggested to input the planned randomization ratio to ratio;
- If not, simtrial takes the empirical randomization ratio.

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

# ---------------------- #
#      Example 1         #
#  Use dataset generated #
#     by simtrial        #
# ---------------------- #
x <- sim_pw_surv(n = 200) |> cut_data_by_event(100)

# Example 1A: WLR test with FH wights
x |> wlr(weight = fh(rho = 0, gamma = 0.5))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -10.39261
#> 
#> $se
#> [1] 2.578534
#> 
#> $z
#> [1] 4.030433
#> 
#> $info
#> [1] 6.706301
#> 
#> $info0
#> [1] 7.185266
#> 
x |> wlr(weight = fh(rho = 0, gamma = 0.5), return_variance = TRUE)
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -10.39261
#> 
#> $se
#> [1] 2.578534
#> 
#> $z
#> [1] 4.030433
#> 
#> $info
#> [1] 6.706301
#> 
#> $info0
#> [1] 7.185266
#> 

# Example 1B: 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] -22.64058
#> 
#> $se
#> [1] 6.064475
#> 
#> $z
#> [1] 3.733313
#> 
#> $info
#> [1] 35.70777
#> 
#> $info0
#> [1] 38.37037
#> 

# Example 1C: 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] -16.83342
#> 
#> $se
#> [1] 3.511878
#> 
#> $z
#> [1] 4.793281
#> 
#> $info
#> [1] 10.75472
#> 
#> $info0
#> [1] 13.25
#> 

# Example 1D
# For increased computational speed when running many WLR tests, you can
# pre-compute the counting_process() step first, and then pass the result of
# counting_process() directly to wlr()
x <- x |> counting_process(arm = "experimental")
x |> wlr(weight = fh(rho = 0, gamma = 1))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=1)"
#> 
#> $estimate
#> [1] -6.606821
#> 
#> $se
#> [1] 1.631434
#> 
#> $z
#> [1] 4.049702
#> 
#> $info
#> [1] 2.854797
#> 
#> $info0
#> [1] 2.982862
#> 
x |> wlr(weight = mb(delay = 4, w_max = 2))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "MB(delay = 4, max_weight = 2)"
#> 
#> $estimate
#> [1] -22.64058
#> 
#> $se
#> [1] 6.064475
#> 
#> $z
#> [1] 3.733313
#> 
#> $info
#> [1] 35.70777
#> 
#> $info0
#> [1] 38.37037
#> 
x |> wlr(weight = early_zero(early_period = 4))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "Xu 2017 with first 4 months of 0 weights"
#> 
#> $estimate
#> [1] -16.83342
#> 
#> $se
#> [1] 3.511878
#> 
#> $z
#> [1] 4.793281
#> 
#> $info
#> [1] 10.75472
#> 
#> $info0
#> [1] 13.25
#> 

# ---------------------- #
#      Example 2         #
#  Use cumsum dataset    #
# ---------------------- #
x <- data.frame(treatment = ifelse(ex1_delayed_effect$trt == 1, "experimental", "control"),
                stratum = rep("All", nrow(ex1_delayed_effect)),
                tte = ex1_delayed_effect$month,
                event = ex1_delayed_effect$evntd)
class(x) <- c("tte_data", class(x))

# Users can specify the randomization ratio to calculate the statistical information under H0
x |> wlr(weight = fh(rho = 0, gamma = 0.5), ratio = 2)
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -12.28665
#> 
#> $se
#> [1] 3.716574
#> 
#> $z
#> [1] 3.305908
#> 
#> $info
#> [1] 16.60727
#> 
#> $info0
#> [1] 15.27192
#> 

x |>
  counting_process(arm = "experimental") |>
  wlr(weight = fh(rho = 0, gamma = 0.5), ratio = 2)
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -12.28665
#> 
#> $se
#> [1] 3.716574
#> 
#> $z
#> [1] 3.305908
#> 
#> $info
#> [1] 16.60727
#> 
#> $info0
#> [1] 15.27192
#> 

# If users don't provide the randomization ratio, we will calculate the emperical ratio
x |> wlr(weight = fh(rho = 0, gamma = 0.5))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -12.28665
#> 
#> $se
#> [1] 3.716574
#> 
#> $z
#> [1] 3.305908
#> 
#> $info
#> [1] 16.60727
#> 
#> $info0
#> [1] 15.31399
#> 

x |>
  counting_process(arm = "experimental") |>
  wlr(weight = fh(rho = 0, gamma = 0.5))
#> $method
#> [1] "WLR"
#> 
#> $parameter
#> [1] "FH(rho=0, gamma=0.5)"
#> 
#> $estimate
#> [1] -12.28665
#> 
#> $se
#> [1] 3.716574
#> 
#> $z
#> [1] 3.305908
#> 
#> $info
#> [1] 16.60727
#> 
#> $info0
#> [1] 15.31399
#>