Computing p-values for Fleming-Harrington weighted logrank tests and the MaxCombo test
Keaven Anderson, Yujie Zhao
Source:vignettes/maxcombo.Rmd
maxcombo.RmdIntroduction
This vignette demonstrates use of a simple routine to do simulations and testing using Fleming-Harrington weighted logrank tests and the MaxCombo test. In addition, we demonstrate how to perform these tests with a dataset not generated by simulation routines within the package. Note that all p-values computed here are one-sided with small values indicating that the experimental treatment is favored.
Defining the test
The MaxCombo test has been posed as the maximum of multiple Fleming-Harrington weighted logrank tests (Harrington and Fleming (1982), Fleming and Harrington (2011)). Combination tests looking at a maximum of selected tests in this class have also been proposed; see Lee (2007), Roychoudhury et al. (2021), and Lin et al. (2020). The Fleming-Harrington class is indexed by the parameters \rho \geq 0 and \gamma \geq 0. We will denote these as FH(\rho, \gamma). This class includes the logrank test as FH(0, 0). Other tests of interest here include:
- FH(0, 1): a test that down-weights early events
- FH(1, 0): a test that down-weights late events
- FH(1, 1): a test that down-weights events increasingly as their quantiles differ from the median
Executing for a single dataset
Generating test statistics with sim_fixed_n()
We begin with a single trial simulation generated by the routine
sim_fixed_n() using default arguments for that routine.
sim_fixed_n() produces one record per test and data cutoff
method per simulation. Here we choose 3 tests (logrank = FH(0, 0), FH(0,
1) and FH(1, 1)). When more than one test is chosen the correlation
between tests is computed as shown by Karrison
(2016), in this case in the columns V1,
V2, V3. The columns rho,
gamma indicate \rho and
\gamma used to compute the test.
z is the FH(\rho, \gamma)
normal test statistic with variance 1 with a negative value favoring
experimental treatment. The variable cut indicates how the
data were cut for analysis, in this case at the maximum of the targeted
minimum follow-up after last enrollment and the date at which the
targeted event count was reached. Sim is a sequential index
of the simulations performed.
set.seed(123)
x <- sim_fixed_n(
n_sim = 1,
timing_type = 5,
rho_gamma = data.frame(rho = c(0, 0, 1), gamma = c(0, 1, 1))
)
#> Backend uses sequential processing.
#> Loading required package: foreach
#> Loading required package: future
x |>
gt() |>
fmt_number(columns = c("ln_hr", "z", "duration", "v1", "v2", "v3"), decimals = 2)| method | parameter | estimate | se | z | p_value | v1 | v2 | v3 | event | ln_hr | cut | duration | sim |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| MaxCombo | FH(0, 0) + FH(0, 1) + FH(1, 1) | - | - | −4.06 | 3.70532e-07 | 1.00 | 0.85 | 0.93 | 350 | −0.44 | Max(min follow-up, event cut) | 77.17 | 1 |
| MaxCombo | FH(0, 0) + FH(0, 1) + FH(1, 1) | - | - | −4.04 | 3.70532e-07 | 0.85 | 1.00 | 0.94 | 350 | −0.44 | Max(min follow-up, event cut) | 77.17 | 1 |
| MaxCombo | FH(0, 0) + FH(0, 1) + FH(1, 1) | - | - | −4.95 | 3.70532e-07 | 0.93 | 0.94 | 1.00 | 350 | −0.44 | Max(min follow-up, event cut) | 77.17 | 1 |
Generating data with sim_pw_surv()
We begin with another simulation generated by
sim_pw_surv(). Again, we use defaults for that routine.
set.seed(123)
s <- sim_pw_surv(n = 100)
s |>
head() |>
gt() |>
fmt_number(columns = c("enroll_time", "fail_time", "dropout_time", "cte"), decimals = 2)| stratum | enroll_time | treatment | fail_time | dropout_time | cte | fail |
|---|---|---|---|---|---|---|
| All | 0.02 | experimental | 23.29 | 1,287.17 | 23.32 | 1 |
| All | 0.14 | control | 6.96 | 306.66 | 7.10 | 1 |
| All | 0.25 | control | 16.96 | 1,761.75 | 17.21 | 1 |
| All | 0.28 | experimental | 3.32 | 1,650.14 | 3.60 | 1 |
| All | 0.46 | control | 19.08 | 787.98 | 19.53 | 1 |
| All | 0.46 | experimental | 39.67 | 50.64 | 40.13 | 1 |
Once generated, we need to cut the data for analysis. Here we cut after 75 events.
x <- s |> cut_data_by_event(75)
x |>
head() |>
gt() |>
fmt_number(columns = "tte", decimals = 2)| tte | event | stratum | treatment |
|---|---|---|---|
| 23.29 | 1 | All | experimental |
| 6.96 | 1 | All | control |
| 16.96 | 1 | All | control |
| 3.32 | 1 | All | experimental |
| 19.08 | 1 | All | control |
| 33.29 | 0 | All | experimental |
Now we can analyze this data. We begin with s to show
how this can be done in a single line. In this case, we use the 4 test
combination suggested in Lin et al.
(2020), Roychoudhury et al.
(2021).
z <- s |>
cut_data_by_event(75) |>
maxcombo(rho = c(0, 0, 1, 1), gamma = c(0, 1, 0, 1))
z
#> $method
#> [1] "MaxCombo"
#>
#> $parameter
#> [1] "FH(0, 0) + FH(0, 1) + FH(1, 0) + FH(1, 1)"
#>
#> $z
#> [1] -2.511925 -2.907093 -1.899871 -3.119549
#>
#> $p_value
#> [1] 0.00204688Suppose we want the p-value just
based on the logrank and FH(0, 1) and FH(1, 0) as suggested by Lee (2007). We remove the rows and columns
associated with FH(0, 0) and FH(1, 1) and then apply
pvalue_maxcombo().
z <- s |>
cut_data_by_event(75) |>
maxcombo(rho = c(0, 1), gamma = c(1, 0))
z
#> $method
#> [1] "MaxCombo"
#>
#> $parameter
#> [1] "FH(0, 1) + FH(1, 0)"
#>
#> $z
#> [1] -2.907093 -1.899871
#>
#> $p_value
#> [1] 0.003395849Using survival data in another format
For a trial not generated by sim_fixed_n(), the process
is slightly more involved. We consider survival data not in the simtrial
format and show the transformation needed. In this case we use the small
aml dataset from the survival package.
| time | status | x |
|---|---|---|
| 9 | 1 | Maintained |
| 13 | 1 | Maintained |
| 13 | 0 | Maintained |
| 18 | 1 | Maintained |
| 23 | 1 | Maintained |
| 28 | 0 | Maintained |
We rename variables and create a stratum variable as follows:
x <- aml |> transmute(
tte = time,
event = status,
stratum = "All",
treatment = case_when(
x == "Maintained" ~ "experimental",
x == "Nonmaintained" ~ "control"
)
)
x |>
head() |>
gt()| tte | event | stratum | treatment |
|---|---|---|---|
| 9 | 1 | All | experimental |
| 13 | 1 | All | experimental |
| 13 | 0 | All | experimental |
| 18 | 1 | All | experimental |
| 23 | 1 | All | experimental |
| 28 | 0 | All | experimental |
Now we analyze the data with a MaxCombo with the logrank and FH(0, 1) and compute a p-value.
Simulation
We now consider the example simulation from the
pvalue_maxcombo() help file to demonstrate how to simulate
power for the MaxCombo test. However, we increase the number of
simulations to 100 in this case; a larger number should be used (e.g.,
1000) for a better estimate of design properties. Here we will test at
the \alpha=0.001 level.
set.seed(123)
# Only use cut events + min follow-up
x <- sim_fixed_n(
n_sim = 100,
timing_type = 5,
rho_gamma = data.frame(rho = c(0, 0, 1), gamma = c(0, 1, 1))
)
# MaxCombo power estimate for cutoff at max of targeted events, minimum follow-up
x |>
group_by(sim) |>
filter(row_number() == 1) |>
ungroup() |>
summarize(power = mean(p_value < .001))
#> # A tibble: 1 × 1
#> power
#> <dbl>
#> 1 0.79We note the use of group_map in the above produces a
list of p-values for each simulation.
It would be nice to have something that worked more like
dplyr::summarize() to avoid unlist() and to
allow evaluating, say, multiple data cutoff methods. The latter can be
done without having to re-run all simulations as follows, demonstrated
with a smaller number of simulations.
# Only use cuts for events and events + min follow-up
set.seed(123)
x <- sim_fixed_n(
n_sim = 100,
timing_type = c(2, 5),
rho_gamma = data.frame(rho = 0, gamma = c(0, 1))
)Now we compute a p-value separately for each cut type, first for targeted event count.
# Subset to targeted events cutoff tests
# This chunk will be updated after the development of sim_gs_n and sim_fixed_n
x |>
filter(cut == "Targeted events") |>
group_by(sim) |>
filter(row_number() == 1) |>
ungroup() |>
summarize(power = mean(p_value < .025))
#> # A tibble: 1 × 1
#> power
#> <dbl>
#> 1 0.95Now we use the later of targeted events and minimum follow-up cutoffs.